| Title:
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M. Kontsevich's graph complexes and universal structures on graded symplectic manifolds (English) |
| Author:
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Morand, Kevin |
| Language:
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English |
| Journal:
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Higher Structures |
| ISSN:
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2209-0606 |
| Volume:
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7 |
| Issue:
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1 |
| Year:
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2023 |
| Pages:
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182-233 |
| Summary lang:
|
English |
| . |
| Category:
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math |
| . |
| Summary:
|
In the formulation of his celebrated Formality conjecture, M. Kontsevich introduced a universal version of the deformation theory for the Schouten algebra of polyvector fields on affine manifolds. This universal deformation complex takes the form of a differential graded Lie algebra of graphs, denoted fGC$_2$, together with an injective morphism towards the Chevalley-Eilenberg complex associated with the Schouten algebra. The latter morphism is given by explicit local formulas making implicit use of the supergeometric interpretation of the Schouten algebra as the algebra of functions on a graded symplectic manifold of degree 1. The ambition of the present work is to generalise Kontsevich’s construction to graded symplectic manifolds of arbitrary degree $n \ge 1$. The corresponding graph model is given by the full Kontsevich graph complex fGC$_d$ where $d=n+1$ stands for the dimension of the associated AKSZ type $\sigma$-model. This generalisation is instrumental to classify universal structures on graded symplectic manifolds. In particular, the zeroth cohomology of the full graph complex fGC$_d$ is shown to act via Lie$_{\infty}$-automorphisms on the algebra of functions on graded symplectic manifolds of degree $n$. This generalises the known action of the Grothendieck-Teichmüller algebra $grt_1 \simeq H^0(\rm{fGC}_2)$ on the space of polyvector fields. This extended action can in turn be used to generate new universal deformations of Hamiltonian functions, generalising Kontsevich flows on the space of Poisson manifolds to differential graded manifolds of higher degrees. As an application of the general formalism, universal deformations of Courant algebroids via trivalent graphs are presented. (English) |
| Keyword:
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Deformation quantization |
| Keyword:
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Graph complexes |
| Keyword:
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Courant algebroids |
| Keyword:
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Graded geometry |
| MSC:
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18G85 |
| MSC:
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37J39 |
| idZBL:
|
Zbl 1539.18015 |
| idMR:
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MR4600460 |
| DOI:
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10.21136/HS.2023.06 |
