Abstract
The infrared formula relates the Schur index of a 4d \( \mathcal{N} \) = 2 theory to its wall-crossing invariant, a.k.a. BPS monodromy. A further extension of this formula, proposed by Córdova, Gaiotto and Shao, includes contributions by various types of line and surface defects. We study BPS monodromies in the presence of vortex surface defects of arbitrary vorticity for general class \( \mathcal{S} \) theories of type A1 engineered by UV curves with at least one regular puncture. The trace of these defect BPS monodromies is shown to coincide with the action of certain q-difference operators acting on the trace of the (pure) 4d BPS monodromy. We use these operators to develop a “bootstrap” (of traces) of BPS monodromies, relying only on their infrared properties, thereby reproducing the very general ultraviolet characterization of the Schur index.
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References
N. Seiberg and E. Witten, Electric-magnetic duality, monopole condensation and confinement in N = 2 supersymmetric Yang-Mills theory, Nucl. Phys.B 426 (1994) 19 [Erratum ibid.B 430 (1994) 485] [hep-th/9407087] [INSPIRE].
N. Seiberg and E. Witten, Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD, Nucl. Phys.B 431 (1994) 484 [hep-th/9408099] [INSPIRE].
D. Gaiotto, \( \mathcal{N} \) = 2 dualities, JHEP08 (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].
L.F. Alday, D. Gaiotto and Y. Tachikawa, Liouville Correlation Functions from Four-dimensional Gauge Theories, Lett. Math. Phys. 91 (2010) 167 [arXiv:0906.3219] [INSPIRE].
N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys.7 (2003) 831 [hep-th/0206161] [INSPIRE].
N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, Prog. Math.244 (2006) 525 [hep-th/0306238] [INSPIRE].
V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys.313 (2012) 71 [arXiv:0712.2824] [INSPIRE].
N. Drukker, D.R. Morrison and T. Okuda, Loop operators and S-duality from curves on Riemann surfaces, JHEP09 (2009) 031 [arXiv:0907.2593] [INSPIRE].
L.F. Alday, D. Gaiotto, S. Gukov, Y. Tachikawa and H. Verlinde, Loop and surface operators in N = 2 gauge theory and Liouville modular geometry, JHEP01 (2010) 113 [arXiv:0909.0945] [INSPIRE].
N. Drukker, J. Gomis, T. Okuda and J. Teschner, Gauge Theory Loop Operators and Liouville Theory, JHEP02 (2010) 057 [arXiv:0909.1105] [INSPIRE].
D. Gaiotto, Surface Operators in N = 2 4d Gauge Theories, JHEP11 (2012) 090 [arXiv:0911.1316] [INSPIRE].
D. Gaiotto, G.W. Moore and A. Neitzke, Spectral networks, Annales Henri Poincaré14 (2013) 1643 [arXiv:1204.4824] [INSPIRE].
O. Chacaltana and J. Distler, Tinkertoys for Gaiotto Duality, JHEP11 (2010) 099 [arXiv:1008.5203] [INSPIRE].
O. Chacaltana, J. Distler and Y. Tachikawa, Nilpotent orbits and codimension-two defects of 6d N = (2, 0) theories, Int. J. Mod. Phys.A 28 (2013) 1340006 [arXiv:1203.2930] [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].
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].
M. Kontsevich and Y. Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, arXiv:0811.2435 [INSPIRE].
P. Longhi, Wall-Crossing Invariants from Spectral Networks, Annales Henri Poincaré19 (2018) 775 [arXiv:1611.00150] [INSPIRE].
M. Gabella, P. Longhi, C.Y. Park and M. Yamazaki, BPS Graphs: From Spectral Networks to BPS Quivers, JHEP07 (2017) 032 [arXiv:1704.04204] [INSPIRE].
C. Cordova and S.-H. Shao, Schur Indices, BPS Particles and Argyres-Douglas Theories, JHEP01 (2016) 040 [arXiv:1506.00265] [INSPIRE].
S. Cecotti and C. Vafa, On classification of N = 2 supersymmetric theories, Commun. Math. Phys.158 (1993) 569 [hep-th/9211097] [INSPIRE].
S. Cecotti and C. Vafa, 2d Wall-Crossing, R-Twisting and a Supersymmetric Index, arXiv:1002.3638 [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].
T.T. Dumitrescu and G. Festuccia, in progress.
