Abstract
We study the correlation functions of Coulomb branch operators of four-dimensional \( \mathcal{N} \) = 2 Superconformal Field Theories (SCFTs). We focus on rank-one theories, such as the SU(2) gauge theory with four fundamental hypermultiplets. “Extremal” correlation functions, involving exactly one anti-chiral operator, are perhaps the simplest nontrivial correlation functions in four-dimensional Quantum Field Theory. We show that the large charge limit of extremal correlators is captured by a “dual” description which is a chiral random matrix model of the Wishart-Laguerre type. This gives an analytic handle on the physics in some particular excited states. In the limit of large random matrices we find the physics of a non-relativistic axion-dilaton effective theory. The random matrix model also admits a ’t Hooft expansion in which the matrix is taken to be large and simultaneously the coupling is taken to zero. This explains why the extremal correlators of SU(2) gauge theory obey a nontrivial double scaling limit in states of large charge. We give an exact solution for the first two orders in the ’t Hooft expansion of the random matrix model and compare with expectations from effective field theory, previous weak coupling results, and we analyze the non-perturbative terms in the strong ’t Hooft coupling limit. Finally, we apply the random matrix theory techniques to study extremal correlators in rank-1 Argyres-Douglas theories. We compare our results with effective field theory and with some available numerical bootstrap bounds.
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R. Mondaini and M. Rigol, Eigenstate thermalization in the two-dimensional transverse field ising model. II. off-diagonal matrix elements of observables, Phys. Rev. E 96 (2017) 012157 [arXiv:1705.08058].
A. Dymarsky and H. Liu, New characteristic of quantum many-body chaotic systems, Phys. Rev. E 99 (2019) 010102 [arXiv:1702.07722] [INSPIRE].
A. Altland and M.R. Zirnbauer, Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures, Phys. Rev. B 55 (1997) 1142 [cond-mat/9602137] [INSPIRE].
E.V. Shuryak and J.J.M. Verbaarschot, Random matrix theory and spectral sum rules for the Dirac operator in QCD, Nucl. Phys. A 560 (1993) 306 [hep-th/9212088] [INSPIRE].
J.J.M. Verbaarschot, The Spectrum of the QCD Dirac operator and chiral random matrix theory: The Threefold way, Phys. Rev. Lett. 72 (1994) 2531 [hep-th/9401059] [INSPIRE].
J.J.M. Verbaarschot and T. Wettig, Random matrix theory and chiral symmetry in QCD, Ann. Rev. Nucl. Part. Sci. 50 (2000) 343 [hep-ph/0003017] [INSPIRE].
M.R. Douglas, I.R. Klebanov, D. Kutasov, J.M. Maldacena, E.J. Martinec and N. Seiberg, A New hat for the c = 1 matrix model, in From Fields to Strings: Circumnavigating Theoretical Physics: A Conference in Tribute to Ian Kogan, (2003) [hep-th/0307195] [INSPIRE].
S. Hellerman, S. Maeda and M. Watanabe, Operator Dimensions from Moduli, JHEP 10 (2017) 089 [arXiv:1706.05743] [INSPIRE].
S. Hellerman and S. Maeda, On the Large R-charge Expansion in \( \mathcal{N} \) = 2 Superconformal Field Theories, JHEP 12 (2017) 135 [arXiv:1710.07336] [INSPIRE].
S. Hellerman, S. Maeda, D. Orlando, S. Reffert and M. Watanabe, Universal correlation functions in rank 1 SCFTs, JHEP 12 (2019) 047 [arXiv:1804.01535] [INSPIRE].
S. Hellerman, D. Orlando, S. Reffert and M. Watanabe, On the CFT Operator Spectrum at Large Global Charge, JHEP 12 (2015) 071 [arXiv:1505.01537] [INSPIRE].
A. Monin, D. Pirtskhalava, R. Rattazzi and F.K. Seibold, Semiclassics, Goldstone Bosons and CFT data, JHEP 06 (2017) 011 [arXiv:1611.02912] [INSPIRE].
L. Álvarez-Gaumé, O. Loukas, D. Orlando and S. Reffert, Compensating strong coupling with large charge, JHEP 04 (2017) 059 [arXiv:1610.04495] [INSPIRE].
