Ellipsitomic associators. (Associateurs ellipsitomiques.) (English. French summary) Zbl 1540.18001
Mémoires de la Société Mathématique de France. Nouvelle Série 179. Paris: Société Mathématique de France (SMF) (ISBN 978-2-85629-981-4). x, 79 p. (2023).
This work fits in a general program that studies associators for an oriented surface together with a finite group acting on it. The aim of this paper is two-fold:
- ●
- to provide an operadic interpretation of elliptic associators
- ●
- to show that holonomies of the universal ellipsitomic KZB connection along suitable paths produce examples of ellipsitomic associators
Reviewer: Nicolas Blanco (Birmingham)
MSC:
| 18-02 | Research exposition (monographs, survey articles) pertaining to category theory |
| 18M60 | Operads (general) |
| 14H52 | Elliptic curves |
| 20F36 | Braid groups; Artin groups |
| 57K20 | 2-dimensional topology (including mapping class groups of surfaces, Teichmüller theory, curve complexes, etc.) |
| 32G34 | Moduli and deformations for ordinary differential equations (e.g., Knizhnik-Zamolodchikov equation) |
| 11F11 | Holomorphic modular forms of integral weight |
Keywords:
associators; operads; groupoids; configuration spaces; braids; chord diagrams; elliptic curves; Grothendieck-Teichmüller; Eisenstein seriesReferences:
| [1] | A. Alekseev, B. Enriquez & C. Torossian -“Drinfeld associators, braid groups and explicit solutions of the Kashiwara-Vergne equations”, Publ. Math. Inst. Hautes Études Sci. 112 (2010), p. 143-189. · Zbl 1238.17008 |
| [2] | A. Alekseev & C. Torossian -“The Kashiwara-Vergne conjecture and Drin-feld”s associators”, Ann. of Math. 175 (2012), p. 415-463. · Zbl 1243.22009 |
| [3] | S. Axelrod & I. M. Singer -“Chern-Simons perturbation theory. II”, J. Dif-ferential Geom. 39 (1994), p. 173-213. · Zbl 0827.53057 |
| [4] | D. Bar-Natan -“On associators and the Grothendieck-Teichmuller group. I”, Selecta Math. (N.S.) 4 (1998), p. 183-212. · Zbl 0974.16028 |
| [5] | D. Bernard -“On the Wess-Zumino-Witten models on Riemann surfaces”, Nu-clear Phys. B 309 (1988), p. 145-174. |
| [6] | ”On the Wess-Zumino-Witten models on the torus”, Nuclear Phys. B 303 (1988), p. 77-93. |
| [7] | J. S. Birman -“On braid groups”, Comm. Pure Appl. Math. 22 (1969), p. 41-72. · Zbl 0157.30904 |
| [8] | R. Bott & C. Taubes -“On the self-linking of knots”, vol. 35, 1994, Topology and physics, p. 5247-5287. · Zbl 0863.57004 |
| [9] | A. Brochier -“Cyclotomic associators and finite type invariants for tangles in the solid torus”, Algebr. Geom. Topol. 13 (2013), p. 3365-3409. · Zbl 1321.57005 |
| [10] | J. Broedel, N. Matthes, G. Richter & O. Schlotterer -“Twisted elliptic multiple zeta values and non-planar one-loop open-string amplitudes”, J. Phys. A 51 (2018), 285401. · Zbl 1401.81071 |
| [11] | F. Brown -“Single-valued motivic periods and multiple zeta values”, Forum Math. Sigma 2 (2014), Paper No. e25. · Zbl 1377.11099 |
| [12] | D. Calaque, B. Enriquez & P. Etingof -“Universal KZB equations: the elliptic case”, in Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. I, Progr. Math., vol. 269, Birkhäuser, 2009, p. 165-266. · Zbl 1241.32011 |
| [13] | D. Calaque & M. Gonzalez -“On the universal ellipsitomic KZB connection”, Selecta Math. (N.S.) 26 (2020), Paper No. 73. · Zbl 1458.32014 |
