Abstract
We use Coulomb branch indices of Argyres-Douglas theories on S1 × L(k, 1) to quantize moduli spaces \( {\mathrm{\mathcal{M}}}_H \) of wild/irregular Hitchin systems. In particular, we obtain formulae for the “wild Hitchin characters” — the graded dimensions of the Hilbert spaces from quantization — for four infinite families of \( {\mathrm{\mathcal{M}}}_H \), giving access to many interesting geometric and topological data of these moduli spaces. We observe that the wild Hitchin characters can always be written as a sum over fixed points in \( {\mathrm{\mathcal{M}}}_H \) under the U(1) Hitchin action, and a limit of them can be identified with matrix elements of the modular transform ST kS in certain two-dimensional chiral algebras. Although naturally fitting into the geometric Langlands program, the appearance of chiral algebras, which was known previously to be associated with Schur operators but not Coulomb branch operators, is somewhat surprising.
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S. Gukov, D. Pei, W. Yan and K. Ye, Equivariant Verlinde algebra from superconformal index and Argyres-Seiberg duality, arXiv:1605.06528 [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].
D. Gaiotto, \( \mathcal{N} \) = 2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [INSPIRE].
D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing, Hitchin Systems and the WKB Approximation, arXiv:0907.3987 [INSPIRE].
M. Bershadsky, A. Johansen, V. Sadov and C. Vafa, Topological reduction of 4-D SYM to 2-D σ-models, Nucl. Phys. B 448 (1995) 166 [hep-th/9501096] [INSPIRE].
J.A. Harvey, G.W. Moore and A. Strominger, Reducing S duality to T duality, Phys. Rev. D 52 (1995) 7161 [hep-th/9501022] [INSPIRE].
N. Seiberg and E. Witten, Gauge dynamics and compactification to three-dimensions, hep-th/9607163 [INSPIRE].
A. Beilinson and V. Drinfeld, Quantization of Hitchin’s integrable system and Hecke eigensheaves, http://www.math.uchicago.edu/∼mitya/langlands/hitchin/BD-hitchin.pdf (1991).
T. Hausel and M. Thaddeus, Mirror symmetry, Langlands duality and the Hitchin system, Invent. Math. 153 (2003) 197 [math/0205236] [INSPIRE].
A. Kapustin and E. Witten, Electric-Magnetic Duality And The Geometric Langlands Program, Commun. Num. Theor. Phys. 1 (2007) 1 [hep-th/0604151] [INSPIRE].
S. Gukov and E. Witten, Gauge Theory, Ramification, And The Geometric Langlands Program, hep-th/0612073 [INSPIRE].
N.J. Hitchin, The self-duality equations on a Riemann surface, Proc. Lond. Math. Soc. 55 (1987) 59 [INSPIRE].
S. Gukov and D. Pei, Equivariant Verlinde formula from fivebranes and vortices, Commun. Math. Phys. 355 (2017) 1 [arXiv:1501.01310] [INSPIRE].
J.E. Andersen, S. Gukov and D. Pei, The Verlinde formula for Higgs bundles, arXiv:1608.01761 [INSPIRE].
D. Halpern-Leistner, The equivariant Verlinde formula on the moduli of Higgs bundles, arXiv:1608.01754 [INSPIRE].
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].
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].
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 \( \mathcal{N} \) = 2 superconformal field theories in four-dimensions, Nucl. Phys. B 461 (1996) 71 [hep-th/9511154] [INSPIRE].
T. Eguchi, K. Hori, K. Ito and S.-K. Yang, Study of \( \mathcal{N} \) = 2 superconformal field theories in four-dimensions, Nucl. Phys. B 471 (1996) 430 [hep-th/9603002] [INSPIRE].
G. Bonelli, K. Maruyoshi and A. Tanzini, Wild Quiver Gauge Theories, JHEP 02 (2012) 031 [arXiv:1112.1691] [INSPIRE].
D. Xie, General Argyres-Douglas Theory, JHEP 01 (2013) 100 [arXiv:1204.2270] [INSPIRE].
Y. Wang and D. Xie, Classification of Argyres-Douglas theories from M5 branes, Phys. Rev. D 94 (2016) 065012 [arXiv:1509.00847] [INSPIRE].
