# Transmutation of a trans-series: the Gross-Witten-Wadia phase transition

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## Abstract

We study the change in the resurgent asymptotic properties of a trans-series in two parameters, a coupling *g*^{2} and a gauge index *N*, as a system passes through a large *N* phase transition, using the universal example of the Gross-Witten-Wadia third-order phase transition in the unitary matrix model. This transition is well-studied in the immediate vicinity of the transition point, where it is characterized by a double-scaling limit Painlevé II equation, and also away from the transition point using the pre-string difference equation. Here we present a complementary analysis of the transition at all coupling and all finite *N*, in terms of a differential equation, using the explicit Tracy-Widom mapping of the Gross-Witten-Wadia partition function to a solution of a Painlevé III equation. This mapping provides a simple method to generate trans-series expansions in all parameter regimes, and to study their transmutation as the parameters are varied. For example, at any finite *N* the weak coupling expansion is divergent, with a non-perturbative trans-series completion; on the other hand, the strong coupling expansion is convergent, and yet there is still a non-perturbative trans-series completion. We show how the different instanton terms ‘condense’ at the transition point to match with the double-scaling limit trans-series. We also define a uniform large *N* strong-coupling expansion (a non-linear analogue of uniform WKB), which is much more precise than the conventional large *N* expansion through the transition region, and apply it to the evaluation of Wilson loops.

## Keywords

Nonperturbative Effects 1/N Expansion Matrix Models## Notes

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## References

- [1]D.J. Gross and E. Witten,
*Possible third order phase transition in the large-N lattice gauge theory, Phys. Rev.***D 21**(1980) 446 [INSPIRE].ADSGoogle Scholar - [2]
- [3]S.R. Wadia,
*N*= ∞*phase transition in a class of exactly soluble model lattice gauge theories, Phys. Lett.***93B**(1980) 403 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [4]
- [5]A.A. Migdal,
*Loop equations and*1/*N expansion, Phys. Rept.***102**(1983) 199 [INSPIRE].ADSCrossRefGoogle Scholar - [6]E. Brézin and S.R. Wadia,
*The large N expansion in quantum field theory and statistical physics: from spin systems to*2-*dimensional gravity*, World Scientific, Singapore (1993).Google Scholar - [7]P. Di Francesco, P.H. Ginsparg and J. Zinn-Justin, 2
*D gravity and random matrices, Phys. Rept.***254**(1995) 1 [hep-th/9306153] [INSPIRE].ADSCrossRefGoogle Scholar - [8]P. Rossi, M. Campostrini and E. Vicari,
*The large-N expansion of unitary matrix models, Phys. Rept.***302**(1998) 143 [hep-lat/9609003] [INSPIRE]. - [9]M. Moshe and J. Zinn-Justin,
*Quantum field theory in the large-N limit: a review, Phys. Rept.***385**(2003) 69 [hep-th/0306133] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar - [10]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]. - [11]P.J. Forrester,
*Log-gases and random matrices*, Princeton University Press, Princeton U.S.A. (2010).Google Scholar - [12]G. Akemann, J. Baik and Ph. Di Francesco,
