×

Wall-crossing holomorphic anomaly and mock modularity of multiple M5-branes. (English) Zbl 1406.81070

Summary: Using wall-crossing formulae and the theory of mock modular forms we derive a holomorphic anomaly equation for the modified elliptic genus of two M5-branes wrapping a rigid divisor inside a Calabi-Yau manifold. The anomaly originates from restoring modularity of an indefinite theta-function capturing the wall-crossing of BPS invariants associated to D4-D2-D0 brane systems. We show the compatibility of this equation with anomaly equations previously observed in the context of \(\mathcal N=4\) topological Yang-Mills theory on \(\mathbb P^2\) and E-strings obtained from wrapping M5-branes on a del Pezzo surface. The non-holomorphic part is related to the contribution originating from bound-states of singly wrapped M5-branes on the divisor. We show in examples that the information provided by the anomaly is enough to compute the BPS degeneracies for certain charges. We further speculate on a natural extension of the anomaly to higher D4-brane charge.

MSC:

81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
PDFBibTeX XMLCite
Full Text: DOI arXiv Link

References:

[1] Witten E.: Topological sigma models. Commun. Math. Phys. 118, 411 (1988) · Zbl 0674.58047 · doi:10.1007/BF01466725
[2] Gopakumar, R., Vafa, C.: M-theory and topological strings. I (1998). arXiv:hep-th/9809187 · Zbl 0922.32015
[3] Gopakumar, R., Vafa, C.: M-theory and topological strings. II. arXiv:hep-th/9812127 · Zbl 0922.32015
[4] Minahan, J.A., Nemeschansky, D., Vafa, C., Warner, N.P.: E-strings and N = 4 topological Yang-Mills theories. Nucl. Phys. B527, 581-623 (1998). arXiv:hep-th/9802168 · Zbl 0951.81025
[5] Vafa, C., Witten, E.: A Strong coupling test of S duality. Nucl. Phys. B431, 3-77 (1994). arXiv:hep-th/9408074 · Zbl 0964.81522
[6] Gaiotto, D., Strominger, A., Yin, X.: The M5-brane elliptic genus: modularity and BPS states. JHEP 08, 070 (2007). arXiv:hep-th/0607010 · Zbl 1326.81189
[7] Maldacena, J.M., Strominger, A., Witten, E.: Black hole entropy in M-theory. JHEP 12, 002 (1997). arXiv:hep-th/9711053 · Zbl 0951.83034
[8] Bershadsky, M., Cecotti, S., Ooguri, H., Vafa, C.: Holomorphic anomalies in topological field theories. Nucl. Phys. B405, 279-304 (1993). arXiv:hep-th/9302103 · Zbl 0908.58074
[9] Bershadsky, M., Cecotti, S., Ooguri, H., Vafa, C.: Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes. Commun. Math. Phys. 165, 311-428 (1994). arXiv:hep-th/9309140 · Zbl 0815.53082
[10] Yamaguchi, S., Yau, S.-T.: Topological string partition functions as polynomials. JHEP 07, 047 (2004). arXiv:hep-th/0406078 · Zbl 0924.57025
[11] Grimm, T.W., Klemm, A., Marino, M., Weiss, M.: Direct integration of the topological string. JHEP 08, 058 (2007). arXiv:hep-th/0702187 · Zbl 1326.81191
[12] Alim, M., Lange, J.D.: Polynomial structure of the (open) topological string partition function. JHEP 10 045 (2007). arXiv:0708.2886