| [14] | ”A moperadic approach to cyclotomic associators”, preprint arXiv:2004.00572. |
| [15] | M. Ching -“A note on the composition product of symmetric sequences”, J. Ho-motopy Relat. Struct. 7 (2012), p. 237-254. · Zbl 1266.18015 |
| [16] | F. Diamond & J. Shurman -A first course in modular forms, Graduate Texts in Math., vol. 228, Springer, 2005. · Zbl 1062.11022 |
| [17] | V. G. Drinfel ′ d -“On quasitriangular quasi-Hopf algebras and on a group that is closely connected with Gal(Q/Q)”, Algebra i Analiz 2 (1990), p. 149-181; translation in Leningrad Math. J. 2 (1991), p. 829-860. · Zbl 0728.16021 |
| [18] | B. Enriquez -“Quasi-reflection algebras and cyclotomic associators”, Selecta Math. (N.S.) 13 (2008), p. 391-463. · Zbl 1221.17014 |
| [19] | ”Elliptic associators”, Selecta Math. (N.S.) 20 (2014), p. 491-584. · Zbl 1294.17012 |
| [20] | ”Flat connections on configuration spaces and braid groups of surfaces”, Adv. Math. 252 (2014), p. 204-226. · Zbl 1286.53058 |
| [21] | ”Analogues elliptiques des nombres multizétas”, Bull. Soc. Math. France 144 (2016), p. 395-427. · Zbl 1407.11101 |
| [22] | P. Etingof & D. Kazhdan -“Quantization of Lie bialgebras. II, III”, Selecta Math. (N.S.) 4 (1998), p. 213-231, 233-269. · Zbl 0915.17009 |
| [23] | B. Fresse -Homotopy of operads and Grothendieck-Teichmüller groups. Part 1, Mathematical Surveys and Monographs, vol. 217, Amer. Math. Soc., 2017. · Zbl 1373.55014 |
| [24] | W. Fulton & R. MacPherson -“A compactification of configuration spaces”, Ann. of Math. 139 (1994), p. 183-225. · Zbl 0820.14037 |
| [25] | A. B. Goncharov -“Multiple ζ-values, Galois groups, and geometry of modular varieties”, in European Congress of Mathematics, Vol. I (Barcelona, 2000), Progr. Math., vol. 201, Birkhäuser, 2001, p. 361-392. · Zbl 1042.11042 |
| [26] | M. Gonzalez -“Contributions to the theory of KZB associators”, Thèse de doctorat, Sorbonne Université, 2018, https://hal.archives-ouvertes.fr/ tel-02541848. |
| [27] | ”Surface Drinfeld Torsors I : Higher Genus Associators”, preprint arXiv:2004.07303. |
| [28] | R. M. Hain -“The Hodge de Rham theory of relative Malcev completion”, Ann. Sci. École Norm. Sup. 31 (1998), p. 47-92. · Zbl 0911.14008 |
| [29] | P. Humbert -“Intégrale de Kontsevich elliptique et enchevêtrements en genre supérieur”, Thèse de doctorat, Université de Strasbourg, 2012, https://tel. archives-ouvertes.fr/tel-00762209. |
| [30] | E. Jurisich & R. Wilson -“A generalization of Lazard”s elimination theorem”, Comm. Algebra 32 (2004), p. 4037-4041. · Zbl 1094.17002 |
| [31] | M. Kaneko & D. Zagier -“A generalized Jacobi theta function and quasimod-ular forms”, in The moduli space of curves (Texel Island, 1994), Progr. Math., vol. 129, Birkhäuser, 1995, p. 165-172. · Zbl 0892.11015 |
| [32] | M. Kontsevich -“Vassiliev”s knot invariants”, in I. M. Gelfand Seminar, Adv. Soviet Math., vol. 16, Amer. Math. Soc., 1993, p. 137-150. · Zbl 0839.57006 |
| [33] | ”Operads and motives in deformation quantization”, vol. 48, 1999, Moshé Flato (1937-1998), p. 35-72. · Zbl 0945.18008 |
| [34] | T. T. Q. Le & J. Murakami -“Kontsevich”s integral for the Kauffman poly-nomial”, Nagoya Math. J. 142 (1996), p. 39-65. · Zbl 0866.57008 |
| [35] | A. Levin & G. Racinet -“Towards multiple elliptic polylogarithms”, preprint arXiv:math/0703237. |
