Abstract
For any 4d \( \mathcal{N} \) = 2 SCFT, there is a subsector described by a 2d chiral algebra. The vacuum character of the chiral algebra reproduces the Schur index of the corresponding 4d theory. The Macdonald index counts the same set of operators as the Schur index, but the former has one more fugacity than the latter. We conjecture a prescription to obtain the Macdonald index from the chiral algebra. The vacuum module admits a filtration, from which we construct an associated graded vector space. From this grading, we conjecture a notion of refined character for the vacuum module of a chiral algebra, which reproduces the Macdonald index. We test this prescription for the Argyres-Douglas theories of type (A 1 , A 2n ) and (A 1 , D 2n+1) where the chiral algebras are given by Virasoro and \( \widehat{\mathfrak{su}}(2) \) affine Kac-Moody algebra. When the chiral algebra has more than one family of generators, our prescription requires a knowledge of the generators from the 4d.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
C. Beem et al., Infinite chiral symmetry in four dimensions, Commun. Math. Phys. 336 (2015) 1359 [arXiv:1312.5344] [INSPIRE].
C. Beem, W. Peelaers, L. Rastelli and B.C. van Rees, Chiral algebras of class S, JHEP 05 (2015) 020 [arXiv:1408.6522] [INSPIRE].
M. Lemos and W. Peelaers, Chiral algebras for trinion theories, JHEP 02 (2015) 113 [arXiv:1411.3252] [INSPIRE].
C. Cordova and S.-H. Shao, Schur indices, BPS particles and Argyres-Douglas theories, JHEP 01 (2016) 040 [arXiv:1506.00265] [INSPIRE].
M. Buican and T. Nishinaka, On the superconformal index of Argyres-Douglas theories, J. Phys. A 49 (2016) 015401 [arXiv:1505.05884] [INSPIRE].
D. Xie, W. Yan and S.-T. Yau, Chiral algebra of Argyres-Douglas theory from M5 brane, arXiv:1604.02155 [INSPIRE].
T. Nishinaka and Y. Tachikawa, On 4D rank-one \( \mathcal{N} \) = 3 superconformal field theories, JHEP 09 (2016) 116 [arXiv:1602.01503] [INSPIRE].
M. Lemos, P. Liendo, C. Meneghelli and V. Mitev, Bootstrapping \( \mathcal{N} \) = 3 superconformal theories, JHEP 04 (2017) 032 [arXiv:1612.01536] [INSPIRE].
F. Bonetti and L. Rastelli, Supersymmetric localization in AdS 5 and the protected chiral algebra, arXiv:1612.06514 [INSPIRE].
P. Liendo, I. Ramirez and J. Seo, Stress-tensor OPE in \( \mathcal{N} \) = 2 superconformal theories, JHEP 02 (2016) 019 [arXiv:1509.00033] [INSPIRE].
M. Lemos and P. Liendo, \( \mathcal{N} \) = 2 central charge bounds from 2d chiral algebras, JHEP 04 (2016) 004 [arXiv:1511.07449] [INSPIRE].
M. Buican and T. Nishinaka, Conformal manifolds in four dimensions and chiral algebras, J. Phys. A 49 (2016) 465401 [arXiv:1603.00887] [INSPIRE].
C. Beem, L. Rastelli and B.C. van Rees, \( \mathcal{W} \) symmetry in six dimensions, JHEP 05 (2015) 017 [arXiv:1404.1079] [INSPIRE].
J. Kinney, J.M. Maldacena, S. Minwalla and S. Raju, An index for 4 dimensional super conformal theories, Commun. Math. Phys. 275 (2007) 209 [hep-th/0510251] [INSPIRE].
C. Romelsberger, Counting chiral primaries in N = 1, D = 4 superconformal field theories, Nucl. Phys. B 747 (2006) 329 [hep-th/0510060] [INSPIRE].
A. Gadde, L. Rastelli, S.S. Razamat and W. Yan, The 4D superconformal index from q-deformed 2D Yang-Mills, Phys. Rev. Lett. 106 (2011) 241602 [arXiv:1104.3850] [INSPIRE].
A. Gadde, L. Rastelli, S.S. Razamat and W. Yan, Gauge theories and macdonald Polynomials, Commun. Math. Phys. 319 (2013) 147 [arXiv:1110.3740] [INSPIRE].
P.C. Argyres and M.R. Douglas, New phenomena in SU(3) supersymmetric gauge theory, Nucl. Phys. B 448 (1995) 93 [hep-th/9505062] [INSPIRE].
P.C. Argyres, M.R. Plesser, N. Seiberg and E. Witten, New N = 2 superconformal field theories in four-dimensions, Nucl. Phys. B 461 (1996) 71 [hep-th/9511154] [INSPIRE].
E. Feigin, The PBW filtration, Repr. Theor. Amer. Math. Soc. 13 (2009) 165.
D. Xie, General Argyres-Douglas theory, JHEP 01 (2013) 100 [arXiv:1204.2270] [INSPIRE].
D. Xie and P. Zhao, Central charges and RG flow of strongly-coupled N = 2 theory, JHEP 03 (2013) 006 [arXiv:1301.0210] [INSPIRE].
Y. Wang and D. Xie, Classification of Argyres-Douglas theories from M 5 branes, Phys. Rev. D 94 (2016) 065012 [arXiv:1509.00847].
