@article{abad2012,
  title = {Representations up to Homotopy of {{Lie}} Algebroids},
  author = {Abad, Camilo Arias and Crainic, Marius},
  year = 2012,
  month = feb,
  journal = {Journal f\"ur die reine und angewandte Mathematik (Crelles Journal)},
  volume = {2012},
  number = {663},
  pages = {91--126},
  publisher = {De Gruyter},
  issn = {1435-5345},
  doi = {10.1515/CRELLE.2011.095},
  urldate = {2026-03-09},
  abstract = {We introduce and study the notion of representation up to homotopy of a Lie algebroid, paying special attention to examples. We use representations up to homotopy to define the adjoint representation of a Lie algebroid and show that the resulting cohomology controls the deformations of the structure. The Weil algebra of a Lie algebroid is defined and shown to coincide with Kalkman's BRST model for equivariant cohomology in the case of group actions. The relation of this algebra with the integration of Poisson and Dirac structures is explained in [3].},
  chapter = {Journal f\"ur die reine und angewandte Mathematik},
  langid = {english},
  file = {/home/alexk/Zotero/storage/T9NVRKXB/Abad and Crainic - 2012 - Representations up to homotopy of Lie algebroids.pdf}
}

@article{alekseev1998,
  title = {Lie Group Valued Moment Maps},
  author = {Alekseev, Anton and Malkin, Anton and Meinrenken, Eckhard},
  year = 1998,
  journal = {Journal of Differential Geometry},
  volume = {48},
  number = {3},
  pages = {445--495},
  issn = {0022-040X,1945-743X},
  mrnumber = {1638045},
  file = {/home/alexk/Zotero/storage/H63HM2S2/article.html}
}

@article{alexandrov1997,
  title = {The Geometry of the Master Equation and Topological Quantum Field Theory},
  author = {Alexandrov, M. and Schwarz, A. and Zaboronsky, O. and Kontsevich, M.},
  year = 1997,
  journal = {International Journal of Modern Physics A. Particles and Fields. Gravitation. Cosmology. Astrophysics. Accelerator Physics},
  volume = {12},
  number = {7},
  pages = {1405--1429},
  issn = {0217-751X,1793-656X},
  doi = {10.1142/S0217751X97001031},
  mrnumber = {1432574},
  file = {/home/alexk/Zotero/storage/B8QMHZ4S/Alexandrov et al. - 1997 - The geometry of the master equation and topologica.pdf;/home/alexk/Zotero/storage/4QVMEX89/article.html}
}

@misc{antieau2024,
  title = {Spectral Sequences, D\'ecalage, and the {{Beilinson}} t-Structure},
  author = {Antieau, Benjamin},
  year = 2024,
  month = nov,
  number = {arXiv:2411.09115},
  eprint = {2411.09115},
  primaryclass = {math},
  publisher = {arXiv},
  doi = {10.48550/arXiv.2411.09115},
  urldate = {2025-12-12},
  abstract = {This paper explains the theory of spectral sequences via d\'ecalage and the Beilinson t-structure.},
  archiveprefix = {arXiv},
  keywords = {Mathematics - Algebraic Topology},
  file = {/home/alexk/Zotero/storage/48DK9778/Antieau - 2024 - Spectral sequences, décalage, and the Beilinson t-structure.pdf;/home/alexk/Zotero/storage/TGTS5SGL/2411.html}
}

@misc{antieau2025,
  title = {Filtrations and Cohomology {{I}}: Crystallization},
  shorttitle = {Filtrations and Cohomology {{I}}},
  author = {Antieau, Benjamin},
  year = 2025,
  month = nov,
  number = {arXiv:2511.01567},
  eprint = {2511.01567},
  primaryclass = {math},
  publisher = {arXiv},
  doi = {10.48550/arXiv.2511.01567},
  urldate = {2025-12-12},
  abstract = {We compare several different notions of filtered derived commutative ring, discussing HKR-filtered Hochschild homology, Hodge-filtered de Rham cohomology, and the lesser-known Hodge-filtered infinitesimal cohomology. Our main result is that de Rham cohomology is the crystallization of infinitesimal cohomology.},
  archiveprefix = {arXiv},
  keywords = {Mathematics - Algebraic Geometry,Mathematics - Algebraic Topology},
  file = {/home/alexk/Zotero/storage/ZSETJYKL/Antieau - 2025 - Filtrations and cohomology I crystallization.pdf;/home/alexk/Zotero/storage/SDLCRP59/2511.html}
}

@misc{antieau2025a,
  title = {Filtrations and Cohomology {{II}}: The {{Gauss-Manin}} Connection},
  shorttitle = {Filtrations and Cohomology {{II}}},
  author = {Antieau, Benjamin},
  year = 2025,
  month = nov,
  number = {arXiv:2511.14731},
  eprint = {2511.14731},
  primaryclass = {math},
  publisher = {arXiv},
  doi = {10.48550/arXiv.2511.14731},
  urldate = {2025-12-12},
  abstract = {We use derived methods to study the Gauss-Manin connection in Hochschild homology, infinitesimal cohomology, and derived de Rham cohomology. As applications, we give new approaches to nilinvariance, the Quillen spectral sequence, and the HKR filtration. We extend the results of Bhatt's work on de Rham cohomology in characteristic zero to infinitesimal cohomology in mixed characteristic and show that the comparison to Hartshorne's algebraic de Rham complex "is" the Gauss-Manin connection. Finally, we explain the main features of prismatic cohomology in characteristic zero via the Gauss-Manin connection.},
  archiveprefix = {arXiv},
  keywords = {Mathematics - Algebraic Geometry,Mathematics - K-Theory and Homology},
  file = {/home/alexk/Zotero/storage/Y3P96XD3/Antieau - 2025 - Filtrations and cohomology II the Gauss-Manin connection.pdf;/home/alexk/Zotero/storage/TF34XSR3/2511.html}
}

@misc{antieau2025b,
  title = {Filtrations and Cohomology {{III}}: Cohomology of \${{E}}\_\textbackslash infty\$ Rings},
  shorttitle = {Filtrations and Cohomology {{III}}},
  author = {Antieau, Benjamin},
  year = 2025,
  month = dec,
  journal = {arXiv.org},
  urldate = {2025-12-18},
  abstract = {We discuss filtrations arising from de Rham-type cohomology theories for \$E\_\textbackslash infty\$ rings and \$E\_n\$ rings. Examples include the HKR filtration on relative topological Hochschild homology, the Hodge filtration on \$E\_\textbackslash infty\$ infinitesimal cohomology, and the Hodge filtration on \$E\_\textbackslash infty\$ de Rham cohomology.},
  howpublished = {https://arxiv.org/abs/2512.15509v1},
  langid = {english},
  file = {/home/alexk/Zotero/storage/RW2FIDC5/Antieau - 2025 - Filtrations and cohomology III cohomology of $E_infty$ rings.pdf}
}

@misc{ariotta2021,
  title = {Coherent Cochain Complexes and {{Beilinson}} T-Structures, with an Appendix by {{Achim Krause}}},
  author = {Ariotta, Stefano},
  year = 2021,
  month = sep,
  number = {arXiv:2109.01017},
  eprint = {2109.01017},
  primaryclass = {math},
  publisher = {arXiv},
  doi = {10.48550/arXiv.2109.01017},
  urldate = {2026-03-30},
  abstract = {We define and study coherent cochain complexes in arbitrary stable \$\textbackslash infty\$-categories, following Joyal. Our main result is that the \$\textbackslash infty\$-category of coherent cochain complexes in a stable \$\textbackslash infty\$-category \$\textbackslash mathscr C\$ is equivalent to the \$\textbackslash infty\$-category of complete filtered objects in \$\textbackslash mathscr C\$. We then show how the Beilinson t-structure can be interpreted in light of such equivalence, and analyze its behavior in the presence of symmetric monoidal structures. We also examine the relationship between the notion of (higher) Toda brackets and coherent cochain complexes. Finally, we prove how every coherent cochain complex gives rise to a spectral sequence and illustrate some examples.},
  archiveprefix = {arXiv},
  keywords = {Mathematics - Algebraic Topology,Mathematics - Category Theory},
  file = {/home/alexk/Zotero/storage/7H3GUI9F/Ariotta - 2021 - Coherent cochain complexes and Beilinson t-structures, with an appendix by Achim Krause.pdf;/home/alexk/Zotero/storage/G8T6UQ6F/2109.html}
}

@article{atiyah1983,
  title = {The {{Yang-Mills}} Equations over {{Riemann}} Surfaces},
  author = {Atiyah, M. F. and Bott, R.},
  year = 1983,
  journal = {Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences},
  volume = {308},
  number = {1505},
  pages = {523--615},
  issn = {0080-4614},
  doi = {10.1098/rsta.1983.0017},
  mrnumber = {702806},
  file = {/home/alexk/Zotero/storage/NZ9L6DD9/article.html}
}

@inproceedings{axelrod1992,
  title = {Chern-{{Simons}} Perturbation Theory},
  booktitle = {Proceedings of the {{XXth International Conference}} on {{Differential Geometric Methods}} in {{Theoretical Physics}}, {{Vol}}. 1, 2 ({{New York}}, 1991)},
  author = {Axelrod, Scott and Singer, I. M.},
  year = 1992,
  pages = {3--45},
  publisher = {World Sci. Publ., River Edge, NJ},
  urldate = {2022-10-07},
  mrnumber = {1225107},
  file = {/home/alexk/Zotero/storage/RUKG4V6J/publdoc.html}
}

@article{baranovsky2010,
  title = {Gerstenhaber-{{Batalin-Vilkoviski}} Structures on Coisotropic Intersections},
  author = {Baranovsky, Vladimir and Ginzburg, Victor},
  year = 2010,
  journal = {Mathematical Research Letters},
  volume = {17},
  number = {2},
  pages = {211--229},
  issn = {10732780, 1945001X},
  doi = {10.4310/MRL.2010.v17.n2.a2},
  urldate = {2022-11-09},
  abstract = {Let Y, Z be a pair of smooth coisotropic subvarieties in a smooth algebraic Poisson variety X. We show that any data of first order deformation of the structure sheaf OX to a sheaf of noncommutative algebras and of the sheaves OY and OZ to sheaves of right and left modules over the deformed algebra, respectively, gives rise to a Batalin-Vilkoviski algebra structure on the Tor-sheaf T orOq X (OY , OZ ). The induced Gerstenhaber bracket on the Tor-sheaf turns out to be canonically defined; it is independent of the choices of deformations involved. There are similar results for Ext-sheaves as well.},
  langid = {english},
  file = {/home/alexk/Zotero/storage/CBEQNV2V/Baranovsky and Ginzburg - 2010 - Gerstenhaber-Batalin-Vilkoviski structures on cois.pdf}
}

@article{behrend1997,
  title = {The Intrinsic Normal Cone},
  author = {Behrend, K. and Fantechi, B.},
  year = 1997,
  journal = {Inventiones Mathematicae},
  volume = {128},
  number = {1},
  pages = {45--88},
  issn = {0020-9910,1432-1297},
  doi = {10.1007/s002220050136},
  mrnumber = {1437495},
  file = {/home/alexk/Zotero/storage/SM7GL5C6/article.html}
}

@article{behrend2002,
  title = {Differential {{Graded Schemes I}}: {{Perfect Resolving Algebras}}},
  shorttitle = {Differential {{Graded Schemes I}}},
  author = {Behrend, Kai},
  year = 2002,
  month = dec,
  journal = {arXiv:math/0212225},
  eprint = {math/0212225},
  urldate = {2021-10-07},
  abstract = {We introduce perfect resolving algebras and study their fundamental properties. These algebras are basic for our theory of differential graded schemes, as they give rise to affine differential graded schemes. We also introduce etale morphisms. The purpose for studying these, is that they will be used to glue differential graded schemes from affine ones with respect to an etale topology.},
  archiveprefix = {arXiv},
  keywords = {Mathematics - Algebraic Geometry,Mathematics - Commutative Algebra},
  file = {/home/alexk/Zotero/storage/CELEN5XW/Behrend - 2002 - Differential Graded Schemes I Perfect Resolving A.pdf;/home/alexk/Zotero/storage/P3I5XCRZ/0212225.html}
}

@incollection{behrend2017,
  title = {Geometric Higher Groupoids and Categories},
  booktitle = {Geometry, Analysis and Probability},
  author = {Behrend, Kai and Getzler, Ezra},
  year = 2017,
  series = {Progr. {{Math}}.},
  volume = {310},
  pages = {1--45},
  publisher = {Birkh\"auser/Springer, Cham},
  doi = {10.1007/978-3-319-49638-2_1},
  isbn = {978-3-319-49636-8 978-3-319-49638-2},
  mrnumber = {3821920},
  file = {/home/alexk/Zotero/storage/6KFR75WS/Behrend and Getzler - 2017 - Geometric higher groupoids and categories.pdf;/home/alexk/Zotero/storage/3G5QHWCN/article.html}
}

@article{ben-bassat2015,
  title = {A `{{Darboux}} Theorem' for Shifted Symplectic Structures on Derived {{Artin}} Stacks, with Applications},
  author = {{Ben-Bassat}, Oren and Brav, Christopher and Bussi, Vittoria and Joyce, Dominic},
  year = 2015,
  month = may,
  journal = {Geometry \& Topology},
  volume = {19},
  number = {3},
  pages = {1287--1359},
  publisher = {Mathematical Sciences Publishers},
  issn = {1364-0380},
  doi = {10.2140/gt.2015.19.1287},
  urldate = {2023-02-25},
  file = {/home/alexk/Zotero/storage/D7KABVNN/Ben-Bassat et al. - 2015 - A ‘Darboux theorem’ for shifted symplectic structu.pdf}
}

@article{ben-zvi2012,
  title = {Loop Spaces and Connections},
  author = {{Ben-Zvi}, David and Nadler, David},
  year = 2012,
  journal = {Journal of Topology},
  volume = {5},
  number = {2},
  pages = {377--430},
  issn = {1753-8424},
  doi = {10.1112/jtopol/jts007},
  urldate = {2024-08-22},
  abstract = {We examine the geometry of loop spaces in derived algebraic geometry and extend in several directions the well-known connection between rotation of loops and the de Rham differential. Our main result, a categorification of the geometric description of cyclic homology, relates S1-equivariant quasicoherent sheaves on the loop space of a smooth scheme or geometric stack X in characteristic zero with sheaves on X with flat connection, or equivalently DX-modules. By deducing the Hodge filtration on de Rham modules from the formality of cochains on the circle, we are able to recover DX-modules precisely rather than a periodic version. More generally, we consider the rotated Hopf fibration \textohm{} S3{$\rightarrow\Omega$} S2{$\rightarrow$}S1, and relate \textohm{} S2-equivariant sheaves on the loop space with sheaves on X with arbitrary connection, with curvature given by their \textohm{} S3-equivariance.},
  copyright = {\copyright{} 2012 London Mathematical Society},
  langid = {english},
  file = {/home/alexk/Zotero/storage/UXT7KXBC/Ben-Zvi and Nadler - 2012 - Loop spaces and connections.pdf;/home/alexk/Zotero/storage/XN6LBXDG/jts007.html}
}

@article{beraldo2021,
  title = {The Center of the Categorified Ring of Differential Operators},
  author = {Beraldo, Dario},
  year = 2021,
  month = feb,
  journal = {Journal of the European Mathematical Society},
  volume = {23},
  number = {6},
  pages = {1999--2049},
  issn = {1435-9855},
  doi = {10.4171/jems/1048},
  urldate = {2026-04-07},
  abstract = {Dario Beraldo},
  langid = {english},
  file = {/home/alexk/Zotero/storage/J994GI9D/Beraldo - 2021 - The center of the categorified ring of differential operators.pdf}
}

@article{bezrukavnikov2004,
  title = {Fedosov {{Quantization}} in {{Algebraic Context}}},
  author = {Bezrukavnikov, R. and Kaledin, D.},
  year = 2004,
  journal = {Moscow Mathematical Journal},
  volume = {4},
  number = {3},
  pages = {559--592},
  issn = {16093321, 16094514},
  doi = {10.17323/1609-4514-2004-4-3-559-592},
  urldate = {2025-08-29},
  abstract = {We consider the problem of quantization of smooth symplectic varieties in the algebro-geometric setting. We show that, under appropriate cohomological assumptions, the Fedosov quantization procedure goes through with minimal changes. The assumptions are satisfied, for example, for affine and for projective varieties. We also give a classification of all possible quantizations.},
  langid = {english},
  file = {/home/alexk/Zotero/storage/IZWSSZFQ/Bezrukavnikov and Kaledin - 2004 - Fedosov Quantization in Algebraic Context.pdf}
}