[13] Huang, M.-X., Klemm, A.: Holomorphic anomaly in gauge theories and matrix models. JHEP 09, 054 (2007). arXiv:hep-th/0605195 · Zbl 0961.14007
[14] Huang, M.-X., Klemm, A., Quackenbush, S.: Topological string theory on compact Calabi-Yau: modularity and boundary conditions. Lect. Notes Phys. 757, 45-102 (2009). arXiv:hep-th/0612125 · Zbl 1166.81358
[15] Haghighat, B., Klemm, A., Rauch, M.: Integrability of the holomorphic anomaly equations. JHEP 10, 097 (2008). arXiv:0809.1674 · Zbl 1245.81173
[16] Alim, M., Lange, J.D., Mayr, P.: Global properties of topological string amplitudes and orbifold invariants. JHEP 03, 113 (2010). arXiv:0809.4253 · Zbl 1271.81117
[17] Witten, E.: Quantum background independence in string theory (1993). arXiv:hep-th/9306122
[18] Aganagic, M., Bouchard, V., Klemm, A.: Topological strings and (almost) modular forms. Commun. Math. Phys. 277, 771-819 (2008). arXiv:hep-th/0607100 · Zbl 1165.81037
[19] Dijkgraaf, R., Verlinde, E.P., Vonk, M.: On the partition sum of the NS five-brane (2002). arXiv:hep-th/0205281 · Zbl 0731.14009
[20] Verlinde, E.P.: Attractors and the holomorphic anomaly (2004). arXiv:hep-th/0412139
[21] Gunaydin, M., Neitzke, A., Pioline, B.: Topological wave functions and heat equations. JHEP 12, 070 (2006). arXiv:hep-th/0607200 · Zbl 1226.81194
[22] Zagier D.: Nombres de classes et formes modulaires de poids 3/2. C. R. Acad. Sci. Paris 21, A883-A886 (1975) · Zbl 0323.10021
[23] Minahan, J.A., Nemeschansky, D., Warner, N.P.: Partition functions for BPS states of the non-critical E(8) string. Adv. Theor. Math. Phys. 1, 167-183 (1998). arXiv:hep-th/9707149 · Zbl 0898.32013
[24] Zwegers, S.P.: Mock theta functions, Proefschrift Universiteit Utrecht (2002) · Zbl 1194.11058
[25] Zagier, D.: Ramanujan’s mock theta functions and their applications d’après Zwegers and Bringmann-Ono. Séminaire BOURBAKI 986 (2007) · Zbl 1198.11046
[26] Ono K.: Unearthing the visions of a master: harmonic maass forms and number theory. Curr. Dev. Math. 2008, 347-454 (2009) · Zbl 1229.11074 · doi:10.4310/CDM.2008.v2008.n1.a5
[27] Gottsche L., Zagier D.: Jacobi forms and the structure of Donaldson invariants for 4-manifolds with \[{b_+=1}\] b+=1. Sel. Math. New Ser. 4, 69-115 (1998) · Zbl 0924.57025 · doi:10.1007/s000290050025
[28] Moore, G.W., Witten, E.: Integration over the u-plane in Donaldson theory. Adv. Theor. Math. Phys. 1, 298-387 (1998). arXiv:hep-th/9709193 · Zbl 0899.57021
[29] Losev, A., Nekrasov, N., Shatashvili, S.L.: Issues in topological gauge theory. Nucl. Phys. B534, 549-611 (1998). arXiv:hep-th/9711108 · Zbl 0954.57013
[30] Cecotti, S., Vafa, C.: On classification of N = 2 supersymmetric theories. Commun. Math. Phys. 158, 569-644 (1993). arXiv:hep-th/9211097 · Zbl 0787.58049
[31] Seiberg, N., Witten, E.: Monopole condensation, and confinement in N = 2 supersymmetric Yang-Mills theory. Nucl. Phys. B426, 19-52 (1994). arXiv:hep-th/9407087 · Zbl 0996.81510