*The Oxford handbook of random matrix theory*, Oxford University Press, Oxford U.K. (2011).Google Scholar - [13]R.J. Szabo and M. Tierz,
*Two-dimensional Yang-Mills theory, Painlevé equations and the six-vertex model, J. Phys.***A 45**(2012) 085401 [arXiv:1102.3640] [INSPIRE]. - [14]M. Mariño,
*Lectures on non-perturbative effects in large-N gauge theories, matrix models and strings, Fortsch. Phys.***62**(2014) 455 [arXiv:1206.6272] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar - [15]M. Mariño,
*Instantons and large N*:*an introduction to non-perturbative methods in quantum field theory*, Cambridge University Press, Cambridge U.K. (2015).Google Scholar - [16]H. Neuberger,
*Nonperturbative contributions in models with a nonanalytic behavior at infinite N*,*Nucl. Phys.***B 179**(1981) 253 [INSPIRE]. - [17]E. Witten,
*On quantum gauge theories in two-dimensions, Commun. Math. Phys.***141**(1991) 153 [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar - [18]E. Witten,
*Two-dimensional gauge theories revisited, J. Geom. Phys.***9**(1992) 303 [hep-th/9204083] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar - [19]M.R. Douglas and V.A. Kazakov,
*Large-N phase transition in continuum QCD in two-dimensions, Phys. Lett.***B 319**(1993) 219 [hep-th/9305047] [INSPIRE]. - [20]D.J. Gross and A. Matytsin,
*Instanton induced large-N phase transitions in two-dimensional and four-dimensional QCD, Nucl. Phys.***B 429**(1994) 50 [hep-th/9404004] [INSPIRE]. - [21]D.J. Gross and A. Matytsin,
*Some properties of large-N two-dimensional Yang-Mills theory, Nucl. Phys.***B 437**(1995) 541 [hep-th/9410054] [INSPIRE]. - [22]P.J. Forrester, S.N. Majumdar and G. Schehr,
*Non-intersecting Brownian walkers and Yang-Mills theory on the sphere, Nucl. Phys.***B 844**(2011) 500 [*Erratum ibid.***B 857**(2012) 424] [arXiv:1009.2362] [INSPIRE]. - [23]K. Johansson,
*The longest increasing subsequence in a random permutation and a unitary random matrix model, Math. Res. Lett.***5**(1998) 63.ADSMathSciNetCrossRefMATHGoogle Scholar - [24]J. Baik, P. Deift and K. Johansson,
*On the distribution of the length of the longest increasing subsequence of random permutations, J. Amer. Math. Soc.***12**(1999) 1119.MathSciNetCrossRefMATHGoogle Scholar - [25]V. Dotsenko,
*Universal randomness, Phys. Usp.***54**(2011) 259 [arXiv:1009.3116].ADSCrossRefGoogle Scholar - [26]S.N. Majumdar and G. Schehr,
*Top eigenvalue of a random matrix: large deviations and third order phase transition, JSTAT*(2014) P01012 [arXiv:1311.0580]. - [27]C.A. Tracy and H. Widom,
*Level spacing distributions and the Airy kernel, Commun. Math. Phys.***159**(1994) 151 [hep-th/9211141] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar - [28]C.A. Tracy and H. Widom,
*Level spacing distributions and the Bessel kernel, Commun. Math. Phys.***161**(1994) 289 [hep-th/9304063] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar - [29]C.A. Tracy and H. Widom,
*Random unitary matrices, permutations and Painlevé, Commun. Math. Phys.***207**(1999) 665 [math/9811154]. - [30]J. Écalle,
*Les Fonctions Resurgentes*, volsumes I-III, Publ. Math. Orsay, France (1981).Google Scholar - [31]O. Costin,
*Asymptotics and Borel summability*, Chapman & Hall/CRC, U.S.A. (2009.Google Scholar - [32]I. Aniceto and R. Schiappa,
*Nonperturbative ambiguities and the reality of resurgent transseries, Commun. Math. Phys.***335**(2015) 183 [arXiv:1308.1115] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar - [33]D. Dorigoni,