@misc{bhatt2012,
  title = {Completions and Derived de {{Rham}} Cohomology},
  author = {Bhatt, Bhargav},
  year = 2012,
  month = jul,
  number = {arXiv:1207.6193},
  eprint = {1207.6193},
  primaryclass = {math},
  publisher = {arXiv},
  doi = {10.48550/arXiv.1207.6193},
  urldate = {2024-09-16},
  abstract = {We show that Illusie's derived de Rham cohomology (Hodge-completed) coincides with Hartshorne's algebraic de Rham cohomology for a finite type map of noetherian schemes in characteristic 0; the case of lci morphisms was a result of Illusie. In particular, the E\_1-differentials in the derived Hodge-to-de Rham spectral sequence for singular varieties are often non-zero. Another consequence is a completely elementary description of Hartshorne's algebraic de Rham cohomology: it is computed by the completed Amitsur complex for any variety in characteristic 0.},
  archiveprefix = {arXiv},
  keywords = {Mathematics - Algebraic Geometry,Mathematics - Commutative Algebra},
  file = {/home/alexk/Zotero/storage/N65KX9BE/Bhatt - 2012 - Completions and derived de Rham cohomology.pdf;/home/alexk/Zotero/storage/6V4PAVM3/1207.html}
}

@article{bischoff2022,
  title = {Brane Quantization of Toric {{Poisson}} Varieties},
  author = {Bischoff, Francis and Gualtieri, Marco},
  year = 2022,
  journal = {Communications in Mathematical Physics},
  volume = {391},
  number = {2},
  pages = {357--400},
  issn = {0010-3616,1432-0916},
  doi = {10.1007/s00220-022-04315-y},
  mrnumber = {4397176},
  file = {/home/alexk/Zotero/storage/PQ89I7L2/Bischoff and Gualtieri - 2022 - Brane quantization of toric Poisson varieties.pdf;/home/alexk/Zotero/storage/A24JPK6V/article.html}
}

@article{bott1976,
  title = {On the de {{Rham}} Theory of Certain Classifying Spaces},
  author = {Bott, R and Shulman, H and Stasheff, J},
  year = 1976,
  month = apr,
  journal = {Advances in Mathematics},
  volume = {20},
  number = {1},
  pages = {43--56},
  issn = {0001-8708},
  doi = {10.1016/0001-8708(76)90169-9},
  urldate = {2025-09-01},
  file = {/home/alexk/Zotero/storage/A9PQUISA/0001870876901699.html}
}

@article{bott1998,
  title = {Integral Invariants of \textbraceleft 3\textbraceright -Manifolds},
  author = {Bott, Raoul and Cattaneo, Alberto S.},
  year = 1998,
  month = jan,
  journal = {Journal of Differential Geometry},
  volume = {48},
  number = {1},
  pages = {91--133},
  publisher = {Lehigh University},
  issn = {0022-040X},
  doi = {10.4310/jdg/1214460608},
  urldate = {2022-10-07},
  abstract = {Journal of Differential Geometry},
  keywords = {57M27,57N10,58J28},
  file = {/home/alexk/Zotero/storage/VNSPYQ6Z/Bott and Cattaneo - 1998 - Integral invariants of 3 -manifolds.pdf;/home/alexk/Zotero/storage/WV9XBCTQ/1214460608.html}
}

@article{bottacin1995,
  title = {Poisson Structures on Moduli Spaces of Sheaves over {{Poisson}} Surfaces},
  author = {Bottacin, Francesco},
  year = 1995,
  month = dec,
  journal = {Inventiones Mathematicae},
  volume = {121},
  number = {1},
  pages = {421--436},
  issn = {0020-9910, 1432-1297},
  doi = {10.1007/BF01884307},
  urldate = {2025-09-01},
  abstract = {We introduce and study the notion of Poisson surface. We prove that the choice of a Poisson structure on a surface S canonically determines a Poisson structure on the moduli space M of stable sheaves on S. This result generalizes previous results obtained by Mukai [14], for abelian or K3 surfaces, and by Tyurin [16].},
  copyright = {http://www.springer.com/tdm},
  langid = {english},
  file = {/home/alexk/Zotero/storage/IVDFPPGV/Bottacin - 1995 - Poisson structures on moduli spaces of sheaves ove.pdf}
}

@article{bottacin2000,
  title = {Poisson Structures on Moduli Spaces of Framed Vector Bundles on Surfaces},
  author = {Bottacin, Francesco},
  year = 2000,
  journal = {Mathematische Nachrichten},
  volume = {220},
  pages = {33--44},
  issn = {0025-584X,1522-2616},
  doi = {10.1002/1522-2616(200012)220:1<33::AID-MANA33>3.3.CO;2-1},
  mrnumber = {1800199},
  file = {/home/alexk/Zotero/storage/DKIJQIS6/article.html}
}

@article{bouaziz2013,
  title = {A D-Shifted {{Darboux}} Theorem},
  author = {Bouaziz, E. and Grojnowski, I.},
  year = 2013,
  month = sep,
  journal = {arXiv:1309.2197},
  eprint = {1309.2197},
  urldate = {2021-09-20},
  abstract = {We give a local model for d-shifted symplectic dg-schemes, or Deligne-Mumford dg-stacks [PTVV]. Locally any such is a product of a "twisted shifted cotangent bundle", where the twist is given by an element df, with \$f \textbackslash in H\textasciicircum\textbraceleft 1-d\textbraceright (\textbackslash Cal O)\$, and a quadratic bundle in middle degree. The latter only occurs if d = 4r+2.},
  archiveprefix = {arXiv},
  keywords = {Mathematics - Algebraic Geometry},
  file = {/home/alexk/Zotero/storage/HTJ8AUKI/Bouaziz and Grojnowski - 2013 - A d-shifted Darboux theorem.pdf;/home/alexk/Zotero/storage/FB6SMAMM/1309.html}
}

@misc{bozec2022,
  title = {Relative Critical Loci and Quiver Moduli},
  author = {Bozec, Tristan and Calaque, Damien and Scherotzke, Sarah},
  year = 2022,
  month = sep,
  number = {arXiv:2006.01069},
  eprint = {2006.01069},
  primaryclass = {math},
  publisher = {arXiv},
  doi = {10.48550/arXiv.2006.01069},
  urldate = {2022-10-30},
  abstract = {In this paper we identify the cotangent to the derived stack of representations of a quiver \$Q\$ with the derived moduli stack of modules over the Ginzburg dg-algebra associated with \$Q\$. More generally, we extend this result to finite type dg-categories, to a relative setting as well, and to deformations of these. It allows us to recover and generalize some results of Yeung, and leads us to the discovery of seemingly new lagrangian subvarieties in the Hilbert scheme of points in the plane.},
  archiveprefix = {arXiv},
  keywords = {Mathematics - Algebraic Geometry,Mathematics - Algebraic Topology,Mathematics - Category Theory,Mathematics - Representation Theory,Mathematics - Symplectic Geometry},
  file = {/home/alexk/Zotero/storage/5LXPAKJ7/Bozec et al. - 2022 - Relative critical loci and quiver moduli.pdf;/home/alexk/Zotero/storage/DAXSMSAZ/2006.html}
}

@article{brantner2025,
  title = {{{PD}} Operads and Explicit Partition {{Lie}} Algebras},
  author = {Brantner, D. Lukas B. and Campos, Ricardo and Nuiten, Joost},
  year = 2025,
  journal = {Memoirs of the American Mathematical Society},
  volume = {315},
  number = {1597},
  pages = {v+125},
  issn = {0065-9266,1947-6221},
  doi = {10.1090/memo/1597},
  isbn = {9781470478377 9781470485085},
  mrnumber = {5003477},
  file = {/home/alexk/Zotero/storage/ATUEB9LK/Brantner et al. - 2025 - PD operads and explicit partition Lie algebras.pdf;/home/alexk/Zotero/storage/KLGPUBX8/article.html}
}

@article{brav2015,
  title = {Symmetries and Stabilization for Sheaves of Vanishing Cycles},
  author = {Brav, C. and Bussi, V. and Dupont, D. and Joyce, D. and Szendr{\H o}i, B.},
  year = 2015,
  journal = {Journal of Singularities},
  volume = {11},
  pages = {85--151},
  issn = {1949-2006},
  doi = {10.5427/jsing.2015.11e},
  mrnumber = {3353002},
  file = {/home/alexk/Zotero/storage/SRQM4H6D/Brav et al. - 2015 - Symmetries and stabilization for sheaves of vanishing cycles.pdf;/home/alexk/Zotero/storage/NMZ6M5NB/article.html}
}

@article{brav2018,
  title = {A {{Darboux}} Theorem for Derived Schemes with Shifted Symplectic Structure},
  author = {Brav, Christopher and Bussi, Vittoria and Joyce, Dominic},
  year = 2018,
  month = oct,
  journal = {Journal of the American Mathematical Society},
  volume = {32},
  number = {2},
  pages = {399--443},
  issn = {0894-0347, 1088-6834},
  doi = {10.1090/jams/910},
  urldate = {2021-09-20},
  abstract = {We prove a Darboux theorem for derived schemes with symplectic forms of degree                                                                                               k                       {$>$}                       0                                          k{$>$}0                                                                  , in the sense of Pantev, To\"en, Vaqui\'e, and Vezzosi. More precisely, we show that a derived scheme                                                                                               X                                          \textbackslash mathbfit \textbraceleft X\textbraceright{}                                                                  with symplectic form                                                                                                                                                   {$\omega$}                                                                                                        \textasciitilde{}                                                                                                                     \textbackslash tilde \textbraceleft\textbackslash omega \textbraceright{}                                                                  of degree                                                                        k                     k                                                                  is locally equivalent to                                                                                               (                       Spec                                                ⁡                                                                       A                       ,                                                {$\omega$}                                                                       )                                          (\textbackslash operatorname \textbraceleft Spec\textbraceright{} A,\textbackslash omega )                                                                  for                                                                                               Spec                                                ⁡                                                                       A                                          \textbackslash operatorname \textbraceleft Spec\textbraceright{} A                                                                  an affine derived scheme in which the cdga                                                                        A                     A                                                                  has Darboux-like coordinates with respect to which the symplectic form                                                                                               {$\omega$}                                                                 \textbackslash omega                                                                  is standard, and in which the differential in                                                                        A                     A                                                                  is given by a Poisson bracket with a Hamiltonian function                                                                                               {$\Phi$}                                                                 \textbackslash Phi                                                                  of degree                                                                                               k                       +                       1                                          k+1                                                                  .                                         When                                                                                               k                       =                                                -                                                                       1                                          k=-1                                                                  , this implies that a                                                                                                                        -                                                                       1                                          -1                                                                  -shifted symplectic derived scheme                                                                                               (                                                X                                              ,                                                                                                        {$\omega$}                                                                                                                \textasciitilde{}                                                                                                                               )                                          (\textbackslash mathbfit \textbraceleft X\textbraceright, \textbackslash tilde \textbraceleft\textbackslash omega \textbraceright )                                                                  is Zariski locally equivalent to the derived critical locus                                                                                               Crit                                                ⁡                                                                       (                                                {$\Phi$}                                                                       )                                          \textbackslash operatorname \textbraceleft Crit\textbraceright (\textbackslash Phi )                                                                  of a regular function                                                                                                                        {$\Phi$}                                                                       :                       U                                                {$\rightarrow$}                                                                                                                           A                                                  1                                                                 \textbackslash Phi :U\textbackslash rightarrow \textbackslash mathbb \textbraceleft A\textbraceright\textasciicircum 1                                                                  on a smooth scheme                                                                        U                     U                                                                  . We use this to show that the classical scheme                                                                                               X                       =                                                t                         0                                              (                                                X                                              )                                          X=t\_0(\textbackslash mathbfit \textbraceleft X\textbraceright )                                                                  has the structure of an               algebraic d-critical locus               , in the sense of Joyce.                          In a series of works, the authors and their collaborators extend these results to (derived) Artin stacks, and discuss a Lagrangian neighbourhood theorem for shifted symplectic derived schemes, and applications to categorified and motivic Donaldson--Thomas theory of Calabi--Yau 3-folds, and to defining new Donaldson--Thomas type invariants of Calabi--Yau 4-folds, and to defining Fukaya categories of Lagrangians in algebraic symplectic manifolds using perverse sheaves.},
  langid = {english},
  file = {/home/alexk/Zotero/storage/KNVWMMFI/Brav et al. - 2018 - A Darboux theorem for derived schemes with shifted.pdf}
}

@article{brav2019,
  title = {Relative {{Calabi-Yau}} Structures},
  author = {Brav, Christopher and Dyckerhoff, Tobias},
  year = 2019,
  journal = {Compositio Mathematica},
  volume = {155},
  number = {2},
  pages = {372--412},
  issn = {0010-437X,1570-5846},
  doi = {10.1112/s0010437x19007024},
  mrnumber = {3911626},
  file = {/home/alexk/Zotero/storage/952K9VZN/Brav and Dyckerhoff - 2019 - Relative Calabi-Yau structures.pdf;/home/alexk/Zotero/storage/6P8BW48Q/article.html}
}

@article{brav2021,
  title = {Relative {{Calabi-Yau}} Structures {{II}}: Shifted {{Lagrangians}} in the Moduli of Objects},
  shorttitle = {Relative {{Calabi-Yau}} Structures {{II}}},
  author = {Brav, Christopher and Dyckerhoff, Tobias},
  year = 2021,
  journal = {Selecta Mathematica. New Series},
  volume = {27},
  number = {4},
  pages = {Paper No. 63, 45},
  issn = {1022-1824,1420-9020},
  doi = {10.1007/s00029-021-00642-5},
  mrnumber = {4281260},
  file = {/home/alexk/Zotero/storage/DQ2CANS4/Brav and Dyckerhoff - 2021 - Relative Calabi-Yau structures II shifted Lagrang.pdf;/home/alexk/Zotero/storage/X8KXC3JR/article.html}
}

@misc{brav2023,
  title = {The Cyclic {{Deligne}} Conjecture and {{Calabi-Yau}} Structures},
  author = {Brav, Christopher and Rozenblyum, Nick},
  year = 2023,
  month = may,
  number = {arXiv:2305.10323},
  eprint = {2305.10323},
  primaryclass = {math},
  publisher = {arXiv},
  doi = {10.48550/arXiv.2305.10323},
  urldate = {2025-09-01},
  abstract = {The Deligne conjecture (many times a theorem) endows Hochschild cochains of a linear category with the structure of an \$E\_2\$-algebra, that is, of an algebra over the little 2-disks operad. In this paper, we prove the cyclic Deligne conjecture, stating that for a linear category equipped with a Calabi-Yau structure (a kind of non-commutative orientation), the Hochschild cochains is endowed with the finer structure of a framed \$E\_2\$-algebra, that is, of a circle-equivariant algebra over the little 2-disks operad. Our approach applies simultaneously to both smooth and proper linear categories, as well as to linear functors equipped with a relative Calabi-Yau structure, and works for a very general notion of linear category, including any dualizable presentable \$\textbackslash infty\$-category. As a particular application, given a compact oriented manifold with boundary \$\textbackslash partial M \textbackslash subset M\$, our construction gives chain-level genus zero string topology operations on the relative loop homology \$H\_\textbraceleft *\textbraceright (LM,L\textbackslash partial M)\$.},
  archiveprefix = {arXiv},
  keywords = {Mathematics - Algebraic Geometry,Mathematics - Algebraic Topology},
  file = {/home/alexk/Zotero/storage/TB79649L/Brav and Rozenblyum - 2023 - The cyclic Deligne conjecture and Calabi-Yau struc.pdf;/home/alexk/Zotero/storage/IZU5LPE8/2305.html}
}

@misc{buccisano2025,
  title = {On Derived {{D-modules}} and Their Several Definitions},
  author = {Buccisano, Carlo},
  year = 2025,
  month = oct,
  number = {arXiv:2510.15665},
  eprint = {2510.15665},
  primaryclass = {math},
  publisher = {arXiv},
  doi = {10.48550/arXiv.2510.15665},
  urldate = {2025-12-12},
  abstract = {The so called theory of derived D-modules is an extension of classical D-modules to derived algebraic geometry, which uses the derived information of the base scheme. We prove that the three different definitions of derived D-modules, given by Beraldo, Nuiten and To\"en-Vezzosi, on a (nice) derived scheme yield equivalent symmetric monoidal \$\textbackslash infty\$-categories. We deduce this as a corollary of more general statements about Chevalley-Eilenberg cohomology of dg-Lie algebroids, proving a conjecture by E. Pavia, and about the relation between representations of a dg-Lie algebroid and some class of ind-coherent sheaves on the associated formal moduli problem, which can be of independent interest.},
  archiveprefix = {arXiv},
  keywords = {Mathematics - Algebraic Geometry,Mathematics - Algebraic Topology},
  file = {/home/alexk/Zotero/storage/L3N7BIQ3/Buccisano - 2025 - On derived D-modules and their several definitions.pdf;/home/alexk/Zotero/storage/VPS9TUWB/2510.html}
}

