Abstract
Every four-dimensional \( \mathcal{N}=2 \) superconformal field theory comes equipped with an intricate algebraic invariant, the associated vertex operator algebra. The relationships between this invariant and more conventional protected quantities in the same theories have yet to be completely understood. In this work, we aim to characterize the connection between the Higgs branch of the moduli space of vacua (as an algebraic geometric entity) and the associated vertex operator algebra. Ultimately our proposal is simple, but its correctness requires the existence of a number of nontrivial null vectors in the vacuum Verma module of the vertex operator algebra. Of particular interest is one such null vector whose presence suggests that the Schur index of any \( \mathcal{N}=2 \) SCFT should obey a finite order modular differential equation. By way of the “high temperature” limit of the superconformal index, this allows the Weyl anomaly coefficient a to be reinterpreted in terms of the representation theory of the associated vertex operator algebra. We illustrate these ideas in a number of examples including a series of rank-one theories associated with the “Deligne-Cvitanović exceptional series” of simple Lie algebras, several families of Argyres-Douglas theories, an assortment of class \( \mathcal{S} \) theories, and \( \mathcal{N}=2 \) super Yang-Mills with \( \mathfrak{s}\mathfrak{u}(n) \) gauge group for small-to-moderate values of n.
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References
C. Beem, M. Lemos, P. Liendo, W. Peelaers, L. Rastelli and B.C. van Rees, Infinite chiral symmetry in four dimensions, Commun. Math. Phys. 336 (2015) 1359 [arXiv:1312.5344] [INSPIRE].
P. Liendo, I. Ramirez and J. Seo, Stress-tensor OPE in N = 2 superconformal theories, JHEP 02 (2016) 019 [arXiv:1509.00033] [INSPIRE].
M. Lemos and P. Liendo, N = 2 central charge bounds from 2d chiral algebras, JHEP 04 (2016) 004 [arXiv:1511.07449] [INSPIRE].
M. Lemos, P. Liendo, C. Meneghelli and V. Mitev, Bootstrapping N = 3 superconformal theories, JHEP 04 (2017) 032 [arXiv:1612.01536] [INSPIRE].
M. Cornagliotto, M. Lemos and V. Schomerus, Long multiplet bootstrap, JHEP 10 (2017) 119 [arXiv:1702.05101] [INSPIRE].
C. Beem, L. Rastelli and B.C. van Rees, The N = 4 superconformal bootstrap, Phys. Rev. Lett. 111 (2013) 071601 [arXiv:1304.1803] [INSPIRE].
C. Beem, M. Lemos, P. Liendo, L. Rastelli and B.C. van Rees, The N = 2 superconformal bootstrap, JHEP 03 (2016) 183 [arXiv:1412.7541] [INSPIRE].
C. Beem, M. Lemos, L. Rastelli and B.C. van Rees, The (2, 0) superconformal bootstrap, Phys. Rev. D 93 (2016) 025016 [arXiv:1507.05637] [INSPIRE].
C. Beem, L. Rastelli and B.C. van Rees, More N = 4 superconformal bootstrap, Phys. Rev. D 96 (2017) 046014 [arXiv:1612.02363] [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].
C. Cordova and S.-H. Shao, Schur indices, BPS particles and Argyres-Douglas theories, JHEP 01 (2016) 040 [arXiv:1506.00265] [INSPIRE].
C. Cordova, D. Gaiotto and S.-H. Shao, Infrared computations of defect Schur indices, JHEP 11 (2016) 106 [arXiv:1606.08429] [INSPIRE].
C. Cordova, D. Gaiotto and S.-H. Shao, Surface defects and chiral algebras, JHEP 05 (2017) 140 [arXiv:1704.01955] [INSPIRE].
Y. Zhu, Vertex operator algebras, elliptic functions and modular forms, Ph.D. thesis, Yale University, ProQuest LLC, Ann Arbor, MI, U.S.A., (1990).
T. Arakawa, A remark on the c 2 -cofiniteness condition on vertex algebras, arXiv:1004.1492.
L. Rastelli, Higgs branches, vertex operator algebras and modular differential equations, in String Math 2016, Collège de France, Paris, France, June 2016.
T. Arakawa and K. Kawasetsu, Quasi-lisse vertex algebras and modular linear differential equations, arXiv:1610.05865 [INSPIRE].
J. Song, D. Xie and W. Yan, Vertex operator algebras of Argyres-Douglas theories from M5-branes, JHEP 12 (2017) 123 [arXiv:1706.01607] [INSPIRE].