*An introduction to resurgence, trans-series and alien calculus*, arXiv:1411.3585 [INSPIRE]. - [34]M. Mariño,
*Nonperturbative effects and nonperturbative definitions in matrix models and topological strings*,*JHEP***12**(2008) 114 [arXiv:0805.3033] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar - [35]I. Aniceto, R. Schiappa and M. Vonk,
*The resurgence of instantons in string theory, Commun. Num. Theor. Phys.***6**(2012) 339 [arXiv:1106.5922] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar - [36]Y. Hatsuda, M. Mariño, S. Moriyama and K. Okuyama,
*Non-perturbative effects and the refined topological string, JHEP***09**(2014) 168 [arXiv:1306.1734] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar - [37]S.M. Nishigaki and F. Sugino,
*Tracy-Widom distribution as instanton sum of*2*D*IIA*superstrings*,*JHEP***09**(2014) 104 [arXiv:1405.1633] [INSPIRE].ADSCrossRefGoogle Scholar - [38]S. Codesido, A. Grassi and M. Mariño,
*Exact results in*\( \mathcal{N}=8 \)*Chern-Simons-matter theories and quantum geometry, JHEP***07**(2015) 011 [arXiv:1409.1799] [INSPIRE]. - [39]J.G. Russo, G.A. Silva and M. Tierz,
*Supersymmetric*U(*N*)*Chern-Simons-Matter theory and phase transitions, Commun. Math. Phys.***338**(2015) 1411 [arXiv:1407.4794] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar - [40]A. Grassi, Y. Hatsuda and M. Mariño,
*Topological strings from quantum mechanics, Annales Henri Poincaré***17**(2016) 3177 [arXiv:1410.3382] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar - [41]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].ADSMathSciNetCrossRefMATHGoogle Scholar - [42]R. Couso-Santamaría, R. Schiappa and R. Vaz,
*Finite N from resurgent large*-*N*,*Annals Phys.***356**(2015) 1 [arXiv:1501.01007] [INSPIRE]. - [43]G. Basar and G.V. Dunne,
*Resurgence and the Nekrasov-Shatashvili limit: connecting weak and strong coupling in the Mathieu and Lamé systems, JHEP***02**(2015) 160 [arXiv:1501.05671] [INSPIRE].ADSCrossRefGoogle Scholar - [44]Y. Hatsuda and K. Okuyama,
*Resummations and non-perturbative corrections, JHEP***09**(2015) 051 [arXiv:1505.07460] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar - [45]G.V. Dunne and M. Ünsal,
*WKB and resurgence in the Mathieu equation, to be published by Scuola Normale Superiore, Pisa*, arXiv:1603.04924 [INSPIRE]. - [46]M. Honda,
*Borel summability of perturbative series in*4*D N*= 2*and*5*D N*= 1*supersymmetric theories, Phys. Rev. Lett.***116**(2016) 211601 [arXiv:1603.06207] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [47]M. Honda,
*How to resum perturbative series in*3*d N*= 2*Chern-Simons matter theories, Phys. Rev.***D 94**(2016) 025039 [arXiv:1604.08653] [INSPIRE]. - [48]S. Gukov, M. Mariño and P. Putrov,
*Resurgence in complex Chern-Simons theory*, arXiv:1605.07615 [INSPIRE]. - [49]R. Couso-Santamaría, R. Schiappa and R. Vaz,
*On asymptotics and resurgent structures of enumerative Gromov-Witten invariants*, arXiv:1605.07473 [INSPIRE]. - [50]P.V. Buividovich, G.V. Dunne and S.N. Valgushev,
*Complex path integrals and saddles in two-dimensional gauge theory, Phys. Rev. Lett.***116**(2016) 132001 [arXiv:1512.09021] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [51]G. Álvarez, L. Martínez Alonso and E. Medina,