[32] Denef, F.: Supergravity flows and D-brane stability. JHEP 08, 050 (2000). arXiv:hep-th/0005049 · Zbl 0990.83553
[33] Denef, F., Moore, G.W.: Split states, entropy enigmas, holes and halos. JHEP 1111, 129 (2011). arXiv:hep-th/0702146 · Zbl 1306.81213
[34] Kontsevich, M., Soibelman, Y.: Stability structures, motivic Donaldson-Thomas invariants and cluster transformations (2008). arXiv:0811.2435 · Zbl 1248.14060
[35] Gaiotto, D., Moore, G.W., Neitzke, A.: Four-dimensional wall-crossing via three-dimensional field theory. Commun. Math. Phys. 299, 163-224 (2010). arXiv:0807.4723 · Zbl 1225.81135
[36] Cecotti, S., Vafa, C.: BPS wall crossing and topological strings (2009). arXiv:0910.2615
[37] Gaiotto, D., Moore, G.W., Neitzke, A.: Wall-crossing, Hitchin systems, and the WKB approximation (2009). arXiv:0907.3987 · Zbl 1358.81150
[38] Gaiotto, D., Moore, G.W., Neitzke, A.: Framed BPS States. Adv. Theor. Math. Phys. 17(2), 241-397 (2013). arXiv:1006.0146 · Zbl 1290.81146
[39] Cecotti, S., Neitzke, A., Vafa, C.: R-twisting and 4d/2d correspondences (2010). arXiv:1006.3435 · Zbl 1355.81120
[40] Dabholkar A.: Cargese lectures on black holes, dyons, and modular forms. Nucl. Phys. Proc. Suppl. 171, 2-15 (2007) · doi:10.1016/j.nuclphysbps.2007.06.003
[41] Dabholkar, A., Murthy, S., Zagier, D.: Quantum black holes and mock modular forms. In: Talks at ASC workshop on Interfaces and Wall crossing, 2009 in Munich, workshop on Automorphic Forms, Kac-Moody Algebras and Strings, 2010 in Bonn and at the conference on Topological String Theory, Modularity and Non-perturbative Physics, 2010 in Vienna
[42] Manschot, J.: Stability and duality in N = 2 supergravity. Commun. Math. Phys. 299(3), 651-676 (2010). arXiv:0906.1767 · Zbl 1201.83045
[43] Manschot, J.: Wall-crossing of D4-branes using flow trees. Adv. Theor. Math. Phys. 15, 1-42 (2011). arXiv:1003.1570 · Zbl 1352.81051
[44] Manschot, J.: The Betti numbers of the moduli space of stable sheaves of rank 3 on P2. Lett. Math. Phys. 98(1), 65-78 (2011). arXiv:1009.1775 · Zbl 1262.14052
[45] Bringmann, K., Manschot, J.: From sheaves on P2 to a generalization of the Rademacher expansion. Am. J. Math. 135(4), 1039-1065 (2013). arXiv:1006.0915 · Zbl 1335.14011
[46] Eguchi, T., Hikami, K.: Superconformal algebras and mock theta functions. J. Phys. A42, 304010 (2009). arXiv:0812.1151 · Zbl 1176.81100
[47] Eguchi, T., Hikami, K.: Superconformal algebras and mock theta functions 2. Rademacher expansion for K3 surface. Commun. Number Theory 3, 531-554 (2009). arXiv:0904.0911 · Zbl 1189.81191
[48] Eguchi, T., Ooguri, H., Tachikawa, Y.: Notes on the K3 Surface and the Mathieu group M24. Exp. Math. 20(1), 91-96 (2011). arXiv:1004.0956 · Zbl 1266.58008
[49] Cheng, M.C.N.: K3 Surfaces, N = 4 dyons, and the Mathieu group M24 (2010). arXiv:1005.5415
[50] Troost, J.: The non-compact elliptic genus: mock or modular. JHEP 06, 104 (2010). arXiv:1004.3649 · Zbl 1288.81124
