The Liouville geometry of N = 2 instantons and the moduli of punctured spheres

Journal of High Energy Physics

Volumn 8, Issue 5, 2004, Pages 1765-1815

  Bertoldi, Gaetano ^a   Bolognesi, Stefane ^b,c   Matone, Marco ^c,d   Mazzucato, Luca ^c,e   Nakayama, Yu ^f  

^a INSTITUTE FOR ADVANCED STUDY   (United States)

^b SCUOLA NORMALE SUPERIORE   (Italy)

^c INFN SEZIONE DI PISA   (Italy)

^d UNIVERSITY OF PADOVA   (Italy)

^e SISSA   (Italy)

^f UNIVERSITY OF TOKYO   (Japan)

Author keywords

Differential and Algebraic Geometry; Solitons Monopoles and Instantons; String Duality; Supersymmetry and Duality

Indexed keywords

EID: 23044444196     PISSN: 10298479     EISSN: 10298479     Source Type: Journal    
DOI: 10.1088/1126-6708/2004/05/075     Document Type: Article

References (146)

1. 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 [hep-th/9407087];
2. Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD, Nucl. Phys. B 431 (1994) 484 [hep-th/9408099].
3. M. Matone, Instantons and recursion relations in N = 2 SUSY gauge theory, Phys. Lett. B 357 (1995) 342 [hep-th/9506102];
4. Koebe 1/4 theorem and inequalities in N = 2 superqcd, Phys. Rev. D 53 (1996) 7354 [hep-th/9506181].
5. J.D. Edelstein, M. Mariño and J. Mas, Whitham hierarchies, instanton corrections and soft supersymmetry breaking in N = 2 SU(N) super Yang-Mills theory, Nucl. Phys. B 541 (1999) 671 [hep-th/9805172];
6. J.D. Edelstein, M. Gomez-Reino and J. Mas, Instanton corrections in N = 2 supersymmetric theories with classical gauge groups and fundamental matter hypermultiplets, Nucl. Phys. B 561 (1999) 273 [hep-th/9904087];
7. J.D. Edelstein, M. Gomez-Reino, M. Mariño and J. Mas, N = 2 supersymmetric gauge theories with massive hypermultiplets and the Whitham hierarchy, Nucl. Phys. B 574 (2000) 587 [hep-th/9911115].
8. G. Bonelli, M. Matone and M. Tonin, Solving N = 2 SYM by reflection symmetry of quantum vacua, Phys. Rev. D 55 (1997) 6466 [hep-th/9610026];
9. R. Flume, M. Magro, L. O'Raifeartaigh, I. Sachs and O. Schnetz, Uniqueness of the Seiberg-Witten effective lagrangian, Nucl. Phys. B 494 (1997) 331 [hep-th/9611123].
10. N. Dorey, T.J. Hollowood, V.V. Khoze and M.P. Mattis, The calculus of many instantons, Phys. Kept. 371 (2002) 231 [hep-th/0206063].
11. M.F. Atiyah, N.J. Hitchin, V.G. Drinfeld and Y.I. Manin, Construction of instantons, Phys. Lett. A 65 (1978) 185.
12. D. Bellisai, F. Fucito, A. Tanzini and G. Travaglini, Instanton calculus, topological field theories and N = 2 super Yang-Mills theories, J. High Energy Phys. 07 (2000) 017 [hep-th/0003272];
13. Multi-instantons, supersymmetry and topological field theories, Phys. Lett. B 480 (2000) 365 [hep-th/0002110];
14. U. Bruzzo, F. Fucito, A. Tanzini and G. Travaglini, On the multi-instanton measure for super Yang-Mills theories, Nucl. Phys. B 611 (2001) 205 [hep-th/0008225];
15. U. Bruzzo, F. Fucito, J.F. Morales and A. Tanzini, Multi-instanton calculus and equivariant cohomology, J. High Energy Phys. 05 (2003) 054 [hep-th/0211108];
16. U. Bruzzo and F. Fucito, Superlocalization formulas and supersymmetric Yang-Mills theories, Nucl. Phys. B 678 (2004) 638 [math-ph/0310036].
17. R. Flume, R. Poghossian and H. Storch, The coefficients of the Seiberg- Witten prepotential as intersection numbers (?), hep-th/0110240;
