Abstract
We investigate quantum corrections to the hypermultiplet moduli space \( \mathcal{M} \) in Calabi-Yau compactifications of type II string theories, with particular emphasis on instanton effects from Euclidean NS5-branes. Based on the consistency of D- and NS5-instanton corrections, we determine the topology of the hypermultiplet moduli space at fixed string coupling, as previewed in [1]. On the type IIB side, we compute corrections from (p, k)-fivebrane instantons to the metric on \( \mathcal{M} \) (specifically, the correction to the complex contact structure on its twistor space \( \mathcal{Z} \)) by applying S-duality to the D-instanton sum. For fixed fivebrane charge k, the corrections can be written as a non-Gaussian theta series, whose summand for k = 1 reduces to the topological A-model amplitude. By mirror symmetry, instanton corrections induced from the chiral type IIA NS5-brane are similarly governed by the wave function of the topological B-model. In the course of this investigation we clarify charge quantization for coherent sheaves and find hitherto unnoticed corrections to the Heisenberg, monodromy and S-duality actions on \( \mathcal{M} \), as well as to the mirror map for Ramond-Ramond fields and D-brane charges.
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References
S. Alexandrov, D. Persson and B. Pioline, On the topology of the hypermultiplet moduli space in type-II/CY string vacua, Phys. Rev. D 83 (2011) 026001 [arXiv:1009.3026] [SPIRES].
M. Günaydin, A. Neitzke, B. Pioline and A. Waldron, BPS black holes, quantum attractor flows and automorphic forms, Phys. Rev. D 73 (2006) 084019 [hep-th/0512296] [SPIRES].
K. Becker, M. Becker and A. Strominger, Five-branes, membranes and nonperturbative string theory, Nucl. Phys. B 456 (1995) 130 [hep-th/9507158] [SPIRES].
J. Bagger and E. Witten, Matter Couplings in \( \mathcal{N} = 2 \) Supergravity, Nucl. Phys. B 222 (1983) 1 [SPIRES].
C.G. Callan Jr., J.A. Harvey and A. Strominger, Worldbrane actions for string solitons, Nucl. Phys. B 367 (1991) 60 [SPIRES].
S. Cecotti, S. Ferrara and L. Girardello, Geometry of Type II Superstrings and the Moduli of Superconformal Field Theories, Int. J. Mod. Phys. A 4 (1989) 2475 [SPIRES].
S. Ferrara and S. Sabharwal, Quaternionic Manifolds for Type II Superstring Vacua of Calabi-Yau Spaces, Nucl. Phys. B 332 (1990) 317 [SPIRES].
D.R. Morrison, Mirror symmetry and the type-II string, Nucl. Phys. Proc. Suppl. 46 (1996) 146 [hep-th/9512016] [SPIRES].
M. Roček, C. Vafa and S. Vandoren, Quaternion-Kähler spaces, hyperKähler cones and the c-map, math/0603048. MATH/0603048;
A. Neitzke, B. Pioline and S. Vandoren, Twistors and Black Holes, JHEP 04 (2007) 038 [hep-th/0701214] [SPIRES].
S. Alexandrov, B. Pioline, F. Saueressig and S. Vandoren, Linear perturbations of quaternionic metrics. I. The HyperKähler case, Lett. Math. Phys. 87 (2009) 225 [arXiv:0806.4620] [SPIRES].
S. Alexandrov, B. Pioline, F. Saueressig and S. Vandoren, Linear perturbations of quaternionic metrics. II. The quaternionic-Kähler case, Commun. Math. Phys. 296 (2010) 353 [arXiv:0810.1675] [SPIRES].
I. Antoniadis, S. Ferrara, R. Minasian and K.S. Narain, R 4 couplings in M-and type-II theories on Calabi-Yau spaces, Nucl. Phys. B 507 (1997) 571 [hep-th/9707013] [SPIRES].
H. Günther, C. Herrmann and J. Louis, Quantum corrections in the hypermultiplet moduli space, Fortsch. Phys. 48 (2000) 119 [hep-th/9901137] [SPIRES].