S.S. Razamat, On a modular property of N = 2 superconformal theories in four dimensions, JHEP 10 (2012) 191 [arXiv:1208.5056] [INSPIRE].
C. Beem and W. Peelaers, work in progress.
M.R. Gaberdiel, Constraints on extremal self-dual CFTs, JHEP 11 (2007) 087 [arXiv:0707.4073] [INSPIRE].
M.R. Gaberdiel and C.A. Keller, Modular differential equations and null vectors, JHEP 09 (2008) 079 [arXiv:0804.0489] [INSPIRE].
M.R. Gaberdiel and S. Lang, Modular differential equations for torus one-point functions, J. Phys. A 42 (2009) 045405 [arXiv:0810.0106] [INSPIRE].
L. Di Pietro and Z. Komargodski, Cardy formulae for SUSY theories in d = 4 and d = 6, JHEP 12 (2014) 031 [arXiv:1407.6061] [INSPIRE].
M. Buican and T. Nishinaka, On the superconformal index of Argyres-Douglas theories, J. Phys. A 49 (2016) 015401 [arXiv:1505.05884] [INSPIRE].
L. Di Pietro, Z. Komargodski and L. Rastelli, unpublished notes, (2015).
A. Arabi Ardehali, High-temperature asymptotics of supersymmetric partition functions, JHEP 07 (2016) 025 [arXiv:1512.03376] [INSPIRE].
S. Cecotti, J. Song, C. Vafa and W. Yan, Superconformal index, BPS monodromy and chiral algebras, JHEP 11 (2017) 013 [arXiv:1511.01516] [INSPIRE].
T. Arakawa and A. Moreau, Joseph ideals and lisse minimal W-algebras, arXiv:1506.00710 [INSPIRE].
C. Beem and L. Rastelli, Infinite chiral symmetry in four and six dimensions, seminar by L. Rastelli, Harvard University, U.S.A., November 2014.
T. Arakawa, Associated varieties of modules over Kac-Moody algebras and C 2 -cofiniteness of W-algebras, arXiv:1004.1554 [INSPIRE].
M. Bershadsky, Conformal field theories via Hamiltonian reduction, Commun. Math. Phys. 139 (1991) 71 [INSPIRE].
A.M. Polyakov, Gauge transformations and diffeomorphisms, Int. J. Mod. Phys. A 5 (1990) 833 [INSPIRE].
B.L. Feigin and A.M. Semikhatov, W (2) n algebras, Nucl. Phys. B 698 (2004) 409 [math/0401164] [INSPIRE].
T. Creutzig, D. Ridout and S. Wood, Coset constructions of logarithmic (1, p) models, Lett. Math. Phys. 104 (2014) 553 [arXiv:1305.2665] [INSPIRE].
T. Creutzig, W-algebras for Argyres-Douglas theories, arXiv:1701.05926 [INSPIRE].
A. Maffei, Quiver varieties of type A, Comment. Math. Helv. 80 (2005) 1 [math.AG/9812142].
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].
E. Frenkel and D. Ben-Zvi, Vertex algebras and algebraic curves, second ed., Math. Surv. Monogr. 88, American Mathematical Society, Providence, RI, U.S.A., (2004) [INSPIRE].
C. Beem, W. Peelaers and L. Rastelli, Deformation quantization and superconformal symmetry in three dimensions, Commun. Math. Phys. 354 (2017) 345 [arXiv:1601.05378] [INSPIRE].
A. Kapustin, Holomorphic reduction of N = 2 gauge theories, Wilson-’t Hooft operators and S-duality, hep-th/0612119 [INSPIRE].
K. Costello, private communication.
J. Song, Macdonald index and chiral algebra, JHEP 08 (2017) 044 [arXiv:1612.08956] [INSPIRE].
D.M. Hofman and J. Maldacena, Conformal collider physics: energy and charge correlations, JHEP 05 (2008) 012 [arXiv:0803.1467] [INSPIRE].
T. Hartman, S. Jain and S. Kundu, A new spin on causality constraints, JHEP 10 (2016) 141 [arXiv:1601.07904] [INSPIRE].
D.M. Hofman, D. Li, D. Meltzer, D. Poland and F. Rejon-Barrera, A proof of the conformal collider bounds, JHEP 06 (2016) 111 [arXiv:1603.03771] [INSPIRE].
S.D. Mathur, S. Mukhi and A. Sen, On the classification of rational conformal field theories, Phys. Lett. B 213 (1988) 303 [INSPIRE].
A.D. Shapere and Y. Tachikawa, Central charges of N = 2 superconformal field theories in four dimensions, JHEP 09 (2008) 109 [arXiv:0804.1957] [INSPIRE].