E. Witten, Gauge theory and wild ramification, arXiv:0710.0631 [INSPIRE].
O. Biquard and P. Boalch, Wild non-abelian Hodge theory on curves, Compos. Math. 140 (2004) 179 [math/0111098].
P. Boalch and D. Yamakawa, Twisted wild character varieties, arXiv:1512.08091.
P.P. Boalch, Geometry and braiding of Stokes data; fission and wild character varieties, arXiv:1111.6228.
P. Boalch, Hyperkahler manifolds and nonabelian Hodge theory of (irregular) curves, arXiv:1203.6607.
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. Buican and T. Nishinaka, Argyres-Douglas theories, S 1 reductions and topological symmetries, J. Phys. A 49 (2016) 045401 [arXiv:1505.06205] [INSPIRE].
O. Chacaltana, J. Distler and Y. Tachikawa, Nilpotent orbits and codimension-two defects of 6d \( \mathcal{N} \) = (2, 0) theories, Int. J. Mod. Phys. A 28 (2013) 1340006 [arXiv:1203.2930] [INSPIRE].
D. Nanopoulos and D. Xie, Hitchin Equation, Singularity and \( \mathcal{N} \) = 2 Superconformal Field Theories, JHEP 03 (2010) 043 [arXiv:0911.1990] [INSPIRE].
W. Wasow, Asymptotic expansions for ordinary differential equations, Courier Corporation (2002).
C.T. Simpson, Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization, J. Am. Math. Soc. 1 (1988) 867.
S.K. Donaldson, Twisted harmonic maps and the self-duality equations, Proc. Lond. Math. Soc. 55 (1987) 127.
K. Corlette, Flat bundles with canonical metrics, J. Diff. Geom. 28 (1988) 361.
C. Sabbah, Harmonic metrics and connections with irregular singularities, Ann. Inst. Fourier 49 (1999) 1265 [math/9905039].
P. Boalch, Quasi-Hamiltonian geometry of meromorphic connections, math/0203161 [INSPIRE].
P. Boalch, Through the analytic halo: Fission via irregular singularities, Ann. Inst. Fourier 59 (2009) 2669 [arXiv:1305.6465] [INSPIRE].
N.J. Hitchin, Stable bundles and integrable systems, Duke Math. J. 54 (1987) 91 [INSPIRE].
P.C. Argyres, K. Maruyoshi and Y. Tachikawa, Quantum Higgs branches of isolated \( \mathcal{N} \) = 2 superconformal field theories, JHEP 10 (2012) 054 [arXiv:1206.4700] [INSPIRE].
L. Fredrickson and A. Neitzke, From S 1 -fixed points to \( \mathcal{W} \) -algebra representations, arXiv:1709.06142 [INSPIRE].
T. Mochizuki, Wild harmonic bundles and wild pure twistor d-modules, arXiv:0803.1344.
A. Alekseev, Notes on equivariant localization, in Geometry and Quantum Physics, Springer (2000), pp. 1-24.
G.W. Moore, N. Nekrasov and S. Shatashvili, Integrating over Higgs branches, Commun. Math. Phys. 209 (2000) 97 [hep-th/9712241] [INSPIRE].
D. Gaiotto, N. Seiberg and Y. Tachikawa, Comments on scaling limits of 4d \( \mathcal{N} \) = 2 theories, JHEP 01 (2011) 078 [arXiv:1011.4568] [INSPIRE].
M. Buican, S. Giacomelli, T. Nishinaka and C. Papageorgakis, Argyres-Douglas Theories and S-duality, JHEP 02 (2015) 185 [arXiv:1411.6026] [INSPIRE].
D. Xie and S.-T. Yau, New \( \mathcal{N} \) = 2 dualities, arXiv:1602.03529 [INSPIRE].
S. Cecotti and M. Del Zotto, Higher S-dualities and Shephard-Todd groups, JHEP 09 (2015) 035 [arXiv:1507.01799] [INSPIRE].
D. Xie and S.-T. Yau, Argyres-Douglas matter and \( \mathcal{N} \) = 2 dualities, arXiv:1701.01123 [INSPIRE].
K.A. Intriligator and B. Wecht, The exact superconformal R symmetry maximizes a, Nucl. Phys. B 667 (2003) 183 [hep-th/0304128] [INSPIRE].
D. Kutasov, A. Parnachev and D.A. Sahakyan, Central charges and U(1)R symmetries in \( \mathcal{N} \) = 1 super Yang-Mills, JHEP 11 (2003) 013 [hep-th/0308071] [INSPIRE].