I. Antoniadis, R. Minasian, S. Theisen and P. Vanhove, String loop corrections to the universal hypermultiplet, Class. Quant. Grav. 20 (2003) 5079 [hep-th/0307268] [SPIRES].
D. Robles-Llana, F. Saueressig and S. Vandoren, String loop corrected hypermultiplet moduli spaces, JHEP 03 (2006) 081 [hep-th/0602164] [SPIRES].
S. Alexandrov, Quantum covariant c-map, JHEP 05 (2007) 094 [hep-th/0702203] [SPIRES].
S. Alexandrov, B. Pioline, F. Saueressig and S. Vandoren, D-instantons and twistors, JHEP 03 (2009) 044 [arXiv:0812.4219] [SPIRES].
S. Alexandrov, D-instantons and twistors: some exact results, J. Phys. A 42 (2009) 335402 [arXiv:0902.2761] [SPIRES].
S. Alexandrov and F. Saueressig, Quantum mirror symmetry and twistors, JHEP 09 (2009) 108 [arXiv:0906.3743] [SPIRES].
D. Robles-Llana, M. Roček, F. Saueressig, U. Theis and S. Vandoren, Nonperturbative corrections to 4D string theory effective actions from SL(2,Z) duality and supersymmetry, Phys. Rev. Lett. 98 (2007) 211602 [hep-th/0612027] [SPIRES].
F. Saueressig and S. Vandoren, Conifold singularities, resumming instantons and non-perturbative mirror symmetry, JHEP 07 (2007) 018 [arXiv:0704.2229] [SPIRES].
D. Robles-Llana, F. Saueressig, U. Theis and S. Vandoren, Membrane instantons from mirror symmetry, Commun. Num. Theor. Phys. 1 (2007) 681 [arXiv:0707.0838] [SPIRES].
D. Gaiotto, G.W. Moore and A. Neitzke, Four-dimensional wall-crossing via three-dimensional field theory, Commun. Math. Phys. 299 (2010) 163 [arXiv:0807.4723] [SPIRES].
M. Kontsevich and Y. Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, arXiv:0811.2435.
D. J oyce, Generalized Donaldson-Thomas invariants, arXiv:0910.0105 [SPIRES].
B. Pioline and S. Vandoren, Large D-instanton effects in string theory, JHEP 07 (2009) 008 [arXiv:0904.2303] [SPIRES].
E. Witten, Five-brane effective action in M-theory, J. Geom. Phys. 22 (1997) 103 [hep-th/9610234] [SPIRES].
L. Dolan and C.R. Nappi, A modular invariant partition function for the fivebrane, Nucl. Phys. B 530 (1998) 683 [hep-th/9806016] [SPIRES].
M. Henningson, B.E.W. Nilsson and P. Salomonson, Holomorphic factorization of correlation functions in (4k+2)-dimensional (2k)-form gauge theory, JHEP 09 (1999) 008 [hep-th/9908107] [SPIRES].
M. Henningson, The quantum Hilbert space of a chiral two-form in D = 5+1 dimensions, JHEP 03 (2002) 021 [hep-th/0111150] [SPIRES].
R. Dijkgraaf, E.P. Verlinde and M. Vonk, On the partition sum of the NS five-brane, hep-th/0205281 [SPIRES].
E. Diaconescu, G.W. Moore and D.S. Freed, The M-theory 3-form and E 8 gauge theory, hep-th/0312069 [SPIRES].
G.W. Moore, Anomalies, Gauss laws and page charges in M-theory, Comptes Rendus Physique 6 (2005) 251 [hep-th/0409158] [SPIRES].
D. Belov and G.W. Moore, Holographic action for the self-dual field, hep-th/0605038 [SPIRES].
D.M. Belov and G.W. Moore, Type II actions from 11-dimensional Chern-Simons theories, hep-th/0611020 [SPIRES].
I.A. Bandos et al., Covariant action for the super-five-brane of M-theory, Phys. Rev. Lett. 78 (1997) 4332 [hep-th/9701149] [SPIRES].