V.G. Kac, Contravariant form for infinite dimensional Lie algebras and superalgebras, in Group theoretical methods in physics. Proceedings, 7th International Colloquium And Integrative Conference, Austin, TX, U.S.A., 11–16 September 1978, Lect. Notes Phys. 94, Springer, Germany, (1979), pg. 441 [INSPIRE].
L. Deka and A. Schilling, Non-unitary minimal models, Bailey’s lemma and N = 1, 2 superconformal algebras, Commun. Math. Phys. 260 (2005) 711 [math-ph/0412084] [INSPIRE].
M. Miyamoto, Pseudo-trace functions and modular invariance of vertex operator algebra, eConf C 0306234 (2003) 1145 [INSPIRE].
S. Mukhi and S. Panda, Fractional level current algebras and the classification of characters, Nucl. Phys. B 338 (1990) 263 [INSPIRE].
T. Creutzig, private communication.
C. Beem, W. Peelaers and L. Rastelli, work in progress.
C. Beem, work in progress.
P. Deligne, La série exceptionnelle de groupes de Lie (in French), C. R. Acad. Sci. Paris Sér. I Math. 322 (1996) 321.
P. Cvitanovic, Group theory: birdtracks, Lie’s and exceptional groups, Princeton Univ. Pr., Princeton, U.S.A., (2008) [INSPIRE].
J. Maldacena and A. Zhiboedov, Constraining conformal field theories with a higher spin symmetry, J. Phys. A 46 (2013) 214011 [arXiv:1112.1016] [INSPIRE].
P. Argyres, M. Lotito, Y. Lü and M. Martone, Geometric constraints on the space of N = 2 SCFTs. Part I: physical constraints on relevant deformations, JHEP 02 (2018) 001 [arXiv:1505.04814] [INSPIRE].
P.C. Argyres, M. Lotito, Y. Lü and M. Martone, Geometric constraints on the space of N = 2 SCFTs. Part II: construction of special Kähler geometries and RG flows, JHEP 02 (2018) 002 [arXiv:1601.00011] [INSPIRE].
P. Argyres, M. Lotito, Y. Lü and M. Martone, Geometric constraints on the space of N = 2 SCFTs. Part III: enhanced Coulomb branches and central charges, JHEP 02 (2018) 003 [arXiv:1609.04404] [INSPIRE].
P.C. Argyres, Y. Lü and M. Martone, Seiberg-Witten geometries for Coulomb branch chiral rings which are not freely generated, JHEP 06 (2017) 144 [arXiv:1704.05110] [INSPIRE].
H. Shimizu, Y. Tachikawa and G. Zafrir, Anomaly matching on the Higgs branch, JHEP 12 (2017) 127 [arXiv:1703.01013] [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].
D. Xie, General Argyres-Douglas theory, JHEP 01 (2013) 100 [arXiv:1204.2270] [INSPIRE].
M. Buican and T. Nishinaka, Argyres-Douglas theories, S 1 reductions and topological symmetries, J. Phys. A 49 (2016) 045401 [arXiv:1505.06205] [INSPIRE].
M. Buican and T. Nishinaka, Argyres-Douglas theories, the Macdonald index and an RG inequality, JHEP 02 (2016) 159 [arXiv:1509.05402] [INSPIRE].
J. Song, Superconformal indices of generalized Argyres-Douglas theories from 2d TQFT, JHEP 02 (2016) 045 [arXiv:1509.06730] [INSPIRE].
M. Buican and T. Nishinaka, On irregular singularity wave functions and superconformal indices, JHEP 09 (2017) 066 [arXiv:1705.07173] [INSPIRE].
M. Buican, S. Giacomelli, T. Nishinaka and C. Papageorgakis, Argyres-Douglas theories and S-duality, JHEP 02 (2015) 185 [arXiv:1411.6026] [INSPIRE].
M. Buican and T. Nishinaka, Conformal manifolds in four dimensions and chiral algebras, J. Phys. A 49 (2016) 465401 [arXiv:1603.00887] [INSPIRE].
L. Fredrickson, D. Pei, W. Yan and K. Ye, Argyres-Douglas theories, chiral algebras and wild Hitchin characters, JHEP 01 (2018) 150 [arXiv:1701.08782] [INSPIRE].
M. Buican, Z. Laczko and T. Nishinaka, N = 2 S-duality revisited, JHEP 09 (2017) 087 [arXiv:1706.03797] [INSPIRE].
K. Maruyoshi and J. Song, N = 1 deformations and RG flows of N = 2 SCFTs, JHEP 02 (2017) 075 [arXiv:1607.04281] [INSPIRE].