D. Jafferis, B. Mukhametzhanov and A. Zhiboedov, Conformal Bootstrap At Large Charge, JHEP 05 (2018) 043 [arXiv:1710.11161] [INSPIRE].
M.V. Libanov, V.A. Rubakov, D.T. Son and S.V. Troitsky, Exponentiation of multiparticle amplitudes in scalar theories, Phys. Rev. D 50 (1994) 7553 [hep-ph/9407381] [INSPIRE].
M.V. Libanov, D.T. Son and S.V. Troitsky, Exponentiation of multiparticle amplitudes in scalar theories. 2. Universality of the exponent, Phys. Rev. D 52 (1995) 3679 [hep-ph/9503412] [INSPIRE].
D.T. Son, Semiclassical approach for multiparticle production in scalar theories, Nucl. Phys. B 477 (1996) 378 [hep-ph/9505338] [INSPIRE].
G. Badel, G. Cuomo, A. Monin and R. Rattazzi, Semiclassics and Multi-Leg Amplitudes, https://indico.in2p3.fr/event/18200/contributions/67400/attachments/53882/70308/saclay_rattazzi.pdf.
M. Dedushenko, S.S. Pufu and R. Yacoby, A one-dimensional theory for Higgs branch operators, JHEP 03 (2018) 138 [arXiv:1610.00740] [INSPIRE].
J. Chen, On exact correlation functions of chiral ring operators in 2d \( \mathcal{N} \) = (2, 2) SCFTs via localization, JHEP 03 (2018) 065 [arXiv:1712.01164] [INSPIRE].
N. Ishtiaque, 2D BPS Rings from Sphere Partition Functions, JHEP 04 (2018) 124 [arXiv:1712.02551] [INSPIRE].
M. Dedushenko, Y. Fan, S.S. Pufu and R. Yacoby, Coulomb Branch Operators and Mirror Symmetry in Three Dimensions, JHEP 04 (2018) 037 [arXiv:1712.09384] [INSPIRE].
D. Rodriguez-Gomez and J.G. Russo, Large N Correlation Functions in Superconformal Field Theories, JHEP 06 (2016) 109 [arXiv:1604.07416] [INSPIRE].
M. Baggio, V. Niarchos, K. Papadodimas and G. Vos, Large-N correlation functions in \( \mathcal{N} \) = 2 superconformal QCD, JHEP 01 (2017) 101 [arXiv:1610.07612] [INSPIRE].
M. Billó, F. Fucito, A. Lerda, J.F. Morales, Y.S. Stanev and C. Wen, Two-point correlators in N = 2 gauge theories, Nucl. Phys. B 926 (2018) 427 [arXiv:1705.02909] [INSPIRE].
M. Billó, F. Fucito, G.P. Korchemsky, A. Lerda and J.F. Morales, Two-point correlators in non-conformal \( \mathcal{N} \) = 2 gauge theories, JHEP 05 (2019) 199 [arXiv:1901.09693] [INSPIRE].
F.A. Dolan and H. Osborn, On short and semi-short representations for four-dimensional superconformal symmetry, Annals Phys. 307 (2003) 41 [hep-th/0209056] [INSPIRE].
K. Papadodimas, Topological Anti-Topological Fusion in Four-Dimensional Superconformal Field Theories, JHEP 08 (2010) 118 [arXiv:0910.4963] [INSPIRE].
S. Lee, S. Minwalla, M. Rangamani and N. Seiberg, Three point functions of chiral operators in D = 4, N = 4 SYM at large N, Adv. Theor. Math. Phys. 2 (1998) 697 [hep-th/9806074] [INSPIRE].
M. Baggio, V. Niarchos and K. Papadodimas, tt* equations, localization and exact chiral rings in 4d \( \mathcal{N} \)= 2 SCFTs, JHEP 02 (2015) 122 [arXiv:1409.4212] [INSPIRE].
M. Baggio, V. Niarchos and K. Papadodimas, Exact correlation functions in SU(2) \( \mathcal{N} \) = 2 superconformal QCD, Phys. Rev. Lett. 113 (2014) 251601 [arXiv:1409.4217] [INSPIRE].
M. Baggio, V. Niarchos and K. Papadodimas, On exact correlation functions in SU(N) \( \mathcal{N} \) = 2 superconformal QCD, JHEP 11 (2015) 198 [arXiv:1508.03077] [INSPIRE].