18. The Seiberg-Witten prepotential and the euler class of the reduced moduli space of instantons, Mod. Phys. Lett. A 17 (2002) 327 [hep-th/0112211].
19. T. J. Hollowood, Calculating the prepotential by localization on the moduli space of instantons, J. High Energy Phys. 03 (2002) 038 [hep-th/0201075];
20. Testing Seiberg-Witten theory to all orders in the instanton expansion, Nucl. Phys. B 639 (2002) 66 [hep-th/0202197].
21. N.A. Nekrasov, Seiberg- Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2004) 831 [hep-th/0206161].
22. N. Nekrasov and A. Okounkov, Seiberg- Witten theory and random partitions, hep-th/0306238.
23. G. Bonelli and M. Matone, Nonperturbative relations in N = 2 SUSY Yang-Mills and wdvv equation, Phys. Rev. Lett. 77 (1996) 4712 [hep-th/9605090];
24. N = 2 SYM RG scale as modulus for WDVV equations, Phys. Rev. D 57 (1998) 6483 [hep-th/9712109].
25. A. Marshakov, A. Mironov and A. Morozov, WDVV-like equations in N = 2 SUSY Yang-Mills theory, Phys. Lett. B 389 (1996) 43 [hep-th/9607109];
26. More evidence for the WDVV equations in N = 2 SUSY Yang-Mills theories, Int. J. Mod. Phys. A 15 (2000) 1157 [hep-th/9701123].
27. K. Ito and S.-K. Yang, The wdvv equations in N = 2 supersymmetric Yang-Mills theory, Phys. Lett. B 433 (1998) 56 [hep-th/9803126].
28. G. Chan and E. D'Hoker, Instanton recursion relations for the effective prepotential in N = 2 super Yang-Mills, Nucl. Phys. B 564 (2000) 503 [hep-th/9906193].
29. A. Losev, N. Nekrasov and S.L. Shatashvili, Issues in topological gauge theory, Nucl. Phys. B 534 (1998) 549 [hep-th/9711108];
30. Testing Seiberg-Witten solution, hep-th/9801061.
31. M. Mariño and G. W. Moore, Integrating over the Coulomb branch in N = 2 gauge theory, Nucl. Phys. 68 (Proc. Suppl.) (1998) 336 [hep-th/9712062];
32. The Donaldson-Witten function for gauge groups of rank larger than one, Commun. Math. Phys. 199 (1998) 25 [hep-th/9802185];
33. M. Mariño, The uses of Whitham hierarchies, Prog. Theor. Phys. Suppl. 135 (1999) 29 [hep-th/9905053];
34. J.D. Edelstein, M. Gomez-Reino and M. Mariño, Blowup formulae in Donaldson- Witten theory and integrable hierarchies, Adv. Theor. Math. Phys. 4 (2000) 503 [hep-th/0006113];
35. J.D. Edelstein, M. Gomez-Reino and M. Mariño, Remarks on twisted theories with matter, J. High Energy Phys. 01 (2001) 004 [hep-th/0011227];
36. J.D. Edelstein and M. Gomez-Reino, Integrable hierarchies in Donaldson- Witten and Seiberg-Witten theories, hep-th/0010061.
37. M. Matone, Nonperturbative model of Liouville gravity, J. Geom. Phys. 21 (1997) 381 [hep-th/9402081].
38. M. Matone, Seiberg- Witten duality in Dijkgraaf-Vafa theory, Nucl. Phys. B 656 (2003) 78 [hep-th/0212253].
39. R. Dijkgraaf and C. Vafa, Matrix models, topological strings and supersymmetric gauge theories, Nucl. Phys. B 644 (2002) 3 [hep-th/0206255];
40. On geometry and matrix models, Nucl. Phys. B 644 (2002) 21 [hep-th/0207106];
41. A perturbative window into non-perturbative physics, hep-th/0208048.
42. R. Dijkgraaf, S. Gukov, V.A. Kazakov and C. Vafa, Perturbative analysis of gauged matrix models, Phys. Rev. D 68 (2003) 045007 [hep-th/0210238].
43. T. Ekedahl, S.K. Lando, M. Shapiro and A. Vainshtein, Hurwitz numbers and intersections on moduli spaces of curves, Invent. Math. 146 (2001) 297;
44. S.K. Lando, Ramified coverings of the two-dimensional sphere and intersection theory in spaces of meromorphic functions on algebraic curves, Russ. Math. Surv. 57 (2002) 463.
