Borel and Stokes Nonperturbative Phenomena in Topological String Theory and c = 1 Matrix Models

Di Francesco P., Ginsparg P.H., Zinn-Justin J.: 2D gravity and random matrices. Phys. Rept. 254, 1 (1995) arXiv:hep-th/9306153

David F.: Phases of the large N matrix model and nonperturbative effects in 2D gravity. Nucl. Phys. B 348, 507 (1991)

David F.: Nonperturbative effects in matrix models and vacua of two-dimensional gravity. Phys. Lett. B 302, 403 (1993) arXiv:hep-th/9212106

Shenker, S.H.: The strength of nonperturbative effects in string theory. In: Cargèse Workshop on Random Surfaces, Quantum Gravity and Strings (1990)

Polchinski J.: Combinatorics of boundaries in string theory. Phys. Rev. D 50, 6041 (1994) arXiv:hep-th/9407031

Alexandrov S.Yu., Kazakov V.A., Kutasov D.: Nonperturbative effects in matrix models and D-Branes. JHEP 0309, 057 (2003) arXiv:hep-th/0306177

Dijkgraaf R., Vafa C.: Matrix models, topological strings, and supersymmetric gauge theories. Nucl. Phys. B 644, 3 (2002) arXiv:hep-th/0206255

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

Aganagic M., Klemm A., Mariño M., Vafa C.: Matrix model as a mirror of Chern–Simons theory. JHEP 0402, 010 (2004) arXiv:hep-th/0211098

Mariño M.: Open string amplitudes and large-order behavior in topological string theory. JHEP 0803, 060 (2008) arXiv:hep-th/0612127

Bouchard V., Klemm A., Mariño M., Pasquetti S.: Remodeling the B-model. Commun. Math. Phys. 287, 117 (2009) arXiv:0709.1453[hep-th]

Bouchard, V., Klemm, A., Mariño, M., Pasquetti, S.: Topological open strings on orbifolds, arXiv:0807.0597[hep-th]

Mariño M., Schiappa R., Weiss M.: Nonperturbative effects and the large-order behavior of matrix models and topological strings. Commun. Number Theor. Phys. 2, 349 (2008) arXiv:0711.1954[hep-th]

Mariño M.: Nonperturbative effects and nonperturbative definitions in matrix models and topological strings. JHEP 0812, 114 (2008) arXiv:0805.3033[hep-th]

Mariño, M., Schiappa, R., Weiss, M.: Multi-instantons and multi-cuts. J. Math. Phys. (2010, in press). arXiv:0809.2619[hep-th]

Hanada M., Hayakawa M., Ishibashi N., Kawai H., Kuroki T., Matsuo Y., Tada T.: Loops versus matrices: the nonperturbative aspects of noncritical string. Prog. Theor. Phys. 112, 131 (2004) arXiv:hep-th/0405076

Ishibashi N., Yamaguchi A.: On the chemical potential of D-instantons in c = 0 noncritical string theory. JHEP 0506, 082 (2005) arXiv:hep-th/0503199

Bessis D., Itzykson C., Zuber J.B.: Quantum field theory techniques in graphical enumeration. Adv. Appl. Math. 1, 109 (1980)

Eynard, B., Mariño, M.: A holomorphic and background independent partition function for matrix models and topological strings. arXiv:0810.4273[hep-th]

Caporaso N., Griguolo L., Mariño M., Pasquetti S., Seminara D.: Phase transitions, double-scaling limit, and topological strings. Phys. Rev. D 75, 046004 (2007) arXiv:hep-th/0606120

Ooguri H., Strominger A., Vafa C.: Black hole attractors and the topological string. Phys. Rev. D 70, 106007 (2004) arXiv:hep-th/0405146

Zinn-Justin J.: Perturbation series at large orders in quantum mechanics and field theories: application to the problem of resummation. Phys. Rept. 70, 109 (1981)