T.T. Dumitrescu, Some Tools for Exploring Supersymmetric RG Flows, talk at NatiFest, 15 September 2016 [https://video.ias.edu/NatiFest/2016/0915-ThomasDumitrescu].
D. Gaiotto, G.W. Moore and A. Neitzke, Four-dimensional wall-crossing via three-dimensional field theory, Commun. Math. Phys.299 (2010) 163 [arXiv:0807.4723] [INSPIRE].
T. Dimofte, S. Gukov and Y. Soibelman, Quantum Wall Crossing in N = 2 Gauge Theories, Lett. Math. Phys.95 (2011) 1 [arXiv:0912.1346] [INSPIRE].
D. Gaiotto, L. Rastelli and S.S. Razamat, Bootstrapping the superconformal index with surface defects, JHEP01 (2013) 022 [arXiv:1207.3577] [INSPIRE].
L.F. Alday, M. Bullimore, M. Fluder and L. Hollands, Surface defects, the superconformal index and q-deformed Yang-Mills, JHEP10 (2013) 018 [arXiv:1303.4460] [INSPIRE].
D. Gaiotto, G.W. Moore and A. Neitzke, Framed BPS States, Adv. Theor. Math. Phys.17 (2013) 241 [arXiv:1006.0146] [INSPIRE].
D. Gaiotto, G.W. Moore and A. Neitzke, Wall-Crossing in Coupled 2d-4d Systems, JHEP12 (2012) 082 [arXiv:1103.2598] [INSPIRE].
C. Cordova, D. Gaiotto and S.-H. Shao, Infrared Computations of Defect Schur Indices, JHEP11 (2016) 106 [arXiv:1606.08429] [INSPIRE].
C. Cordova, D. Gaiotto and S.-H. Shao, Surface Defect Indices and 2d-4d BPS States, JHEP12 (2017) 078 [arXiv:1703.02525] [INSPIRE].
C. Cordova, D. Gaiotto and S.-H. Shao, Surface Defects and Chiral Algebras, JHEP05 (2017) 140 [arXiv:1704.01955] [INSPIRE].
A. Neitzke and F. Yan, Line defect Schur indices, Verlinde algebras and U(1)rfixed points, JHEP11 (2017) 035 [arXiv:1708.05323] [INSPIRE].
P. Longhi, The BPS Spectrum Generator In 2d-4d Systems, JHEP11 (2012) 107 [arXiv:1205.2512] [INSPIRE].
L. Hollands and A. Neitzke, Spectral Networks and Fenchel-Nielsen Coordinates, Lett. Math. Phys.106 (2016) 811 [arXiv:1312.2979] [INSPIRE].
D. Galakhov, P. Longhi and G.W. Moore, Spectral Networks with Spin, Commun. Math. Phys.340 (2015) 171 [arXiv:1408.0207] [INSPIRE].
K. Hori, C.Y. Park and Y. Tachikawa, 2d SCFTs from M2-branes, JHEP11 (2013) 147 [arXiv:1309.3036] [INSPIRE].
P. Longhi and C.Y. Park, ADE Spectral Networks, JHEP08 (2016) 087 [arXiv:1601.02633] [INSPIRE].
P. Longhi and C.Y. Park, ADE Spectral Networks and Decoupling Limits of Surface Defects, JHEP02 (2017) 011 [arXiv:1611.09409] [INSPIRE].
Y. Imamura, Orbifold Schur Index and IR formula, PTEP2018 (2018) 043B01 [arXiv:1710.08853] [INSPIRE].
D. Gaiotto, G.W. Moore and E. Witten, Algebra of the Infrared: String Field Theoretic Structures in Massive \( \mathcal{N} \) = (2, 2) Field Theory In Two Dimensions, arXiv:1506.04087 [INSPIRE].
S. Cecotti, P. Fendley, K.A. Intriligator and C. Vafa, A New supersymmetric index, Nucl. Phys.B 386 (1992) 405 [hep-th/9204102] [INSPIRE].
S. Cecotti and C. Vafa, Ising model and N = 2 supersymmetric theories, Commun. Math. Phys.157 (1993) 139 [hep-th/9209085] [INSPIRE].
T. Dimofte and S. Gukov, Refined, Motivic and Quantum, Lett. Math. Phys.91 (2010) 1 [arXiv:0904.1420] [INSPIRE].
N.J. Hitchin, The Selfduality equations on a Riemann surface, Proc. Lond. Math. Soc.55 (1987)59 [INSPIRE].
