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Exact Schur index in closed form for non-Lagrangian theories

¹School of Physics, Sun Yat-sen University, No. 135 Xingangxi Road, Guangzhou, Guangdong, China
²Center for High Energy Physics, Peking University, Yiheyuan Road, Beijing 100871, China

^*Contact author: panyw5@mail.sysu.edu.cn
^†Contact author: peiheyang@pku.edu.cn

Phys. Rev. D 113, 045007 – Published 10 February, 2026

DOI: https://doi.org/10.1103/4n8q-cpmb

Abstract

The Schur index is a powerful tool to probe the spectrum and dualities of four-dimensional $N = 2$ superconformal field theories, deeply related to two-dimensional vertex operator algebras (VOAs). In this paper, we compute the Schur index in closed form for two series of non-Lagrangian theories. We explore and classify the Argyres-Douglas (AD) theories $D_{p}^{b} ({sl}_{N}, [Y])$ realized as the $S U (2)$ gauging of two AD matter theories, where we identify several infinite families with interesting central charge relations analogous to the $a_{4 d} = c_{4 d}$ of $N = 4$ theories. We focus on $D_{N - 4} (sl (N), [N - 4, 4])$ and $D_{N - 2} (sl (N), [N - 3, 3])$ and compute their flavored and unflavored Schur and Wilson line indices in compact form. We also explore their large- $N$ behavior and show that they arise as special limits of the $S U (2)$ SQCD flavored index, also analogous to the relation among the $a_{4 d} = c_{4 d}$ theories. We also generalize the elliptic function integration formula in the presence of higher-order poles to compute in closed form the partially flavored indices of the Minahan-Nemeschansky $E_{6}$ and $E_{7}$ theories. Our results point to a universal structure underlying the residues of elliptic integrands, Wilson loop indices, and nonvacuum modules of the corresponding VOAs.

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References (88)