Gopakumar R., Vafa C.: Topological gravity as large N topological gauge theory. Adv. Theor. Math. Phys. 2, 413 (1998) arXiv:hep-th/9802016

Gopakumar, R., Vafa, C.: M-theory and topological strings 1. arXiv:hep-th/9809187

Gross D.J., Miljkovic N.: A nonperturbative solution of d = 1 string theory. Phys. Lett. B 238, 217 (1990)

Matsuo Y.: Nonperturbative effect in c = 1 noncritical string theory and Penner model. Nucl. Phys. B 740, 222 (2006) arXiv:hep-th/0512176

Schwinger J.H.: On gauge invariance and vacuum polarization. Phys. Rev. 82, 664 (1951)

Chadha S., Olesen P.: On Borel singularities in quantum field theory. Phys. Lett. B 72, 87 (1977)

Bender C.M., Wu T.T.: Anharmonic oscillator 2: a study of perturbation theory in large order. Phys. Rev. D 7, 1620 (1973)

Collins J.C., Soper D.E.: Large order expansion in perturbation theory. Ann. Phys. 112, 209 (1978)

Dunne, G.V.: Heisenberg–Euler effective Lagrangians: basics and extensions. In: Ian Kogan Memorial Collection “From Fields to Strings: Circumnavigating Theoretical Physics” (2004). arXiv:hep-th/0406216

Kim S.P., Page D.N. (2006) Schwinger pair production in electric and magnetic fields. Phys. Rev. D 73, 065020 (2006) arXiv:hep-th/0301132

Dunne G.V., Schubert C.: Closed form two loop Euler–Heisenberg Lagrangian in a selfdual background. Phys. Lett. B 526, 55 (2002) arXiv:hep-th/0111134

Dunne G.V., Schubert C.: Two loop selfdual Euler–Heisenberg Lagrangians 1, real part and helicity amplitudes. JHEP 0208, 053 (2002) arXiv:hep-th/0205004

Dunne G.V., Schubert C.: Two loop selfdual Euler–Heisenberg Lagrangians 2, imaginary part and Borel analysis. JHEP 0206, 042 (2002) arXiv:hep-th/0205005

Popov V.S.: Pair production in a variable external field (quasiclassical approximation). Sov. Phys. JETP 34, 709 (1972)

Vonk, M.: A mini-course on topological strings. arXiv:hep-th/0504147

Bershadsky M., Cecotti S., Ooguri H., Vafa C.: Kodaira–Spencer theory of gravity and exact results for quantum string amplitudes. Commun. Math. Phys. 165, 311 (1994) arXiv:hep-th/9309140

Ghoshal D., Vafa C.: c =  1 String as the Topological Theory of the Conifold. Nucl. Phys. B 453, 121 (1995) arXiv:hep-th/9506122

Klebanov, I.R.: String theory in two-dimensions. In: Trieste Spring School on string theory and quantum gravity (1991). arXiv:hep-th/9108019

Klemm, A., Zaslow, E.: Local mirror symmetry at higher genus. arXiv:hep-th/9906046

Katz S., Klemm A., Vafa C.: M-theory, topological strings and spinning black holes. Adv. Theor. Math. Phys. 3, 1445 (1999) arXiv:hep-th/9910181

Gross D.J., Klebanov I.R.: One-dimensional string theory on a circle. Nucl. Phys. B 344, 475 (1990)

Gopakumar, R., Vafa, C.: M-theory and topological strings 2. arXiv:hep-th/9812127

Mariño M., Moore G.W.: Counting higher genus curves in a Calabi–Yau manifold. Nucl. Phys. B 543, 592 (1999) arXiv:hep-th/9808131

Faber C., Pandharipande R.: Hodge integrals and Gromov–Witten theory. Invent. Math. 139, 173 (2000) arXiv:math/9810173

Gopakumar R., Vafa C. (1999) On the gauge theory/geometry correspondence. Adv. Theor. Math. Phys. 3, 1415 (1999) arXiv:hep-th/9811131