E. Gerchkovitz, J. Gomis, N. Ishtiaque, A. Karasik, Z. Komargodski and S.S. Pufu, Correlation Functions of Coulomb Branch Operators, JHEP 01 (2017) 103 [arXiv:1602.05971] [INSPIRE].
I. Aniceto, J.G. Russo and R. Schiappa, Resurgent Analysis of Localizable Observables in Supersymmetric Gauge Theories, JHEP 03 (2015) 172 [arXiv:1410.5834] [INSPIRE].
M. Honda, Borel Summability of Perturbative Series in 4D N = 2 and 5D N =1 Supersymmetric Theories, Phys. Rev. Lett. 116 (2016) 211601 [arXiv:1603.06207] [INSPIRE].
A. Bourget, D. Rodriguez-Gomez and J.G. Russo, A limit for large R-charge correlators in \( \mathcal{N} \) = 2 theories, JHEP 05 (2018) 074 [arXiv:1803.00580] [INSPIRE].
M. Beccaria, On the large R-charge \( \mathcal{N} \) = 2 chiral correlators and the Toda equation, JHEP 02 (2019) 009 [arXiv:1809.06280] [INSPIRE].
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].
M. Cornagliotto, M. Lemos and P. Liendo, Bootstrapping the (A1, A2) Argyres-Douglas theory, JHEP 03 (2018) 033 [arXiv:1711.00016] [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].
J. Gomis, P.-S. Hsin, Z. Komargodski, A. Schwimmer, N. Seiberg and S. Theisen, Anomalies, Conformal Manifolds, and Spheres, JHEP 03 (2016) 022 [arXiv:1509.08511] [INSPIRE].
Y. Tachikawa and K. Yonekura, Anomalies involving the space of couplings and the Zamolodchikov metric, JHEP 12 (2017) 140 [arXiv:1710.03934] [INSPIRE].
A. Schwimmer and S. Theisen, Moduli Anomalies and Local Terms in the Operator Product Expansion, JHEP 07 (2018) 110 [arXiv:1805.04202] [INSPIRE].
A. Schwimmer and S. Theisen, Osborn Equation and Irrelevant Operators, J. Stat. Mech. 1908 (2019) 084011 [arXiv:1902.04473] [INSPIRE].
Y. Nakayama, Conformal Contact Terms and Semi-Local Terms, Annales Henri Poincaré 21 (2020) 3201 [arXiv:1906.07914] [INSPIRE].
E. Gerchkovitz, J. Gomis and Z. Komargodski, Sphere Partition Functions and the Zamolodchikov Metric, JHEP 11 (2014) 001 [arXiv:1405.7271] [INSPIRE].
J. Gomis and N. Ishtiaque, Kähler potential and ambiguities in 4d \( \mathcal{N} \) = 2 SCFTs, JHEP 04 (2015) 169 [arXiv:1409.5325] [INSPIRE].
N. Seiberg, Y. Tachikawa and K. Yonekura, Anomalies of Duality Groups and Extended Conformal Manifolds, PTEP 2018 (2018) 073B04 [arXiv:1803.07366] [INSPIRE].
G.W. Moore, N. Nekrasov and S. Shatashvili, Integrating over Higgs branches, Commun. Math. Phys. 209 (2000) 97 [hep-th/9712241] [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].
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].
A. Klemm, M. Mariño and S. Theisen, Gravitational corrections in supersymmetric gauge theory and matrix models, JHEP 03 (2003) 051 [hep-th/0211216] [INSPIRE].
R. Hirota, Y. Ohta and J. Satsuma, Solutions of the Kadomtsev-Petviashvili Equation and the Two-Dimensional Toda Equations, J. Phys. Soc. Jap. 57 (1988) 1901.
S. Cecotti and C. Vafa, Topological antitopological fusion, Nucl. Phys. B 367 (1991) 359 [INSPIRE].
M. Henningson, Extended superspace, higher derivatives and SL(2, ℤ) duality, Nucl. Phys. B 458 (1996) 445 [hep-th/9507135] [INSPIRE].
B. de Wit, M.T. Grisaru and M. Roček, Nonholomorphic corrections to the one loop N = 2 superYang-Mills action, Phys. Lett. B 374 (1996) 297 [hep-th/9601115] [INSPIRE].