M. Aganagic, J. Park, C. Popescu and J.H. Schwarz, World-volume action of the M-theory five-brane, Nucl. Phys. B 496 (1997) 191 [hep-th/9701166] [SPIRES].
P.S. Howe, E. Sezgin and P.C. West, Covariant field equations of the M-theory five-brane, Phys. Lett. B 399 (1997) 49 [hep-th/9702008] [SPIRES].
I.A. Bandos, A. Nurmagambetov and D.P. Sorokin, The type IIA NS5-brane, Nucl. Phys. B 586 (2000) 315 [hep-th/0003169] [SPIRES].
M. Cederwall, B.E.W. Nilsson and P. Sundell, An action for the super-5-brane in D = 11 supergravity, JHEP 04 (1998) 007 [hep-th/9712059] [SPIRES].
E. Sezgin and P. Sundell, Aspects of the M5-brane, hep-th/9902171 [SPIRES].
M. Berkooz, M. Rozali and N. Seiberg, Matrix description of M-theory on T 4 and T 5, Phys. Lett. B 408 (1997) 105 [hep-th/9704089] [SPIRES].
N. Seiberg, New theories in six dimensions and matrix description of M-theory on T 5 and \( {{{{T^5}}} \left/ {{{\mathbb{Z}_2}}} \right.} \), Phys. Lett. B 408 (1997) 98 [hep-th/9705221] [SPIRES].
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] [SPIRES].
M. Mariño, R. Minasian, G.W. Moore and A. Strominger, Nonlinear instantons from supersymmetric p-branes, JHEP 01 (2000) 005 [hep-th/9911206] [SPIRES].
L. Bao, M. Cederwall and B.E.W. Nilsson, A note on topological M5-branes and string-fivebrane duality, JHEP 06 (2008) 100 [hep-th/0603120] [SPIRES].
V. Bengtsson and M. Cederwall, The B-model on the A-model NS5-brane, arXiv:0805.1782 [SPIRES].
M. de Vroome and S. Vandoren, Supergravity description of spacetime instantons, Class. Quant. Grav. 24 (2007) 509 [hep-th/0607055] [SPIRES].
J.D. Blum and J.A. Harvey, Anomaly inflow for gauge defects, Nucl. Phys. B 416 (1994) 119 [hep-th/9310035] [SPIRES].
M.J. Duff, J.T. Liu and R. Minasian, Eleven-dimensional origin of string/string duality: A one-loop test, Nucl. Phys. B 452 (1995) 261 [hep-th/9506126] [SPIRES].
L. Bao, A. Kleinschmidt, B.E.W. Nilsson, D. Persson and B. Pioline, Instanton Corrections to the Universal Hypermultiplet and Automorphic Forms on SU(2, 1), Commun. Num. Theor. Phys. 4 (2010) 187 [arXiv:0909.4299] [SPIRES].
L. Bao, A. Kleinschmidt, B.E.W. Nilsson, D. Persson and B. Pioline, Rigid Calabi-Yau threefolds, Picard Eisenstein series and instantons, arXiv:1005.4848 [SPIRES].
B. Pioline and D. Persson, The automorphic NS5-brane, Commun. Num. Theor. Phys. 3 (2009) 697 [arXiv:0902.3274] [SPIRES].
P. Candelas, X.C. De La Ossa, P.S. Green and L. Parkes, A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, Nucl. Phys. B 359 (1991) 21 [SPIRES].
A. Klemm and S. Theisen, Considerations of one modulus Calabi-Yau compactifications: Picard-Fuchs equations, Kähler potentials and mirror maps, Nucl. Phys. B 389 (1993) 153 [hep-th/9205041] [SPIRES].
P. Candelas, X. De La Ossa, A. Font, S.H. Katz and D.R. Morrison, Mirror symmetry for two parameter models. I, Nucl. Phys. B 416 (1994) 481 [hep-th/9308083] [SPIRES].
P. Candelas, A. Font, S.H. Katz and D.R. Morrison, Mirror symmetry for two parameter models. 2, Nucl. Phys. B 429 (1994) 626 [hep-th/9403187] [SPIRES].