M. Buican and T. Nishinaka, Argyres-Douglas theories, the Macdonald index, and an RG inequality, JHEP 02 (2016) 159 [arXiv:1509.05402].
J. Song, Superconformal indices of generalized Argyres-Douglas theories from 2d TQFT, JHEP 02 (2016) 045 [arXiv:1509.06730].
J. Song, D. Xie and W. Yan, Vertex operator algebras of Argyres-Douglas theories from M5-branes, arXiv:1706.01607.
S. Cecotti, J. Song, C. Vafa and W. Yan, Superconformal Index, BPS Monodromy and Chiral Algebras, arXiv:1511.01516 [INSPIRE].
C. Cordova, D. Gaiotto and S.-H. Shao, Infrared computations of defect Schur indices, JHEP 11 (2016) 106 [arXiv:1606.08429] [INSPIRE].
S. Cecotti, A. Neitzke and C. Vafa, R-Twisting and 4d/2d correspondences, arXiv:1006.3435 [INSPIRE].
A. Iqbal and C. Vafa, BPS degeneracies and superconformal index in diverse dimensions, Phys. Rev. D 90 (2014) 105031 [arXiv:1210.3605] [INSPIRE].
K. Maruyoshi and J. Song, Enhancement of supersymmetry via renormalization group flow and the superconformal index, Phys. Rev. Lett. 118 (2017) 151602 [arXiv:1606.05632] [INSPIRE].
K. Maruyoshi and J. Song, \( \mathcal{N} \) = 1 deformations and RG flows of \( \mathcal{N} \) = 2 SCFTs, JHEP 02 (2017) 075 [arXiv:1607.04281] [INSPIRE].
P. Agarwal, K. Maruyoshi and J. Song, \( \mathcal{N} \) = 1 deformations and RG flows of \( \mathcal{N} \) = 2 SCFTs, part II: non-principal deformations, JHEP 12 (2016) 103 [arXiv:1610.05311] [INSPIRE].
M.R. Gaberdiel, Fusion rules and logarithmic representations of a WZW model at fractional level, Nucl. Phys. B 618 (2001) 407 [hep-th/0105046] [INSPIRE].
V.G. Kac and M. Wakimoto, Modular invariant representations of infinite dimensional Lie algebras and superalgebras, Proc. Nat. Acad. Sci. 85 (1988) 4956 [INSPIRE].
D. Ridout, ŝl(2)− 1/2 : a case study, Nucl. Phys. B 814 (2009) 485 [arXiv:0810.3532] [INSPIRE].
C. Cordova, D. Gaiotto and S.-H. Shao, Surface defects and chiral algebras, JHEP 05 (2017) 140 [arXiv:1704.01955] [INSPIRE].
C. Beem, W. Peelaers and L. Rastelli, work in progress.
C. Beem and L. Rastelli, Vertex operator algebras, higgs branches and modular differential equations, arXiv:1707.07679.
T. Arakawa and K. Kawasetsu, Quasi-lisse vertex algebras and modular linear differential equations, arXiv:1610.05865 [INSPIRE].
T. Arakawa, Associated varieties of modules over Kac-Moody algebras and C 2 -cofiniteness of W -algebras, Int. Math. Res. Not. (2015) 11605.
T. Arakawa, Introduction to W-algebras and their representation theory, arXiv:1605.00138 [INSPIRE].
H. Li, Abelianizing vertex algebras, Commun. Math. Phys. 259 (2005) 391.
Y. Zhu, Vertex operator algebras, elliptic functions and modular forms, Dissertation, Yale University, U.S.A. (1990).
Y. Zhu, Modular invariance of characters of vertex operator algebras, J. Amer. Math. Soc. 9 (1996) 237.
C. Beem, M. Lemos, P. Liendo, L. Rastelli and B.C. van Rees, The \( \mathcal{N} \) = 2 superconformal bootstrap, JHEP 03 (2016) 183 [arXiv:1412.7541] [INSPIRE].
M. Lemos and P. Liendo, Bootstrapping \( \mathcal{N} \) = 2 chiral correlators, JHEP 01 (2016) 025 [arXiv:1510.03866] [INSPIRE].
F.G. Malikov, B.L. Feigin and D.B. Fuks, Singular vectors in verma modules over Kac-Moody algebras, Funct. Anal. Appl. 20 (1986) 103.
M. Bauer and N. Sochen, Fusion and singular vectors in A 1(1) highest weight cyclic modules, Commun. Math. Phys. 152 (1993) 127 [hep-th/9201079] [INSPIRE].
P. Mathieu and M.A. Walton, On principal admissible representations and conformal field theory, Nucl. Phys. B 553 (1999) 533 [hep-th/9812192] [INSPIRE].
T. Creutzig and D. Ridout, Modular data and Verlinde formulae for fractional level WZW models II, Nucl. Phys. B 875 (2013) 423 [arXiv:1306.4388] [INSPIRE].
C. Cordova, D. Gaiotto and S.-H. Shao, Surface defect indices and 2d-4d BPS states, arXiv:1703.02525 [INSPIRE].
G.E. Andrews, A. Schilling and S.O. Warnaar, An A 2 Bailey lemma and Rogers-Ramanujan-type identities, J. Amer. Math. Soc. 12 (1999) 677.
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1612.08956
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Song, J. Macdonald index and chiral algebra. J. High Energ. Phys. 2017, 44 (2017). https://doi.org/10.1007/JHEP08(2017)044
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2017)044