| . |
| Date available:
|
2026-03-13T10:14:22Z |
| Last updated:
|
2026-03-13 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/153461 |
| . |
| Reference:
|
[1] Alekseev, A., Enriquez, B., Torossian, C.: Drinfeld associators, braid groups and explicit solutions of the Kashiwara-Vergne equations.Publications Mathématiques de l’IHéS 112 143-189 arXiv:0903.4067 MR 2737979, 10.1007/s10240-010-0029-4 |
| Reference:
|
[2] Alekseev, A., Meinrenken, E.: On the Kashiwara Vergne conjecture.Inventiones mathematicae 164 3 615-634 arXiv:math/0506499 MR 2221133, 10.1007/s00222-005-0486-4 |
| Reference:
|
[3] Alekseev, A., Rossi, C. A., Torossian, C., Willwacher, T.: Logarithms and deformation quantization.Inventiones mathematicae 206 1 1-28 arXiv:1401.3200 MR 3556523, 10.1007/s00222-016-0647-7 |
| Reference:
|
[4] Alekseev, A., Torossian, C.: The Kashiwara-Vergne conjecture and Drinfeld’s associators.Annals of mathematics 175 2 415-463 arXiv:0802.4300 MR 2877064, 10.4007/annals.2012.175.2.1 |
| Reference:
|
[5] Alekseev, A., Torossian, C.: Kontsevich Deformation Quantization and Flat Connections.C. Commun. Math. Phys. 300 1 47-64 arXiv:0906.0187 MR 2725182, 10.1007/s00220-010-1106-8 |
| Reference:
|
[6] Alexandrov, M., Schwarz, A., Zaboronsky, O., Kontsevich, M.: The Geometry of the master equation and topological quantum field theory.Int. J. Mod. Phys. A 12 1405 arXiv:hep-th/9502010 |
| Reference:
|
[7] Alm, J.: Universal algebraic structures on polyvector fields.Ph.D. Thesis, Stockholm University, Faculty of Science, Department of Mathematics |
| Reference:
|
[8] Alm, J., Merkulov, S.: Grothendieck-Teichmueller group and Poisson cohomologies.Journal of Noncommutative Geometry 9 1 185-214 arXiv:1203.5933 MR 3337958, 10.4171/jncg/191 |
| Reference:
|
[9] Arnal, D., Amar, N. B., Masmoudi, M.: Cohomology of Good Graphs and Kontsevich Linear State Products.Letters in Mathematical Physics 48 4 291-306 |
| Reference:
|
[10] Bar-Natan, D.: On the Vassiliev knot invariants.Topology 34 2 423-472 10.1016/0040-9383(95)93237-2 |
| Reference:
|
[11] Bar-Natan, D., McKay, B.: Graph cohomology-an overview and some computations.Unpublished |
| Reference:
|
[12] Batalin, I. A., Vilkovisky, G. A.: Gauge Algebra and Quantization.Physics Letters B 102 1 27-31 |
| Reference:
|
[13] Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., Sternheimer, D.: Deformation Theory and Quantization. 1. Deformations of Symplectic Structures.Annals Phys. 111 1 61-110 |
| Reference:
|
[14] Berezin, F. A.: General concept of quantization.Comm. Math. Phys. 40 2 153-174 10.1007/BF01609397 |
| Reference:
|
[15] Bouisaghouane, A., Buring, R., Kiselev, A. V.: The Kontsevich tetrahedral flow revisited.J. Geom. Phys. 119 272-285 arXiv:1608.01710 MR 3661536, 10.1016/j.geomphys.2017.04.014 |
| Reference:
|
[16] Bouisaghouane, A., Kiselev, A. V.: Do the Kontsevich tetrahedral flows preserve or destroy the space of Poisson bi-vectors?.Journal of Physics: Conf.Series 804 Paper 012008 arXiv:1609.06677 |
| Reference:
|
[17] Bressler, P.: The first Pontryagin class.Compositio Mathematica 143 5 1127-1163 arXiv:math/0509563 MR 2360313, 10.1112/S0010437X07002710 |
| Reference:
|