A. Gadde, L. Rastelli, S. S. Razamat, and W. Yan, Gauge theories and Macdonald polynomials, Commun. Math. Phys. 319, 147 (2013).
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, 1359 (2015).
M. Lemos and W. Peelaers, Chiral algebras for trinion theories, J. High Energy Phys. 02 (2015) 113.
M. Buican and T. Nishinaka, On the superconformal index of Argyres–Douglas theories, J. Phys. A 49, 015401 (2016).
J. Song, Superconformal indices of generalized Argyres-Douglas theories from 2d TQFT, J. High Energy Phys. 02 (2016) 045.
A. Gadde, L. Rastelli, S. S. Razamat, and W. Yan, The 4d Superconformal Index from q-deformed 2d Yang-Mills, Phys. Rev. Lett. 106, 241602 (2011).
M. Lemos, W. Peelaers, and L. Rastelli, The superconformal index of class $S$ theories of type $D$ , J. High Energy Phys. 05 (2014) 120.
N. Mekareeya, J. Song, and Y. Tachikawa, 2d TQFT structure of the superconformal indices with outer-automorphism twists, J. High Energy Phys. 03 (2013) 171.
M. Buican and T. Nishinaka, On irregular singularity wave functions and superconformal indices, J. High Energy Phys. 09 (2017) 066.
M. Buican and T. Nishinaka, Argyres–Douglas theories, $S^{1}$ reductions, and topological symmetries, J. Phys. A 49, 045401 (2016).
K. Maruyoshi and J. Song, Enhancement of supersymmetry via renormalization group flow and the superconformal index, Phys. Rev. Lett. 118, 151602 (2017).
K. Maruyoshi and J. Song, $N = 1$ deformations and RG flows of $N = 2$ SCFTs, J. High Energy Phys. 02 (2017) 075.
P. Agarwal, K. Maruyoshi, and J. Song, $N = 1$ deformations and RG flows of $N = 2$ SCFTs, part II: Non-principal deformations, J. High Energy Phys. 12 (2016) 103.
P. Agarwal, A. Sciarappa, and J. Song, $N = 1$ Lagrangians for generalized Argyres-Douglas theories, J. High Energy Phys. 10 (2017) 211.
P. Agarwal, K. Maruyoshi, and J. Song, A “Lagrangian” for the $E_{7}$ superconformal theory, J. High Energy Phys. 05 (2018) 193.
S. Benvenuti and S. Giacomelli, Lagrangians for generalized Argyres-Douglas theories, J. High Energy Phys. 10 (2017) 106.
G. Zafrir, An $N = 1$ Lagrangian for the rank 1 $E_{6}$ superconformal theory, J. High Energy Phys. 12 (2020) 098.
G. Zafrir, An $N = 1$ Lagrangian for an $N = 3$ SCFT, J. High Energy Phys. 01 (2021) 062.
J. Bourdier, N. Drukker, and J. Felix, The exact Schur index of $N = 4$ SYM, J. High Energy Phys. 11 (2015) 210.
J. Bourdier, N. Drukker, and J. Felix, The $N = 2$ Schur index from free fermions, J. High Energy Phys. 01 (2016) 167.
Y. Hatsuda and T. Okazaki, $N = 2^{*}$ Schur indices, J. High Energy Phys. 01 (2023) 029.
B.-n. Du, M.-x. Huang, and X. Wang, Schur indices for $N = 4$ super-Yang-Mills with more general gauge groups, J. High Energy Phys. 03 (2024) 009.
Y. Pan and W. Peelaers, The exact Schur index in closed form, Phys. Rev. D 106, 045017 (2022).
Z. Guo, Y. Li, Y. Pan, and Y. Wang, $N = 2$ $N = 2$ Schur index and line operators, Phys. Rev. D 108, 106002 (2023).
H. Zheng, Y. Pan, and Y. Wang, Surface defects, flavored modular differential equations, and modularity, Phys. Rev. D 106, 105020 (2022).
M.-x. Huang, Modular anomaly equation for Schur index of $N = 4$ Super-Yang-Mills, J. High Energy Phys. 08 (2022) 049.
C. Beem, S. S. Razamat, and P. Singh, Schur indices of class $S$ and quasimodular forms, Phys. Rev. D 105, 085009 (2022).
D. Xie, W. Yan, and S.-T. Yau, Chiral algebra of the Argyres-Douglas theory from M5 branes, Phys. Rev. D 103, 065003 (2021).
J. Song, D. Xie, and W. Yan, Vertex operator algebras of Argyres-Douglas theories from M5-branes, J. High Energy Phys. 12 (2017) 123.
D. Xie and W. Yan, Schur sector of Argyres-Douglas theory and $W$ -algebra, SciPost Phys. 10, 080 (2021).