M. Dine and N. Seiberg, Comments on higher derivative operators in some SUSY field theories, Phys. Lett. B 409 (1997) 239 [hep-th/9705057] [INSPIRE].
Z. Komargodski and A. Schwimmer, On Renormalization Group Flows in Four Dimensions, JHEP 12 (2011) 099 [arXiv:1107.3987] [INSPIRE].
Z. Komargodski, The Constraints of Conformal Symmetry on RG Flows, JHEP 07 (2012) 069 [arXiv:1112.4538] [INSPIRE].
G. Livan, M. Novaes and P. Vivo, Introduction to Random Matrices — Theory and Practice, arXiv:1712.07903.
G. Szegö, Orthogonal Polynomials, vol. 23 in American Mathematical Society colloquium publications, American Mathematical Society (1959).
S. Kumar, Recursion for the smallest eigenvalue density of β-Wishart-Laguerre ensemble, arXiv:1708.08646 [INSPIRE].
V.A. Marčenko and L.A. Pastur, Distribution of eigenvalues for some sets of random matrices, Math. USSR Sb. 1 (1967) 457.
E.L. Basor, Y. Chen and H. Widom, Determinants of hankel matrices, math/0006070.
E.L. Basor, Y. Chen and H. Widom, Hankel Determinants as Fredholm Determinants, Random matrices and their applications, P. Bleher and A. Its eds., Publ. MSRI 40 (2001) 1.
Y. Chen and N. Lawrence, On the linear statistics of hermitian random matrices, J. Phys. A 31 (1998) 1141.
Y. Chen and M.R. McKay, Coulumb Fluid, Painlevé Transcendents and the Information Theory of MIMO Systems, IEEE Trans. Inform. Theory 58 (2012) 4594.
M. Beccaria, Double scaling limit of N = 2 chiral correlators with Maldacena-Wilson loop, JHEP 02 (2019) 095 [arXiv:1810.10483] [INSPIRE].
G. ’t Hooft, A Planar Diagram Theory for Strong Interactions, Nucl. Phys. B 72 (1974) 461 [INSPIRE].
R.B. Paris and D. Kaminski, Asymptotics and Mellin-Barnes Integrals, Encyclopedia of Mathematics and its Applications, Cambridge University Press (2001) [DOI].
D.J. Binder, S.M. Chester, S.S. Pufu and Y. Wang, \( \mathcal{N} \) = 4 Super-Yang-Mills correlators at strong coupling from string theory and localization, JHEP 12 (2019) 119 [arXiv:1902.06263] [INSPIRE].
N. Drukker, M. Mariño and P. Putrov, From weak to strong coupling in ABJM theory, Commun. Math. Phys. 306 (2011) 511 [arXiv:1007.3837] [INSPIRE].
N. Drukker, M. Mariño and P. Putrov, Nonperturbative aspects of ABJM theory, JHEP 11 (2011) 141 [arXiv:1103.4844] [INSPIRE].
A. Grassi and M. Mariño, M-theoretic matrix models, JHEP 02 (2015) 115 [arXiv:1403.4276] [INSPIRE].
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. 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].
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].
T. Masuda and H. Suzuki, Periods and prepotential of N = 2 SU(2) supersymmetric Yang-Mills theory with massive hypermultiplets, Int. J. Mod. Phys. A 12 (1997) 3413 [hep-th/9609066] [INSPIRE].
T. Masuda and H. Suzuki, On explicit evaluations around the conformal point in N = 2 supersymmetric Yang-Mills theories, Nucl. Phys. B 495 (1997) 149 [hep-th/9612240] [INSPIRE].
J.G. Russo, \( \mathcal{N} \) = 2 gauge theories and quantum phases, JHEP 12 (2014) 169 [arXiv:1411.2602] [INSPIRE].
G.W. Moore and I. Nidaiev, The Partition Function Of Argyres-Douglas Theory On A Four-Manifold, arXiv:1711.09257 [INSPIRE].
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Grassi, A., Komargodski, Z. & Tizzano, L. Extremal correlators and random matrix theory. J. High Energ. Phys. 2021, 214 (2021). https://doi.org/10.1007/JHEP04(2021)214
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DOI: https://doi.org/10.1007/JHEP04(2021)214