*Complex saddles in the Gross-Witten-Wadia matrix model, Phys. Rev.***D 94**(2016) 105010 [arXiv:1610.09948] [INSPIRE].ADSMathSciNetGoogle Scholar - [52]K. Okuyama,
*Wilson loops in unitary matrix models at finite N, JHEP***07**(2017) 030 [arXiv:1705.06542] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar - [53]E. Alfinito and M. Beccaria,
*Large-N expansion of Wilson loops in the Gross-Witten-Wadia matrix model*, arXiv:1707.09625 [INSPIRE]. - [54]J.C. Le Guillou and J. Zinn-Justin,
*Large order behavior of perturbation theory*, North-Holland, The Netherlands (1990).Google Scholar - [55]I. Bars and F. Green,
*Complete integration of U (N ) lattice gauge theory in a large-N limit, Phys. Rev.***D 20**(1979) 3311 [INSPIRE].ADSGoogle Scholar - [56]A. Borodin,
*Discrete gap probabilities and discrete Painlevé equations, Duke Math. J.***117**(2003) 489 [math-ph/0111008]. - [57]P. Rossi,
*On the exact evaluation of*<*detU*(*p*) >*in a lattice gauge model, Phys. Lett.***B 117**(1982) 72.Google Scholar - [58]F. Green and S. Samuel,
*Chiral models: their implication for gauge theories and large-N, Nucl. Phys.***B 190**(1981) 113 [INSPIRE]. - [59]
*Asymptotic expansions for large argument: Hankel’s Expansions — NIST DLMF entry*, http://dlmf.nist.gov/10.40.i. - [60]M.V. Berry and C.J. Howls,
*Hyperasymptotics for integrals with saddles, Proc. Roy. Soc.***A 434**(1991) 657.Google Scholar - [61]M. Ünsal,
*Theta dependence, sign problems and topological interference, Phys. Rev.***D 86**(2012) 105012 [arXiv:1201.6426] [INSPIRE]. - [62]G.V. Dunne and M. Ünsal,
*Deconstructing zero: resurgence, supersymmetry and complex saddles, JHEP***12**(2016) 002 [arXiv:1609.05770] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar - [63]
*Barnes Gamma function — NIST DLMF entry*, http://dlmf.nist.gov/5.17. - [64]
*Series expansions of modified Bessel functions — NIST DLMF entry*, http://dlmf.nist.gov/10.25.E2 - [65]N.A. Nekrasov and S.L. Shatashvili,
*Quantization of integrable systems and four dimensional gauge theories, in Proceedings of*16^{th}*International Congress on Mathematical Physics*, P. Exner ed., World Scientific, Singapore (2010), arXiv:0908.4052 [INSPIRE]. - [66]A. Mironov and A. Morozov,
*Nekrasov functions and exact Bohr-Zommerfeld integrals, JHEP***04**(2010) 040 [arXiv:0910.5670] [INSPIRE].ADSCrossRefMATHGoogle Scholar - [67]A.-K. Kashani-Poor and J. Troost,
*The toroidal block and the genus expansion, JHEP***03**(2013) 133 [arXiv:1212.0722] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar - [68]A.-K. Kashani-Poor and J. Troost,
*Pure*\( \mathcal{N}=2 \)*super Yang-Mills and exact WKB, JHEP***08**(2015) 160 [arXiv:1504.08324] [INSPIRE]. - [69]D. Krefl,
*Non-perturbative quantum geometry, JHEP***02**(2014) 084 [arXiv:1311.0584] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar - [70]D. Krefl,
*Non-perturbative quantum geometry II, JHEP***12**(2014) 118 [arXiv:1410.7116] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar - [71]D. Krefl,
*Non-perturbative quantum geometry III, JHEP***08**(2016) 020 [arXiv:1605.00182] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar - [72]J. Zinn-Justin and U.D. Jentschura,
*Multi-instantons and exact results I: conjectures, WKB expansions and instanton interactions, Annals Phys.***313**(2004) 197 [quant-ph/0501136] [INSPIRE]. - [73]J. Zinn-Justin and U.D. Jentschura,