@incollection{burghelea1986,
  title = {Cyclic Homology and the Algebraic {{K-theory}} of Spaces. {{I}}},
  booktitle = {Applications of Algebraic {{K-theory}} to Algebraic Geometry and Number Theory, {{Part I}}, {{II}} ({{Boulder}}, {{Colo}}., 1983)},
  author = {Burghelea, Dan},
  year = 1986,
  series = {Contemp. {{Math}}.},
  volume = {55},
  pages = {89--115},
  publisher = {Amer. Math. Soc., Providence, RI},
  doi = {10.1090/conm/055.1/862632},
  isbn = {978-0-8218-5054-1},
  mrnumber = {862632},
  file = {/home/alexk/Zotero/storage/AWWZQYYD/article.html}
}

@article{calaque2014,
  title = {On the {{Lie}} Algebroid of a Derived Self-Intersection},
  author = {Calaque, Damien and C{\u a}ld{\u a}raru, Andrei and Tu, Junwu},
  year = 2014,
  month = sep,
  journal = {Advances in Mathematics},
  volume = {262},
  pages = {751--783},
  issn = {0001-8708},
  doi = {10.1016/j.aim.2014.06.002},
  urldate = {2024-01-14},
  abstract = {Let i:X{$\hookrightarrow$}Y be a closed embedding of smooth algebraic varieties. Denote by N the normal bundle of X in Y. The present paper contains two constructions of certain Lie structure on the shifted normal bundle N[-1] encoding the information of the formal neighborhood of X in Y. We also present a few applications of these Lie theoretic constructions in understanding the algebraic geometry of embeddings.},
  keywords = {Deformation,Embedding,Lie algebroid},
  file = {/home/alexk/Zotero/storage/SXYW2IUV/Calaque et al. - 2014 - On the Lie algebroid of a derived self-intersectio.pdf;/home/alexk/Zotero/storage/ECZHBHZZ/S0001870814002084.html}
}

@incollection{calaque2015,
  title = {Lagrangian Structures on Mapping Stacks and Semi-Classical {{TFTs}}},
  booktitle = {Stacks and Categories in Geometry, Topology, and Algebra},
  author = {Calaque, Damien},
  year = 2015,
  series = {Contemp. {{Math}}.},
  volume = {643},
  pages = {1--23},
  publisher = {Amer. Math. Soc., Providence, RI},
  doi = {10.1090/conm/643/12894},
  isbn = {978-1-4704-1557-0},
  mrnumber = {3381468},
  file = {/home/alexk/Zotero/storage/8ZXS5MC9/Calaque - 2015 - Lagrangian structures on mapping stacks and semi-c.pdf;/home/alexk/Zotero/storage/8MRFR6NI/article.html}
}

@article{calaque2017,
  title = {Shifted {{Poisson}} Structures and Deformation Quantization},
  author = {Calaque, Damien and Pantev, Tony and To{\"e}n, Bertrand and Vaqui{\'e}, Michel and Vezzosi, Gabriele},
  year = 2017,
  journal = {Journal of Topology},
  volume = {10},
  number = {2},
  pages = {483--584},
  issn = {1753-8424},
  doi = {10.1112/topo.12012},
  urldate = {2021-09-20},
  abstract = {This paper is a sequel to `Shifted symplectic structures' [T. Pantev, B. To\"en, M. Vaqui\'e and G. Vezzosi, Publ. Math. Inst. Hautes E'tudes Sci. 117 (2013) 271--328]. We develop a general and flexible context for differential calculus in derived geometry, including the de Rham algebra and the mixed algebra of polyvector fields. We then introduce the formalism of formal derived stacks and prove formal localization and gluing results. These allow us to define shifted Poisson structures on general derived Artin stacks, and to prove that the non-degenerate Poisson structures correspond exactly to shifted symplectic forms. Shifted deformation quantization for a derived Artin stack endowed with a shifted Poisson structure is discussed in the last section. This paves the road for shifted deformation quantization of many interesting derived moduli spaces, like those studied in our earlier paper and many others.},
  langid = {english},
  keywords = {13D10,14A20,18G55 (primary)},
  file = {/home/alexk/Zotero/storage/3LYYJYYR/Calaque et al. - 2017 - Shifted Poisson structures and deformation quantiz.pdf}
}

@article{calaque2019,
  title = {Shifted Cotangent Stacks Are Shifted Symplectic},
  author = {Calaque, Damien},
  year = 2019,
  journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
  volume = {28},
  number = {1},
  pages = {67--90},
  issn = {2258-7519},
  doi = {10.5802/afst.1593},
  urldate = {2023-03-09},
  langid = {english},
  file = {/home/alexk/Zotero/storage/9EN4K6RV/Calaque - 2019 - Shifted cotangent stacks are shifted symplectic.pdf}
}

@misc{calaque2019a,
  title = {Formal Moduli Problems and Formal Derived Stacks},
  author = {Calaque, Damien and Grivaux, Julien},
  year = 2019,
  month = apr,
  number = {arXiv:1802.09556},
  eprint = {1802.09556},
  primaryclass = {math},
  publisher = {arXiv},
  doi = {10.48550/arXiv.1802.09556},
  urldate = {2024-04-02},
  abstract = {This paper presents a survey on formal moduli problems. It starts with an introduction to pointed formal moduli problems and a sketch of proof of a Theorem (independently proven by Lurie and Pridham) which gives a precise mathematical formulation for Drinfeld's derived deformation theory philosophy, which gives a correspondence between formal moduli problems and differential graded Lie algebras. The second part deals with Lurie's general theory of deformation contexts, which we present in a slightly different way than the original paper, emphasising the (more symmetric) notion of Koszul duality contexts and morphisms thereof. In the third part, we explain how to apply this machinery to the case of non-split formal moduli problems under a given derived affine scheme; this situation has been dealt with recently by Joost Nuiten, and requires to replace differential graded Lie algebras with differential graded Lie algebroids. In the last part, we globalize this to the more general setting of formal thickenings of derived stacks, and suggest an alternative approach to results of Gaitsgory and Rozenblyum.},
  archiveprefix = {arXiv},
  keywords = {Mathematics - Algebraic Geometry,Mathematics - Algebraic Topology},
  file = {/home/alexk/Zotero/storage/3I68AN4W/Calaque and Grivaux - 2019 - Formal moduli problems and formal derived stacks.pdf;/home/alexk/Zotero/storage/9PFQJ7NE/1802.html}
}

@misc{calaque2024,
  title = {Shifted Cotangent Bundles, Symplectic Groupoids and Deformation to the Normal Cone},
  author = {Calaque, Damien and Safronov, Pavel},
  year = 2024,
  month = jul,
  number = {arXiv:2407.08622},
  eprint = {2407.08622},
  primaryclass = {math},
  urldate = {2024-07-13},
  abstract = {This article generalizes the theory of shifted symplectic structures to the relative context and non-geometric stacks. We describe basic constructions that naturally appear in this theory: shifted cotangent bundles and the AKSZ procedure. Along the way, we also develop the theory of shifted symplectic groupoids presenting shifted symplectic structures on quotients and define a deformation to the normal cone for shifted Lagrangian morphisms.},
  archiveprefix = {arXiv},
  keywords = {Mathematics - Algebraic Geometry,Mathematics - Symplectic Geometry},
  file = {/home/alexk/Zotero/storage/HA72GSDC/Calaque and Safronov - 2024 - Shifted cotangent bundles, symplectic groupoids and deformation to the normal cone.pdf}
}

@article{calaque2025,
  title = {The {{AKSZ}} Construction in Derived Algebraic Geometry as an Extended Topological Field Theory},
  author = {Calaque, Damien and Haugseng, Rune and Scheimbauer, Claudia},
  year = 2025,
  journal = {Memoirs of the American Mathematical Society},
  volume = {308},
  number = {1555},
  pages = {v+173},
  issn = {0065-9266,1947-6221},
  doi = {10.1090/memo/1555},
  isbn = {9781470472726 9781470481421},
  mrnumber = {4896164},
  file = {/home/alexk/Zotero/storage/SCXRIE4F/Calaque et al. - 2025 - The AKSZ construction in derived algebraic geometr.pdf;/home/alexk/Zotero/storage/FEL2US34/article.html}
}

@inproceedings{cattaneo2001,
  title = {Poisson Sigma Models and Symplectic Groupoids},
  booktitle = {Quantization of {{Singular Symplectic Quotients}}},
  author = {Cattaneo, Alberto S. and Felder, Giovanni},
  editor = {Landsman, N. P. and Pflaum, M. and Schlichenmaier, M.},
  year = 2001,
  pages = {61--93},
  publisher = {Birkh\"auser},
  address = {Basel},
  doi = {10.1007/978-3-0348-8364-1_4},
  abstract = {We consider the Poisson sigma model associated to a Poisson manifold. The perturbative quantization of this model yields the Kontsevich star product formula. We study here the classical model in the Hamiltonian formalism. The phase space is the space of leaves of a Hamiltonian foliation and has a natural groupoid structure. If it is a manifold then it is a symplectic groupoid for the given Poisson manifold. We study various families of examples. In particular, a global symplectic groupoid for a general class of two-dimensional Poisson domains is constructed.},
  isbn = {978-3-0348-8364-1},
  langid = {english},
  keywords = {Cotangent Bundle,Gauge Transformation,Poisson Manifold,Poisson Structure,Symplectic Structure},
  file = {/home/alexk/Zotero/storage/ZJ4S8Z3Z/Cattaneo and Felder - 2001 - Poisson sigma models and symplectic groupoids.pdf}
}

@misc{chen2025,
  title = {\${{A}}\_\textbackslash infty\$ {{Sabloff Duality}} via the {{LSFT Algebra}}},
  author = {Chen, Zhenyi},
  year = 2025,
  month = may,
  number = {arXiv:2410.20523},
  eprint = {2410.20523},
  primaryclass = {math},
  publisher = {arXiv},
  doi = {10.48550/arXiv.2410.20523},
  urldate = {2025-09-04},
  abstract = {We use Ng's LSFT algebra to upgrade Sabloff duality of Legendrian knots to a quasi-isomorphism of \$A\_\textbackslash infty\$ bimodules over the positive augmentation category \$\textbackslash mathcal\textbraceleft A\textbraceright ug\_+\$. We also extend the Ekholm-Etnyre-Sabloff exact sequence to an exact sequence of \$\textbackslash mathcal\textbraceleft A\textbraceright ug\_+\$-bimodules, using a quotient category \$\textbackslash mathcal\textbraceleft C\textbraceright\$ of short Reeb chords. In addition, we define curved augmentations of the LSFT algebra and show that they can be used to construct a homotopy inverse of the \$A\_\textbackslash infty\$ Sabloff map, together with all higher homotopies. The above results suggest a conjectural recipe for an explicit weak relative Calabi-Yau structure on the quotient \$A\_\textbackslash infty\$ functor \$\textbackslash pi:\textbackslash mathcal\textbraceleft A\textbraceright ug\_+\textbackslash to \textbackslash mathcal\textbraceleft C\textbraceright\$.},
  archiveprefix = {arXiv},
  keywords = {Mathematics - Geometric Topology,Mathematics - Symplectic Geometry},
  file = {/home/alexk/Zotero/storage/X8ZPSMHR/Chen - 2025 - $A_infty$ Sabloff Duality via the LSFT Algebra.pdf;/home/alexk/Zotero/storage/GCRVKRLA/2410.html}
}

@article{ciocan-fontanine2001,
  title = {Derived Quot Schemes},
  author = {{Ciocan-Fontanine}, Ionu{\c t} and Kapranov, Mikhail},
  year = 2001,
  month = may,
  journal = {Annales Scientifiques de l'\'Ecole Normale Sup\'erieure},
  volume = {34},
  number = {3},
  pages = {403--440},
  issn = {0012-9593},
  doi = {10.1016/S0012-9593(01)01064-3},
  urldate = {2021-09-20},
  abstract = {We construct a ``derived'' version of Grothendieck's Quot scheme which is a dg-scheme, i.e., an object RQuot of a certain nonabelian right derived category of schemes. It has the property of being manifestly smooth in an appropriate sense (whereas the usual Quot scheme is often singular). The usual scheme Quot is obtained from RQuot by degree 0 truncation. The construction of RQuot can be seen as realization of a part of the Derived Deformation Theory program, which proposes to replace all the moduli spaces arising in geometry by their derived versions by retaining the information about all the higher cohomology instead of H1 in the classical theory. R\'esum\'e On construit une version ``d\'eriv\'ee'' du sch\'ema Quot de Grothendieck. C'est un dg-sch\'ema, i.e., un objet RQuot d'une certaine cat\'egorie d\'eriv\'ee non-ab\'elienne \`a droite des sch\'emas usuels. Il est toujours lisse en un sens convenable (tandis que le sch\'ema Quot classique peut \^etre singulier). On peut obtenir le sch\'ema Quot de RQuot par troncature en degr\'e 0. La construction de RQuot peut \^etre vue comme r\'ealisation d'une part du programme des d\'eformations d\'eriv\'ees qui cherche \`a remplacer tous les espaces de modules en g\'eom\'etrie par leurs versions d\'eriv\'ees qui font usage de la cohomologie sup\'erieure, m\^eme au niveau des espaces tangents qui en th\'eorie traditionnelle s'interpr\`etent comme certains H1.},
  langid = {english},
  file = {/home/alexk/Zotero/storage/XLBHDDSJ/Ciocan-Fontanine and Kapranov - 2001 - Derived quot schemes.pdf;/home/alexk/Zotero/storage/EQTJYV3J/S0012959301010643.html}
}

@article{ciocan-fontanine2002,
  title = {Derived {{Hilbert}} Schemes},
  author = {{Ciocan-Fontanine}, Ionu{\c t} and Kapranov, Mikhail},
  year = 2002,
  journal = {Journal of the American Mathematical Society},
  volume = {15},
  number = {4},
  pages = {787--815},
  issn = {0894-0347, 1088-6834},
  doi = {10.1090/S0894-0347-02-00399-5},
  urldate = {2021-09-20},
  abstract = {We construct the derived version of the Hilbert scheme parametrizing subschemes in a given projective scheme X with given Hilbert polynomial h. This is a dg-manifold (smooth dg-scheme) R⁢H⁢i⁢l⁢bh⁡(X) which carries a natural family of commutative (up to homotopy) dg-algebras, which over the usual Hilbert scheme is given by truncations of the homogeneous coordinate rings of subschemes in X. In particular, R⁢H⁢i⁢l⁢bh⁡(X) differs from R⁢Q⁢u⁢o⁢tn⁡(OX), the derived Quot scheme constructed in our previous paper, which carries only a family of A{$\infty$}-modules over the coordinate algebra of X. As an application, we construct the derived version of the moduli stack of stable maps of algebraic curves to a given projective variety Y, thus realizing the original suggestion of M. Kontsevich.},
  langid = {english},
  file = {/home/alexk/Zotero/storage/7X7GZVX2/Ciocan-Fontanine and Kapranov - 2002 - Derived Hilbert schemes.pdf;/home/alexk/Zotero/storage/DSKUCWSC/S0894-0347-02-00399-5.html}
}

@book{costello2021,
  title = {Factorization Algebras in Quantum Field Theory. {{Vol}}. 2},
  author = {Costello, Kevin and Gwilliam, Owen},
  year = 2021,
  series = {New {{Mathematical Monographs}}},
  volume = {41},
  publisher = {Cambridge University Press, Cambridge},
  doi = {10.1017/9781316678664},
  isbn = {978-1-107-16315-7 978-1-009-00616-3},
  mrnumber = {4300181},
  file = {/home/alexk/Zotero/storage/IAZWS93S/article.html}
}

@article{crainic2003,
  title = {Integrability of {{Lie}} Brackets},
  author = {Crainic, Marius and Fernandes, Rui Loja},
  year = 2003,
  journal = {Annals of Mathematics. Second Series},
  volume = {157},
  number = {2},
  pages = {575--620},
  issn = {0003-486X,1939-8980},
  doi = {10.4007/annals.2003.157.575},
  mrnumber = {1973056},
  file = {/home/alexk/Zotero/storage/I78NWURT/Crainic and Fernandes - 2003 - Integrability of Lie brackets.pdf;/home/alexk/Zotero/storage/LINBG56B/article.html}
}