| [36] | M. Maassarani -“Sur certains espaces de configuration associés aux sous-groupes finis de PSL 2 (C)”, Bull. Soc. Math. France 147 (2019), p. 123-157. · Zbl 1477.55013 |
| [37] | J. D. Stasheff -“Homotopy associativity of H-spaces. I, II”, Trans. Amer. Math. Soc. 108 (1963), p. 275-292, 293-312. · Zbl 0114.39402 |
| [38] | D. E. Tamarkin -“Formality of chain operad of little discs”, Lett. Math. Phys. 66 (2003), p. 65-72. · Zbl 1048.18007 |
| [39] | ”Another proof of M. Kontsevich formality theorem”, preprint arXiv:math.QA/9803025. |
| [40] | H. Tsunogai -“The stable derivation algebras for higher genera”, Israel J. Math. 136 (2003), p. 221-250. · Zbl 1051.14025 |
| [41] | D. Zagier -“Elliptic modular forms and their applications”, in The 1-2-3 of modular forms, Universitext, Springer, 2008, p. 1-103. · Zbl 1259.11042 |
| [42] | G. LELLOUCH -Sur les ensembles de rotation des homéomorphismes de surface de genre ≥ 2 |
| [43] | C. ARHANCET & C. KRIEGLER -Projections, multipliers and decomposable maps on noncommutative L p -spaces |
| [44] | É. GAUDRON & G. RÉMOND -Nouveaux théorèmes d’isogénie 2022 |
| [45] | R. OLLIVIER & P. SCHNEIDER -On the pro-p Iwahori Hecke Ext-algebra of SL 2 (Qp) |
| [46] | R. CARLES & C. CHEVERRY -Constructive and destructive interferences in nonlinear hyperbolic equations |
| [47] | C. PARK & Z. QIAN -On mod p local-global compatibility for GLn(Qp) in the ordinary case |
| [48] | L. BIGORGNE -Asymptotic properties of small data solutions of the Vlasov-Maxwell system in high dimensions 2021 |
| [49] | K. COULIBALY-PASQUIER & L. MICLO -On the evolution by duality of domains on manifolds · Zbl 1482.58019 |
| [50] | A. ARABIA -Espaces de configuration généralisés. Espaces topologiques i-acycliques. Suites spectrales basiques |
| [51] | C. ERIGNOU -Hydrodynamic limit for an active exclusion process |
| [52] | V. A. DOLGUSHEV -Stable Formality Quasi-isomorphisms for Hochschild Cochains 2020 |
| [53] | D. BENOIS -p-adic height and p-adic Hodge theory |
| [54] | Y. ALMOG & B. HELFFER -The spectrum of a Schrödinger operator in a wire-like domain with a purely imaginary degenerate potential in the semiclassical limit · Zbl 1458.35285 |
| [55] | D. ARA & G. MALTSINIOTIS -Joint et tranches pour les ∞-catégories strictes |
| [56] | S. GHAZOUANI & L. PIRIO -Moduli spaces of flat tori and elliptic hypergeometric functions 2019 |
| [57] | D. XU -Lifting the Cartier transform of Ogus-Vologodsky module p n 162. J.-H. CHIENG, C.-Y. HSIAO & I-H. TSAI -Heat kernel asymptotics, local index theorem and trace integrals for Cauchy-Riemann manifolds with S 1 action |
| [58] | F. JAUBERTEAU, Y. ROLLIN & S. TAPIE -Discrete geometry and isotropic surfaces · Zbl 1432.53110 |
| [59] | P. VIDOTTO -Ergodic properties of some negatively curved manifolds with infinite measure 2018 |
| [60] | L. POSITSELSKI -Weakly curved A∞-algebras over a topological local ring |
| [61] | T. LUPU -Poisson ensembles of loops of one-dimensional diffusions |
| [62] | M. SPITZWECK -A commutative P 1 -spectrum representing motivic cohomology over Dedekind domains |
| [63] | C. SABBAH -Irregular Hodge Theory 2017 |
| [64] | Y. DING -Formes modulaires p-adiques sur les courbes de Shimura unitaires et compatibilité local-global |