M. Buican and T. Nishinaka, $N = 4$ SYM, Argyres-Douglas theories, and an exact graded vector space isomorphism, J. High Energy Phys. 04 (2022) 028.
M. J. Kang, C. Lawrie, and J. Song, Infinitely many 4d $N = 2$ SCFTs with $a = c$ and beyond, Phys. Rev. D 104, 105005 (2021).
H. Jiang, Modularity in Argyres-Douglas theories with $a = c$ , J. High Energy Phys. 06 (2024) 131.
Y. Wang and D. Xie, Classification of Argyres-Douglas theories from M5 branes, Phys. Rev. D 94, 065012 (2016).
D. Xie and S.-T. Yau, New $N = 2$ dualities, arXiv:1602.03529.
A. Deb and S. S. Razamat, Generalized Schur partition functions and RG flows, arXiv:2506.13764.
D. Xie and S.-T. Yau, Argyres-Douglas matter and $N = 2$ dualities, arXiv:1701.01123.
D. Xie and K. Ye, Argyres-Douglas matter and S-duality: Part II, J. High Energy Phys. 03 (2018) 186.
Y. Wang and D. Xie, Codimension-two defects and Argyres-Douglas theories from outer-automorphism twist in 6d (2,0) theories, Phys. Rev. D 100, 025001 (2019).
D. Xie and W. Yan, $W$ algebra, Cosets and VOAs for 4d $N = 2$ SCFT from M5 branes, J. High Energy Phys. 04 (2021) 076.
D. Xie and W. Yan, 4d $N = 2$ SCFTs and lisse W-algebras, J. High Energy Phys. 04 (2021) 271.
P. Shan, D. Xie, and W. Yan, Mirror symmetry for circle compactified 4d $N = 2$ SCFTs, arXiv:2306.15214.
S. Cecotti, M. Del Zotto, and S. Giacomelli, More on the $N = 2$ superconformal systems of type $D_{p} (G)$ , J. High Energy Phys. 04 (2013) 153.
S. Cecotti and M. Del Zotto, Infinitely many $N = 2$ SCFT with ADE flavor symmetry, J. High Energy Phys. 01 (2013) 191.
P. C. Argyres and M. R. Douglas, New phenomena in SU(3) supersymmetric gauge theory, Nucl. Phys. B448, 93 (1995).
S. Cecotti, A. Neitzke, and C. Vafa, R-twisting and 4d/2d correspondences, arXiv:1006.3435.
A. D. Shapere and C. Vafa, BPS structure of Argyres-Douglas superconformal theories, arXiv:hep-th/9910182.
D. Gaiotto, $N = 2$ dualities, J. High Energy Phys. 08 (2012) 034.
D. Xie, General Argyres-Douglas theory, J. High Energy Phys. 01 (2013) 100.
C. Beem, M. Martone, M. Sacchi, P. Singh, and J. Stedman, Simplifying the Type $A$ Argyres-Douglas landscape, J. High Energy Phys. 12 (2025) 144.
Simone Giacomelli, Noppadol Mekareeya, and Matteo Sacchi. New aspects of Argyres–Douglas theories and their dimensional reduction, J. High Energy Phys. 03 (2021) 242.
Cyril Closset, Simone Giacomelli, Sakura Schafer-Nameki, and Yi-Nan Wang, 5d and 4d SCFTs: Canonical singularities, trinions and $S$ -dualities. J. High Energy Phys. 05 (2021) 274.
Y. Pan and P. Yang, Modularity, 4d mirror symmetry, and VOA modules of 4d $N = 2$ SCFTs with $a = c$ , Phys. Rev. D 112, 105011 (2025).
C. Cordova, D. Gaiotto, and S.-H. Shao, Infrared computations of defect Schur indices, J. High Energy Phys. 11 (2016) 106.
C. Cordova, D. Gaiotto, and S.-H. Shao, Surface defects and Chiral algebras, J. High Energy Phys. 05 (2017) 140.
L. Bianchi and M. Lemos, Superconformal surfaces in four dimensions, J. High Energy Phys. 06 (2020) 056.
Y. Pan, Y. Wang, and H. Zheng, Defects, modular differential equations, and free field realization of $N = 4$ vertex operator algebras, Phys. Rev. D 105, 085005 (2022).
Y. Pan and W. Yan, Mirror symmetry for 4d $A_{1}$ class- $S$ theories: Modularity, defects and Coulomb branch, arXiv:2412.03155.
Y. Pan and W. Yan, Mirror symmetry for circle compactified 4d $A_{1}$ class- $S$ theories, arXiv:2410.15695.
S. D. Mathur, S. Mukhi, and A. Sen, On the classification of rational conformal field theories, Phys. Lett. B 213, 303 (1988).