S.S. Razamat and B. Willett, Global Properties of Supersymmetric Theories and the Lens Space, Commun. Math. Phys. 334 (2015) 661 [arXiv:1307.4381] [INSPIRE].
F. Nieri and S. Pasquetti, Factorisation and holomorphic blocks in 4d, JHEP 11 (2015) 155 [arXiv:1507.00261] [INSPIRE].
F. Benini, T. Nishioka and M. Yamazaki, 4d Index to 3d Index and 2d TQFT, Phys. Rev. D 86 (2012) 065015 [arXiv:1109.0283] [INSPIRE].
S.S. Razamat and M. Yamazaki, S-duality and the \( \mathcal{N} \) = 2 Lens Space Index, JHEP 10 (2013) 048 [arXiv:1306.1543] [INSPIRE].
S. Gukov, P. Putrov and C. Vafa, Fivebranes and 3-manifold homology, JHEP 07 (2017) 071 [arXiv:1602.05302] [INSPIRE].
S. Gukov, D. Pei, P. Putrov and C. Vafa, BPS spectra and 3-manifold invariants, arXiv:1701.06567 [INSPIRE].
P. Boalch, Irregular connections and Kac-Moody root systems, arXiv:0806.1050.
P. Argyres, M. Lotito, Y. Lü and M. Martone, Geometric constraints on the space of \( \mathcal{N} \) = 2 SCFTs I: physical constraints on relevant deformations, arXiv:1505.04814 [INSPIRE].
S. Gukov, Quantization via Mirror Symmetry, arXiv:1011.2218 [INSPIRE].
F. Benini, Y. Tachikawa and D. Xie, Mirrors of 3d Sicilian theories, JHEP 09 (2010) 063 [arXiv:1007.0992] [INSPIRE].
D. Nanopoulos and D. Xie, More Three Dimensional Mirror Pairs, JHEP 05 (2011) 071 [arXiv:1011.1911] [INSPIRE].
S.S. Razamat and B. Willett, Down the rabbit hole with theories of class S, JHEP 10 (2014) 99 [arXiv:1403.6107] [INSPIRE].
W. Peelaers, Higgs branch localization of \( \mathcal{N} \) = 1 theories on S 3 × S 1, JHEP 08 (2014) 060 [arXiv:1403.2711] [INSPIRE].
E. Frenkel, Lectures on the Langlands program and conformal field theory, hep-th/0512172 [INSPIRE].
E. Frenkel, Gauge Theory and Langlands Duality, arXiv:0906.2747 [INSPIRE].
S. Gukov and E. Witten, Branes and Quantization, Adv. Theor. Math. Phys. 13 (2009) 1445 [arXiv:0809.0305] [INSPIRE].
E. Witten, More On Gauge Theory And Geometric Langlands, arXiv:1506.04293 [INSPIRE].
D. Nadler and E. Zaslow, Constructible Sheaves and the Fukaya Category, math/0604379.
D. Nadler, Microlocal branes are constructible sheaves, math/0612399.
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].
C. Cordova, D. Gaiotto and S.-H. Shao, Infrared Computations of Defect Schur Indices, JHEP 11 (2016) 106 [arXiv:1606.08429] [INSPIRE].
V.P. Spiridonov and G.S. Vartanov, Elliptic hypergeometric integrals and ’t Hooft anomaly matching conditions, JHEP 06 (2012) 016 [arXiv:1203.5677] [INSPIRE].
S.S. Razamat, On a modular property of N = 2 superconformal theories in four dimensions, JHEP 10 (2012) 191 [arXiv:1208.5056] [INSPIRE].
T. Creutzig, W-algebras for Argyres-Douglas theories, arXiv:1701.05926 [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].
V.G. Kac and M. Wakimoto, Quantum reduction and representation theory of superconformal algebras, Adv. Math. 185 (2004) 400 [math-ph/0304011] [INSPIRE].
T. Creutzig and T. Gannon, Logarithmic conformal field theory, log-modular tensor categories and modular forms, J. Phys. A 50 (2017) 404004 [arXiv:1605.04630] [INSPIRE].
H. Haahr Andersen and J. Paradowski, Fusion categories arising from semisimple Lie algebras, Commun. Math. Phys. 169 (1995) 563.
