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Supersymmetric U(N) Chern–Simons-Matter Theory and Phase Transitions

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Abstract

We study \({{\mathcal{N}} = 2}\) supersymmetric U(N) Chern–Simons with \({N_{f}}\) fundamental and \({N_{f}}\) antifundamental chiral multiplets of mass m in the parameter space spanned by (g, m, N, N f ), where g denotes the coupling constant. In particular, we analyze the matrix model description of its partition function, both at finite N using the method of orthogonal polynomials together with Mordell integrals and, at large N with fixed g, using the theory of Toeplitz determinants. We show for the massless case that there is an explicit realization of the Giveon–Kutasov duality. For finite N, with \({N > N_{f}}\), three regimes that exactly correspond to the known three large N phases of the theory are identified and characterized.

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References

  1. Mordell L.J.: The definite integral \({\int_{-\infty}^{\infty}\frac{e^{at^{2}+bt}}{e^{ct}+d}dt}\) and the analytic theory of numbers. Acta Math. 61, 322–360 (1933)

    Article  MathSciNet  Google Scholar 

  2. Zwegers, S.: Mock Theta Functions, Ph.D. thesis. Utrecht University (2002). arXiv:0807.4834

  3. Eguchi, T., Sugawara, Y.: Non-holomorphic modular forms and SL(2,R)/U(1) superconformal field theory. JHEP 1103, 107 (2011). arXiv:1012.5721 [hep-th]

  4. Kapustin, A., Willett, B., Yaakov, I.: Exact results for Wilson loops in superconformal Chern–Simons theories with matter. JHEP 1003, 089 (2010). arXiv:0909.4559 [hep-th]

  5. Kapustin, A., Willett, B., Yaakov, I.: Nonperturbative tests of three-dimensional dualities. JHEP 1010, 013 (2010). arXiv:1003.5694 [hep-th]

  6. Jafferis, D.L.: The exact superconformal R-symmetry extremizes Z. JHEP 1205, 159 (2012). arXiv:1012.3210 [hep-th]

  7. Hama, N., Hosomichi, K., Lee, S.: Notes on SUSY gauge theories on three-sphere. JHEP 1103, 127 (2011). arXiv:1012.3512 [hep-th]

  8. Barranco, A., Russo, J.G.: Large N phase transitions in supersymmetric Chern–Simons theory with massive matter. JHEP 1403, 012 (2014). arXiv:1401.3672 [hep-th]

  9. Russo, J.G., Zarembo, K.: Evidence for large-N phase transitions in N = 2* theory. JHEP 1304, 065 (2013). arXiv:1302.6968 [hep-th]

  10. Russo, J.G., Zarembo, K.: Massive N = 2 gauge theories at large N. JHEP 1311, 130 (2013). arXiv:1309.1004 [hep-th]

  11. Russo, J.G., Zarembo, K.: Localization at large N. arXiv:1312.1214 [hep-th]

  12. Anderson, L., Zarembo, K.: Quantum phase transitions in mass-deformed ABJM matrix model. JHEP 1409, 021 (2014). arXiv:1406.3366 [hep-th]

  13. Szegö, G.: Orthogonal Polynomials, Colloquium Publications of the American Mathematical Society, vol. XXIII, 4th edn, Section 2.7

  14. Mariño, M.: Les Houches lectures on matrix models and topological strings. arXiv:hep-th/0410165

  15. Deift, P., Its, A., Krasovsky, I.: Toeplitz matrices and Toeplitz determinants under the impetus of the Ising model. Some history and some recent results. Commun. Pure Appl. Math. 66, 1360–1438 (2013). arXiv:1207.4990v3 [math.FA]

  16. Simon B.: Szegő’s Theorem and its Descendants: Spectral Theory for \({L^{2}}\) Perturbations of Orthogonal Polynomials. Princeton Univeristy Press, Princeton (2010)

    Book  Google Scholar 

  17. Tierz, M.: Soft matrix models and Chern–Simons partition functions. Mod. Phys. Lett. A 19, 1365–1378 (2004). arXiv:hep-th/0212128

  18. Mariño, M.: Chern–Simons theory, matrix integrals, and perturbative three-manifold invariants. Commun. Math. Phys. 253, 25–49 (2004). arXiv:hep-th/0207096

  19. Giveon, A., Kutasov, D.: Seiberg duality in Chern–Simons theory. Nucl. Phys. B 812, 1 (2009). arXiv:0808.0360 [hep-th]

  20. Kapustin, A., Willett, B., Yaakov, I.: Tests of Seiberg-like duality in three dimensions. arXiv:1012.4021 [hep-th]

  21. Romo, M., Tierz, M.: Unitary Chern–Simons matrix model and the Villain lattice action. Phys. Rev. D 86, 045027 (2012). arXiv:1103.2421 [hep-th]

  22. Szabo, R.J., Tierz, M.: Matrix models and stochastic growth in Donaldson–Thomas theory. J. Math. Phys. 53, 103502 (2012). arXiv:1005.5643 [hep-th]

  23. Szabo, R.J., Tierz, M.: Two-dimensional Yang–Mills theory, Painleve equations and the six-vertex model. J. Phys. A 45, 085401 (2012). arXiv:1102.3640 [hep-th]

  24. Szabo, R.J., Tierz, M.: Q-deformations of two-dimensional Yang–Mills theory: classification, categorification and refinement. Nucl. Phys. B 876, 234 (2013). arXiv:1305.1580 [hep-th]

  25. Fisher M.E., Hartwig R.E.: Toeplitz determinants: some applications, theorems, and conjectures. Adv. Chem. Phys. 15, 333 (1968)