@article{crainic2004,
  title = {Integrability of {{Poisson}} Brackets},
  author = {Crainic, Marius and Fernandes, Rui Loja},
  year = 2004,
  journal = {Journal of Differential Geometry},
  volume = {66},
  number = {1},
  pages = {71--137},
  issn = {0022-040X,1945-743X},
  mrnumber = {2128714},
  file = {/home/alexk/Zotero/storage/NMMDTDSK/article.html}
}

@article{darboux1882,
  title = {{Sur le probl\`eme de Pfaff}},
  author = {Darboux, G.},
  year = 1882,
  journal = {Bulletin des Sciences Math\'ematiques et Astronomiques},
  volume = {6},
  number = {1},
  pages = {14--36},
  publisher = {Gauthier-Villars},
  issn = {1155-8431},
  urldate = {2025-09-01},
  langid = {fra},
  file = {/home/alexk/Zotero/storage/CERLALZ8/85135.html}
}

@article{deligne1971,
  title = {Th\'eorie de {{Hodge}}. {{II}}},
  author = {Deligne, Pierre},
  year = 1971,
  journal = {Institut des Hautes \'Etudes Scientifiques. Publications Math\'ematiques},
  number = {40},
  pages = {5--57},
  issn = {0073-8301,1618-1913},
  mrnumber = {498551},
  file = {/home/alexk/Zotero/storage/N9ZC7M3V/article.html}
}

@article{deligne1974,
  title = {Th\'eorie de {{Hodge}}. {{III}}},
  author = {Deligne, Pierre},
  year = 1974,
  journal = {Institut des Hautes \'Etudes Scientifiques. Publications Math\'ematiques},
  number = {44},
  pages = {5--77},
  issn = {0073-8301,1618-1913},
  mrnumber = {498552},
  file = {/home/alexk/Zotero/storage/A8QN2L75/article.html}
}

@article{dwyer2002,
  title = {Complete {{Modules}} and {{Torsion Modules}}},
  author = {Dwyer, W. G. and Greenless, J. P. C.},
  year = 2002,
  journal = {American Journal of Mathematics},
  volume = {124},
  number = {1},
  eprint = {25099110},
  eprinttype = {jstor},
  pages = {199--220},
  publisher = {Johns Hopkins University Press},
  issn = {0002-9327},
  urldate = {2026-08-11},
  abstract = {Suppose that R is a ring and that A is a chain complex over R. Inside the derived category of differential graded R-modules there are naturally defined subcategories of A-torsion objects and of A-complete objects. Under a finiteness condition on A, we develop a Morita theory for these subcategories, find conceptual interpretations for some associated algebraic functors, and, in appropriate commutative situations, identify the associated functors as local homology or local cohomology. Some of the results are suprising even in the case \$R=\textbraceleft\textbackslash Bbb Z\textbraceright\$ and \$A=\textbraceleft\textbackslash Bbb Z\textbraceright/p\$.},
  file = {/home/alexk/Zotero/storage/I8HRG6HK/Dwyer and Greenless - 2002 - Complete Modules and Torsion Modules.pdf}
}

@incollection{felder2014,
  title = {The Classical Master Equation},
  booktitle = {Perspectives in Representation Theory},
  author = {Felder, Giovanni and Kazhdan, David},
  year = 2014,
  series = {Contemp. {{Math}}.},
  volume = {610},
  pages = {79--137},
  publisher = {Amer. Math. Soc., Providence, RI},
  doi = {10.1090/conm/610/12124},
  isbn = {978-0-8218-9170-4},
  mrnumber = {3220627},
  file = {/home/alexk/Zotero/storage/Z98PUFQK/Felder and Kazhdan - 2014 - The classical master equation.pdf;/home/alexk/Zotero/storage/B7CCRGUK/article.html}
}

@misc{fu2024,
  title = {A Duality between {{Lie}} Algebroids and Infinitesimal Foliations},
  author = {Fu, Jiaqi},
  year = 2024,
  month = dec,
  number = {arXiv:2410.04950},
  eprint = {2410.04950},
  primaryclass = {math},
  publisher = {arXiv},
  doi = {10.48550/arXiv.2410.04950},
  urldate = {2026-03-05},
  abstract = {There are two natural analogues of algebraic foliations in derived algebraic geometry, called partition Lie algebroids and infinitesimal derived foliations, and both make sense in general characteristics. We construct an equivalence between these two notions under some finiteness conditions. Our method is refining the PD Koszul duality in \textbackslash cite\textbraceleft BM\textbraceright\textbackslash cite\textbraceleft BCN\textbraceright{} using the (completed) Hodge filtration.},
  archiveprefix = {arXiv},
  keywords = {Mathematics - Algebraic Geometry,Mathematics - Algebraic Topology},
  file = {/home/alexk/Zotero/storage/CLR83Y2U/Fu - 2024 - A duality between Lie algebroids and infinitesimal foliations.pdf}
}

@article{gaitsgory,
  title = {A Study in Derived Algebraic Geometry {{Volume II}}: {{Deformations}}, {{Lie}} Theory and Formal Geometry},
  author = {Gaitsgory, Dennis and Rozenblyum, Nick},
  langid = {english},
  file = {/home/alexk/Zotero/storage/VJAHSTSZ/Gaitsgory and Rozenblyum - A study in derived algebraic geometry Volume II Deformations, Lie theory and formal geometry.pdf}
}

@article{gaitsgory2014,
  title = {Crystals and {{D-modules}}},
  author = {Gaitsgory, Dennis and Rozenblyum, Nick},
  year = 2014,
  month = jan,
  journal = {Pure and Applied Mathematics Quarterly},
  volume = {10},
  number = {1},
  pages = {57--154},
  publisher = {International Press of Boston},
  issn = {1558-8602},
  doi = {10.4310/PAMQ.2014.v10.n1.a2},
  urldate = {2024-02-13},
  abstract = {International Press of Boston - publishers of scholarly mathematical and scientific journals and books},
  langid = {english},
  file = {/home/alexk/Zotero/storage/KBSXXME4/Gaitsgory and Rozenblyum - 2014 - Crystals and D-modules.pdf}
}

@book{gaitsgory2017,
  title = {A {{Study}} in {{Derived Algebraic Geometry}}, {{Part}} 1: {{Volume I}}: {{Correspondences}} and {{Duality}}},
  shorttitle = {A {{Study}} in {{Derived Algebraic Geometry}}, {{Part}} 1},
  author = {Gaitsgory, Dennis and Rozenblyum, Nick},
  year = 2017,
  month = jul,
  series = {Mathematical {{Surveys}} and {{Monographs}}},
  volume = {221.1},
  publisher = {American Mathematical Society},
  issn = {0076-5376, 2331-7159},
  doi = {10.1090/surv/221.1},
  urldate = {2024-06-06},
  abstract = {Advancing research. Creating connections.},
  isbn = {978-1-4704-3569-1 978-1-4704-4085-5 978-1-4704-4086-2},
  langid = {english}
}

@book{gaitsgory2017a,
  title = {A {{Study}} in {{Derived Algebraic Geometry}}, {{Part}} 2: {{Volume II}}: {{Deformations}}, {{Lie Theory}} and {{Formal Geometry}}},
  shorttitle = {A {{Study}} in {{Derived Algebraic Geometry}}, {{Part}} 2},
  author = {Gaitsgory, Dennis and Rozenblyum, Nick},
  year = 2017,
  month = aug,
  series = {Mathematical {{Surveys}} and {{Monographs}}},
  volume = {221.2},
  publisher = {American Mathematical Society},
  issn = {0076-5376, 2331-7159},
  doi = {10.1090/surv/221.2},
  urldate = {2024-06-06},
  abstract = {Advancing research. Creating connections.},
  isbn = {978-1-4704-4087-9 978-1-4704-4088-6 978-1-4704-3570-7},
  langid = {english}
}

@misc{gaitsgory2023,
  title = {{{DG Indschemes}}},
  author = {Gaitsgory, Dennis and Rozenblyum, Nick},
  year = 2023,
  month = mar,
  number = {arXiv:1108.1738},
  eprint = {1108.1738},
  publisher = {arXiv},
  doi = {10.48550/arXiv.1108.1738},
  urldate = {2024-11-08},
  abstract = {We develop the notion of indscheme in the context of derived algebraic geometry, and study the categories of quasi-coherent sheaves and ind-coherent sheaves on indschemes. The main results concern the relation between classical and derived indschemes and the notion of formal smoothness.},
  archiveprefix = {arXiv},
  keywords = {Mathematics - Algebraic Geometry},
  file = {/home/alexk/Zotero/storage/LW447XMA/Gaitsgory and Rozenblyum - 2023 - DG Indschemes.pdf;/home/alexk/Zotero/storage/CUL35MIC/1108.html}
}

@article{gelfand1971,
  title = {{Some problems of differential geometry and the calculation of cohomologies of Lie algebras of vector fields}},
  author = {Gel'fand, I M and Kazhdan, D A},
  year = 1971,
  journal = {Doklady Akademii Nauk SSSR},
  volume = {200},
  number = {2},
  pages = {269--272},
  langid = {russian},
  file = {/home/alexk/Zotero/storage/LDYN5QMX/Gel'fand and Kazhdan - Some problems of differential geometry and the cal.pdf}
}

@misc{getzler,
  title = {In Preparation},
  author = {Getzler, Ezra}
}

@article{getzler2009,
  title = {Lie Theory for Nilpotent {{L}} {\textsubscript{{$\infty$}}} -Algebras},
  author = {Getzler, Ezra},
  year = 2009,
  month = jul,
  journal = {Annals of Mathematics},
  volume = {170},
  number = {1},
  pages = {271--301},
  issn = {0003-486X},
  doi = {10.4007/annals.2009.170.271},
  urldate = {2022-10-12},
  abstract = {The Deligne groupoid is a functor from nilpotent differential graded Lie algebras concentrated in positive degrees to groupoids; in the special case of Lie algebras over a field of characteristic zero, it gives the associated simply connected Lie group. We generalize the Deligne groupoid to a functor from L1-algebras concentrated in degree {$>$} n to n-groupoids. (We actually construct the nerve of the n-groupoid, which is an enriched Kan complex.) The construction of gamma is quite explicit (it is based on Dupont's proof of the de Rham theorem) and yields higher dimensional analogues of holonomy and of the Campbell-Hausdorff formula.},
  langid = {english},
  file = {/home/alexk/Zotero/storage/7378D39G/Getzler - 2009 - Lie theory for nilpotent L ∞ -algebras.pdf}
}

@article{getzler2017,
  title = {The Derived {{Maurer}}--{{Cartan}} Locus},
  author = {Getzler, Ezra},
  year = 2017,
  month = jan,
  journal = {L'Enseignement Math\'ematique},
  volume = {62},
  number = {1},
  pages = {261--284},
  issn = {0013-8584},
  doi = {10.4171/lem/62-1/2-14},
  urldate = {2022-10-12},
  abstract = {The derived Maurer-Cartan locus is a functor \$\textbackslash mathrm \textbraceleft MC\textbraceright\textasciicircum\textbackslash bullet\$ from differential graded Lie algebras to cosimplicial schemes. If \$L\$ is a differential graded Lie algebra, let \$L\textbackslash\_+\$ be the truncation of \$L\$ in positive degrees \$i {$>$} 0\$. We prove that the differential graded algebra of functions on the cosimplicial scheme \$\textbackslash mathrm \textbraceleft MC\textbraceright\textasciicircum\textbackslash bullet(L)\$ is quasi-isomorphic to the Chevalley-Eilenberg complex of \$L\textbackslash\_+\$.},
  langid = {english},
  file = {/home/alexk/Zotero/storage/BHETQ95D/Getzler - 2017 - The derived Maurer–Cartan locus.pdf;/home/alexk/Zotero/storage/223LLQ8B/14585.html}
}

@misc{ginzburg2018,
  title = {Gaiotto's {{Lagrangian}} Subvarieties via Derived Symplectic Geometry},
  author = {Ginzburg, Victor and Rozenblyum, Nick},
  year = 2018,
  month = may,
  number = {arXiv:1703.08578},
  eprint = {1703.08578},
  primaryclass = {math},
  publisher = {arXiv},
  doi = {10.48550/arXiv.1703.08578},
  urldate = {2025-09-05},
  abstract = {Let Bun\_G be the moduli space of G-bundles on a smooth complex projective curve. Motivated by a study of boundary conditions in mirror symmetry, D. Gaiotto associated to any symplectic representation of G a Lagrangian subvariety of the cotangent bundle of Bun\_G. We give a simple interpretation of (a generalization of) Gaiotto's construction in terms of derived symplectic geometry. This allows to consider a more general setting where symplectic G-representations are replaced by arbitrary symplectic manifolds equipped with a Hamiltonian G-action and with an action of the multiplicative group that rescales the symplectic form with positive weight.},
  archiveprefix = {arXiv},
  keywords = {Mathematical Physics,Mathematics - Algebraic Geometry,Mathematics - Mathematical Physics},
  file = {/home/alexk/Zotero/storage/KLCBDFK8/Ginzburg and Rozenblyum - 2018 - Gaiotto's Lagrangian subvarieties via derived symplectic geometry.pdf;/home/alexk/Zotero/storage/GHWX6GBM/1703.html}
}

@article{goldman1984,
  title = {The Symplectic Nature of Fundamental Groups of Surfaces},
  author = {Goldman, William M.},
  year = 1984,
  journal = {Advances in Mathematics},
  volume = {54},
  number = {2},
  pages = {200--225},
  issn = {0001-8708},
  doi = {10.1016/0001-8708(84)90040-9},
  mrnumber = {762512},
  file = {/home/alexk/Zotero/storage/GTZGRHDE/Goldman - 1984 - The symplectic nature of fundamental groups of sur.pdf;/home/alexk/Zotero/storage/ZIFFQSQK/article.html}
}

@phdthesis{grataloup2022,
  title = {Derived {{Symplectic Reduction}} and {{Equivariant Geometry}}},
  author = {Grataloup, Albin},
  year = 2022,
  month = dec,
  urldate = {2025-09-01},
  abstract = {This thesis aims to generalize the procedure of symplectic reduction to the setting of derived geometry, for infinitesimal actions (actions by Lie algebroids) and for actions by Segal groupoids.One of the goals is to relate this geometric construction to the BV formalism.},
  langid = {english},
  school = {Universit\'e de Montpellier},
  file = {/home/alexk/Zotero/storage/YX5PZK3A/Grataloup - 2022 - Derived Symplectic Reduction and Equivariant Geome.pdf}
}

@article{greenlees1992,
  title = {Derived Functors of {{{\emph{I}}}}-Adic Completion and Local Homology},
  author = {Greenlees, J. P. C and May, J. P},
  year = 1992,
  month = jul,
  journal = {Journal of Algebra},
  volume = {149},
  number = {2},
  pages = {438--453},
  issn = {0021-8693},
  doi = {10.1016/0021-8693(92)90026-I},
  urldate = {2026-08-11},
  file = {/home/alexk/Zotero/storage/3NLJ7NIM/Greenlees and May - 1992 - Derived functors of I-adic completion and local homology.pdf;/home/alexk/Zotero/storage/IY2DDPAZ/002186939290026I.html}
}

@article{gukov2009,
  title = {Branes and Quantization},
  author = {Gukov, Sergei and Witten, Edward},
  year = 2009,
  journal = {Advances in Theoretical and Mathematical Physics},
  volume = {13},
  number = {5},
  pages = {1445--1518},
  issn = {1095-0761,1095-0753},
  doi = {10.4310/atmp.2009.v13.n5.a5},
  mrnumber = {2672467},
  file = {/home/alexk/Zotero/storage/M8MN487C/Gukov and Witten - 2009 - Branes and quantization.pdf;/home/alexk/Zotero/storage/2EMKNWVM/article.html}
}