Witten E.: Chern–Simons gauge theory as a string theory. Prog. Math. 133, 637 (1995) arXiv:hep-th/9207094

Brézin E., Itzykson C., Parisi G., Zuber J.B.: Planar diagrams. Commun. Math. Phys. 59, 35 (1978)

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

Eynard B., Orantin N.: Invariants of algebraic curves and topological expansion. Commun. Number Theor. Phys. 1, 347 (2007) arXiv:math-ph/0702045

Eynard, B., Orantin, N.: Algebraic methods in random matrices and enumerative geometry. arXiv:0811.3531[math-ph]

Macdonald I.G.: The volume of a compact Lie group. Invent. Math. 56, 93 (1980)

Penner R.C.: Perturbative series and the moduli space of Riemann surfaces. J. Differ. Geom. 27, 35 (1988)

Distler, J., Vafa, C.: The Penner model and d = 1 string theory. In: Cargèse workshop on random surfaces, quantum gravity and strings (1990)

Distler J., Vafa C.: Critical matrix model at c = 1. Mod. Phys. Lett. A 6, 259 (1991)

Chaudhuri S., Dykstra H., Lykken J.D.: The Penner matrix model and c = 1 strings. Mod. Phys. Lett. A 6, 1665 (1991)

Ambjørn J., Kristjansen C.F., Makeenko Yu.: Generalized Penner models to all genera. Phys. Rev. D 50, 5193 (1994) arXiv:hep-th/9403024

Mariño M.: Chern–Simons theory and topological strings. Rev. Mod. Phys. 77, 675 (2005) arXiv:hep-th/0406005

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

Aganagic, M., Vafa, C.: Mirror symmetry, D-Branes and counting holomorphic discs. arXiv:hep-th/0012041

Aganagic M., Klemm A., Vafa C.: Disk instantons, mirror symmetry and the duality web. Z. Naturforsch. A 57, 1 (2002) arXiv:hep-th/0105045

Fateev, V., Zamolodchikov, A.B., Zamolodchikov, A.B.: Boundary Liouville field theory I: boundary state and boundary two-point function. arXiv:hep-th/0001012

Zamolodchikov, A.B., Zamolodchikov, A.B.: Liouville field theory on a pseudosphere. arXiv:hep-th/0101152

Hosomichi K.: Bulk-boundary propagator in Liouville theory on a disc. JHEP 0111, 044 (2001) arXiv:hep-th/0108093

Martinec, E.J.: The annular report on non-critical string theory. arXiv:hep-th/0305148

Seiberg N., Shih D.: Branes, rings and matrix models in minimal (Super)String Theory. JHEP 0402, 021 (2004) arXiv:hep-th/0312170

Alexandrov S.Y., Kostov I.K.: Time-dependent backgrounds of 2D string theory: nonperturbative effects. JHEP 0502, 023 (2005) arXiv:hep-th/0412223

Alexandrov S.: D-Branes and complex curves in c =  1 string theory. JHEP 0405, 025 (2004) arXiv:hep-th/0403116

Kazakov, V.A., Kostov, I.K.: Instantons in non-critical strings from the two-matrix model. arXiv:hep-th/0403152

Alexandrov S.: (m, n) ZZ Branes and the c = 1 Matrix Model. Phys. Lett. B 604, 115 (2004) arXiv:hep-th/0310135

Bertoldi G., Hollowood T.J.: Large N gauge theories and topological cigars. JHEP 0704, 078 (2007) arXiv:hep-th/0611016

Adamchik, V.S.: Contributions to the theory of the Barnes function. arXiv: math/0308086[math.CA]

Gradshteyn I.S., Ryzhik I.M.: Table of Integrals, Series, and Products. Academic Press, London (2007)

Berry M.V., Howls C.J.: Hyperasymptotics. Proc. R. Soc. Lond. A 430, 653 (1990)