[18] Brown, F.: Mixed Tate motives over ℤ.Annals of mathematics 175 2 949-976 arXiv:1102.1312 MR 2993755, 10.4007/annals.2012.175.2.10 |
| Reference:
|
[19] Buring, R., Kiselev, A. V.: The orientation morphism: from graph cocycles to deformations of Poisson structures.Journal of Physics: Conference Series 1194 Paper 012017 1-10 arXiv:1811.07878 |
| Reference:
|
[20] Buring, R., Kiselev, A. V., Rutten, N. J.: The heptagon-wheel cocycle in the Kontsevich graph complex.J. Nonlin. Math. Phys. 24 Suppl. 1 157-173 arXiv:1710.00658 MR 3750843, 10.1080/14029251.2017.1418060 |
| Reference:
|
[21] Buring, R., Kiselev, A. V., Rutten, N. J.: Infinitesimal deformations of Poisson bi-vectors using the Kontsevich graph calculus.Journal of Physics: Conf. Series 965 Paper 012010 arXiv:1710.02405 |
| Reference:
|
[22] Buring, R., Kiselev, A. V., Rutten, N. J.: Poisson Brackets Symmetry from the Pentagon-Wheel Cocycle in the Graph Complex.Physics of Particles and Nuclei 49 5 924-928 arXiv:1712.05259 10.1134/S1063779618050118 |
| Reference:
|
[23] Bursztyn, H., Dolgushev, V. A., Waldmann, S.: Morita equivalence and characteristic classes of star products.Journal für die reine und angewandte Mathematik 2012 662 95-163 arXiv:0909.4259 MR 2876262 |
| Reference:
|
[24] Campos, R.: BV Formality.Advances in Mathematics 306 807-851 arXiv:1506.07715 MR 3581318, 10.1016/j.aim.2016.10.034 |
| Reference:
|
[25] Cattaneo, A. S., Felder, G.: A Path integral approach to the Kontsevich quantization formula.Commun. Math. Phys. 212 591 arXiv:math/9902090 |
| Reference:
|
[26] Cattaneo, A. S., Felder, G.: On the AKSZ formulation of the Poisson sigma model.Letters in Mathematical Physics 56 163 arXiv:math/0102108 MR 1854134 |
| Reference:
|
[27] Cattaneo, A. S., Felder, G.: Poisson sigma models and deformation quantization.Mod. Phys. Lett. A 16 179-190 arXiv:hep-th/0102208 MR 1832088, 10.1142/S0217732301003255 |
| Reference:
|
[28] Cattaneo, A., Keller, B., Torossian, C., Bruguières, A.: Déformation, Quantification, Théorie de Lie.Panoramas et Synthèses SMF 20 MR 2274222 |
| Reference:
|
[29] Cattaneo, A., Schaetz, F.: Introduction to supergeometry.Rev. Math. Phys. 23 669-690 arXiv:1011.3401 MR 2819233, 10.1142/S0129055X11004400 |
| Reference:
|
[30] Conant, J., Vogtmann, K.: On a theorem of Kontsevich.Algebr. Geom. Topol. 3 1167-1224 arXiv:math/0208169 Zbl 1063.18007, MR 2026331, 10.2140/agt.2003.3.1167 |
| Reference:
|
[31] Courant, T., Weinstein, A.: Beyond Poisson structures.in Actions hamiltoniennes de groupes. Troisième théorème de Lie (Lyon), Travaux en Cours 27 Hermann Paris 39-49 |
| Reference:
|
[32] Courant, T.: Dirac manifolds.Trans. Amer. Math. Soc. 319 631-661 10.1090/S0002-9947-1990-0998124-1 |
| Reference:
|
[33] Culler, M., Vogtmann, K.: Moduli of graphs and automorphisms of free groups.Inventiones mathematicae 84 1 91-119 10.1007/BF01388734 |
| Reference:
|
[34] Dito, G.: Kontsevich star-product on the dual of a Lie algebra.Letters in Mathematical Physics 48 4 307-322 arXiv:math/9905080 10.1023/A:1007643618406 |
| Reference:
|
[35] Dolgushev, V. A.: Erratum to: A Proof of Tsygan’s Formality Conjecture for an Arbitrary Smooth Manifold (2007).arXiv:math/0703113 MR 2717256 |
| Reference:
|
[36] Dolgushev, V. A.: Stable Formality Quasi-isomorphisms for Hochschild Cochains.Mémoires de la Société Mathématique de France 168 SMF arXiv:1109.6031 MR 4234594 |