45. R. Flume, F. Fucito, J.F. Morales and R. Poghossian, Matone's relation in the presence of gravitational couplings, J. High Energy Phys. 04 (2004) 008 [hep-th/0403057].
46. Y. Nakayama, Liouville field theory: a decade after the revolution, hep-th/0402009.
47. M. Goulian and M. Li, Correlation functions in Liouville theory, Phys. Rev. Lett. 66 (1991) 2051.
48. H. Dorn and H. J. Otto, Two and three point functions in Liouville theory, Nucl. Phys. B 429 (1994) 375 [hep-th/9403141].
49. A.B. Zamolodchikov and A.B. Zamolodchikov, Structure constants and conformal bootstrap in Liouville field theory, Nucl. Phys. B 477 (1996) 577 [hep-th/9506136].
50. Y. Nakayama, Tadpole cancellation in unoriented Liouville theory, J. High Energy Phys. 11 (2003) 017 [hep-th/0309063].
51. K. Hori and A. Kapustin, Duality of the fermionic 2D black hole and N = 2 Liouville theory as mirror symmetry, J. High Energy Phys. 08 (2001) 045 [hep-th/0104202].
52. M. Aganagic, R. Dijkgraaf, A. Klemm, M. Mariño and C. Vafa, Topological strings and integrable hierarchies, hep-th/0312085.
53. E. Witten, Two-dimensional gravity and intersection theory on moduli space, Surveys Diff. Geom. 1 (1991) 243.
54. P. Pasti, D.P. Sorokin and M. Tonin, Note on manifest Lorentz and general coordinate invariance in duality symmetric models, Phys. Lett. B 352 (1995) 59 [hep-th/9503182].
55. S. Katz, A. Klemm and C. Vafa, Geometric engineering of quantum field theories, Nucl. Phys. B 497 (1997) 173 [hep-th/9609239].
56. M. Matone, Uniformization theory and 2-D gravity, 1. Liouville action and intersection numbers, Int. J. Mod. Phys. A 10 (1995) 289 [hep-th/9306150].
57. P.G. Zograf and L.A. Takhtajan, On Liouville equation, accessory parameters, and the geometry of Teichmüller space for Riemann surfaces of genus 0, Math. USSR Sb. 60 (1988) 143;
58. On the uniformization of Riemann surfaces an on the Weil-Petersson metric on the Teichmüller and Schottky spaces, Math. USSR Sb. 60 (1988) 297;
59. Hyperbolic 2-spheres with conical singularities, accessory parameters and Kaehler metrics on M0,n, Trans. Am. Math. Soc. 355 (2002) 1857 [math.CV/0112170];
60. L.A. Takhtajan and L.P. Teo, Liouville action and Weil-Petersson metric on deformation spaces, global Kleinian reciprocity and holography, Commun. Math. Phys. 239 (2003) 183 [math.CV/0204318];
61. Weil-Petersson metric on the universal Teichmüller space I: curvature properties and Chern forms, math.CV/0312172.
62. L. Cantini, P. Menotti and D. Seminara, Proof of polyakov conjecture for general elliptic singularities, Phys. Lett. B 517 (2001) 203 [hep-th/0105081];
63. Liouville theory, accessory parameters and 2+1 dimensional gravity, Nucl. Phys. B 638 (2002) 351 [hep-th/0203103].
64. P. Deligne and D. Mumford, On the irreducibility of the space of curves of given genus, IHES Publ. Math. 36 (1969) 75;
65. F. Knudsen and D. Mumford, The projectivity of the moduli space of stable curves I: preliminaries on "det" and "div", Math. Scand. 39 (1976) 19;
66. F. Knudsen, Math. Scand. 52 (1983) 200
67. P.G. Zograf, The Weil-Petersson volume of the moduli space of punctured spheres, Cont. Math. Amer. AMS 150 (1993) 267.
68. S.A. Wolpert, On the homology of the moduli space of stable curves, Ann. Math. 118 (1983) 491;