R. Donagi and E. Witten, Supersymmetric Yang-Mills theory and integrable systems, Nucl. Phys.B 460 (1996) 299 [hep-th/9510101] [INSPIRE].
E.J. Martinec and N.P. Warner, Integrable systems and supersymmetric gauge theory, Nucl. Phys.B 459 (1996) 97 [hep-th/9509161] [INSPIRE].
C. Córdova and A. Neitzke, Line Defects, Tropicalization and Multi-Centered Quiver Quantum Mechanics, JHEP09 (2014) 099 [arXiv:1308.6829] [INSPIRE].
W.-y. Chuang, D.-E. Diaconescu, J. Manschot, G.W. Moore and Y. Soibelman, Geometric engineering of (framed) BPS states, Adv. Theor. Math. Phys.18 (2014) 1063 [arXiv:1301.3065] [INSPIRE].
M. Del Zotto and A. Sen, About the Absence of Exotics and the Coulomb Branch Formula, Commun. Math. Phys.357 (2018) 1113 [arXiv:1409.5442] [INSPIRE].
T. Bridgeland and I. Smith, Quadratic differentials as stability conditions, arXiv:1302.7030.
S. Cecotti and C. Vafa, Classification of complete N = 2 supersymmetric theories in 4 dimensions, Surveys in differential geometry18 (2013) [arXiv:1103.5832] [INSPIRE].
M. Alim, S. Cecotti, C. Cordova, S. Espahbodi, A. Rastogi and C. Vafa, BPS Quivers and Spectra of Complete N = 2 Quantum Field Theories, Commun. Math. Phys.323 (2013) 1185 [arXiv:1109.4941] [INSPIRE].
M. Alim, S. Cecotti, C. Cordova, S. Espahbodi, A. Rastogi and C. Vafa, \( \mathcal{N} \) = 2 quantum field theories and their BPS quivers, Adv. Theor. Math. Phys.18 (2014) 27 [arXiv:1112.3984] [INSPIRE].
K.G. Wilson, Confinement of Quarks, Phys. Rev.D 10 (1974) 2445 [INSPIRE].
G. ’t Hooft, On the Phase Transition Towards Permanent Quark Confinement, Nucl. Phys.B 138 (1978) 1 [INSPIRE].
A. Kapustin, Wilson-’t Hooft operators in four-dimensional gauge theories and S-duality, Phys. Rev.D 74 (2006) 025005 [hep-th/0501015] [INSPIRE].
O. Aharony, N. Seiberg and Y. Tachikawa, Reading between the lines of four-dimensional gauge theories, JHEP08 (2013) 115 [arXiv:1305.0318] [INSPIRE].
S. Gukov and E. Witten, Gauge Theory, Ramification, And The Geometric Langlands Program, hep-th/0612073 [INSPIRE].
S. Gukov and E. Witten, Rigid Surface Operators, Adv. Theor. Math. Phys. 14 (2010) 87 [arXiv:0804.1561] [INSPIRE].
E. Frenkel, S. Gukov and J. Teschner, Surface Operators and Separation of Variables, JHEP01 (2016) 179 [arXiv:1506.07508] [INSPIRE].
S.K. Ashok, M. Billó, E. Dell’Aquila, M. Frau, R.R. John and A. Lerda, Modular and duality properties of surface operators in N = 2∗gauge theories, JHEP07 (2017) 068 [arXiv:1702.02833] [INSPIRE].
A. Gorsky, B. Le Floch, A. Milekhin and N. Sopenko, Surface defects and instanton-vortex interaction, Nucl. Phys.B 920 (2017) 122 [arXiv:1702.03330] [INSPIRE].
A. Balasubramanian and J. Teschner, Supersymmetric field theories and geometric Langlands: The other side of the coin, Proc. Symp. Pure Math.98 (2018) 79 [arXiv:1702.06499] [INSPIRE].
S. Jeong and N. Nekrasov, Opers, surface defects and Yang-Yang functional, arXiv:1806.08270 [INSPIRE].
S. Gukov, Surface Operators, in New Dualities of Supersymmetric Gauge Theories, J. Teschner ed., pp. 223-259 (2016), [DOI:10.1007/978-3-319-18769-3 8] [arXiv:1412.7127] [INSPIRE].