| Reference:
|
[37] Dolgushev, V. A.: A Formality quasi-isomorphism for Hochschild cochains over rationals can be constructed recursively.International Mathematics Research Notices 18 5729-5785 arXiv:1306.6733 MR 3862118 |
| Reference:
|
[38] Dolgushev, V. A., Paljug, B.: Tamarkin’s construction is equivariant with respect to the action of the Grothendieck-Teichmueller group.Journal of Homotopy and Related Structures 11 3 503-552 arXiv:1402.7356 MR 3542096, 10.1007/s40062-015-0115-x |
| Reference:
|
[39] Dolgushev, V. A., Rogers, C. L.: Notes on Algebraic Operads, Graph Complexes, and Willwacher’s Construction.Contemporary Mathematics 583 “Mathematical Aspects of Quantization” arXiv:1202.2937 MR 3013092 |
| Reference:
|
[40] Dolgushev, V. A., Rogers, C. L.: The cohomology of the full directed graph complex.Algebras and Representation Theory 1-45 arXiv:1711.04701 MR 4109144 |
| Reference:
|
[41] Dolgushev, V. A., Rogers, C. L., Willwacher, T.: Kontsevich’s graph complex, GRT, and the deformation complex of the sheaf of polyvector fields.Annals of Mathematics Second Series 182 3 855-943 arXiv:1211.4230 MR 3418532 |
| Reference:
|
[42] Dolgushev, V. A., Schneider, G. E.: When can a formality quasi-isomorphism over rationals be constructed recursively?.Journal of Pure and Applied Algebra 223 5 2145-2172 arXiv:1610.04879 MR 3906544, 10.1016/j.jpaa.2018.07.012 |
| Reference:
|
[43] Dorfman, I. Y.: Dirac structures of integrable evolution equations.Physics Letters A 125 5 240-246 Dirac structures and integrability of nonlinear evolution equations Nonlinear science : theory and applications John Wiley & Sons Ltd., Chichester 10.1016/0375-9601(87)90201-5 |
| Reference:
|
[44] Drinfel’d, V. G.: On quasitriangular quasi-Hopf algebras and on a group that is closely connected with Gal(ℚ-bar/ℚ).Algebra i Analiz 2 4 149-181 ; Leningrad Math. J. 2 4 829-860 |
| Reference:
|
[45] Esposito, C., Stapor, P., Waldmann, S.: Convergence of the Gutt Star Product.Journal of Lie Theory 27 2 579-622 arXiv:1509.09160 MR 3589272 |
| Reference:
|
[46] Etingof, P., Kazhdan, D.: Quantization of Lie bialgebras, I.Selecta Mathematica 2 1 1-41 arXiv:q-alg/9506005 |
| Reference:
|
[47] Felder, G., Willwacher, T.: On the (ir)rationality of Kontsevich weights.International Mathematics Research Notices 2010 4 701-716 arXiv:0808.2762 MR 2595009 |
| Reference:
|
[48] Fresse, B.: Homotopy of operads & Grothendieck-Teichmüller groups.Mathematical Surveys and Monographs 217 1236pp MR 3616816 |
| Reference:
|
[49] Fresse, B., Turchin, V., Willwacher, T.: The rational homotopy of mapping spaces of E_(n) operads (2017).arXiv:1703.06123 MR 3870769 |
| Reference:
|
[50] Furusho, H.: Double shuffle relation for associators.Annals of mathematics 174 1 341-360 arXiv:0808.0319 MR 2811601, 10.4007/annals.2011.174.1.9 |
| Reference:
|
[51] Gerstenhaber, M.: The Cohomology Structure of an Associative Ring.Annals of Mathematics Second Series 78 2 267-288 10.2307/1970343 |
| Reference:
|
[52] Getzler, E., Kapranov, M. M.: Modular operads.Compositio Mathematica 110 1 65-125 arXiv:dg-ga/9408003 10.1023/A:1000245600345 |
| Reference:
|