69. On the Weil-Petersson geometry of the moduli space of curves, Amer. J. Math. 107 (1985) 969.
70. S.A. Wolpert, Geometry of the Weil-Petersson completion of the Teichmüller space, http://www.math.umd.edu/~saw/preprints/WPcompletion.pdf
71. Y.I. Manin, Frobenius manifolds, quantum cohomology, and moduli spaces, AMS Colloquium Publications 47 (1999).
72. G. Bonelli, P.A. Marchetti and M. Matone, Nonperturbative 2-D gravity, punctured spheres and theta vacua in string theories, Phys. Lett. B 339 (1994) 49 [hep-th/9407091];
73. Algebraic-geometrical formulation of two-dimensional quantum gravity, Lett. Math. Phys. 36 (1996) 189 [hep-th/9502089].
74. M. Kontsevich and Y.I. Manin (with appendix by R. Kaufmann), Quantum cohomology of a product, Invent. Math. 124 (1996) 313 [q-alg/9502009];
75. R. Kaufmann, Y. Manin and D. Zagier, Higher weil-petersson volumes of moduli spaces of stable n-pointed curves, alg-geom/9604001.
76. A. Givental, Semisimple Frobenius structures at higher genus, math.AG/0008067.
77. E. Brezin and V.A. Kazakov, Exactly solvable field theories of closed strings, Phys. Lett. B 236 (1990) 144;
78. M.R. Douglas and S.H. Shenker, Strings in less than one-dimension, Nucl. Phys. B 335 (1990) 635;
79. D.J. Gross and A.A. Migdal, Nonperturbative two-dimensional quantum gravity, Phys. Rev. Lett. 64 (1990) 127;
80. Nonperturbative solution of the ising model on a random surface, Phys. Rev. Lett. 64 (1990) 717.
81. H. Nakajima and K. Yoshioka, Instanton counting on blowup, I, math. AG/0306198;
82. Lectures on instanton counting, math.AG/0311058.
83. M. Mariño and N. Wyllard, A note on instanton counting for N = 2 gauge theories with classical gauge groups, J. High Energy Phys. 05 (2004) 021 [hep-th/0404125].
84. N. Nekrasov and S. Shadchin, Abcd of instantons, hep-th/0404225.
85. M. Matone, The instanton universal moduli space of N = 2 supersymmetric Yang-Mills theory, Phys. Lett. A 514 (2001) 161 [hep-th/0105041].
86. M. Matone, Multi-instanton measure from recursion relations in N = 2 supersymmetric Yang-Mills theory, J. High Energy Phys. 04 (2001) 041 [hep-th/0103246].
87. A. Maciocia, Metrics on the moduli spaces of instantons over euclidean four space, Commun. Math. Phys. 135 (1991) 467.
88. B. Chen, A smooth compactification of the moduli space of instantons and its application, math.GT/0204287.
89. T.H. Parker and J.G. Wolfeon, Pseudo-holomorphic maps and bubble trees, J. Geom. Phys. 3 (1993) 63.
90. L.A. Takhtajan, Liouville theory: quantum geometry of Riemann surfaces, Mod. Phys. Lett. A 8 (1993) 3529 [hep-th/9308125].
91. M. Matone, Quantum riemann surfaces, 2-D gravity and the geometrical origin of minimal models, Mod. Phys. Lett. A 9 (1994) 2871 [hep-th/9309096].
92. M. Matone, The Higgs model for anyons and Liouville action: chaotic spectrum, energy gap and exclusion principle, Mod. Phys. Lett. A 9 (1994) 1673 [hep-th/9403079];
93. The Higgs model for anyons and Liouville action, J. Geom. Phys. 17 (1995) 49 [hep-th/9310149].
94. N.A. Nekrasov, Seiberg-Witten prepotential from instanton counting, hep-th/0306211.
95. S.K. Lando and D. Zvonkine, Counting ramified coverings and intersection theory on spaces of rational functions I, math.AG/0303218.
96. M.E. Kazaryan and S.K. Lando, Towards the intersection theory on Hurwitz spaces, MPI preprint.