*Multi-instantons and exact results II: specific cases, higher-order effects and numerical calculations, Annals Phys.***313**(2004) 269 [quant-ph/0501137] [INSPIRE]. - [74]U.D. Jentschura and J. Zinn-Justin,
*Instantons in quantum mechanics and resurgent expansions, Phys. Lett.***B 596**(2004) 138 [hep-ph/0405279] [INSPIRE]. - [75]G. Álvarez,
*Langer-Cherry derivation of the multi-instanton expansion for the symmetric double well, J. Math. Phys.***45**(2004) 3095.ADSMathSciNetCrossRefMATHGoogle Scholar - [76]G.V. Dunne and M. Ünsal,
*Generating nonperturbative physics from perturbation theory, Phys. Rev.***D 89**(2014) 041701 [arXiv:1306.4405] [INSPIRE]. - [77]G.V. Dunne and M. Ünsal,
*Uniform WKB, multi-instantons and resurgent trans-series, Phys. Rev.***D 89**(2014) 105009 [arXiv:1401.5202] [INSPIRE].ADSGoogle Scholar - [78]M. Hisakado,
*Unitary matrix models and Painlevé III, Mod. Phys. Lett.***A 11**(1996) 3001 [hep-th/9609214] [INSPIRE]. - [79]M. Hisakado,
*Unitary matrix models and phase transition, Phys. Lett.***B 416**(1998) 179 [hep-th/9705121] [INSPIRE]. - [80]P.J. Forrester and N.S. Witte,
*Application of the tau-function theory of Painlevé equations to random matrices: PV, PIII, the LUE, JUE, and CUE, Commun. Pure Appl. Math.***55**(2002) 0679.CrossRefGoogle Scholar - [81]M. Mariño, R. Schiappa and M. Weiss,
*Nonperturbative effects and the large-order behavior of matrix models and topological strings, Commun. Num. Theor. Phys.***2**(2008) 349 [arXiv:0711.1954] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar - [82]K. Okamoto,
*Studies on the Painlevé equations IV: Third Painlevé equation PIII, Funkc. Ekvacioj***30**(1987) 305.MATHGoogle Scholar - [83]Y. Ohyama, H. Kawamuko, H. Sakai and K. Okamoto,
*Studies on the Painlevé equations V: third Painlevé equations of special type PIII (D7) and PIII (D8), J. Math. Sci. Univ. Tokyo***13**(2006) 204.Google Scholar - [84]
*Painlevé equations — NIST DMLF entry*, http://dlmf.nist.gov/32.2.i. - [85]
*Coalescence cascades — NIST DMLF entry*, http://dlmf.nist.gov/32.2.vi. - [86]R. Rosales,
*The similarity solution for the Korteweg-de Vries equation and the related Painlevé transcendant, Proc. Roy. Soc. Lond.***A 361**(1978) 265.Google Scholar - [87]S.P. Hastings and J.B. McLeod,
*A boundary value problem associated with the second Painlevé transcendent and the Korteweg-de Vries equation, Arch. Rat. Mech. Anal.***73**(1980) 31.CrossRefMATHGoogle Scholar - [88]H. Neuberger,
*Complex Burgers’ equation in*2*D*SU(*N*)*YM*,*Phys. Lett.***B 670**(2008) 235 [arXiv:0809.1238] [INSPIRE]. - [89]O. Costin,
*On Borel summation and Stokes phenomena of nonlinear differential systems, Duke Math. J*.**93**(1998) 289.MathSciNetCrossRefMATHGoogle Scholar - [90]
*Asymptotic expansions for large order: Debye expansion — NIST DLMF entry*, http://dlmf.nist.gov/10.19.ii, - [91]
*Uniform large N approximations to Bessel functions — NIST DLMF entry*, http://dlmf.nist.gov/10.20. - [92]
- [93]
*Asymptotic Expansions for Large Order: transition region — NIST DLMF entry*, http://dlmf.nist.gov/10.19.iii. - [94]
- [95]M. Prähofer and H. Spohn,
*Exact scaling functions for one-dimensional stationary KPZ growth, J. Stat. Phys.***115**(2004) 255 [cond-mat/0212519]. - [96]B. Fornberg and J.A.C. Weideman,
*A computational exploration of the second Painlevé equation, Found. Comput. Math.***14**(2014) 985.MathSciNetCrossRefMATHGoogle Scholar