[53] Goldman, W. M., Millson, J. J.: The deformation theory of representations of fundamental groups of compact Kähler manifolds.Publications Mathématiques de l’IHéS 67 43-96 10.1007/BF02699127 |
| Reference:
|
[54] Gorbounov, V., Malikov, F., Schechtman, V.: Gerbes of chiral differential operators. II. Vertex algebroids.Inventiones mathematicae 155 605-680 arXiv:math/0003170 MR 2038198, 10.1007/s00222-003-0333-4 |
| Reference:
|
[55] Gorbounov, V., Malikov, F., Schechtman, V.: Gerbes of chiral differential operators. III.C. Duval, V. Ovsienko, L. Guieu (eds) The Orbit Method in Geometry and Physics: In Honor of A.A. Kirillov. Progress in Mathematics 213 73–100 Birkhäuser, Boston, MA. arXiv:math/0005201 MR 1995376 |
| Reference:
|
[56] Groenewold, H. J.: On the Principles of elementary quantum mechanics.Physica 12 7 405-460 10.1016/S0031-8914(46)80059-4 |
| Reference:
|
[57] Grothendieck, A.: Esquisse d’un Programme.London Math. Soc. Lect. Note Ser. 1 7-48 |
| Reference:
|
[58] Grützmann, M.: H-twisted Lie algebroids.Journal of Geometry and Physics 61 2 476-484 arXiv:1005.5680 MR 2746131, 10.1016/j.geomphys.2010.10.016 |
| Reference:
|
[59] Gualtieri, M.: Generalized complex geometry.Annals of Mathematics 174 75-123 arXiv:math/0401221 MR 2811595, 10.4007/annals.2011.174.1.3 |
| Reference:
|
[60] Gutt, S.: An explicit *-product on the cotangent bundle of a Lie group.Letters in Mathematical Physics 7 3 249-258 10.1007/BF00400441 |
| Reference:
|
[61] Hinich, V.: Tamarkin’s proof of Kontsevich formality theorem.Forum Mathematicum 15 4 591-614 arXiv:math/0003052 MR 1978336 |
| Reference:
|
[62] Hitchin, N.: Generalized Calabi-Yau manifolds.Quarterly Journal of Mathematics 54 3 arXiv:math/0209099 MR 2013140 |
| Reference:
|
[63] Hofman, C., Park, J. S.: Topological open membranes (2002).arXiv:hep-th/0209148 |
| Reference:
|
[64] Hofman, C., Park, J. S.: BV quantization of topological open membranes.Commun. Math. Phys. 249 2 249-271 arXiv:hep-th/0209214 MR 2080953 |
| Reference:
|
[65] Ikeda, N.: Two-dimensional gravity and nonlinear gauge theory.Annals Phys. 235 435 arXiv:hep-th/9312059 |
| Reference:
|
[66] Ikeda, N.: Deformation of BF theories, topological open membrane and a generalization of the star deformation.JHEP 0107 037 arXiv:hep-th/0105286 MR 1850860 |
| Reference:
|
[67] Ikeda, N.: Chern–Simons gauge theory coupled with BF theory.Int. J. Mod. Phys. A 18 2689 arXiv:hep-th/0203043 MR 1985761 |
| Reference:
|
[68] Ikeda, N., Izawa, K. I.: General form of dilaton gravity and nonlinear gauge theory.Prog. Theor. Phys. 90 237 arXiv:hep-th/9304012 |
| Reference:
|
[69] Ikeda, N., Uchino, K.: QP-Structures of Degree 3 and 4D Topological Field Theory.Commun. Math. Phys. 303 317 arXiv:1004.0601 MR 2782617 |
| Reference:
|
[70] Jost, C.: Globalizing L-infinity automorphisms of the Schouten algebra of polyvector fields.Differential Geometry and its Applications 31 2 239-247 arXiv:1201.1392 MR 3032646, 10.1016/j.difgeo.2012.12.002 |
| Reference:
|
[71] Jurco, B., Vysoky, J.: Courant Algebroid Connections and String Effective Actions.Noncommutative Geometry and Physics 4 211-265 arXiv:1612.01540 MR 3674835 |
| Reference:
|
[72] Kathotia, V.: Kontsevich’s Universal Formula for Deformation Quantization and the Campbell-Baker-Hausdorff Formula, I.International Journal of Mathematics 11 04 523-551 arXiv:math/9811174 10.1142/S0129167X0000026X |