[51] Gottsche L.: Theta functions and Hodge numbers of moduli spaces of sheaves on rational surfaces. Commun. Math. Phys. 206, 105-136 (1999) · Zbl 0961.14022 · doi:10.1007/s002200050699
[52] Maldacena, J.M., Moore, G.W., Strominger, A.: Counting BPS black holes in toroidal type II string theory (1999). arXiv:hep-th/9903163
[53] de Boer, J., Cheng, M.C.N., Dijkgraaf, R., Manschot, J., Verlinde, E.: A farey tail for attractor black holes. JHEP 11, 024 (2006). arXiv:hep-th/0608059
[54] Kraus, P., Larsen, F.: Partition functions and elliptic genera from supergravity. JHEP 01, 002 (2007). arXiv:hep-th/0607138
[55] Gaiotto, D., Yin, X.: Examples of M5-brane elliptic genera. JHEP 11, 004 (2007). arXiv:hep-th/0702012 · Zbl 1245.81165
[56] Manschot, J., Moore, G.W.: A modern fareytail. Commun. Num. Theor. Phys. 4, 103-159 (2010). arXiv:0712.0573 · Zbl 1259.58005
[57] Dabholkar, A., Denef, F., Moore, G.W., Pioline, B.: Precision counting of small black holes. JHEP 10, 096 (2005). arXiv:hep-th/0507014
[58] Minasian, R., Moore, G.W., Tsimpis, D.: Calabi-Yau black holes and (0, 4) sigma models. Commun. Math. Phys. 209, 325-352 (2000). arXiv:hep-th/9904217 · Zbl 0960.83022
[59] Guica, M., Strominger, A.: Cargese lectures on string theory with eight supercharges. Nucl. Phys. Proc. Suppl. 171, 39-68 (2007). arXiv:0704.3295
[60] Weist, T.: Torus fixed points of moduli spaces of stable bundles of rank three. J. Pure Appl. Algebra 215(10), 2406-2422 (2011). arXiv:0903.0723 · Zbl 1233.14008
[61] Freed, D.S., Witten, E.: Anomalies in string theory with D-branes. Asian. J. Math. 3, 819 (1999). arXiv:hep-th/9907189 · Zbl 1028.81052
[62] Minasian, R., Moore, G.W.: K-theory and Ramond-Ramond charge. JHEP 11, 002 (1997). arXiv:hep-th/9710230 · Zbl 0949.81511
[63] Gottsche L.: The Betti numbers of the Hilbert schemes of points on a smooth projective surface. Math. Ann. 286, 193-207 (1990) · Zbl 0679.14007 · doi:10.1007/BF01453572
[64] Joyce: Holomorphic generating functions for invariants counting coherent sheaves on Calabi-Yau 3-folds. Geom. Topol. 11, 667-725 (2007). arXiv:hep-th/0607039 · Zbl 1141.14023
[65] Joyce, S.: A theory of generalized Donaldson-Thomas invariants (2008). arXiv:0810.5645 · Zbl 1259.14054
[66] Dabholkar, A., Denef, F., Moore, G.W., Pioline, B.: Exact and asymptotic degeneracies of small black holes. JHEP 0508, 021 (2005). arXiv:hep-th/0502157
[67] Ganor, O.J., Hanany, A.: Small \[{E_8}\] E8 instantons and tensionless non-critical strings. Nucl. Phys. B474, 122-140 (1996). arXiv:hep-th/9602120 · Zbl 0925.81170
[68] Seiberg, N., Witten, E.: Comments on string dynamics in six dimensions. Nucl. Phys. B471, 121-134 (1996). arXiv:hep-th/9603003 · Zbl 1003.81535
[69] Morrison, D.R., Vafa, C.: Compactifications of F-theory on Calabi-Yau threefolds—I. Nucl. Phys. B473, 74-92 (1996). arXiv:hep-th/9602114 · Zbl 0925.14005
[70] Morrison, D.R., Vafa, C.: Compactifications of F-theory on Calabi-Yau threefolds—II. Nucl. Phys. B476, 437-469 (1996). arXiv:hep-th/9603161 · Zbl 0925.14007