A. Kapustin, Gauge theory, topological strings and S-duality, JHEP 09 (2004) 034 [hep-th/0404041] [SPIRES].
D. Maulik, N. Nekrasov, A. Okounkov and R. Pandharipande, Gromov-Witten theory and Donaldson-Thomas theory. I, Compos. Math. 142 (2006) 1263.
D. Maulik, N. Nekrasov, A. Okounkov, and R. Pandharipande, Gromov-Witten theory and Donaldson-Thomas theory. II, Compos. Math. 142 (2006) 1286 [math/0406092].
S. Alexandrov, B. Pioline and S. Vandoren, Self-dual Einstein Spaces, Heavenly Metrics and Twistors, J. Math. Phys. 51 (2010) 073510 [arXiv:0912.3406] [SPIRES].
L. Álvarez-Gaumé, G.W. Moore, P.C. Nelson, C. Vafa and J.b. Bost, Bosonization in arbitrary genus, Phys. Lett. B 178 (1986) 41 [SPIRES].
L. Álvarez-Gaumé, J.B. Bost, G.W. Moore, P.C. Nelson and C. Vafa, Bosonization on higher genus Riemann surfaces, Commun. Math. Phys. 112 (1987) 503 [SPIRES].
C. Birkenhake and H. Lange, Fundamental Principles of Mathematical Sciences. Vol. 302: Complex abelian varieties, second edition, Springer-Verlag, Berlin Germany (2004).
E.P. Verlinde, Attractors and the holomorphic anomaly, hep-th/0412139 [SPIRES].
M. Günaydin, A. Neitzke and B. Pioline, Topological wave functions and heat equations, JHEP 12 (2006) 070 [hep-th/0607200] [SPIRES].
E. Witten, Quantum background independence in string theory, hep-th/9306122 [SPIRES].
M. Aganagic, V. Bouchard and A. Klemm, Topological Strings and (Almost) Modular Forms, Commun. Math. Phys. 277 (2008) 771 [hep-th/0607100] [SPIRES].
A. Schwarz and X. Tang, Quantization and holomorphic anomaly, JHEP 03 (2007) 062 [hep-th/0611281] [SPIRES].
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] [SPIRES].
A. Dabholkar, F. Denef, G.W. Moore and B. Pioline, Precision counting of small black holes, JHEP 10 (2005) 096 [hep-th/0507014] [SPIRES].
F. Denef and G.W. Moore, Split states, entropy enigmas, holes and halos, hep-th/0702146 [SPIRES].
T. Eguchi, P.B. Gilkey and A.J. Hanson, Gravitation, Gauge Theories and Differential Geometry, Phys. Rept. 66 (1980) 213 [SPIRES].
S. Sethi, C. Vafa and E. Witten, Constraints on low-dimensional string compactifications, Nucl. Phys. B 480 (1996) 213 [hep-th/9606122] [SPIRES].
I. Brunner and R. Schimmrigk, F-theory on Calabi-Yau fourfolds, Phys. Lett. B 387 (1996) 750 [hep-th/9606148] [SPIRES].
J.A. Harvey and G.W. Moore, Superpotentials and membrane instantons, hep-th/9907026 [SPIRES].
D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing, Hitchin Systems and the WKB Approximation, arXiv:0907.3987 [SPIRES].
E. Witten, Non-Perturbative Superpotentials In String Theory, Nucl. Phys. B 474 (1996) 343 [hep-th/9604030] [SPIRES].
M. Atiyah, The logarithm of the Dedekind η-function, Math. Ann. 278 (1987) 335.
S. Monnier, Geometric quantization and the metric dependence of the self-dual field, arXiv:1011.5890 [SPIRES].
A. Baarsma, J. Stienstra, T. van der Aalst and S. Vandoren, Special quaternionic manifolds from the variation of Hodge structures, in preparation.