E. Witten, Phases of N = 2 theories in two-dimensions, Nucl. Phys. B 403 (1993) 159 [hep-th/9301042] [INSPIRE].
A. Hanany and K. Hori, Branes and N = 2 theories in two-dimensions, Nucl. Phys.B 513 (1998) 119 [hep-th/9707192] [INSPIRE].
K. Iwaki and T. Nakanishi, Exact WKB analysis and cluster algebras, J. Phys.A 47 (2014) 474009.
E. Witten, Solutions of four-dimensional field theories via M-theory, Nucl. Phys.B 500 (1997) 3 [hep-th/9703166] [INSPIRE].
A. Gadde and S. Gukov, 2d Index and Surface operators, JHEP03 (2014) 080 [arXiv:1305.0266] [INSPIRE].
D. Gaiotto, S. Gukov and N. Seiberg, Surface Defects and Resolvents, JHEP09 (2013) 070 [arXiv:1307.2578] [INSPIRE].
M. Bullimore, M. Fluder, L. Hollands and P. Richmond, The superconformal index and an elliptic algebra of surface defects, JHEP10 (2014) 062 [arXiv:1401.3379] [INSPIRE].
S. Cecotti and C. Vafa, Topological antitopological fusion, Nucl. Phys.B 367 (1991) 359 [INSPIRE].
B. Dubrovin, Geometry and integrability of topological-antitopological fusion, Commun. Math. Phys.152 (1993) 539 [hep-th/9206037] [INSPIRE].
K. Strebel, Quadratic differentials, Springer-Verlag Berlin Heidelberg (1984) [https://doi.org/10.1007/978-3-662-02414-0].
J. Liu, Jenkins-Strebel differentials with poles, Comment. Math. Helv. 83 (2008) 211.
R. Donagi, Decomposition of spectral covers, in Journées de Geometrie Algébrique d’Orsay, Astérisque218 (1993) 145.
A. Klemm, W. Lerche, P. Mayr, C. Vafa and N.P. Warner, Selfdual strings and N = 2 supersymmetric field theory, Nucl. Phys.B 477 (1996) 746 [hep-th/9604034] [INSPIRE].
N. Dorey, The BPS spectra of two-dimensional supersymmetric gauge theories with twisted mass terms, JHEP11 (1998) 005 [hep-th/9806056] [INSPIRE].
P. Longhi, The Structure of BPS Spectra, Ph.D. Thesis, Rutgers University Library (2015) [https://doi.org/10.7282/T3FQ9ZMF].
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].
G. Bonelli, K. Maruyoshi and A. Tanzini, Wild Quiver Gauge Theories, JHEP02 (2012) 031 [arXiv:1112.1691] [INSPIRE].
D. Xie, General Argyres-Douglas Theory, JHEP01 (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].
D. Xie and P. Zhao, Central charges and RG flow of strongly-coupled N = 2 theory, JHEP03 (2013) 006 [arXiv:1301.0210] [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].
D. Gang, P. Longhi and M. Yamazaki, S duality and Framed BPS States via BPS Graphs, arXiv:1711.04038 [INSPIRE].
I.G. Macdonald, Symmetric Functions and Hall Polynomials, Clarendon Press, Oxford (1995).
L. Lapointe and L. Vinet, Creation operators for the Macdonald and Jack polynomials, Lett. Math. Phys.40 (1997) 269.
M. Dedushenko and M. Fluder, Chiral Algebra, Localization, Modularity, Surface defects, And All That, arXiv:1904.02704 [INSPIRE].
M. Gabella and P. Longhi, unpublished (2017).
C. Papageorgakis, A. Pini and D. Rodriguez-Gomez, Nekrasov-Shatashvili limit of the 5D superconformal index, Phys. Rev.D 94 (2016) 045007 [arXiv:1602.02647] [INSPIRE].
R. Eager, S.A. Selmani and J. Walcher, Exponential Networks and Representations of Quivers, JHEP08 (2017) 063 [arXiv:1611.06177] [INSPIRE].
S. Banerjee, P. Longhi and M. Romo, Exploring 5d BPS Spectra with Exponential Networks, arXiv:1811.02875 [INSPIRE].
L. Rastelli and S.S. Razamat, The Superconformal Index of Theories of Class \( \mathcal{S} \), in New Dualities of Supersymmetric Gauge Theories, J. Teschner ed., pp. 261-305 (2016), [DOI:10.1007/978-3-319-18769-3 9] [arXiv:1412.7131] [INSPIRE].