[71] Witten, E.: Phase transitions in M-theory and F-theory. Nucl. Phys. B471, 195-216 (1996). arXiv:hep-th/9603150 · Zbl 1003.81537
[72] Klemm, A., Mayr, P., Vafa, C.: BPS states of exceptional non-critical strings. Nucl. Phys. B Proc. Suppl 58, 177-194 (1997). arXiv:hep-th/9607139 · Zbl 0976.81503
[73] Lerche, W., Mayr, P., Warner, N.P.: Non-critical strings, del Pezzo singularities and Seiberg- Witten curves. Nucl. Phys. B499, 125-148 (1997). arXiv:hep-th/9612085 · Zbl 0934.81036
[74] Minahan, J.A., Nemeschansky, D., Warner, N.P.: Investigating the BPS spectrum of non-critical E(n) strings. Nucl. Phys. B508, 64-106 (1997). arXiv:hep-th/9705237 · Zbl 0925.81282
[75] Bruenier J., van der Geer G., Harder G., Zagier D.: The 1-2-3 of Modular Forms. Springer, Universitext, Berlin (2008) · Zbl 1197.11047 · doi:10.1007/978-3-540-74119-0
[76] Yau, S.-T., Zaslow, E.: BPS states, string duality, and nodal curves on K3. Nucl. Phys. B471, 503-512 (1996). arXiv:hep-th/9512121 · Zbl 0964.81521
[77] Klemm, A., Maulik, D., Pandharipande, R., Scheidegger, E.: Noether-lefschetz theory and the yau-zaslow conjecture. J. Am. Math. Soc. 23, 1013-1040 (2010). arXiv:0807.2477 · Zbl 1207.14057
[78] Hosono, S., Saito, M.H., Takahashi, A.: Holomorphic anomaly equation and BPS state counting of rational elliptic surface. Adv. Theor. Math. Phys. 3, 177-208 (1999). arXiv:hep-th/9901151 · Zbl 1062.14504
[79] Hosono, S.: Counting BPS states via holomorphic anomaly equations (2002). arXiv:hep-th/0206206 · Zbl 1046.81086
[80] Diaconescu, E., Moore, G.W.: Crossing the wall: branes vs. bundles. Adv. Theor. Math. Phys. 14(6), 1621-1650. (2010) arXiv:0706.3193 · Zbl 1245.81159
[81] Maruyama M.: Moduli of stable sheaves. II. J. Math. Kyoto Univ. 18, 557 (1977) · Zbl 0395.14006
[82] Manschot, J.: BPS invariants of \[{{\mathcal{N}}=4}N=4\] gauge theory on Hirzebruch surfaces. Commun. Number Theory Phys. 6, 497-516 (2012). arXiv:1103.0012 · Zbl 1270.81138
[83] Yoshioka, K.: The betti numbers of the moduli space of stable sheaves of rank 2 on a ruled surface. Math. Ann (1995) · Zbl 0828.14006
[84] Yoshioka, K.: The chamber structure of polarizations and the moduli of stable sheaves on a ruled surface. Int. J. Math. 7, 411-431 (1996). alg-geom/9409008 · Zbl 0883.14016
[85] Li, W.-P., Qin, Z.: On blowup formulae for the s-duality conjecture of Vafa and Witten. Invent. Math. 136, 451-482 (1999). math/9805054 · Zbl 0952.14036
[86] Yoshioka, K.: The betti numbers of the moduli space of stable sheaves of rank 2 on \[{{\mathbbm{P}}^2}\] P2. J. Reine Angew. Math 453 (1994) · Zbl 0806.14017
[87] Kool, M.: Euler characteristics of moduli spaces of torsion free sheaves on toric surfaces. Geometriae Dedicata 176(1), 241-269 (2015). arXiv:0906.3393 · Zbl 1331.14023
[88] Yoshioka K.: Euler characteristics of su(2) instanton moduli spaces on rational elliptic surfaces. Commun. Math. Phys. 205, 501-517 (1999) · Zbl 1191.35189 · doi:10.1007/s002200050687
