Remote Access Transactions of the Moscow Mathematical Society

Transactions of the Moscow Mathematical Society

ISSN 1547-738X(online) ISSN 0077-1554(print)

   
 
 

 

Representations of superconformal algebras and mock theta functions


Authors: V. G. Kac and M. Wakimoto
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 78 (2017), vypusk 1.
Journal: Trans. Moscow Math. Soc. 2017, 9-74
MSC (2010): Primary 17B67, 17B10, 17B68, 11F50, 33E05
DOI: https://doi.org/10.1090/mosc/268
Published electronically: December 1, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: It is well known that the normalized characters of integrable highest weight modules of given level over an affine Lie algebra $ \hat {\mathfrak{g}}$ span an $ \textup {SL}_2(\mathbb{Z})$-invariant space. This result extends to admissible $ \hat {\mathfrak{g}}$-modules, where $ \mathfrak{g}$ is a simple Lie algebra or $ \textup {osp}_{1\vert n}$. Applying the quantum Hamiltonian reduction (QHR) to admissible $ \hat {\mathfrak{g}}$-modules when $ \mathfrak{g} =s\ell _2$ (resp. $ =\textup {osp}_{1\vert 2}$) one obtains minimal series modules over the Virasoro (resp. $ N=1$ superconformal algebras), which form modular invariant families.

Another instance of modular invariance occurs for boundary level admissible modules, including when $ \mathfrak{g}$ is a basic Lie superalgebra. For example, if $ \mathfrak{g}=s\ell _{2\vert 1}$ (resp. $ =\textup {osp}_{3\vert 2}$), we thus obtain modular invariant families of $ \hat {\mathfrak{g}}$-modules, whose QHR produces the minimal series modules for the $ N=2$ superconformal algebras (resp. a modular invariant family of $ N=3$ superconformal algebra modules).

However, in the case when $ \mathfrak{g}$ is a basic Lie superalgebra different from a simple Lie algebra or $ \textup {osp}_{1\vert n}$, modular invariance of normalized supercharacters of admissible $ \hat {\mathfrak{g}}$-modules holds outside of boundary levels only after their modification in the spirit of Zwegers' modification of mock theta functions. Applying the QHR, we obtain families of representations of $ N=2,3,4$ and big $ N=4$ superconformal algebras, whose modified (super)characters span an $ \textup {SL}_2(\mathbb{Z})$-invariant space.


References [Enhancements On Off] (What's this?)