@misc{heine2025,
  title = {A Derived {{Milnor-Moore}} Theorem},
  author = {Heine, Hadrian},
  year = 2025,
  month = aug,
  number = {arXiv:2408.06917},
  eprint = {2408.06917},
  primaryclass = {math},
  publisher = {arXiv},
  doi = {10.48550/arXiv.2408.06917},
  urldate = {2026-03-25},
  abstract = {For every stable presentably symmetric monoidal \$\textbackslash infty\$-category \$\textbackslash mathcal\textbraceleft C\textbraceright\$ we use the Koszul duality between the spectral Lie operad and the cocommutative cooperad to construct an enveloping Hopf algebra functor \$\textbackslash mathcal\textbraceleft U\textbraceright : \textbackslash mathrm\textbraceleft Alg\textbraceright\_\textbraceleft\textbackslash mathrm\textbraceleft Lie\textbraceright\textbraceright (\textbackslash mathcal\textbraceleft C\textbraceright ) \textbackslash to \textbackslash mathrm\textbraceleft Hopf\textbraceright (\textbackslash mathcal\textbraceleft C\textbraceright )\$ from Lie algebras in \$\textbackslash mathcal\textbraceleft C\textbraceright\$ to cocommutative Hopf algebras in \$\textbackslash mathcal\textbraceleft C\textbraceright\$ left adjoint to a functor of derived primitive elements \$\textbackslash mathrm\textbraceleft Prim\textbraceright\$. We study the unit of this adjunction in rational and chromatic homotopy theory: we prove that if \$\textbackslash mathcal\textbraceleft C\textbraceright\$ is a rational stable presentably symmetric monoidal \$\textbackslash infty\$-category, the enveloping Hopf algebra functor \$\textbackslash mathcal\textbraceleft U\textbraceright : \textbackslash mathrm\textbraceleft Alg\textbraceright\_\textbraceleft\textbackslash mathrm\textbraceleft Lie\textbraceright\textbraceright (\textbackslash mathcal\textbraceleft C\textbraceright ) \textbackslash to \textbackslash mathrm\textbraceleft Hopf\textbraceright (\textbackslash mathcal\textbraceleft C\textbraceright )\$ is fully faithful reproving a result of Gaitsgory-Rozenblyum. Let \$n \textbackslash geq 1 \$ be a natural and \${$\Phi$}[-1]: \textbackslash mathcal\textbraceleft S\textbraceright\_\textbraceleft v\_n\textbraceright{} \textbackslash to \textbackslash mathrm\textbraceleft Alg\textbraceright\_\textbraceleft\textbackslash mathrm\textbraceleft Lie\textbraceright\textbraceright (\textbackslash mathrm\textbraceleft Sp\textbraceright\_\textbraceleft T\_n\textbraceright )\$ the shifted Bousfield-Kuhn functor from \$v\_n\$-periodic homotopy types to spectral Lie algebras in \$T\_n\$-local spectra. We prove that for every \$v\_n\$-periodic homotopy type \$X\$ the unit \${$\Phi$}(X)[-1] \textbackslash to Prim \textbackslash mathcal\textbraceleft U\textbraceright ({$\Phi$}(X)[-1])\$ identifies with the Goodwillie completion \$ {$\Phi\backslash$}to \textbackslash lim\_\textbraceleft n \textbackslash geq 0\textbraceright{} P\_n({$\Phi$})\$ evaluated at the loop space of \$X.\$},
  archiveprefix = {arXiv},
  keywords = {Mathematics - Algebraic Topology},
  file = {/home/alexk/Zotero/storage/W3JEQ6A2/Heine - 2025 - A derived Milnor-Moore theorem.pdf;/home/alexk/Zotero/storage/TI83K9CK/2408.html}
}

@misc{hennion2025,
  title = {Gluing Invariants of {{Donaldson--Thomas}} Type -- {{Part I}}: The {{Darboux}} Stack},
  shorttitle = {Gluing Invariants of {{Donaldson--Thomas}} Type -- {{Part I}}},
  author = {Hennion, Benjamin and Holstein, Julian and Robalo, Marco},
  year = 2025,
  month = mar,
  number = {arXiv:2407.08471},
  eprint = {2407.08471},
  primaryclass = {math},
  publisher = {arXiv},
  doi = {10.48550/arXiv.2407.08471},
  urldate = {2025-09-01},
  abstract = {Let \$X\$ be a (-1)-shifted symplectic derived Deligne--Mumford stack. In this paper we introduce the Darboux stack of \$X\$, parametrizing local presentations of \$X\$ as a derived critical locus of a function \$f\$ on a smooth formal scheme \$U\$. Local invariants such as the Milnor number \$\textbackslash mu\_f\$, the perverse sheaf of vanishing cycles \$\textbackslash mathsf\textbraceleft P\textbraceright\_\textbraceleft U,f\textbraceright\$ and the category of matrix factorizations \$\textbackslash mathsf\textbraceleft MF\textbraceright (U,f)\$ are naturally defined on the Darboux stack, without ambiguity. The stack of non-degenerate flat quadratic bundles acts on the Darboux stack and our main theorem is the contractibility of the quotient stack when taking a further homotopy quotient identifying isotopic automorphisms. As a corollary we recover the gluing results for vanishing cycles by Brav--Bussi--Dupont--Joyce--Szendr\textbackslash H oi. In a second part (to appear), we will apply this general mechanism to glue the motives of the locally defined categories of matrix factorizations \$\textbackslash mathsf\textbraceleft MF\textbraceright (U,f)\$ under the prescription of additional orientation data, thus answering positively conjectures by Kontsevich--Soibelman and Toda in motivic Donaldson--Thomas theory.},
  archiveprefix = {arXiv},
  keywords = {Mathematical Physics,Mathematics - Algebraic Geometry,Mathematics - Mathematical Physics,Mathematics - Representation Theory},
  file = {/home/alexk/Zotero/storage/Y42UQGHS/Hennion et al. - 2025 - Gluing invariants of Donaldson--Thomas type -- Par.pdf;/home/alexk/Zotero/storage/9HBJNPZ8/2407.html}
}

@article{hua2018,
  title = {Shifted {{Poisson}} Structures and Moduli Spaces of Complexes},
  author = {Hua, Zheng and Polishchuk, Alexander},
  year = 2018,
  journal = {Advances in Mathematics},
  volume = {338},
  pages = {991--1037},
  issn = {0001-8708,1090-2082},
  doi = {10.1016/j.aim.2018.09.018},
  mrnumber = {3861721},
  file = {/home/alexk/Zotero/storage/Z3C9NYEP/Hua and Polishchuk - 2018 - Shifted Poisson structures and moduli spaces of co.pdf;/home/alexk/Zotero/storage/C3Q2SHZV/article.html}
}

@article{huebschmann2011,
  title = {The Sh-{{Lie}} Algebra Perturbation Lemma},
  author = {Huebschmann, Johannes},
  year = 2011,
  month = jul,
  volume = {23},
  number = {4},
  pages = {669--691},
  publisher = {De Gruyter},
  issn = {1435-5337},
  doi = {10.1515/form.2011.023},
  urldate = {2024-08-09},
  abstract = {Let R be a commutative ring which contains the rationals as a subring and let g be a chain complex. Suppose given an sh-Lie algebra structure on g , that is, a coalgebra perturbation of the coalgebra differential on the cofree coaugmented differential graded cocommutative coalgebra T {$\prime$} on the suspension of g and write the perturbed coalgebra as T {${''}$}. Suppose, furthermore, given a contraction of g onto a chain complex M . We show that the data determine an sh-Lie algebra structure on M , that is, a coalgebra perturbation of the coalgebra differential on the cofree coaugmented differential graded cocommutative coalgebra S {$\prime$} on the suspension of M , a Lie algebra twisting cochain from the perturbed coalgebra S {${''}$} to the loop Lie algebra L on the perturbed coalgebra T {${''}$}, and an extension of this Lie algebra twisting cochain to a contraction of chain complexes from the Cartan--Chevalley--Eilenberg coalgebra on L onto S {${''}$} which is natural in the data. For the special case where M and g are connected we also construct an explicit extension of the perturbed retraction to an sh-Lie map. This approach includes a very general solution of the master equation.},
  chapter = {Forum Mathematicum},
  copyright = {De Gruyter expressly reserves the right to use all content for commercial text and data mining within the meaning of Section 44b of the German Copyright Act.},
  langid = {english},
  keywords = {Cartan-Chevalley-Eilenberg coalgebra,contraction,higher homotopies,Lie algebra perturbation,Lie algebra twisting cochain,master equation,perturbation lemma,sh-Lie algebra},
  file = {/home/alexk/Zotero/storage/X8SX22GC/Huebschmann - 2011 - The sh-Lie algebra perturbation lemma.pdf}
}

@article{joyce2019,
  title = {A {{Lagrangian}} Neighbourhood Theorem for Shifted Symplectic Derived Schemes},
  author = {Joyce, Dominic and Safronov, Pavel},
  year = 2019,
  journal = {Annales de la Facult\'e des Sciences de Toulouse. Math\'ematiques. S\'erie 6},
  volume = {28},
  number = {5},
  pages = {831--908},
  issn = {0240-2963,2258-7519},
  doi = {10.5802/afst.1616},
  mrnumber = {4093980},
  file = {/home/alexk/Zotero/storage/4WLV2JWK/Joyce and Safronov - 2019 - A Lagrangian neighbourhood theorem for shifted sym.pdf;/home/alexk/Zotero/storage/QBHGNZXF/article.html}
}

@article{kapranov2001,
  title = {Injective Resolutions of {{BG}} and Derived Moduli Spaces of Local Systems},
  author = {Kapranov, M.},
  year = 2001,
  month = jan,
  journal = {Journal of Pure and Applied Algebra},
  volume = {155},
  number = {2},
  pages = {167--179},
  issn = {0022-4049},
  doi = {10.1016/S0022-4049(99)00109-7},
  urldate = {2021-09-20},
  abstract = {It was suggested on several occasions by Deligne, Drinfeld and Kontsevich that all the moduli spaces arising in the classical problems of deformation theory should be extended to natural ``derived'' moduli spaces which are always smooth in an appropriate sense and whose tangent spaces involve the entire cohomology of the sheaf of infinitesimal automorphisms, not just H1. In this note we give an algebraic construction of such an extension for the simplest class of moduli spaces, namely for moduli of local systems (representations of the fundamental group).},
  langid = {english},
  file = {/home/alexk/Zotero/storage/4DVX3IPK/Kapranov - 2001 - Injective resolutions of BG and derived moduli spa.pdf}
}

@phdthesis{karapetyan,
  title = {Darboux Theorems in Derived Algebraic Geometry, {{In}} Preparation},
  author = {Karapetyan, Alex}
}

@misc{karapetyana,
  title = {Formal Groups and Shifted {{Poisson}} Geometry, {{In}} Preparation},
  author = {Karapetyan, Alex}
}

@article{karasev1986,
  title = {Analogues of Objects of the Theory of {{Lie}} Groups for Nonlinear {{Poisson}} Brackets},
  author = {Karas{\"e}v, M. V.},
  year = 1986,
  journal = {Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya},
  volume = {50},
  number = {3},
  pages = {508--538, 638},
  issn = {0373-2436},
  mrnumber = {854594},
  file = {/home/alexk/Zotero/storage/A8ZGSB9P/article.html}
}

@article{khudaverdian2008,
  title = {Higher {{Poisson Brackets}} and {{Differential Forms}}},
  author = {Khudaverdian, H. M. and Voronov, {\relax Th}. {\relax Th}.},
  year = 2008,
  month = nov,
  journal = {AIP Conference Proceedings},
  volume = {1079},
  number = {1},
  pages = {203--215},
  publisher = {American Institute of Physics},
  issn = {0094-243X},
  doi = {10.1063/1.3043861},
  urldate = {2022-12-04},
  file = {/home/alexk/Zotero/storage/83TPKN35/Khudaverdian and Voronov - 2008 - Higher Poisson Brackets and Differential Forms.pdf}
}

@article{kirillov1962,
  title = {Unitary Representations of Nilpotent {{Lie}} Groups},
  author = {Kirillov, A. A.},
  year = 1962,
  month = aug,
  journal = {Russian Mathematical Surveys},
  volume = {17},
  number = {4},
  pages = {53},
  publisher = {IOP Publishing},
  issn = {0036-0279},
  doi = {10.1070/RM1962v017n04ABEH004118},
  urldate = {2025-09-03},
  abstract = {Unitary representations of nilpotent Lie groups, A A Kirillov},
  langid = {english}
}

@misc{kochan2004,
  title = {Differential Gorms, Differential Worms},
  author = {Kochan, Denis and Severa, Pavol},
  year = 2004,
  month = jan,
  number = {arXiv:math/0307303},
  eprint = {math/0307303},
  publisher = {arXiv},
  doi = {10.48550/arXiv.math/0307303},
  urldate = {2026-03-31},
  abstract = {We study "higher-dimensional" generalizations of differential forms. Just as differential forms can be defined as the universal commutative differential algebra containing C\textasciicircum\textbackslash infty(M), we can define differential gorms as the universal commutative bidifferential algebra. From a more conceptual point of view, differential forms are functions on the superspace of maps from the odd line to M and the action of Diff(the odd line) on forms is equivalent to deRham differential and to degrees of forms. Gorms are functions on the superspace of maps from the odd plane to M and we study the action of Diff(the odd plane) on gorms; it contains more than just degrees and differentials. By replacing 2 with arbitrary n, we get differential worms. We also study a generalization of homological algebra that uses Diff(the odd plane) or higher instead of Diff(the odd line), and a closely related question of forms (and gorms and worms) on some generalized spaces (contravariant functors and stacks) and of approximations of such "spaces" in terms of worms. Clearly, this is not a gormless paper.},
  archiveprefix = {arXiv},
  keywords = {Mathematics - Differential Geometry},
  file = {/home/alexk/Zotero/storage/K4XTSHNL/Kochan and Severa - 2004 - Differential gorms, differential worms.pdf;/home/alexk/Zotero/storage/SEJI8N57/0307303.html}
}

@incollection{kontsevich1994,
  title = {Feynman Diagrams and Low-Dimensional Topology},
  booktitle = {First {{European Congress}} of {{Mathematics}}, {{Vol}}.\textbackslash{} {{II}} ({{Paris}}, 1992)},
  author = {Kontsevich, Maxim},
  year = 1994,
  series = {Progr. {{Math}}.},
  volume = {120},
  pages = {97--121},
  publisher = {Birkh\"auser, Basel},
  isbn = {978-3-7643-2799-6},
  mrnumber = {1341841},
  file = {/home/alexk/Zotero/storage/EREICMMM/article.html}
}

@misc{kuo2024,
  title = {Relative {{Calabi-Yau}} Structure on Microlocalization},
  author = {Kuo, Christopher and Li, Wenyuan},
  year = 2024,
  month = aug,
  number = {arXiv:2408.04085},
  eprint = {2408.04085},
  primaryclass = {math},
  publisher = {arXiv},
  doi = {10.48550/arXiv.2408.04085},
  urldate = {2025-09-04},
  abstract = {For an oriented manifold \$M\$ and a compact subanalytic Legendrian \$\textbackslash Lambda \textbackslash subseteq S\textasciicircum *M\$, we construct a canonical strong smooth relative Calabi--Yau structure on the microlocalization at infinity and its left adjoint \$m\_\textbackslash Lambda\textasciicircum l: \textbackslash operatorname\textbraceleft\textbackslash mu sh\textbraceright\_\textbackslash Lambda(\textbackslash Lambda) \textbackslash rightleftharpoons \textbackslash operatorname\textbraceleft Sh\textbraceright\_\textbackslash Lambda(M)\_0 : m\_\textbackslash Lambda\$ between compactly supported sheaves on \$M\$ with singular support on \$\textbackslash Lambda\$ and microsheaves on \$\textbackslash Lambda\$. We also construct a canonical strong Calabi-Yau structure on microsheaves \$\textbackslash operatorname\textbraceleft\textbackslash mu sh\textbraceright\_\textbackslash Lambda(\textbackslash Lambda)\$. Our approach does not require local properness and hence does not depend on arborealization. We thus obtain a canonical smooth relative Calabi-Yau structure on the Orlov functor for wrapped Fukaya categories of cotangent bundles with Weinstein stops, such that the wrap-once functor is the inverse dualizing bimodule.},
  archiveprefix = {arXiv},
  keywords = {Mathematics - Algebraic Topology,Mathematics - K-Theory and Homology,Mathematics - Symplectic Geometry},
  file = {/home/alexk/Zotero/storage/RP32U5BX/Kuo and Li - 2024 - Relative Calabi-Yau structure on microlocalization.pdf;/home/alexk/Zotero/storage/WF9G5WBY/2408.html}
}

@phdthesis{lurie2004,
  title = {Derived Algebraic Geometry},
  author = {Lurie, Jacob},
  year = 2004,
  address = {United States -- Massachusetts},
  urldate = {2021-09-20},
  abstract = {The purpose of this document is to establish the foundations for a theory of derived algebraic geometry based upon simplicial commutative rings. We define derived versions of schemes, algebraic spaces, and algebraic stacks. Our main result is a derived analogue of Artin's representability theorem, which provides a precise criteria for the representability of a moduli functor by geometric objects of these types. (Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617-253-5668; Fax 617-253-1690.)},
  copyright = {Database copyright ProQuest LLC; ProQuest does not claim copyright in the individual underlying works.},
  langid = {english},
  school = {Massachusetts Institute of Technology},
  keywords = {Algebraic geometry,Commutative rings,Geometric objects,Pure sciences}
}