97. C.G. Callan Jr. and F. Wilczek, Infrared behavior at negative curvature, Nucl. Phys. B 340 (1990) 366.
98. F. Fucito and G. Travaglini, Instanton calculus and nonperturbative relations in N = 2 supersymmetric gauge theories, Phys. Rev. D 55 (1997) 1099 [hep-th/9605215];
99. N. Dorey, V.V. Khoze and M.P. Mattis, Multi-instanton check of the relation between the prepotential f and the modulus u in N = 2 SUSY Yang-Mills theory, Phys. Lett. B 390 (1997) 205 [hep-th/9606199].
100. P.S. Howe and P.C. West, Superconformal ward identities and N = 2 Yang-Mills theory, Nucl. Phys. B 486 (1997) 425 [hep-th/9607239].
101. E. D'Hoker and D.H. Phong, Strong coupling expansions of SU(N) Seiberg- Witten theory, Phys. Lett. B 397 (1997) 94 [hep-th/9701055];
102. K. Ito, C.-S. Xiong and S.-K. Yang, Seiberg- Witten theory as d < 1 topological strings, Phys. Lett. B 441 (1998) 155 [hep-th/9807183];
103. J.D. Edelstein and J. Mas, Strong coupling expansion and Seiberg- Witten- Whitham equations, Phys. Lett. B 452 (1999) 69 [hep-th/9901006].
104. N. Dorey, V.V. Khoze and M.P. Mattis, Multi-instanton check of the relation between the prepotential F and the modulus u in N = 2 SUSY Yang-Mills theory, Phys. Lett. B 390 (1997) 205 [hep-th/9606199].
105. A. Iqbal and A.-K. Kashani-Poor, Instanton counting and Chern-Simons theory, Adv. Theor. Math. Phys. 7 (2004) 457 [hep-th/0212279];
106. SU(N) geometries and topological string amplitudes, hep-th/0306032.
107. T. Eguchi and H. Kanno, Topological strings and Nekrasov's formulas, J. High Energy Phys. 12 (2003) 006 [hep-th/0310235].
108. R. Gopakumar and C. Vafa, M-theory and topological strings, I, hep-th/9809187;
109. M-tkeory and topological strings, II, hep-th/9812127.
110. I. Antoniadis, E. Gava, K.S. Narain and T.R. Taylor, Topological amplitudes in string theory, Nucl. Phys. B 413 (1994) 162 [hep-th/9307158].
111. M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, Kodaira-spencer theory of gravity and exact results for quantum string amplitudes, Commun. Math. Phys. 165 (1994) 311 [hep-th/9309140].
112. A. Zamolodchikov, Higher equations of motion in Liouville field theory, hep-th/0312279.
113. A.M. Polyakov, String theory and quark confinement, Nucl. Phys. 68 (Proc. Suppl.) (1998) 1 [hep-th/9711002].
114. C. Ahn, C. Kim, C. Rim and M. Stanishkov, Duality in N = 2 super-Liouville theory, Phys. Rev. D 69 (2004) 106011 [hep-th/0210208].
115. K. Hori and A. Kapustin, Worldsheet descriptions of wrapped NS five-branes, J. High Energy Phys. 11 (2002) 038 [hep-th/0203147].
116. D. Ghoshal and C. Vafa, C = 1 string as the topological theory of the conifold, Nucl. Phys. B 453 (1995) 121 [hep-th/9506122].
117. J.J. Duistermaat and G.J. Heckman, IN the variation in the cohomology of the symplectic form of the reduced phase space, Invent. Math. 69 (1982) 259.
118. F. David, Conformal field theories coupled to 2-D gravity in the conformal gauge, Mod. Phys. Lett. A 3 (1988) 1651;
119. J. Distler and H. Kawai, Conformal field theory and 2-d quantum gravity or who 's afraid of joseph liouville?, Nucl. Phys. B 321 (1989) 509.
120. D.J. Gross and J. Walcher, Non-perturbative RR potentials in the ĉ = 1 matrix model, hep-th/0312021.
121. M. Kontsevich and Y. Manin, Gromov- Witten classes, quantum cohomology and enumerative geometry, Commun. Math. Phys. 164 (1994) 525 [hep-th/9402147].