S.S. Razamat and M. Yamazaki, S-duality and the N = 2 Lens Space Index, JHEP10 (2013) 048 [arXiv:1306.1543] [INSPIRE].
S.S. Razamat and B. Willett, Down the rabbit hole with theories of classS, JHEP10 (2014) 99 [arXiv:1403.6107] [INSPIRE].
H.-Y. Chen and H.-Y. Chen, Heterotic Surface Defects and Dualities from 2d/4d Indices, JHEP10 (2014) 004 [arXiv:1407.4587] [INSPIRE].
D. Gaiotto and S.S. Razamat, \( \mathcal{N} \) = 1 theories of class \( \mathcal{S} \) k, JHEP07 (2015) 073 [arXiv:1503.05159] [INSPIRE].
K. Maruyoshi and J. Yagi, Surface defects as transfer matrices, PTEP2016 (2016) 113B01 [arXiv:1606.01041] [INSPIRE].
Y. Ito and Y. Yoshida, Superconformal index with surface defects for class \( \mathcal{S} \) k, arXiv:1606.01653 [INSPIRE].
J. Yagi, Surface defects and elliptic quantum groups, JHEP06 (2017) 013 [arXiv:1701.05562] [INSPIRE].
B. Nazzal and S.S. Razamat, Surface Defects in E-String Compactifications and the van Diejen Model, SIGMA14 (2018) 036 [arXiv:1801.00960] [INSPIRE].
S.S. Razamat, Flavored surface defects in 4d \( \mathcal{N} \) = 1 SCFTs, Lett. Math. Phys.109 (2019) 1377 [arXiv:1808.09509] [INSPIRE].
T. Nishinaka, S. Sasa and R.-D. Zhu, On the Correspondence between Surface Operators in Argyres-Douglas Theories and Modules of Chiral Algebra, JHEP03 (2019) 091 [arXiv:1811.11772] [INSPIRE].
A. Gadde, E. Pomoni, L. Rastelli and S.S. Razamat, S-duality and 2d Topological QFT, JHEP03 (2010) 032 [arXiv:0910.2225] [INSPIRE].
T. Kawano and N. Matsumiya, 5D SYM on 3D Sphere and 2D YM, Phys. Lett.B 716 (2012)450 [arXiv:1206.5966] [INSPIRE].
Y. Fukuda, T. Kawano and N. Matsumiya, 5D SYM and 2D q-Deformed YM, Nucl. Phys.B 869 (2013) 493 [arXiv:1210.2855] [INSPIRE].
N. Mekareeya, J. Song and Y. Tachikawa, 2d TQFT structure of the superconformal indices with outer-automorphism twists, JHEP03 (2013) 171 [arXiv:1212.0545] [INSPIRE].
Y. Tachikawa, A brief review of the 2d/4d correspondences, J. Phys.A 50 (2017) 443012 [arXiv:1608.02964] [INSPIRE].
M. Buican and T. Nishinaka, On the superconformal index of Argyres-Douglas theories, J. Phys.A 49 (2016) 015401 [arXiv:1505.05884] [INSPIRE].
M. Buican and T. Nishinaka, Argyres-Douglas Theories, the Macdonald Index and an RG Inequality, JHEP02 (2016) 159 [arXiv:1509.05402] [INSPIRE].
J. Song, Superconformal indices of generalized Argyres-Douglas theories from 2d TQFT, JHEP02 (2016) 045 [arXiv:1509.06730] [INSPIRE].
M. Buican and T. Nishinaka, On Irregular Singularity Wave Functions and Superconformal Indices, JHEP09 (2017) 066 [arXiv:1705.07173] [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, JHEP02 (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, JHEP12 (2016) 103 [arXiv:1610.05311] [INSPIRE].
S. Gupta and M. Wolf, Quadratic differentials, half-plane structures, and harmonic maps to graphs, arXiv:1505.02939.
S. Gupta and M. Wolf, Meromorphic quadratic differentials with complex residues and spiralling foliations, arXiv:1607.06931.
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Fluder, M., Longhi, P. An infrared bootstrap of the Schur index with surface defects. J. High Energ. Phys. 2019, 62 (2019). https://doi.org/10.1007/JHEP09(2019)062
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DOI: https://doi.org/10.1007/JHEP09(2019)062