Y. Zhu, Modular invariance of characters of vertex operator algebras, J. Am. Math. Soc. 9, 237 (1996).
M. R. Gaberdiel and C. A. Keller, Modular differential equations and null vectors, J. High Energy Phys. 09 (2008) 079.
C. Beem and L. Rastelli, Vertex operator algebras, Higgs branches, and modular differential equations, J. High Energy Phys. 08 (2018) 114.
Y. Pan and Y. Wang, Flavored modular differential equations, Phys. Rev. D 108, 085027 (2023).
Y. Pan and C. Zeng, Holomorphic quasi-modular bootstrap, arXiv:2409.01095.
J. Kaidi, M. Martone, L. Rastelli, and M. Weaver, Needles in a haystack. An algorithmic approach to the classification of 4d $N = 2$ SCFTs, J. High Energy Phys. 03 (2022) 210.
B. Li, H. Li, and W. Yan, Spectral flow, twisted modules and MLDE of quasi-lisse vertex algebras, arXiv:2304.09681.
D. Gang, E. Koh, and K. Lee, Line operator index on $S^{1} \times S^{3}$ , J. High Energy Phys. 05 (2012) 007.
Y. Hatsuda and T. Okazaki, Exact $N = 2^{*}$ Schur line defect correlators, J. High Energy Phys. 06 (2023) 169.
D. Gaiotto, G. W. Moore, and A. Neitzke, Wall-crossing, Hitchin systems, and the WKB approximation, Adv. Math. 234, 239 (2013).
D. Xie, 3d mirror for Argyres-Douglas theories, arXiv:2107.05258.
Y. Pan and W. Peelaers, Schur correlation functions on $S^{3} \times S^{1}$ , J. High Energy Phys. 07 (2019) 013.
M. Dedushenko and M. Fluder, Chiral algebra, localization, modularity, surface defects, and all that, J. Math. Phys. (N.Y.) 61, 092302 (2020).
V. G. Kac and M. Wakimoto, A remark on boundary level admissible representations, arXiv:1612.07423.
A. Gadde, E. Pomoni, L. Rastelli, and S. S. Razamat, S-duality and 2d Topological QFT, J. High Energy Phys. 03 (2010) 032.
D. Gaiotto, L. Rastelli, and S. S. Razamat, Bootstrapping the superconformal index with surface defects, J. High Energy Phys. 01 (2013) 022.
T. Creutzig, W-algebras for Argyres-Douglas theories, arXiv:1701.05926.
D. Gaiotto and J. H. Lee, The giant graviton expansion, J. High Energy Phys. 08 (2024) 025.
M. Beccaria and A. Cabo-Bizet, Giant graviton expansion of Schur index and quasimodular forms, J. High Energy Phys. 05 (2024) 282.
J. A. Minahan and D. Nemeschansky, An $N = 2$ superconformal fixed point with E(6) global symmetry, Nucl. Phys. B482, 142 (1996).
C. Beem, W. Peelaers, L. Rastelli, and B. C. van Rees, Chiral algebras of class S, J. High Energy Phys. 05 (2015) 020.
A. Gadde, L. Rastelli, S. S. Razamat, and W. Yan, The superconformal index of the $E_{6}$ SCFT, J. High Energy Phys. 08 (2010) 107.
T. Eguchi and H. Ooguri, Conformal and current algebras on general Riemann surface, Nucl. Phys. B282, 308 (1987).
S. S. Razamat, On a modular property of $N = 2$ superconformal theories in four dimensions, J. High Energy Phys. 10 (2012) 191.
V. P. Spiridonov and S. O. Warnaar, Inversions of integral operators and elliptic beta integrals on root systems, Adv. Math. 207, 91 (2006).
T. Arakawa and K. Kawasetsu, Quasi-lisse vertex algebras and modular linear differential equations, arXiv:1610.05865.
T. Arakawa and A. Moreau, Joseph ideals and lisse minimal W-algebras, J. Inst. Math. Jussieu 17, 397 (2018).
R. Eager, G. Lockhart, and E. Sharpe, Hidden exceptional symmetry in the pure spinor superstring, Phys. Rev. D 101, 026006 (2020).

Phys. Rev. D 113, 045007– Published 10 February, 2026

Vol. 113, Iss. 4 — 15 February 2026

Received 6 December 2025
Accepted 5 January 2026

DOI: https://doi.org/10.1103/4n8q-cpmb

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP³.

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Exact Schur index in closed form for non-Lagrangian theories

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