| Reference:
|
[73] Keller, B.: Introduction to A-infinity algebras and modules.Homology, Homotopy and Applications 3 1 1-35 arXiv:math/9910179 MR 1854636, 10.4310/HHA.2001.v3.n1.a1 |
| Reference:
|
[74] Keller, F., Waldmann, S.: Deformation Theory of Courant Algebroids via the Rothstein Algebra.Journal of Pure and Applied Algebra 219 8 3391-3426 arXiv:0807.0584 MR 3320226, 10.1016/j.jpaa.2014.12.002 |
| Reference:
|
[75] Khoroshkin, A., Willwacher, T., Živković, M.: Differentials on graph complexes.Advances in Mathematics 307 1184-1214 arXiv:1411.2369 MR 3590540, 10.1016/j.aim.2016.05.029 |
| Reference:
|
[76] Kiselev, A. V.: Open problems in the Kontsevich graph construction of Poisson bracket symmetries.Journal of Physics: Conference Series 1416 Issue 1 012018 1184-1214 arXiv:1910.05844 |
| Reference:
|
[77] Kiselev, A. V., Buring, R.: The Kontsevich graph orientation morphism revisited.Banach Center Publications 123 123-139 arXiv:1904.13293 MR 4276046, 10.4064/bc123-5 |
| Reference:
|
[78] Klimčik, C., Strobl, T.: WZW - Poisson manifolds.J. Geom. Phys. 43 341 arXiv:math/0104189 MR 1929911 |
| Reference:
|
[79] Kneissler, J.: On spaces of connected graphs I: Properties of Ladders.Proc. Internat. Conf. “Knots in Hellas ’98”, Series on Knots and Everything 24 252-273 arXiv:math/0301018 MR 1865711 |
| Reference:
|
[80] Kneissler, J.: On spaces of connected graphs II: Relations in the algebra Lambda.Journal of Knot Theory and Its Ramifications 10 5 667-674 arXiv:math/0301019 MR 1839694, 10.1142/S0218216501001074 |
| Reference:
|
[81] Kneissler, J.: On spaces of connected graphs III: The Ladder Filtration.Journal of Knot Theory and Its Ramifications 10 5 675-686 arXiv:math/0301020 MR 1839695, 10.1142/S0218216501001086 |
| Reference:
|
[82] Kontsevich, M.: Feynman Diagrams and Low-Dimensional Topology.First European Congress of Mathematics Paris 97-121 |
| Reference:
|
[83] Kontsevich, M.: Formal (non)-commutative symplectic geometry.The Gelfand Mathematical Seminars 1990-1992 173-187 |
| Reference:
|
[84] Kontsevich, M.: Formality conjecture.Deformation Theory and Symplectic Geometry 20 139-156 |
| Reference:
|
[85] Kontsevich, M.: Deformation quantization of Poisson manifolds. I..Letters in Mathematical Physics 66 157-216 arXiv:q-alg/9709040 MR 2062626, 10.1023/B:MATH.0000027508.00421.bf |
| Reference:
|
[86] Kontsevich, M.: Operads and Motives in Deformation Quantization.Letters in Mathematical Physics 48 35-72 arXiv:math/9904055 10.1023/A:1007555725247 |
| Reference:
|
[87] Kosmann-Schwarzbach, Y.: Courant Algebroids. A Short History.SIGMA 9 014 8p. arXiv:1212.0559 MR 3033556 |
| Reference:
|
[88] Le, T. T. Q., Murakami, J.: Kontsevich’s integral for the Kauffman polynomial.Nagoya Mathematical Journal 142 39-65 |
| Reference:
|
[89] Liu, J., Sheng, Y.: QP-structures of degree 3 and CLWX 2-algebroids.J. Symplectic Geom. 17 6 1853-1891 arXiv:1602.01127 MR 4057729, 10.4310/JSG.2019.v17.n6.a8 |
| Reference:
|
[90] Liu, Z. J., Weinstein, A., Xu, P.: Manin triples for Lie bialgebroids.J. Diff. Geom. 45 no.3, 547 arXiv:dg-ga/9508013 |
| Reference:
|