@book{lurie2009,
  title = {Higher {{Topos Theory}} ({{AM-170}})},
  author = {Lurie, Jacob},
  year = 2009,
  publisher = {Princeton University Press},
  urldate = {2026-03-21},
  abstract = {Higher category theory is generally regarded as technical and forbidding, but part of it is considerably more tractable: the theory of infinity-categories, higher categories in which all higher morphisms are assumed to be invertible. In{$<$}em{$>$}Higher Topos Theory{$<$}/em{$>$}, Jacob Lurie presents the foundations of this theory, using the language of weak Kan complexes introduced by Boardman and Vogt, and shows how existing theorems in algebraic topology can be reformulated and generalized in the theory's new language. The result is a powerful theory with applications in many areas of mathematics.  The book's first five chapters give an exposition of the theory of infinity-categories that emphasizes their role as a generalization of ordinary categories. Many of the fundamental ideas from classical category theory are generalized to the infinity-categorical setting, such as limits and colimits, adjoint functors, ind-objects and pro-objects, locally accessible and presentable categories, Grothendieck fibrations, presheaves, and Yoneda's lemma. A sixth chapter presents an infinity-categorical version of the theory of Grothendieck topoi, introducing the notion of an infinity-topos, an infinity-category that resembles the infinity-category of topological spaces in the sense that it satisfies certain axioms that codify some of the basic principles of algebraic topology. A seventh and final chapter presents applications that illustrate connections between the theory of higher topoi and ideas from classical topology.},
  isbn = {978-0-691-14049-0}
}

@misc{lurie2017,
  title = {Higher {{Algebra}}},
  author = {Lurie, Jacob},
  urldate = {2026-02-06},
  howpublished = {https://people.math.harvard.edu/\textasciitilde lurie/papers/HA.pdf},
  file = {/home/alexk/Zotero/storage/KUQ36JMQ/HA.pdf}
}

@article{marsden1974,
  title = {Reduction of Symplectic Manifolds with Symmetry},
  author = {Marsden, Jerrold and Weinstein, Alan},
  year = 1974,
  journal = {Reports on Mathematical Physics},
  volume = {5},
  number = {1},
  pages = {121--130},
  issn = {0034-4877},
  doi = {10.1016/0034-4877(74)90021-4},
  mrnumber = {402819},
  file = {/home/alexk/Zotero/storage/4FIEBCI7/Marsden and Weinstein - 1974 - Reduction of symplectic manifolds with symmetry.pdf;/home/alexk/Zotero/storage/SFTGT74X/article.html}
}

@article{melani2016,
  title = {Poisson Bivectors and {{Poisson}} Brackets on Affine Derived Stacks},
  author = {Melani, Valerio},
  year = 2016,
  month = jan,
  journal = {Advances in Mathematics},
  volume = {288},
  pages = {1097--1120},
  issn = {0001-8708},
  doi = {10.1016/j.aim.2015.11.008},
  urldate = {2024-01-30},
  abstract = {Let A be a commutative dg algebra concentrated in degrees (-{$\infty$},m], and let SpecA be the associated derived stack. We give two proofs of the existence of a canonical map from the moduli space of shifted Poisson structures (in the sense of [16]) on SpecA to the moduli space of homotopy (shifted) Poisson algebra structures on A. The first makes use of a more general description of the Poisson operad and of its cofibrant models, while the second is more computational and involves an explicit resolution of the Poisson operad.},
  file = {/home/alexk/Zotero/storage/UM3UM6EU/Melani - 2016 - Poisson bivectors and Poisson brackets on affine d.pdf;/home/alexk/Zotero/storage/A7CDH2W6/S0001870815004582.html}
}

@article{melani2018,
  title = {Derived Coisotropic Structures {{I}}: Affine Case},
  shorttitle = {Derived Coisotropic Structures {{I}}},
  author = {Melani, Valerio and Safronov, Pavel},
  year = 2018,
  month = sep,
  journal = {Selecta Mathematica},
  volume = {24},
  number = {4},
  pages = {3061--3118},
  issn = {1420-9020},
  doi = {10.1007/s00029-018-0406-2},
  urldate = {2021-09-20},
  abstract = {We define and study coisotropic structures on morphisms of commutative dg algebras in the context of shifted Poisson geometry, i.e. \$\$\textbackslash mathbb \textbraceleft P\textbraceright\_n\$\$-algebras. Roughly speaking, a coisotropic morphism is given by a \$\$\textbackslash mathbb \textbraceleft P\textbraceright\_\textbraceleft n+1\textbraceright\$\$-algebra acting on a \$\$\textbackslash mathbb \textbraceleft P\textbraceright\_n\$\$-algebra. One of our main results is an identification of the space of such coisotropic structures with the space of Maurer--Cartan elements in a certain dg Lie algebra of relative polyvector fields. To achieve this goal, we construct a cofibrant replacement of the operad controlling coisotropic morphisms by analogy with the Swiss-cheese operad which can be of independent interest. Finally, we show that morphisms of shifted Poisson algebras are identified with coisotropic structures on their graph.},
  langid = {english},
  file = {/home/alexk/Zotero/storage/PZ3USEDR/Melani and Safronov - 2018 - Derived coisotropic structures I affine case.pdf}
}

@article{melani2018a,
  title = {Derived Coisotropic Structures {{II}}: Stacks and Quantization},
  shorttitle = {Derived Coisotropic Structures {{II}}},
  author = {Melani, Valerio and Safronov, Pavel},
  year = 2018,
  month = sep,
  journal = {Selecta Mathematica},
  volume = {24},
  number = {4},
  pages = {3119--3173},
  issn = {1420-9020},
  doi = {10.1007/s00029-018-0407-1},
  urldate = {2021-09-20},
  abstract = {We extend results about n-shifted coisotropic structures from part I of this work to the setting of derived Artin stacks. We show that an intersection of coisotropic morphisms carries a Poisson structure of shift one less. We also compare non-degenerate shifted coisotropic structures and shifted Lagrangian structures and show that there is a natural equivalence between the two spaces in agreement with the classical result. Finally, we define quantizations of n-shifted coisotropic structures and show that they exist for \$\$n{$>$}1\$\$.},
  langid = {english},
  file = {/home/alexk/Zotero/storage/6D5642VP/Melani and Safronov - 2018 - Derived coisotropic structures II stacks and quan.pdf}
}

@article{moulinos2021,
  title = {The Geometry of Filtrations},
  author = {Moulinos, Tasos},
  year = 2021,
  journal = {Bulletin of the London Mathematical Society},
  volume = {53},
  number = {5},
  pages = {1486--1499},
  issn = {1469-2120},
  doi = {10.1112/blms.12512},
  urldate = {2023-08-09},
  abstract = {We display a symmetric monoidal equivalence between the stable {$\infty$}-category of filtered spectra, and quasi-coherent sheaves on A1/Gm, the quotient in the setting of spectral algebraic geometry of the flat affine line by the canonical action of the flat multiplicative group scheme. Via a Tannaka duality argument, we identify the underlying spectrum and associated graded functors with pull-backs of quasi-coherent sheaves along certain morphisms of stacks.},
  copyright = {\copyright{} 2021 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.},
  langid = {english},
  keywords = {14F05,55P42 (primary)},
  file = {/home/alexk/Zotero/storage/SYBD6UAB/Moulinos - 2021 - The geometry of filtrations.pdf}
}

@article{moulinos2022,
  title = {A Universal {{Hochschild}}--{{Kostant}}--{{Rosenberg}} Theorem},
  author = {Moulinos, Tasos and Robalo, Marco and To{\"e}n, Bertrand},
  year = 2022,
  month = jun,
  journal = {Geometry \& Topology},
  volume = {26},
  number = {2},
  pages = {777--874},
  publisher = {Mathematical Sciences Publishers},
  issn = {1364-0380},
  doi = {10.2140/gt.2022.26.777},
  urldate = {2023-05-30},
  file = {/home/alexk/Zotero/storage/YUWKHGM2/Moulinos et al. - 2022 - A universal Hochschild–Kostant–Rosenberg theorem.pdf}
}

@article{moulinos2024,
  title = {Cogroupoid Structures on the Circle and the {{Hodge}} Degeneration},
  author = {Moulinos, Tasos},
  year = 2024,
  month = jan,
  journal = {Forum of Mathematics, Sigma},
  volume = {12},
  pages = {e10},
  issn = {2050-5094},
  doi = {10.1017/fms.2023.122},
  urldate = {2024-08-21},
  abstract = {We exhibit the Hodge degeneration from nonabelian Hodge theory as a 222-fold delooping of the filtered loop space E2E2E\_2-groupoid in formal moduli problems. This is an iterated groupoid object which in degree 111 recovers the filtered circle S1filS1filS\textasciicircum 1\_\textbraceleft fil\textbraceright{} of [MRT22]. This exploits a hitherto unstudied additional piece of structure on the topological circle, that of an E2E2E\_2-cogroupoid object in the {$\infty\infty\backslash$}infty -category of spaces. We relate this cogroupoid structure with the more commonly studied `pinch map' on S1S1S\textasciicircum 1, as well as the Todd class of the Lie algebroid TXTX\textbackslash mathbb \textbraceleft T\textbraceright\_\textbraceleft X\textbraceright; this is an invariant of a smooth and proper scheme X that arises, for example, in the Grothendieck-Riemann-Roch theorem. In particular, we relate the existence of nontrivial Todd classes for schemes to the failure of the pinch map to be formal in the sense of rational homotopy theory. Finally, we record some consequences of this bit of structure at the level of Hochschild cohomology.},
  langid = {english},
  keywords = {14F40,55P35},
  file = {/home/alexk/Zotero/storage/SFMI3PPH/Moulinos - 2024 - Cogroupoid structures on the circle and the Hodge .pdf}
}

@article{mukai1984,
  title = {Symplectic Structure of the Moduli Space of Sheaves on an Abelian or {{K3}} Surface},
  author = {Mukai, Shigeru},
  year = 1984,
  month = feb,
  journal = {Inventiones mathematicae},
  volume = {77},
  number = {1},
  pages = {101--116},
  issn = {1432-1297},
  doi = {10.1007/BF01389137},
  urldate = {2025-09-01},
  langid = {english},
  keywords = {Modulus Space,Symplectic Structure},
  file = {/home/alexk/Zotero/storage/BAHFCG9G/Mukai - 1984 - Symplectic structure of the moduli space of sheave.pdf}
}

@article{nuiten2019,
  title = {Koszul Duality for {{Lie}} Algebroids},
  author = {Nuiten, Joost},
  year = 2019,
  month = oct,
  journal = {Advances in Mathematics},
  volume = {354},
  pages = {106750},
  issn = {0001-8708},
  doi = {10.1016/j.aim.2019.106750},
  urldate = {2024-09-13},
  abstract = {This paper studies the role of dg-Lie algebroids in derived deformation theory. More precisely, we provide an equivalence between the homotopy theories of formal moduli problems and dg-Lie algebroids over a commutative dg-algebra of characteristic zero. At the level of linear objects, we show that the category of representations of a dg-Lie algebroid is an extension of the category of quasi-coherent sheaves on the corresponding formal moduli problem. We describe this extension geometrically in terms of pro-coherent sheaves.},
  keywords = {Formal moduli problem,Koszul duality,Lie algebroid},
  file = {/home/alexk/Zotero/storage/A45MQAAX/Nuiten - 2019 - Koszul duality for Lie algebroids.pdf;/home/alexk/Zotero/storage/9IJAGJ9U/S0001870819303603.html}
}

@article{nuiten2019a,
  title = {Homotopical {{Algebra}} for {{Lie Algebroids}}},
  author = {Nuiten, Joost},
  year = 2019,
  month = oct,
  journal = {Applied Categorical Structures},
  volume = {27},
  number = {5},
  pages = {493--534},
  issn = {1572-9095},
  doi = {10.1007/s10485-019-09563-z},
  urldate = {2026-03-06},
  abstract = {We construct Quillen equivalent semi-model structures on the categories of dg-Lie algebroids and \$\$L\_\textbackslash infty \$\$-algebroids over a commutative dg-algebra in characteristic zero. This allows one to apply the usual methods of homotopical algebra to dg-Lie algebroids: for example, every Lie algebroid can be resolved by dg-Lie algebroids that arise from dg-Lie algebras, i.e. whose anchor map is zero. As an application, we show how Lie algebroid cohomology is represented by an object in the homotopy category of dg-Lie algebroids.},
  langid = {english},
  keywords = {18G55,55U15,55U35,Dg-Lie algebroid,Lie algebroid cohomology,Model category},
  file = {/home/alexk/Zotero/storage/NXK4TP72/Nuiten - 2019 - Homotopical Algebra for Lie Algebroids.pdf}
}

@article{pantev2013,
  title = {Shifted Symplectic Structures},
  author = {Pantev, Tony and To{\"e}n, Bertrand and Vaqui{\'e}, Michel and Vezzosi, Gabriele},
  year = 2013,
  month = jun,
  journal = {Publications math\'ematiques de l'IH\'ES},
  volume = {117},
  number = {1},
  pages = {271--328},
  issn = {1618-1913},
  doi = {10.1007/s10240-013-0054-1},
  urldate = {2021-09-20},
  abstract = {This is the first of a series of papers about quantization in the context of derived algebraic geometry. In this first part, we introduce the notion of n-shifted symplectic structures (n-symplectic structures for short), a generalization of the notion of symplectic structures on smooth varieties and schemes, meaningful in the setting of derived Artin n-stacks (see To\"en and Vezzosi in Mem. Am. Math. Soc. 193, 2008 and To\"en in Proc. Symp. Pure Math. 80:435--487, 2009). We prove that classifying stacks of reductive groups, as well as the derived stack of perfect complexes, carry canonical 2-symplectic structures. Our main existence theorem states that for any derived Artin stack F equipped with an n-symplectic structure, the derived mapping stack Map(X,F) is equipped with a canonical (n-d)-symplectic structure as soon a X satisfies a Calabi-Yau condition in dimension d. These two results imply the existence of many examples of derived moduli stacks equipped with n-symplectic structures, such as the derived moduli of perfect complexes on Calabi-Yau varieties, or the derived moduli stack of perfect complexes of local systems on a compact and oriented topological manifold. We explain how the known symplectic structures on smooth moduli spaces of simple objects (e.g. simple sheaves on Calabi-Yau surfaces, or simple representations of {$\pi$}1 of compact Riemann surfaces) can be recovered from our results, and that they extend canonically as 0-symplectic structures outside of the smooth locus of simple objects. We also deduce new existence statements, such as the existence of a natural (-1)-symplectic structure (whose formal counterpart has been previously constructed in (Costello, arXiv:1111.4234, 2001) and (Costello and Gwilliam, 2011) on the derived mapping scheme Map(E,T{$\ast$}X), for E an elliptic curve and T{$\ast$}X is the total space of the cotangent bundle of a smooth scheme X. Canonical (-1)-symplectic structures are also shown to exist on Lagrangian intersections, on moduli of sheaves (or complexes of sheaves) on Calabi-Yau 3-folds, and on moduli of representations of {$\pi$}1 of compact topological 3-manifolds. More generally, the moduli sheaves on higher dimensional varieties are shown to carry canonical shifted symplectic structures (with a shift depending on the dimension).},
  langid = {english},
  file = {/home/alexk/Zotero/storage/CTXXH53Z/Pantev et al. - 2013 - Shifted symplectic structures.pdf}
}

@article{pantev2021,
  title = {Poisson {{Geometry}} of the {{Moduli}} of {{Local Systems}} on {{Smooth Varieties}}},
  author = {Pantev, Tony and To{\"e}n, Bertrand},
  year = 2021,
  month = oct,
  journal = {Publications of the Research Institute for Mathematical Sciences},
  volume = {57},
  number = {3},
  pages = {959--991},
  issn = {0034-5318},
  doi = {10.4171/prims/57-3-8},
  urldate = {2022-10-30},
  abstract = {We study the moduli of \$G\$-local systems on smooth but not necessarily proper complex algebraic varieties. We show that, when considered as derived algebraic stacks, they carry natural Poisson structures, generalizing the well-known case of curves. We also construct symplectic leaves of this Poisson structure by fixing local monodromies at infinity and show that a new feature, called strictness, appears as soon as the divisor at infinity has nontrivial crossings.},
  langid = {english},
  file = {/home/alexk/Zotero/storage/5UF7DD6S/Pantev and Toën - 2021 - Poisson Geometry of the Moduli of Local Systems on.pdf}
}

@article{pantev2023,
  title = {Private Communication},
  author = {Pantev, Tony},
  year = 2023
}