P. Agarwal, K. Maruyoshi and J. Song, N = 1 deformations and RG flows of N = 2 SCFTs, part II: non-principal deformations, JHEP 12 (2016) 103 [Addendum ibid. 04 (2017) 113] [arXiv:1610.05311] [INSPIRE].
P. Agarwal, A. Sciarappa and J. Song, N = 1 Lagrangians for generalized Argyres-Douglas theories, JHEP 10 (2017) 211 [arXiv:1707.04751] [INSPIRE].
S. Benvenuti and S. Giacomelli, Lagrangians for generalized Argyres-Douglas theories, JHEP 10 (2017) 106 [arXiv:1707.05113] [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].
S. Cecotti, A. Neitzke and C. Vafa, R-twisting and 4d/2d correspondences, arXiv:1006.3435 [INSPIRE].
M. Del Zotto and L. Rastelli, unpublished, November 2014.
V.G. Kac and M. Wakimoto, Modular invariant representations of infinite dimensional Lie algebras and superalgebras, Proc. Nat. Acad. Sci. 85 (1988) 4956 [INSPIRE].
M. Lehn, Y. Namikawa and C. Sorger, Slodowy slices and universal Poisson deformations, Compos. Math. 148 (2012) 121 [arXiv:1002.4107].
P. Slodowy, Four lectures on simple groups and singularities, Commun. Math. Inst. 11, Mathematical Institute, Rijksuniversiteit, Utrecht, Netherlands, (1980).
S. Cremonesi, A. Hanany, N. Mekareeya and A. Zaffaroni, T σ ρ (G) theories and their Hilbert series, JHEP 01 (2015) 150 [arXiv:1410.1548] [INSPIRE].
H. Nakajima, Instantons on ALE spaces, quiver varieties and Kac-Moody algebras, Duke Math. J. 76 (1994) 365 [INSPIRE].
A. Henderson, Singularities of nilpotent orbit closures, Rev. Roumaine Math. Pures Appl. 60 (2015) 441 [arXiv:1408.3888].
D. Gaiotto, L. Rastelli and S.S. Razamat, Bootstrapping the superconformal index with surface defects, JHEP 01 (2013) 022 [arXiv:1207.3577] [INSPIRE].
M. Lemos and W. Peelaers, Chiral algebras for trinion theories, JHEP 02 (2015) 113 [arXiv:1411.3252] [INSPIRE].
A. Gadde, E. Pomoni, L. Rastelli and S.S. Razamat, S-duality and 2d topological QFT, JHEP 03 (2010) 032 [arXiv:0910.2225] [INSPIRE].
F. Benini, S. Benvenuti and Y. Tachikawa, Webs of five-branes and N = 2 superconformal field theories, JHEP 09 (2009) 052 [arXiv:0906.0359] [INSPIRE].
J.H. Bruinier, G. van der Geer, G. Harder and D. Zagier, The 1-2-3 of modular forms, lectures from the Summer school on modular forms and their applications, Nordfjordeid, Norway, June 2004, K. Ranestad ed., Universitext, Springer-Verlag, Berlin, Germany, (2008).
G. Mason, M.P. Tuite and A. Zuevsky, Torus n-point functions for R-graded vertex operator superalgebras and continuous fermion orbifolds, Commun. Math. Phys. 283 (2008) 305 [INSPIRE].
G. Mason, Vector-valued modular forms and linear differential operators, Int. J. Number Theor. 3 (2007) 377.
Y. Zhu, Modular invariance of characters of vertex operator algebras, J. Amer. Math. Soc. 9 (1996) 237.
C. Franc and G. Mason, Fourier coefficients of vector-valued modular forms of dimension 2, Canad. Math. Bull. 57 (2014) 485.
M. Kaneko and M. Koike, On modular forms arising from a differential equation of hypergeometric type, math.NT/0206022.
M. Kaneko, K. Nagatomo and Y. Sakai, Modular forms and second order ordinary differential equations: applications to vertex operator algebras, Lett. Math. Phys. 103 (2013) 439 [INSPIRE].
J. Bourdier, N. Drukker and J. Felix, The exact Schur index of N = 4 SYM, JHEP 11 (2015) 210 [arXiv:1507.08659] [INSPIRE].
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Beem, C., Rastelli, L. Vertex operator algebras, Higgs branches, and modular differential equations. J. High Energ. Phys. 2018, 114 (2018). https://doi.org/10.1007/JHEP08(2018)114
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DOI: https://doi.org/10.1007/JHEP08(2018)114