[91] Loday, J. L., Vallette, B.: Algebraic Operads.Grundlehren der mathematischen Wissenschaften 346 636p MR 2954392 |
| Reference:
|
[92] Markl, M.: Cyclic operads and homology of graph complexes.J. Slovák and M. Čadek (eds.): Proceedings of the 18th Winter School “Geometry and Physics”. Circolo Matematico di Palermo, Palermo 161-170 arXiv:math/9801095 |
| Reference:
|
[93] Mehta, R. A.: Supergroupoids, double structures, and equivariant cohomology.Ph.D. Thesis, University of California, Berkeley arXiv:math/0605356 MR 2709144 |
| Reference:
|
[94] Merkulov, S.: Exotic automorphisms of the Schouten algebra of polyvector fields (2008).arXiv:0809.2385 |
| Reference:
|
[95] Merkulov, S.: Multi-oriented props and homotopy algebras with branes.Letters in Mathematical Physics 110 pages 1425-1475 arXiv:1712.09268 MR 4109803, 10.1007/s11005-019-01248-x |
| Reference:
|
[96] Merkulov, S.: Grothendieck-Teichmueller group, operads and graph complexes: a survey.Integrability, Quantization, and Geometry II. Quantum Theories and Algebraic Geometry, Proc. Sympos. Pure Math. 103 Amer. Math. Soc., Providence, RI 383-445 arXiv:1904.13097 MR 4285704 |
| Reference:
|
[97] Merkulov, S., Vallette, B.: Deformation theory of representations of prop(erad)s.Journal für die reine und angewandte Mathematik 2009 634 51-106 arXiv:0707.0889 MR 2560406 |
| Reference:
|
[98] Merkulov, S., Willwacher, T.: Grothendieck-Teichmüller and Batalin-Vilkovisky.Letters in Mathematical Physics 104 5 625-634 arXiv:1012.2467 MR 3197008, 10.1007/s11005-014-0692-3 |
| Reference:
|
[99] Morand, K.: A Note on Multi-Oriented Graph Complexes and Deformation Quantization of Lie Bialgebroids.SIGMA 18 020 arXiv:2102.07593 MR 4395777 |
| Reference:
|
[100] Moyal, J. E.: Quantum mechanics as a statistical theory.Mathematical Proceedings of the Cambridge Philosophical Society 45 1 99-124 10.1017/S0305004100000487 |
| Reference:
|
[101] Penner, R. C.: The decorated Teichmüller space of punctured surfaces.Commun.Math. Phys. 113 2 299-339 10.1007/BF01223515 |
| Reference:
|
[102] Penner, R. C.: Perturbative series and the moduli space of Riemann surfaces.J. Differential Geom. 27 1 35-53 |
| Reference:
|
[103] Rossi, C. A., Willwacher, T.: P. Etingof’s conjecture about Drinfel’d associators (2014).arXiv:1404.2047 |
| Reference:
|
[104] Roytenberg, D.: Courant algebroids, derived brackets and even symplectic supermanifolds.Ph.D. Thesis, University of California, Berkeley arXiv:math/9910078 |
| Reference:
|
[105] Roytenberg, D.: On the structure of graded symplectic supermanifolds and Courant algebroids.Contemp. Math. 315 Amer. Math. Soc. arXiv:math/0203110 Zbl 1036.53057, MR 1958835 |
| Reference:
|
[106] Roytenberg, D.: AKSZ-BV Formalism and Courant Algebroid-induced Topological Field Theories.Letters in Mathematical Physics 79 143 arXiv:hep-th/0608150 MR 2301393 |
| Reference:
|
[107] Rutten, N. J., Kiselev, A. V.: The defining properties of the Kontsevich unoriented graph complex.Journal of Physics: Conference Series 1194 Paper 012095 1-10 arXiv:1811.10638 |
| Reference:
|
[108] Schaller, P., Strobl, T.: Poisson structure induced (topological) field theories.Mod. Phys. Lett. A 9 3129 arXiv:hep-th/9405110 Zbl 1015.81574 |
| Reference:
|