@misc{park2024,
  title = {Shifted Symplectic Pushforwards},
  author = {Park, Hyeonjun},
  year = 2024,
  month = jun,
  number = {arXiv:2406.19192},
  eprint = {2406.19192},
  primaryclass = {math},
  publisher = {arXiv},
  doi = {10.48550/arXiv.2406.19192},
  urldate = {2025-09-02},
  abstract = {We introduce how to pushforward shifted symplectic fibrations along base changes. This is achieved by considering symplectic forms that are closed in a stronger sense. Examples include: symplectic zero loci and symplectic quotients. Observing that twisted cotangent bundles are symplectic pushforwards, we obtain an equivalence between symplectic fibrations and Lagrangians to critical loci. We provide two local structure theorems for symplectic fibrations: a smooth local structure theorem for higher stacks via symplectic zero loci and twisted cotangents, and an \textbraceleft\textbackslash 'e\textbraceright tale local structure theorem for \$1\$-stacks with reductive stabilizers via symplectic quotients of the smooth local models. We resolve deformation invariance issue in Donaldson-Thomas theory of Calabi-Yau \$4\$-folds. Abstractly, we associate virtual Lagrangian cycles for oriented \$(-2)\$-symplectic fibrations as unique functorial bivariant classes over the exact loci. For moduli of perfect complexes, we show that the exact loci consist of deformations for which the \$(0,4)\$-Hodge pieces of the second Chern characters remain zero.},
  archiveprefix = {arXiv},
  keywords = {Mathematics - Algebraic Geometry},
  file = {/home/alexk/Zotero/storage/SM6PJFRK/Park - 2024 - Shifted symplectic pushforwards.pdf;/home/alexk/Zotero/storage/FXDS77QM/2406.html}
}

@article{porta2014,
  title = {On the {{Homology}} of {{Completion}} and {{Torsion}}},
  author = {Porta, Marco and Shaul, Liran and Yekutieli, Amnon},
  year = 2014,
  month = feb,
  journal = {Algebras and Representation Theory},
  volume = {17},
  number = {1},
  pages = {31--67},
  issn = {1572-9079},
  doi = {10.1007/s10468-012-9385-8},
  urldate = {2026-08-11},
  abstract = {Let A be a commutative ring, and \$\textbraceleft\textbackslash mathfrak\textbraceleft a\textbraceright\textbraceright\$a weakly proregular ideal in A. This includes the noetherian case: if A is noetherian then any ideal in it is weakly proregular; but there are other interesting examples. In this paper we prove the MGM equivalence, which is an equivalence between the category of cohomologically \$\textbraceleft\textbackslash mathfrak\textbraceleft a\textbraceright\textbraceright\$-adically complete complexes and the category of cohomologically \$\textbraceleft\textbackslash mathfrak\textbraceleft a\textbraceright\textbraceright\$-torsion complexes. These are triangulated subcategories of the derived category of A-modules. Our work extends earlier work by Alonso--Jeremias--Lipman, Schenzel and Dwyer--Greenlees.},
  langid = {english},
  keywords = {13C12,13D09,18E30,Adic completion,derived functors,Primary 13D07; Secondary 13B35,torsion},
  file = {/home/alexk/Zotero/storage/XBAH7CL7/Porta et al. - 2014 - On the Homology of Completion and Torsion.pdf}
}

@article{pridham2010,
  title = {Unifying Derived Deformation Theories},
  author = {Pridham, J. P.},
  year = 2010,
  journal = {Advances in Mathematics},
  volume = {224},
  number = {3},
  pages = {772--826},
  issn = {0001-8708,1090-2082},
  doi = {10.1016/j.aim.2009.12.009},
  mrnumber = {2628795},
  file = {/home/alexk/Zotero/storage/U9RH6LLE/Pridham - 2010 - Unifying derived deformation theories.pdf;/home/alexk/Zotero/storage/B2IQ998M/article.html}
}

@article{pridham2017,
  title = {Shifted {{Poisson}} and Symplectic Structures on Derived {{N-stacks}}},
  author = {Pridham, J. P.},
  year = 2017,
  journal = {Journal of Topology},
  volume = {10},
  number = {1},
  pages = {178--210},
  issn = {1753-8424},
  doi = {10.1112/topo.12004},
  urldate = {2022-09-28},
  abstract = {We show that on a derived Artin N-stack, there is a canonical equivalence between the spaces of n-shifted symplectic structures and non-degenerate n-shifted Poisson structures.},
  langid = {english},
  keywords = {14D23 (primary),17B63 (secondary)},
  file = {/home/alexk/Zotero/storage/8VHDNV74/Pridham - 2017 - Shifted Poisson and symplectic structures on deriv.pdf;/home/alexk/Zotero/storage/GUM9SXN6/topo.html}
}

@article{pridham2018,
  title = {Deformation Quantisation for Unshifted Symplectic Structures on Derived {{Artin}} Stacks},
  author = {Pridham, J. P.},
  year = 2018,
  journal = {Selecta Mathematica. New Series},
  volume = {24},
  number = {4},
  pages = {3027--3059},
  issn = {1022-1824,1420-9020},
  doi = {10.1007/s00029-018-0414-2},
  mrnumber = {3848015},
  file = {/home/alexk/Zotero/storage/XBU4JKNY/Pridham - 2018 - Deformation quantisation for unshifted symplectic .pdf;/home/alexk/Zotero/storage/NMDJ9ZKC/article.html}
}

@article{pridham2019,
  title = {Deformation Quantisation for (-1)-Shifted Symplectic Structures and Vanishing Cycles},
  author = {Pridham, J. P.},
  year = 2019,
  journal = {Algebraic Geometry},
  volume = {6},
  number = {6},
  pages = {747--779},
  issn = {2313-1691,2214-2584},
  doi = {10.14231/ag-2019-032},
  mrnumber = {4009180},
  file = {/home/alexk/Zotero/storage/CSBKHZLE/Pridham - 2019 - Deformation quantisation for (-1)-shifted symplect.pdf;/home/alexk/Zotero/storage/IV5DVV97/article.html}
}

@article{pym2020,
  title = {Shifted {{Symplectic Lie Algebroids}}},
  author = {Pym, Brent and Safronov, Pavel},
  year = 2020,
  month = nov,
  journal = {International Mathematics Research Notices},
  volume = {2020},
  number = {21},
  pages = {7489--7557},
  issn = {1073-7928},
  doi = {10.1093/imrn/rny215},
  urldate = {2024-04-15},
  abstract = {Shifted symplectic Lie and \$L\_\textbraceleft\textbackslash infty \textbraceright\$ algebroids model formal neighborhoods of manifolds in shifted symplectic stacks and serve as target spaces for twisted variants of the classical topological field theory defined by Alexandrov--Kontsevich--Schwarz--Zaboronsky. In this paper, we classify zero-, one-, and two-shifted symplectic algebroids and their higher gauge symmetries, in terms of classical geometric ``higher structures'', such as Courant algebroids twisted by \$\textbackslash Omega \textasciicircum\textbraceleft 2\textbraceright\$-gerbes. As applications, we produce new examples of twisted Courant algebroids from codimension-two cycles, and we give symplectic interpretations for several well-known features of higher structures (such as twists, Pontryagin classes, and tensor products). The proofs are valid in the \$C\textasciicircum\textbraceleft\textbackslash infty \textbraceright\$, holomorphic, and algebraic settings and are based on a number of technical results on the homotopy theory of \$L\_\textbraceleft\textbackslash infty \textbraceright\$ algebroids and their differential forms, which may be of independent interest.},
  file = {/home/alexk/Zotero/storage/LC56KL3K/Pym and Safronov - 2020 - Shifted Symplectic Lie Algebroids.pdf;/home/alexk/Zotero/storage/ZBULAJVV/5092455.html}
}

@misc{raksit2020,
  title = {Hochschild Homology and the Derived de {{Rham}} Complex Revisited},
  author = {Raksit, Arpon},
  year = 2020,
  month = sep,
  number = {arXiv:2007.02576},
  eprint = {2007.02576},
  primaryclass = {math},
  publisher = {arXiv},
  doi = {10.48550/arXiv.2007.02576},
  urldate = {2024-02-09},
  abstract = {We characterize two objects by universal property: the derived de Rham complex and Hochschild homology together with its Hochschild-Kostant-Rosenberg filtration. This involves endowing these objects with extra structure, built on notions of "homotopy-coherent cochain complex" and "filtered circle action" that we study here. We use these universal properties to give a conceptual proof of the statements relating Hochschild homology and the derived de Rham complex, in particular giving a new construction of the filtrations on cyclic, negative cyclic, and periodic cyclic homology that relate these invariants to derived de Rham cohomology.},
  archiveprefix = {arXiv},
  keywords = {Mathematics - Algebraic Geometry,Mathematics - K-Theory and Homology},
  file = {/home/alexk/Zotero/storage/H82VPLMG/Raksit - 2020 - Hochschild homology and the derived de Rham comple.pdf;/home/alexk/Zotero/storage/X5Z5KH6C/2007.html}
}

@misc{rozenblyum,
  title = {Upcoming},
  author = {Rozenblyum, Nick}
}

@misc{rozenblyum2017,
  type = {Seminar Talk},
  title = {{{BV}} Formalism and Derived Symplectic Geometry},
  author = {Rozenblyum, Nick},
  year = 2017,
  month = jan,
  address = {Perimeter Institute}
}

@article{safronov2016,
  title = {Quasi-{{Hamiltonian}} Reduction via Classical {{Chern}}--{{Simons}} Theory},
  author = {Safronov, Pavel},
  year = 2016,
  month = jan,
  journal = {Advances in Mathematics},
  volume = {287},
  pages = {733--773},
  issn = {0001-8708},
  doi = {10.1016/j.aim.2015.09.031},
  urldate = {2023-05-08},
  abstract = {This paper puts the theory of quasi-Hamiltonian reduction in the framework of shifted symplectic structures developed by Pantev, To\"en, Vaqui\'e and Vezzosi. We compute the symplectic structures on mapping stacks and show how the AKSZ topological field theory defined by Calaque allows one to neatly package the constructions used in quasi-Hamiltonian reduction. Finally, we explain how a prequantization of character stacks can be obtained purely locally.},
  langid = {english},
  keywords = {Derived algebraic geometry,Hamiltonian reduction,Moment maps,Symplectic geometry},
  file = {/home/alexk/Zotero/storage/KMSP55UR/Safronov - 2016 - Quasi-Hamiltonian reduction via classical Chern–Si.pdf;/home/alexk/Zotero/storage/8M6P5AD5/S0001870815004260.html}
}

@article{safronov2018,
  title = {Braces and {{Poisson}} Additivity},
  author = {Safronov, Pavel},
  year = 2018,
  month = aug,
  journal = {Compositio Mathematica},
  volume = {154},
  number = {8},
  pages = {1698--1745},
  publisher = {London Mathematical Society},
  issn = {0010-437X, 1570-5846},
  doi = {10.1112/S0010437X18007212},
  urldate = {2022-11-05},
  abstract = {We relate the brace construction introduced by Calaque and Willwacher to an additivity functor. That is, we construct a functor from brace algebras associated to an operad O\textbraceleft\textbackslash mathcal\textbraceleft O\textbraceright\textbraceright{} to associative algebras in the category of homotopy O\textbraceleft\textbackslash mathcal\textbraceleft O\textbraceright\textbraceright -algebras. As an example, we identify the category of Pn+1\textbackslash mathbb\textbraceleft P\textbraceright\_\textbraceleft n+1\textbraceright -algebras with the category of associative algebras in Pn\textbackslash mathbb\textbraceleft P\textbraceright\_\textbraceleft n\textbraceright -algebras. We also show that under this identification there is an equivalence of two definitions of derived coisotropic structures in the literature.},
  langid = {english},
  keywords = {18D50 (primary),18G55 (secondary),Koszul duality,operads,shifted Poisson structures},
  file = {/home/alexk/Zotero/storage/6HUDF73X/Safronov - 2018 - Braces and Poisson additivity.pdf}
}

@article{safronov2021,
  title = {Poisson-{{Lie}} Structures as Shifted {{Poisson}} Structures},
  author = {Safronov, Pavel},
  year = 2021,
  month = apr,
  journal = {Advances in Mathematics},
  volume = {381},
  pages = {107633},
  issn = {0001-8708},
  doi = {10.1016/j.aim.2021.107633},
  urldate = {2023-03-29},
  abstract = {Classical limits of quantum groups give rise to multiplicative Poisson structures such as Poisson-Lie and quasi-Poisson structures. We relate them to the notion of a shifted Poisson structure which gives a conceptual framework for understanding classical (dynamical) r-matrices, quasi-Poisson groupoids and so on. We also propose a notion of a symplectic realization of shifted Poisson structures and show that Manin pairs and Manin triples give examples of such.},
  langid = {english},
  keywords = {Classical -matrix,Poisson groupoid,Poisson-Lie group,Shifted Poisson structure},
  file = {/home/alexk/Zotero/storage/URNUEWPM/Safronov - 2021 - Poisson-Lie structures as shifted Poisson structur.pdf}
}

@misc{shende2024,
  title = {Calabi-{{Yau}} Structures on Topological {{Fukaya}} Categories},
  author = {Shende, Vivek and Takeda, Alex},
  year = 2024,
  month = may,
  number = {arXiv:1605.02721},
  eprint = {1605.02721},
  primaryclass = {math},
  publisher = {arXiv},
  doi = {10.48550/arXiv.1605.02721},
  urldate = {2025-09-04},
  abstract = {We develop a local-to-global formalism for constructing Calabi-Yau structures for global sections of constructible sheaves or cosheaves of categories. The required data - an isomorphism of the sheafified Hochschild homology with the topological dualizing sheaf - specializes to the classical notion of orientation when applied to the category of local systems on a manifold. We apply this construction to the cosheaves on arboreal skeleta arising in the microlocal approach to the A-model.},
  archiveprefix = {arXiv},
  keywords = {Mathematics - Algebraic Geometry,Mathematics - Symplectic Geometry},
  file = {/home/alexk/Zotero/storage/TZ4KA3DN/Shende and Takeda - 2024 - Calabi-Yau structures on topological Fukaya categories.pdf;/home/alexk/Zotero/storage/WXES6SFT/1605.html}
}

@book{shulman1972,
  title = {Characteristic Classes and Foliations},
  author = {Shulman, Herbert Byron},
  year = 1972,
  publisher = {ProQuest LLC, Ann Arbor, MI},
  mrnumber = {2940315},
  file = {/home/alexk/Zotero/storage/WNB97636/article.html}
}

@incollection{simpson1996,
  title = {Homotopy over the Complex Numbers and Generalized de {{Rham}} Cohomology},
  booktitle = {Moduli of Vector Bundles ({{Sanda}}, 1994; {{Kyoto}}, 1994)},
  author = {Simpson, Carlos},
  year = 1996,
  series = {Lecture {{Notes}} in {{Pure}} and {{Appl}}. {{Math}}.},
  volume = {179},
  pages = {229--263},
  publisher = {Dekker, New York},
  isbn = {978-0-8247-9738-6},
  mrnumber = {1397992},
  file = {/home/alexk/Zotero/storage/8LHK55UH/article.html}
}

@incollection{simpson2002,
  title = {Algebraic Aspects of Higher Nonabelian {{Hodge}} Theory},
  booktitle = {Motives, Polylogarithms and {{Hodge}} Theory, {{Part II}} ({{Irvine}}, {{CA}}, 1998)},
  author = {Simpson, Carlos},
  year = 2002,
  series = {Int. {{Press Lect}}. {{Ser}}.},
  volume = {3, II},
  pages = {417--604},
  publisher = {Int. Press, Somerville, MA},
  isbn = {978-1-57146-091-2},
  mrnumber = {1978713},
  file = {/home/alexk/Zotero/storage/M2XK959K/article.html}
}

@article{simpson2009,
  title = {Geometricity of the {{Hodge}} Filtration on the Infty-Stack of Perfect Complexes over {{X}}\_{{dR}}},
  author = {Simpson, Carlos},
  year = 2009,
  journal = {Moscow Mathematical Journal},
  volume = {9},
  number = {3},
  pages = {665--721, back matter},
  issn = {1609-3321,1609-4514},
  doi = {10.17323/1609-4514-2009-9-3-665-721},
  mrnumber = {2562796},
  file = {/home/alexk/Zotero/storage/7QGZETMC/article.html}
}

@misc{stacks-project,
  title = {\textit{Stacks Project}},
  shorttitle = {Stacks},
  author = {{The Stacks Project Authors}},
  year = 2018,
  howpublished = {\url{https://stacks.math.columbia.edu}},
  shorthand = {Stacks}
}

@article{toen2005,
  title = {Homotopical Algebraic Geometry {{I}}: Topos Theory},
  shorttitle = {Homotopical Algebraic Geometry {{I}}},
  author = {To{\"e}n, Bertrand and Vezzosi, Gabriele},
  year = 2005,
  month = jun,
  journal = {Advances in Mathematics},
  volume = {193},
  number = {2},
  pages = {257--372},
  issn = {00018708},
  doi = {10.1016/j.aim.2004.05.004},
  urldate = {2021-09-20},
  abstract = {This is the first of a series of papers devoted to lay the foundations of Algebraic Geometry in homotopical and higher categorical contexts. In this first part we investigate a notion of higher topos.},
  langid = {english},
  file = {/home/alexk/Zotero/storage/Z9NFMEMT/Toën and Vezzosi - 2005 - Homotopical algebraic geometry I topos theory.pdf}
}