R. Minasian and G.W. Moore, K-theory and Ramond-Ramond charge, JHEP 11 (1997) 002 [hep-th/9710230] [SPIRES].
M.R. Douglas, R. Reinbacher and S.-T. Yau, Branes, bundles and attractors: Bogomolov and beyond, math/0604597. MATH/0604597;
S. Hosono, A. Klemm and S. Theisen, Lectures on mirror symmetry, hep-th/9403096 [SPIRES].
J.M. Maldacena, A. Strominger and E. Witten, Black hole entropy in M-theory, JHEP 12 (1997) 002 [hep-th/9711053] [SPIRES].
J. de Boer, M.C.N. Cheng, R. Dijkgraaf, J. Manschot and E. Verlinde, A farey tail for attractor black holes, JHEP 11 (2006) 024 [hep-th/0608059] [SPIRES].
D. Gaiotto, A. Strominger and X. Yin, The M5-brane elliptic genus: Modularity and BPS states, JHEP 08 (2007) 070 [hep-th/0607010] [SPIRES].
P.S. Aspinwall, D-branes on Calabi-Yau manifolds, hep-th/0403166 [SPIRES].
D. Gaiotto, A. Strominger and X. Yin, New Connections Between 4D and 5D Black Holes, JHEP 02 (2006) 024 [hep-th/0503217] [SPIRES].
D. Gaiotto, A. Strominger and X. Yin, 5D black rings and 4D black holes, JHEP 02 (2006) 023 [hep-th/0504126] [SPIRES].
P. Gao and B. Pioline, Topological wave functions and the 4D-5D lift, JHEP 07 (2008) 087 [arXiv:0803.0562] [SPIRES].
R. Böhm, H. Günther, C. Herrmann and J. Louis, Compactification of type IIB string theory on Calabi-Yau threefolds, Nucl. Phys. B 569 (2000) 229 [hep-th/9908007] [SPIRES].
J. Manschot, On the space of elliptic genera, Commun. Num. Theor. Phys. 2 (2008) 803 [arXiv:0805.4333] [SPIRES].
J. Manschot, Stability and duality in N = 2 supergravity, Commun. Math. Phys. 299 (2010) 651 [arXiv:0906.1767] [SPIRES].
M. Chiodaroli and M. Gutperle, Instantons and Wormholes in N = 2 supergravity, Phys. Rev. D 79 (2009) 085023 [arXiv:0901.1616] [SPIRES].
S.M. Salamon, Quaternionic Kähler manifolds, Invent. Math. 67 (1982) 143.
C. LeBrun, Quaternionic-Kähler manifolds and conformal geometry, Math. Ann. 284 (1989) 353.
C. LeBrun and S. Salamon, Strong rigidity of positive quaternion-Kähler manifolds, Invent. Math. 118 (1994) 109.
S. Alexandrov and P. Roche, TBA for non-perturbative moduli spaces, JHEP 06 (2010) 066 [arXiv:1003.3964] [SPIRES].
J. Manschot, Wall-crossing of D4-branes using flow trees, arXiv:1003.1570 [SPIRES].
J. Manschot, B. Pioline and A. Sen, Wall-crossing from Boltzmann black hole halos, arXiv:1011.1258 [SPIRES].
S. Alexandrov, J. Manschot and B. Pioline, S-Duality and D-Instantons, in progress.
N. Nekrasov, H. Ooguri and C. Vafa, S-duality and topological strings, JHEP 10 (2004) 009 [hep-th/0403167] [SPIRES].
J.A. Harvey and G.W. Moore, Exact gravitational threshold correction in the FHSV model, Phys. Rev. D 57 (1998) 2329 [hep-th/9611176] [SPIRES].
E. Witten, Duality relations among topological effects in string theory, JHEP 05 (2000) 031 [hep-th/9912086] [SPIRES].
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Alexandrov, S., Persson, D. & Pioline, B. Fivebrane instantons, topological wave functions and hypermultiplet moduli spaces. J. High Energ. Phys. 2011, 111 (2011). https://doi.org/10.1007/JHEP03(2011)111
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DOI: https://doi.org/10.1007/JHEP03(2011)111