[109] Schneps, L.: Double Shuffle and Kashiwara-Vergne Lie algebras.Journal of Algebra 367 54-74 arXiv:1201.5316 MR 2948210, 10.1016/j.jalgebra.2012.04.034 |
| Reference:
|
[110] Ševera, P.: Quantization of Poisson families and of twisted Poisson structures.Letters in Mathematical Physics 63 2 105-113 arXiv:math/0205294 MR 1978549, 10.1023/A:1023077126186 |
| Reference:
|
[111] Ševera, P., Weinstein, A.: Poisson geometry with a 3 form background.Prog. Theor. Phys. Suppl. 144 145 arXiv:math/0107133 MR 2023853 |
| Reference:
|
[112] Ševera, P., Willwacher, T.: Equivalence of formalities of the little discs operad.Duke Math. J. 160 1 175-206 arXiv:0905.1789 MR 2838354 |
| Reference:
|
[113] Shoikhet, B.: On the Kontsevich and the Campbell-Baker-Hausdorff deformation quantizations of a linear Poisson structure (1999).arXiv:math/9903036 |
| Reference:
|
[114] Shoikhet, B.: An L_(∞) algebra structure on polyvector fields.Selecta Mathematica 24 2 1691-1728 arXiv:0805.3363 MR 3782433, 10.1007/s00029-017-0382-y |
| Reference:
|
[115] Tamarkin, D. E.: Another proof of M. Kontsevich formality theorem for $ℝ^(n)$ (1998).arXiv:math/9803025 |
| Reference:
|
[116] Tamarkin, D. E.: Quantization of lie Bialgebras via the Formality of the operad of Little Disks.GAFA Geometric And Functional Analysis 17 2 537-604 MR 2322494, 10.1007/s00039-007-0591-1 |
| Reference:
|
[117] Vogel, P.: Algebraic structures on modules of diagrams.Journal of Pure and Applied Algebra 215 6 1292-1339 MR 2769234, 10.1016/j.jpaa.2010.08.013 |
| Reference:
|
[118] Voronov, A. A.: Quantizing Poisson Manifolds.Perspectives on Quantization (L. A. Coburn and M. A. Rieffel (eds.) Contemp. Math. 214 AMS, Providence, RI 189-195 arXiv:q-alg/9701017 |
| Reference:
|
[119] Willwacher, T.: M. Kontsevich’s graph complex and the Grothendieck-Teichmueller Lie algebra.Inventiones mathematicae 200 671-760 arXiv:1009.1654 MR 3348138, 10.1007/s00222-014-0528-x |
| Reference:
|
[120] Willwacher, T.: The Homotopy Braces Formality Morphism.Duke Math. J. 165 10 1815-1964 arXiv:1109.3520 MR 3522653 |
| Reference:
|
[121] Willwacher, T.: Characteristic classes in deformation quantization.International Mathematics Research Notices 2015 6538-6557 arXiv:1208.4249 MR 3384487, 10.1093/imrn/rnu136 |
| Reference:
|
[122] Willwacher, T.: The Oriented Graph Complexes.Communications in Mathematical Physics 334 3 1649-1666 arXiv:1308.4006 MR 3312446, 10.1007/s00220-014-2168-9 |
| Reference:
|
[123] Willwacher, T.: The Grothendieck–Teichmüller Group.Unpublished notes |
| Reference:
|
[124] Willwacher, T., M.Živković, : Multiple edges in M. Kontsevich’s graph complexes and computations of the dimensions and Euler characteristics.Adv. Math. 272 553-578 arXiv:1401.4974 MR 3303240, 10.1016/j.aim.2014.12.010 |
| Reference:
|
[125] Yekutieli, A.: MC Elements in Pronilpotent DG Lie Algebras.Journal of Pure and Applied Algebra 216 11 2338-2360 arXiv:1103.1035 MR 2927171, 10.1016/j.jpaa.2012.03.002 |
| Reference:
|
[126] Živković, M.: Multi-directed graph complexes and quasi-isomorphisms between them I: oriented graphs.High. Struct. 4 arXiv:1703.09605 MR 4074277 |
| Reference:
|
[127] Živković, M.: Multi-directed graph complexes and quasi-isomorphisms between them II: Sourced graphs.Int. Math. Res. Not. 948-1004 arXiv:1712.01203 MR 4201958 |
| . |