  • [AKMF$^+$17] Dražen Adamović, Victor G. Kac, Pierluigi Möseneder Frajria, Paolo Papi, and Ozren Perše, Conformal embeddings of affine vertex algebras in minimal W-algebras II: decompositions, Jpn. J. Math. 12 (2017), no. 2, 261-315. MR 3694933
  • [App] P. Appell, Sur les fonctions doublement périodique de troisième espèce. with Annals Sci. École Norm. Sup. (3) 1 (1884), 135-164., 2 (1885), 9-36., and 3 (1886), 9-42.
  • [Ara05] Tomoyuki Arakawa, Representation theory of superconformal algebras and the Kac-Roan-Wakimoto conjecture, Duke Math. J. 130 (2005), no. 3, 435-478. MR 2184567
  • [FF90] Boris Feigin and Edward Frenkel, Quantization of the Drinfe'ld-Sokolov reduction, Phys. Lett. B 246 (1990), no. 1-2, 75-81. MR 1071340
  • [FKW92] Edward Frenkel, Victor Kac, and Minoru Wakimoto, Characters and fusion rules for $ W$-algebras via quantized Drinfeld-Sokolov reduction, Comm. Math. Phys. 147 (1992), no. 2, 295-328. MR 1174415
  • [GK15] Maria Gorelik and Victor G. Kac, Characters of (relatively) integrable modules over affine Lie superalgebras, Jpn. J. Math. 10 (2015), no. 2, 135-235. MR 3392530
  • [GS88] Peter Goddard and Adam Schwimmer, Factoring out free fermions and superconformal algebras, Phys. Lett. B 214 (1988), no. 2, 209-214. MR 968813
  • [Kac77] V.G. Kac, Lie superalgebras, Adv. Math. 26 (1977), no. 1, 8-96.
  • [Kac90] Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219
  • [KP84] Victor G. Kac and Dale H. Peterson, Infinite-dimensional Lie algebras, theta functions and modular forms, Adv. in Math. 53 (1984), no. 2, 125-264. MR 750341
  • [KRW03] Victor Kac, Shi-Shyr Roan, and Minoru Wakimoto, Quantum reduction for affine superalgebras, Comm. Math. Phys. 241 (2003), no. 2-3, 307-342. MR 2013802
  • [KW88] Victor G. Kac and Minoru Wakimoto, Modular invariant representations of infinite-dimensional Lie algebras and superalgebras, Proc. Nat. Acad. Sci. U.S.A. 85 (1988), no. 14, 4956-4960. MR 949675
  • [KW89] V. G. Kac and M. Wakimoto, Classification of modular invariant representations of affine algebras, Infinite-dimensional Lie algebras and groups (Luminy-Marseille, 1988) Adv. Ser. Math. Phys., vol. 7, World Sci. Publ., Teaneck, NJ, 1989, pp. 138-177. MR 1026952
  • [KW94] Victor G. Kac and Minoru Wakimoto, Integrable highest weight modules over affine superalgebras and number theory, Lie theory and geometry, Progr. Math., vol. 123, Birkhäuser Boston, Boston, MA, 1994, pp. 415-456. MR 1327543
  • [KW01] Victor G. Kac and Minoru Wakimoto, Integrable highest weight modules over affine superalgebras and Appell's function, Comm. Math. Phys. 215 (2001), no. 3, 631-682. MR 1810948
  • [KW05a] Victor G. Kac and Minoru Wakimoto, Corrigendum to: ``Quantum reduction and representation theory of superconformal algebras'' [Adv. Math. 185 (2004), no. 2, 400-458; MR2060475], Adv. Math. 193 (2005), no. 2, 453-455. MR 2137292
  • [KW05b] Victor G. Kac and Minoru Wakimoto, Quantum reduction in the twisted case, Infinite dimensional algebras and quantum integrable systems, Progr. Math., vol. 237, Birkhäuser, Basel, 2005, pp. 89-131. MR 2160843
  • [KW14] Victor G. Kac and Minoru Wakimoto, Representations of affine superalgebras and mock theta functions, Transform. Groups 19 (2014), no. 2, 383-455. MR 3200431
  • [KW16a] Victor G. Kac and Minoru Wakimoto, Representations of affine superalgebras and mock theta functions II, Adv. Math. 300 (2016), 17-70. MR 3534829
  • [KW16b] V. G. Kac and M. Wakimoto, Representations of affine superalgebras and mock theta functions. III, Izv. Ross. Akad. Nauk Ser. Mat. 80 (2016), no. 4, 65-122; English transl., Izv. Math. 80 (2016), no. 4, 693-750. MR 3535359
  • [KW17] V. G. Kac and M. Wakimoto, A characterization of modified mock theta functions, Transform Groups 22 (2017). arxiv:1510.05683
  • [Mum83] David Mumford, Tata lectures on theta. I, Progress in Mathematics, vol. 28, Birkhäuser Boston, Inc., Boston, MA, 1983. MR 688651
  • [RY87] Francesco Ravanini and Sung-Kil Yang, Modular invariance in $ N=2$ superconformal field theories, Phys. Lett. B 195 (1987), no. 2, 202-208. MR 910261
  • [Zag09] Don Zagier, Ramanujan's mock theta functions and their applications (after Zwegers and Ono-Bringmann), Astérisque 326 (2009), Exp. No. 986, vii-viii, 143-164 (2010). Séminaire Bourbaki. Vol. 2007/2008. MR 2605321
  • [Zwe] S.P. Zwegers, Mock theta functions, arxiv: 0807.4834

Similar Articles

Retrieve articles in Transactions of the Moscow Mathematical Society with MSC (2010): 17B67, 17B10, 17B68, 11F50, 33E05

Retrieve articles in all journals with MSC (2010): 17B67, 17B10, 17B68, 11F50, 33E05


Additional Information

V. G. Kac
Affiliation: Department of Mathematics, M.I.T, Cambridge, Massachusetts 02139
Email: kac@math.mit.edu

M. Wakimoto
Affiliation: 12–4 Karato-Rokkoudai, Kita-ku, Kobe 651–1334, Japan
Email: wakimoto@r6.dion.ne.jp

DOI: https://doi.org/10.1090/mosc/268
Keywords: Basic Lie superalgebra, affine Lie superalgebra, superconformal algebra, integrable and admissible representations of affine Lie superalgebras, quantum Hamiltonian reduction, theta function, mock theta function and its modification, modular invariant family of characters
Published electronically: December 1, 2017
Additional Notes: The first named author was supported in part by an NSF grant. The second named author was supported in part by Department of Mathematics, M.I.T
Dedicated: To Ernest Borisovich Vinberg on his 80th birthday
Article copyright: © Copyright 2017 V.G.Kac, M.Wakimoto

American Mathematical Society