122. P. Di Francesco and C. Itzykson, Quantum intersection rings, hep-th/9412175;
123. B. Dubrovin, Geometry of 2-d topological field theories, hep-th/9407018.
124. M. Aganagic, A. Klemm, M. Mariño and C. Vafa, The topological vertex, hep-th/0305132.
125. A. Klemm, M. Mariño and S. Theisen, Gravitational corrections in supersymmetric gauge theory and matrix models, J. High Energy Phys. 03 (2003) 051 [hep-th/0211216];
126. R. Dijkgraaf, A. Sinkovics and M. Temurhan, Matrix models and gravitational corrections, hep-th/0211241.
127. M. Matone, Eigenfunctions of the Laplacian acting on degree zero bundles over special Riemann surfaces Trans. Am. Math. Soc. 356 (2004) 2989 [math.AG/0105051].
128. B. Eynard, A. Kokotov and D. Korotkin, Genus one contribution to free energy in hermitian two-matrix model, hep-th/0403072.
129. M. Matone, Liouville equation and Schottky problem, Lett. Math. Phys. 33 (1995) 75 [hep-th/9310066].
130. T. Eguchi and H. Ooguri, Chiral bosonization on a Riemann surface, Phys. Lett. B 187 (1987) 127.
131. G. Grignani, P. Orland, L.D. Paniak and G.W. Semenoff, Matrix theory interpretation of dlcq string worldsheets, Phys. Rev. Lett. 85 (2000) 3343 [hep-th/0004194].
132. M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, math.GT/0201292;
133. A. Eskin, H. Masur and A. Zorich, Moduli spaces of abelian differentials: the principal boundary, counting problems and the Siegel-Veech constants, math.DS/0202134;
134. A. Kokotov and D. Korotkin, Tau-functions on spaces of holomorphic differentials over Riemann surfaces and determinants of Laplacians in flat metrics with conic singularities, math.DG/0405042.
135. L.A. Takhtajan, in The verdier memorial conference on integrable systems, Birkhäuser, Boston 1993;
136. M. J. Ablowitz, S. Chakravarty and R.G. Halburd, Integrable systems and reductions of the self-dual Yang-Mills equations, J. Math. Phys. 44 (2003) 3147.
137. J. Harnad and J. McKay, Modular solutions to equations of generalized Halphen type, Proc. Roy. Soc. Land. 456 (2000) 261 [solv-int/9804006].
138. A. Klemm, W. Lerche, S. Yankielowicz and S. Theisen, Simple singularities and N = 2 supersymmetric Yang-Mills theory, Phys. Lett. B 344 (1995) 169 [hep-th/9411048];
139. P.C. Argyres and A.E. Faraggi, The vacuum structure and spectrum of N = 2 supersymmetric SU(N) gauge theory, Phys. Rev. Lett. 74 (1995) 3931 [hep-th/9411057].
140. J. Sonnenschein, S. Theisen and S. Yankielowicz, On the relation between the holomorphic prepotential and the quantum moduli in SUSY gauge theories, Phys. Lett. B 367 (1996) 145 [hep-th/9510129];
141. T. Eguchi and S.-K. Yang, Prepotentials of n = 2 supersymmetric gauge theories and soliton equations, Mod. Phys. Lett. A 11 (1996) 131 [hep-th/9510183];
142. E. D'Hoker, I.M. Krichever and D.H. Phong, The renormalization group equation in N = 2 supersymmetric gauge theories, Nucl. Phys. B 494 (1997) 89 [hep-th/9610156].
143. Y. Imayoshi and M. Taniguchi, An introduction to the Teichmüller spaces, Springer-Verlag, Berlin 1992;
144. P. Buser, Geometry and spectra of compact Riemann surfaces, Birkhäuser, Boston 1992.
145. Ph. Griffiths and J. Harris, Principles of algebraic geometry, Wiley, New York 1978.
146. P.G. Zograf, The Liouville action on moduli spaces, and uniformization of degenerating Riemann surfaces, Leningrad Math. J. 1 (1990) 941.