    Google Scholar 

  26. Armitage, V., Rogers, A.: Gauss sums and quantum mechanics. J. Phys. A Math. Gen. 33, 5993 (2000). arXiv:quant-ph/0003107

  27. Mariño, M., Putrov, P.: Interacting fermions and N = 2 Chern–Simons-matter theories. JHEP 1311, 199 (2013). arXiv:1206.6346 [hep-th]

  28. Balasubramanian, V., Keski-Vakkuri, E., Kraus, P., Naqvi, A.: String scattering from decaying branes. Commun. Math. Phys. 257, 363 (2005). hep-th/0404039

  29. Tracy, C.A., Widom, H.: On the distributions of the lengths of the longest monotone subsequences in random words. Probab. Theory Relat. Fields 119, 350–380 (2001). arXiv:math/9904042

  30. Baxter G.: Polynomials defined by a difference system. J. Math. Anal. Appl. 2, 223–263 (1961)

    Article  MATH  MathSciNet  Google Scholar 

  31. Tierz, M.: To appear

  32. Berele A., Regev A.: Hook Young diagrams with applications to combinatorics and to representations of Lie superalgebras. Adv. Math. 64, 118–175 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  33. Aganagic, M., Ooguri, H., Saulina, N., Vafa, C.: Black holes, q-deformed 2d Yang–Mills, and non-perturbative topological strings. Nucl. Phys. B 715, 304 (2005). hep-th/0411280

  34. Moens E.M., Van der Jeugt J.: On dimension formulas for gl(m|n) representations. J. Lie Theory 14, 523 (2004)

    MATH  MathSciNet  Google Scholar 

  35. Bourdeau M., Mlawer E.J., Riggs H., Schnitzer H.J.: The Quasirational fusion structure of SU(m/n) Chern–Simons and W-Z-W theories. Nucl. Phys. B 372, 303 (1992)

    Article  ADS  MathSciNet  Google Scholar 

  36. Gadde, A., Rastelli, L., Razamat, S.S., Yan, W.: Gauge theories and Macdonald polynomials. Commun. Math. Phys. 319, 147–193 (2013). arXiv:1110.3740 [hep-th]

  37. Gadde, A., Rastelli, L., Razamat, S.S., Yan, W.: The 4d superconformal index from q-deformed 2d Yang–Mills. Phys. Rev. Lett. 106, 241602 (2011). arXiv:1104.3850 [hep-th]

  38. Macdonald I.G.: Symmetric Functions and Hall Polynomials. Oxford University Press, Oxford (1979)

    MATH  Google Scholar 

  39. Bump, D., Gamburd, A.: On the averages of characteristic polynomials from classical groups. Commun. Math. Phys. 265, 227–274 (2006). arXiv:math-ph/0502043

  40. Forrester., P.J., Gamburd, A.: Counting formulas associated with some random matrix averages. J. Combin. Theory A 113, 934–951 (2006). arXiv:math.CO/0503002

  41. Foata D., Han G.N.: The triple, quintuple and septuple product identities revisited. S ém. Lothar. Combin. 42, 1–12 (1999)

    MathSciNet  Google Scholar 

  42. Barnes E.W.: The theory of the G-function. Q. J. Math. 31, 264–314 (1900)

    MATH  Google Scholar 

  43. Safdi, B.R., Klebanov, I.R., Lee, J.: A crack in the conformal window. JHEP 1304, 165 (2013). arXiv:1212.4502 [hep-th]

  44. Böttcher A., Silbermann B.: Toeplitz matrices and determinants with Fisher–Hartwig symbols. J. Funct. Anal. 62, 178–214 (1985)

    Article  Google Scholar 

  45. Böttcher, A., Widom, H.: Two elementary derivations of the pure Fisher–Hartwig determinant. Integr. Equ. Oper. theory 53, 593–596 (2005). arXiv:math/0312198

  46. Gessel I.M.: Symmetric functions and P-recursiveness. J. Combin. Theory A 53, 257–285 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  47. van Moerbeke, P.: Random and integrable models in mathematics and physics, Lectures at Montreal, CRM Program. arXiv:0712.3847]

  48. de Haro, S., Tierz, M.: Discrete and oscillatory matrix models in Chern–Simons theory. Nucl. Phys. B 731, 225 (2005). arXiv:hep-th/0501123

  49. Crouch E., Spiegelman D.: The evaluation of integrals of the form \({\int_{-\infty}^{\infty} dtf(t)\exp (-t^{2})}\): application to logistic-normal models. J. Am. Stat. Assoc. 85, 464 (1990)

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. ECM Department and Institute for Sciences of the Cosmos, Facultat de Física, Universitat de Barcelona, Martí Franqués 1, 08028, Barcelona, Spain

    Jorge G. Russo

  2. Institució Catalana de Recerca i Estudis Avançats (ICREA), Pg. Lluis Companys 23, 08010, Barcelona, Spain

    Jorge G. Russo

  3. Departamento de Física, Universidad Nacional de La Plata and Instituto de Física La Plata, CONICET, C.C. 67, 1900, La Plata, Argentina

    Guillermo A. Silva

  4. Departamento de Análisis Matemático, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040, Madrid, Spain

    Miguel Tierz

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  1. Jorge G. Russo
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  2. Guillermo A. Silva
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  3. Miguel Tierz
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Correspondence to Miguel Tierz.

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Communicated by H. Ooguri

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Russo, J.G., Silva, G.A. & Tierz, M. Supersymmetric U(N) Chern–Simons-Matter Theory and Phase Transitions. Commun. Math. Phys. 338, 1411–1442 (2015). https://doi.org/10.1007/s00220-015-2399-4

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