Boyd J.P.: The Devil’s invention: asymptotic, superasymptotic and hyperasymptotic series. Acta Appl. Math. 56, 1 (1999)

Berry M.V., Howls C.J.: Hyperasymptotics for integrals with saddles. Proc. R. Soc. Lond. A 434, 657 (1991)

Boyd W.G.C.: Error bounds for the method of steepest descents. Proc. R. Soc. Lond. A 440, 493 (1993)

Friot, S., Greynat, D.: Non-perturbative asymptotic improvement of perturbation theory and Mellin–Barnes representation. arXiv:0907.5593[hep-th]

Paris R.B., Kaminski D.: Asymptotics and Mellin–Barnes Integrals. Cambridge University Press, Cambridge (2001)

Boyd W.G.C.: Gamma function asymptotics by an extension of the method of steepest descents. Proc. R. Soc. Lond. A 447, 609 (1994)

Berry M.V.: Infinitely many stokes smoothings in the gamma function. Proc. R. Soc. Lond. A 434, 465 (1991)

Koshkin, S.: Quantum Barnes function as the partition function of the resolved conifold. arXiv:0710.2929[math.AG]

Maldacena J.M., Moore G.W., Seiberg N., Shih D.: Exact vs. semiclassical target space of the minimal string. JHEP 0410, 020 (2004) arXiv:hep-th/0408039

Eynard B.: Large N expansion of convergent matrix integrals, holomorphic anomalies, and background independence. JHEP 0903, 003 (2009) arXiv:0802.1788[math-ph]

Berry M.V.: Uniform asymptotic smoothing of Stokes’ discontinuities. Proc. R. Soc. Lond. A 422, 7 (1989)

Chapman S.J.: On the non-universality of the error function in the smoothing of stokes discontinuities. Proc. R. Soc. Lond. A 452, 2225 (1996)

Bonnet G., David F., Eynard B.: Breakdown of universality in multicut matrix models. J. Phys. A 33, 6739 (2000) arXiv:cond-mat/0003324

Maximon L.C.: The dilogarithm function for complex argument. Proc. R. Soc. Lond. A 459, 2807 (2003)

Vepštas L.: An Efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and Hurwitz zeta functions. Numer. Algorithm 47, 211 (2008) arXiv:math/0702243

Ooguri H., Vafa C.: Summing up D-Instantons. Phys. Rev. Lett. 77, 3296 (1996) arXiv:hep-th/9608079

Collinucci, A., Soler, P., Uranga, A.M.: Nonperturbative effects and wall-crossing from topological strings. arXiv:0904.1133[hep-th]

Dijkgraaf R., Moore G.W., Plesser R.: The partition function of 2D string theory. Nucl. Phys. B 394, 356 (1993) arXiv:hep-th/9208031

Kazakov V., Kostov I.K., Kutasov D.: A matrix model for the two-dimensional black hole. Nucl. Phys. B 622, 141 (2002) arXiv:hep-th/0101011

Sauzin, D.: Resurgent functions and splitting problems. arXiv:0706.0137 [math.DS]

Bryan J., Pandharipande R.: BPS states of curves in Calabi–Yau Threefolds. Geom. Topol. 5, 287 (2001) arXiv:math/0009025

Bryan J., Katz S., Leung N.C.: Multiple covers and the integrality conjecture for rational curves in Calabi–Yau threefolds. J. Algebraic Geom. 10, 549 (2001) arXiv:math/9911056

Karp D., Liu C.-C.M., Mariño M.: The local Gromov–Witten invariants of configurations of rational curves. Geom. Topol. 10, 115 (2006) arXiv:math/0506488

Aganagic M., Jafferis D., Saulina N.: Branes, black holes and topological strings on toric Calabi–Yau manifolds. JHEP 0612, 018 (2006) arXiv:hep-th/0512245

Haghighat B., Klemm A., Rauch M.: Integrability of the holomorphic anomaly equations. JHEP 0810, 097 (2008) arXiv:0809.1674[hep-th]