| [65] | G. MASSUYEAU, V. TURAEV -Brackets in the Pontryagin algebras of manifolds |
| [66] | M.P. GUALDANI, S. MISCHLER, C. MOUHOT -Factorization of non-symmetric operators and exponential H-theorem |
| [67] | M. MACULAN -Diophantine applications of geometric invariant theory · Zbl 1420.14109 |
| [68] | T. SCHOENEBERG -Semisimple Lie algebras and their classification over p-adic fields |
| [69] | P.G. LE FLOCH , Y. MA -The mathematical validity of the f (R) theory of modified gravity 2016 |
| [70] | R. BEUZART-PLESSIS -La conjecture locale de Gross-Prasad pour les représentations tempérées des groupes unitaires · Zbl 1357.22008 |
| [71] | M. MOKHTAR-KHARROUBI -Compactness properties of perturbed sub-stochastic C 0 -semigroups on L 1 (µ) with applications to discreteness and spectral gaps |
| [72] | Y. CHITOUR, P. KOKKONEN -Rolling of manifolds and controllability in dimension three |
| [73] | N. KARALIOLIOS -Global aspects of the reducibility of quasiperiodic cocycles in compact Lie groups |
| [74] | V. BONNAILLIE-NOËL, M. DAUGE, N. POPOFF -Ground state energy of the magnetic Laplacian on corner domains · Zbl 1368.81072 |
| [75] | P. AUSCHER, S. STAHLHUT -Functional calculus for first order systems of Dirac type and boundary value problems 2015 |
| [76] | R. DANCHIN, P.B. MUCHA -Critical functional framework and maximal regularity in action on systems of incompressible flows |
| [77] | J. AYOUB -Motifs des variétés analytiques rigides 140/141. |
| [78] | Y. LU, B. TEXIER -A stability criterion for high-frequency oscillations 2014 138/139. T. MOCHIZUKI -Holonomic D-modules with Betti structures 137. P. SEIDEL -Abstract analogues of flux as symplectic invariants |
| [79] | J. SJÖSTRAND -Weyl law for semi-classical resonances with randomly perturbed potentials 2013 135. L. PRELLI -Microlocalization of subanalytic sheaves |
| [80] | P. BERGER -Persistence of stratification of normally expanded laminations |
| [81] | L. DESIDERI -Problème de Plateau, équations fuchsiennes et problème de Riemann Hilbert |
| [82] | X. BRESSAUD, N. FOURNIER -One-dimensional general forest fire processes 2012 130/131. |
| [83] | Y. NAKKAJIMA -Weight filtration and slope filtration on the rigid cohomology of a variety in characteristic p > 0 |
| [84] | W. A STEINMETZ-ZIKESCH -Algèbres de Lie de dimension infinie et théorie de la descente 128. D. DOLGOPYAT -Repulsion from resonances 2011 |
| [85] | B. LE STUM -The overconvergent site 125/126. J. BERTIN, M. ROMAGNY -Champs de Hurwitz 124. G. HENNIART, B. LEMAIRE -Changement de base et induction automorphe pour GLn en caractéristique non nulle 2010 |
| [86] | C.-H. HSIAO -Projections in several complex variables |
| [87] | H. DE THÉLIN, G. VIGNY -Entropy of meromorphic maps and dynamics of birational maps 121. M. REES -A Fundamental Domain for V 3 |
| [88] | B. DEMANGE -Uncertainty principles associated to non-degenerate quadratic forms |
| [89] | A. SIEGEL, J. M. THUSWALDNER -Topological properties of Rauzy fractals · Zbl 1229.28021 |
| [90] | D. HÄFNER -Creation of fermions by rotating charged black holes |
| [91] | P. BOYER -Faisceaux pervers des cycles évanescents des variétés de Drinfeld et groupes de cohomologie du modèle de Deligne-Carayol |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