B. Bakalov and A.A. Kirillov, Lectures on tensor categories and modular functors, vol. 21, American Mathematical Soc. (2001).
V.G. Kac and M. Wakimoto, Modular invariant representations of infinite dimensional Lie algebras and superalgebras, Proc. Nat. Acad. Sci. 85 (1988) 4956 [INSPIRE].
P. Francesco, P. Mathieu and D. Sénéchal, Conformal field theory, Springer Science & Business Media (2012).
T. Creutzig and D. Ridout, Modular Data and Verlinde Formulae for Fractional Level WZW Models I, Nucl. Phys. B 865 (2012) 83 [arXiv:1205.6513] [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].
B. Pareigis, On braiding and dyslexia, J. Algebra 171 (1995) 413.
C. Dong, Twisted modules for vertex algebras associated with even lattices, J. Algebra 165 (1994) 91.
T. Creutzig, S. Kanade and R. McRae, Tensor categories for vertex operator superalgebra extensions, arXiv:1705.05017 [INSPIRE].
J. Auger, T. Creutzig, S. Kanade and M. Rupert, Semisimplification of a Category of Modules for the Logarithmic B p -Algebras, to appear.
T. Arakawa and A. Moreau, Joseph ideals and lisse minimal W-algebras, arXiv:1506.00710 [INSPIRE].
O. Perse, A note on representations of some affine vertex algebras of type D, arXiv:1205.3003.
J.A. Minahan and D. Nemeschansky, An \( \mathcal{N} \) = 2 superconformal fixed point with E 6 global symmetry, Nucl. Phys. B 482 (1996) 142 [hep-th/9608047] [INSPIRE].
J.A. Minahan and D. Nemeschansky, Superconformal fixed points with E n global symmetry, Nucl. Phys. B 489 (1997) 24 [hep-th/9610076] [INSPIRE].
O. Chacaltana and J. Distler, Tinkertoys for Gaiotto Duality, JHEP 11 (2010) 099 [arXiv:1008.5203] [INSPIRE].
L.F. Alday, M. Bullimore and M. Fluder, On S-duality of the Superconformal Index on Lens Spaces and 2d TQFT, JHEP 05 (2013) 122 [arXiv:1301.7486] [INSPIRE].
D. Nanopoulos and D. Xie, Hitchin Equation, Irregular Singularity and \( \mathcal{N} \) = 2 Asymptotical Free Theories, arXiv:1005.1350 [INSPIRE].
J. Song, Superconformal indices of generalized Argyres-Douglas theories from 2d TQFT, JHEP 02 (2016) 045 [arXiv:1509.06730] [INSPIRE].
D. Gaiotto, Asymptotically free \( \mathcal{N} \) = 2 theories and irregular conformal blocks, J. Phys. Conf. Ser. 462 (2013) 012014 [arXiv:0908.0307] [INSPIRE].
D. Gaiotto and J. Teschner, Irregular singularities in Liouville theory and Argyres-Douglas type gauge theories, I, JHEP 12 (2012) 050 [arXiv:1203.1052] [INSPIRE].
D. Gaiotto and J. Lamy-Poirier, Irregular Singularities in the H +3 WZW Model, arXiv:1301.5342 [INSPIRE].
M. Buican, T. Nishinaka and C. Papageorgakis, Constraints on chiral operators in \( \mathcal{N} \) = 2 SCFTs, JHEP 12 (2014) 095 [arXiv:1407.2835] [INSPIRE].
B.M. McCoy, C.A. Tracy and T.T. Wu, Painlevé Functions of the Third Kind, J. Math. Phys. 18 (1977) 1058 [INSPIRE].
P.B. Gothen, The Betti numbers of the moduli space of stable rank 3 Higgs bundles on a Riemann surface, Int. J. Math. 5 (1994) 861.
L. Fredrickson and S. Rayan, Topology of Twisted Wild Hitchin Moduli Spaces from Morse flow, in preparation.
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ArXiv ePrint: 1701.08782
Primary affiliation: Yau Mathematical Sciences Center, Tsinghua University. (Wenbin Yan)
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Fredrickson, L., Pei, D., Yan, W. et al. Argyres-Douglas theories, chiral algebras and wild Hitchin characters. J. High Energ. Phys. 2018, 150 (2018). https://doi.org/10.1007/JHEP01(2018)150
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DOI: https://doi.org/10.1007/JHEP01(2018)150