[89] Yoshioka, K.: Chamber structure of polarizations and the moduli of stable sheaves on a ruled surface. Int. J. Math. 07, 411 (1996). alg-geom/9409008 · Zbl 0883.14016
[90] Klyachko A.: Moduli of vector bundels and numbers of classes. Funct. Anal. Appl. 25, 67-68 (1991) · Zbl 0731.14009 · doi:10.1007/BF01090685
[91] Zwegers, S.P.: Mock modular forms. In: Talk Given at the Conference “Partitions, q-series and modular forms”, University of Florida, Gainesville, March 12-16, 2008; Talk available under http://mathsci.ucd.ie/zwegers/presentations/001.pdf
[92] Sen, A.: Walls of marginal stability and dyon spectrum in N = 4 supersymmetric string theories. JHEP 05, 039 (2007). arXiv:hep-th/0702141
[93] Cheng, M.C.N., Verlinde, E.: Dying dyons don’t count. JHEP 09, 070 (2007). arXiv:0706.2363 · Zbl 0924.57025
[94] Nekrasov, N.A.: Seiberg-Witten prepotential from instanton counting. Adv. Theor. Math. Phys. 7, 831-864 (2004). arXiv:hep-th/0206161 · Zbl 1056.81068
[95] Iqbal, A., Kozcaz, C., Vafa, C.: The refined topological vertex. JHEP 10, 069 (2009). arXiv:hep-th/0701156 · Zbl 1207.81123
[96] Gottsche, L., Nakajima, H., Yoshioka, K.: K-theoretic donaldson invariants via instanton counting. math/0611945 · Zbl 1192.14011
[97] Ooguri, H., Strominger, A., Vafa, C.: Black hole attractors and the topological string. Phys. Rev. D70 106007 (2004). arXiv:hep-th/0405146
[98] Aganagic, M., Ooguri, H., Vafa, C., Yamazaki, M.: Wall Crossing and M-theory. Publ. Res. Inst. Math. Sci. Kyoto 47, 569 (2011). arXiv:0908.1194 · Zbl 1230.14084
[99] David, J.R., Jatkar, D.P., Sen, A.: Dyon spectrum in N = 4 supersymmetric type II string theories. JHEP 11, 073 (2006). arXiv:hep-th/0607155
[100] Alexeev V., Nikulin V.V.: Del Pezzo and k3 surfaces. Math. Soc. Jpn. Mem. 15, 1-164 (2006) · Zbl 1097.14001
[101] Aspinwall, P.S.: D-branes on Calabi-Yau manifolds (2003). arXiv:hep-th/0403166 · Zbl 1084.81058
[102] Green, M.B., Harvey, J.A., Moore, G.W.: I-brane inflow and anomalous couplings on D-branes. Class. Quantum Gravity 14, 47-52 (1997). arXiv:hep-th/9605033 · Zbl 0867.53063
[103] Witten, E.: D-branes and K-theory. JHEP 12, 019 (1998). arXiv:hep-th/9810188 · Zbl 0959.81070
[104] Diaconescu, D.-E., Romelsberger, C.: D-branes and bundles on elliptic fibrations. Nucl. Phys. B574, 245-262 (2000). arXiv:hep-th/9910172 · Zbl 1056.81546
[105] Douglas, M.R.: D-branes, categories and N = 1 supersymmetry. J. Math. Phys. 42, 2818-2843 (2001). arXiv:hep-th/0011017 · Zbl 1036.81027
[106] Douglas, M.R., Fiol, B., Romelsberger, C.: Stability and BPS branes. JHEP 09, 006 (2005). arXiv:hep-th/0002037
[107] Manschot, J.: On the space of elliptic genera. Commun. Number Theory Phys. 2, 803-833 (2008). arXiv:0805.4333 · Zbl 1214.58009
[108] Hecht, M.: Black Holes in M-theory, BPS states and modularity. Diploma thesis at the Ludwig-Maximilians University of Munich (2008)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.