@article{toen2007,
  title = {Moduli of Objects in Dg-Categories},
  author = {To{\"e}n, Bertrand and Vaqui{\'e}, Michel},
  year = 2007,
  journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
  volume = {40},
  number = {3},
  pages = {387--444},
  issn = {1873-2151},
  doi = {10.1016/j.ansens.2007.05.001},
  urldate = {2023-04-07},
  langid = {english},
  file = {/home/alexk/Zotero/storage/AM5QPAMV/Toën and Vaquié - 2007 - Moduli of objects in dg-categories.pdf}
}

@article{toen2008,
  title = {Homotopical Algebraic Geometry. {{II}}. {{Geometric}} Stacks and Applications},
  author = {To{\"e}n, Bertrand and Vezzosi, Gabriele},
  year = 2008,
  journal = {Memoirs of the American Mathematical Society},
  volume = {193},
  number = {902},
  pages = {0--0},
  issn = {0065-9266, 1947-6221},
  doi = {10.1090/memo/0902},
  urldate = {2021-09-20},
  langid = {english},
  file = {/home/alexk/Zotero/storage/P6X5EYM4/Toën and Vezzosi - 2008 - Homotopical algebraic geometry. II. Geometric stac.pdf}
}

@article{toen2011b,
  title = {Alg\`ebres Simpliciales {{S1-\'equivariantes}}, Th\'eorie de de {{Rham}} et Th\'eor\`emes {{HKR}} Multiplicatifs},
  author = {To{\"e}n, Bertrand and Vezzosi, Gabriele},
  year = 2011,
  month = nov,
  journal = {Compositio Mathematica},
  volume = {147},
  number = {6},
  pages = {1979--2000},
  issn = {1570-5846, 0010-437X},
  doi = {10.1112/S0010437X11005501},
  urldate = {2026-04-01},
  abstract = {This work establishes a comparison between functions on derived loop spaces (To\"en and Vezzosi, Chern character, loop spaces and derived algebraic geometry, in Algebraic topology: the Abel symposium 2007, Abel Symposia, vol.~4, eds N.~Baas, E.~M. Friedlander, B.~Jahren and P.~A.~\O stv\ae r (Springer, 2009), ISBN:978-3-642-01199-3) and de Rham theory. If A is a smooth commutative k-algebra and k has characteristic 0, we show that two objects, S1{$\otimes$}A and {$\epsilon$}(A), determine one another, functorially in A. The object S1{$\otimes$}A is the S1-equivariant simplicial k-algebra obtained by tensoring A by the simplicial group S1 :=B{$\mathbb{Z}$}, while the object {$\epsilon$}(A)is the de Rham algebra of A, endowed with the de Rham differential, and viewed as a {$\epsilon$}-dg-algebra (see the main text). We define an equivalence {$\varphi$} between the homotopy theory of simplicial commutative S1-equivariant k-algebras and the homotopy theory of {$\epsilon$}-dg-algebras, and we show the existence of a functorial equivalence {$\phi$}(S1 {$\otimes$}A){$\sim\epsilon$}(A) . We deduce from this the comparison mentioned above, identifying the S1-equivariant functions on the derived loop space LX of a smooth k-scheme X with the algebraic de Rham cohomology of~X/k. As corollaries, we obtain functorial and multiplicative versions of decomposition theorems for Hochschild homology (in the spirit of Hochschild--Kostant--Rosenberg) for arbitrary semi-separated k-schemes. By construction, these decompositions are moreover compatible with the S1-action on the Hochschild complex, on one hand, and with the de Rham differential, on the other hand., Ce travail a pour objectif d'etablir une comparaison entre fonctions sur les espaces des lacets d\'eriv\'es~(To\"en and Vezzosi, Chern character, loop spaces and derived algebraic geometry, in Algebraic topology: the Abel symposium 2007, Abel Symposia, vol. 4, eds N.~Baas, E. M. Friedlander, B. Jahren and P. A. \O stv\ae r (Springer, 2009), ISBN:978-3-642-01199-3) et th\'eorie de de Rham. Pour une k-alg\`ebre commutative A, lisse sur k de caract\'eristique nulle, nous montrons que deux objets, S1{$\otimes$}A et {$\epsilon$}(A), se d\'eterminent mutuellement, et ce fonctoriellement en A. L'objet S1{$\otimes$}A est la k-alg\`ebre simpliciale S1-\'equivariante obtenue en tensorisant A par le groupe simplicial S1 :=B{$\mathbb{Z}$}. L'objet {$\epsilon$}(A)est l'alg\`ebre de de Rham de A, munie de la diff\'erentielle de de Rham et consid\'er\'ee comme une {$\epsilon$}-dg-alg\`ebre (i.e.~une alg\`ebre dans une certaine cat\'egorie mono\"idale de k[{$\epsilon$}] -dg-modules, o\`u k[{$\epsilon$}]:=H* (S1,k) ). Nous construisons une \'equivalence {$\phi$}, entre la th\'eorie homotopique des k-alg\`ebres simpliciales S1-\'equivariantes et celle des {$\epsilon$}-dg-alg\`ebres, et nous montrons l'existence d'une \'equivalence fonctorielle {$\phi$}(S1 {$\otimes$}A){$\sim\epsilon$}(A) . Nous d\'eduisons de cela la comparaison annonc\'ee, identifiant les fonctions S1-\'equivariantes sur LX, l'espace des lacets d\'eriv\'e d'un k-sch\'ema X lisse, et la cohomologie de de Rham alg\'ebrique de X/k. Cela nous permet aussi de prouver des versions fonctorielles et multiplicatives des th\'eor\`emes de d\'ecomposition de l'homologie de Hochschild (du type Hochschild--Kostant--Rosenberg), pour des k-sch\'emas semi-s\'epar\'es quelconques. Par construction, ces d\'ecompositions sont de plus compatibles avec d'une part l'action naturelle de S1 sur le complexe de Hochschild, et d'autre part la diff\'erentielle de de Rham.},
  langid = {english},
  keywords = {14F40,16E45 (primary),18G55 (secondary),55U10,de Rham algebra,Hochschild homology,homotopical algebra,simplicial algebras},
  file = {/home/alexk/Zotero/storage/9BUP9STJ/Toën and Vezzosi - 2011 - Algèbres simpliciales S1-équivariantes, théorie de de Rham et théorèmes HKR multiplicatifs.pdf}
}

@misc{toen2013,
  title = {Operations on Derived Moduli Spaces of Branes},
  author = {To{\"e}n, B.},
  year = 2013,
  month = oct,
  number = {arXiv:1307.0405},
  eprint = {1307.0405},
  primaryclass = {math},
  publisher = {arXiv},
  doi = {10.48550/arXiv.1307.0405},
  urldate = {2026-03-27},
  abstract = {The main theme of this work is the study of the operations that naturally exist on moduli spaces of maps \$Map(S,X)\$, also called the space of branes of \$X\$ with respect \$S\$. These operations will be constructed as operations on the (quasi-coherent) derived category \$\textbackslash D(Map(S,X))\$, in the particular case where \$S\$ has some close relations with an operad \$\textbackslash OO\$. More precisely, for an \$\textbackslash s\$-operad \$\textbackslash OO\$ and an algebraic variety \$X\$ (or more generally a derived algebraic stack), satisfying some natural conditions, we prove that \$\textbackslash OO\$ acts on the object \$\textbackslash OO(2)\$ by mean cospans. This universal action is used to prove that \$\textbackslash OO\$ acts on the derived category of the space of maps \$Map(\textbackslash OO(2),X)\$, which will call the brane operations. We apply the existence of these operations, as well as their naturality in \$\textbackslash OO\$, in order to propose a sketch for a proof of the \textbackslash emph\textbraceleft higher formality conjecture\textbraceright, a far reaching extension of Konstevich's formality's theorem. By doing so we present a positive answer to a conjecture of Kapustin (see \textbackslash cite[p. 14]\textbraceleft kap\textbraceright ), relating polyvector fields on a variety \$X\$ and deformations of the mono/"i dal derived category \$\textbackslash D(X)\$.},
  archiveprefix = {arXiv},
  keywords = {Mathematics - Algebraic Geometry,Mathematics - Category Theory},
  file = {/home/alexk/Zotero/storage/TQ7JQMLT/Toën - 2013 - Operations on derived moduli spaces of branes.pdf;/home/alexk/Zotero/storage/4RZYEQ2W/1307.html}
}

@misc{toen2020,
  title = {Algebraic Foliations and Derived Geometry {{II}}: The {{Grothendieck-Riemann-Roch}} Theorem},
  shorttitle = {Algebraic Foliations and Derived Geometry {{II}}},
  author = {To{\"e}n, Bertrand and Vezzosi, Gabriele},
  year = 2020,
  month = jul,
  number = {arXiv:2007.09251},
  eprint = {2007.09251},
  primaryclass = {math},
  publisher = {arXiv},
  doi = {10.48550/arXiv.2007.09251},
  urldate = {2023-01-08},
  abstract = {This is the second of series of papers on the study of foliations in the setting of derived algebraic geometry based on the central notion of derived foliation. We introduce sheaf-like coefficients for derived foliations, called quasi-coherent crystals, and construct a certain sheaf of dg-algebras of differential operators along a given derived foliation, with the property that quasi-coherent crystals can be interpreted as modules over this sheaf of differential operators. We use this interpretation in order to introduce the notion of good filtrations on quasi-coherent crystals, and define the notion of characteristic cycle. Finally, we prove a Grothendieck-Riemann-Roch (GRR) formula expressing that formation of characteristic cycles is compatible with push-forwards along proper and quasi-smooth morphisms. Several examples and applications are deduced from this, e.g. a GRR formula for D-modules on possibly singular schemes, and a foliated index formula for weakly Fredholm operators.},
  archiveprefix = {arXiv},
  keywords = {Mathematics - Algebraic Geometry,Mathematics - K-Theory and Homology},
  file = {/home/alexk/Zotero/storage/G9ZZZ783/Toën and Vezzosi - 2020 - Algebraic foliations and derived geometry II the .pdf;/home/alexk/Zotero/storage/VP9PMZGG/2007.html}
}

@article{toen2022,
  title = {Algebraic Foliations and Derived Geometry: The {{Riemann}}--{{Hilbert}} Correspondence},
  shorttitle = {Algebraic Foliations and Derived Geometry},
  author = {To{\"e}n, Bertrand and Vezzosi, Gabriele},
  year = 2022,
  month = nov,
  journal = {Selecta Mathematica},
  volume = {29},
  number = {1},
  pages = {5},
  issn = {1420-9020},
  doi = {10.1007/s00029-022-00808-9},
  urldate = {2023-08-09},
  abstract = {This is the first in a series of papers about foliations in derived geometry. After introducing derived foliations on arbitrary derived stacks, we concentrate on quasi-smooth and rigid derived foliations on smooth complex algebraic varieties and on their associated formal and analytic versions. Their truncations are classical singular foliations defined in terms of differential ideals in the algebra of forms. We prove that a quasi-smooth rigid derived foliation on a smooth complex variety X is formally integrable at any point, and, if we suppose that its singular locus has codimension \$\$\textbackslash ge 2\$\$, its analytification is a locally integrable singular foliation on the associated complex manifold \$\$X\textasciicircum h\$\$. We then introduce the derived category of perfect crystals on a quasi-smooth rigid derived foliation on X, and prove a Riemann-Hilbert correspondence for them when X is proper. We discuss several examples and applications.},
  langid = {english},
  keywords = {14A20,14F40,Primary: 37F75,Secondary: 14F05},
  file = {/home/alexk/Zotero/storage/N6EUE4IQ/Toën and Vezzosi - 2022 - Algebraic foliations and derived geometry the Rie.pdf}
}

@misc{toen2025,
  title = {Derived {{Foliations}}},
  author = {Toen, Bertrand and Vezzosi, Gabriele},
  year = 2025,
  month = jul,
  number = {arXiv:2305.08212},
  eprint = {2305.08212},
  primaryclass = {math},
  publisher = {arXiv},
  doi = {10.48550/arXiv.2305.08212},
  urldate = {2026-04-09},
  abstract = {This is a book on derived foliations, that are a generalisation of classical foliations in the context of derived geometry. The text starts with the basic definitions and constructions, then explore foliated cohomology (with crystal coefficients), formal and analytic integrability problems, existence of the leaf space, characteristic classes, and two different notions of derived foliation in arbitrary characteristics.},
  archiveprefix = {arXiv},
  keywords = {Mathematics - Algebraic Geometry},
  file = {/home/alexk/Zotero/storage/L5BA24D9/Toen and Vezzosi - 2025 - Derived Foliations.pdf;/home/alexk/Zotero/storage/HTCL29KA/2305.html}
}

@misc{tomic2025,
  title = {Shifted {{Lagrangian}} Thickenings of Shifted {{Poisson}} Derived Schemes},
  author = {Tomi{\'c}, Nikola},
  year = 2025,
  month = jun,
  number = {arXiv:2506.23348},
  eprint = {2506.23348},
  primaryclass = {math},
  publisher = {arXiv},
  doi = {10.48550/arXiv.2506.23348},
  urldate = {2025-09-05},
  abstract = {We prove that the space of shifted Poisson structures on a derived scheme \$X\$ locally of finite presentation is equivalent to the space of shifted Lagrangian thickenings out \$X\$, solving a conjecture in shifted Poisson geometry. As a corollary, we show that for \$M\$ a compact oriented \$d\$-dimensional manifold and an \$n\$-shifted Poisson structure on \$X\$, the mapping stack \$\textbackslash mathrm\textbraceleft Map\textbraceright (M,X)\$ has an \$(n-d)\$-shifted Poisson structure. It extends a known theorem for shifted symplectic structures to shifted Poisson structures.},
  archiveprefix = {arXiv},
  keywords = {Mathematics - Algebraic Geometry,Mathematics - Algebraic Topology,Mathematics - Symplectic Geometry},
  file = {/home/alexk/Zotero/storage/VS9BWLY5/Tomić - 2025 - Shifted Lagrangian thickenings of shifted Poisson derived schemes.pdf;/home/alexk/Zotero/storage/6BKWXGK9/2506.html}
}

@misc{tomic2026,
  title = {{{AKSZ}} Construction for Shifted {{Poisson}} Structures},
  author = {Tomi{\'c}, Nikola},
  year = 2026,
  month = jan,
  number = {arXiv:2601.04064},
  eprint = {2601.04064},
  primaryclass = {math},
  publisher = {arXiv},
  doi = {10.48550/arXiv.2601.04064},
  urldate = {2026-03-04},
  abstract = {We prove the AKSZ theorem for shifted Poisson structures: if \$X\$ is an \$n\$-shifted Poisson derived stack, and \$Y\$ a \$d\$-oriented derived stack, then the mapping stack \textbackslash [\textbackslash underline\textbraceleft\textbackslash mathrm\textbraceleft Map\textbraceright\textbraceright (Y,X)\textbackslash ] is naturally endowed with an \$(n-d)\$-shifted Poisson structure. For this, we prove that the data of an \$n\$-shifted Poisson structure on a derived Artin stack is equivalent to the data of an \$(n+1)\$-shifted Lagrangian thickening of it. We also extend the definition of shifted Poisson structures to derived prestacks having a deformation theory and give two applications, one for mapping stacks with a non-proper source and one in BV formalism.},
  archiveprefix = {arXiv},
  keywords = {Mathematics - Algebraic Geometry,Mathematics - Algebraic Topology},
  file = {/home/alexk/Zotero/storage/85HWIAVG/Tomić - 2026 - AKSZ construction for shifted Poisson structures.pdf;/home/alexk/Zotero/storage/FT55IT4L/2601.html}
}

@article{tsygan2007a,
  title = {On the {{Gauss-Manin}} Connection in Cyclic Homology},
  author = {Tsygan, Boris},
  year = 2007,
  journal = {Methods of Functional Analysis and Topology},
  volume = {13},
  number = {1},
  pages = {83--94},
  issn = {1029-3531,2415-7503},
  mrnumber = {2308582},
  file = {/home/alexk/Zotero/storage/HSRZN8AS/article.html}
}

@article{weinstein1971,
  title = {Symplectic Manifolds and Their Lagrangian Submanifolds},
  author = {Weinstein, Alan},
  year = 1971,
  month = jun,
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  urldate = {2021-09-20},
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@article{weinstein1983,
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@article{xu2004,
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}