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Enhanced gauge symmetry in type II string theory. (English) Zbl 0925.81188
Summary: We show how enhanced gauge symmetry in type II string theory compactified on a Calabi-Yau threefold arises from singularities in the geometry of the target space. When the target space of the type IIA string acquires a genus $$g$$ curve $$C$$ of $$A_{N-1}$$ singularities, we find that an $$SU(N)$$ gauge theory with $$g$$ adjoint hypermultiplets appears at the singularity. The new massless states correspond to solitons wrapped about the collapsing cycles, and their dynamics is described by a twisted supersymmetric gauge theory on $$C\times \mathbb{R}^{4}$$. We reproduce this result from an analysis of the $$S$$-dual $$D$$-manifold. We check that the predictions made by this model about the nature of the Higgs branch, the monodromy of period integrals, and the asymptotics of the one-loop topological amplitude are in agreement with geometrical computations. In one of our examples we find that the singularity occurs at strong coupling in the heterotic dual proposed by Kachru and Vafa.

##### MSC:
 81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory 32J81 Applications of compact analytic spaces to the sciences 32S30 Deformations of complex singularities; vanishing cycles 83E30 String and superstring theories in gravitational theory
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##### References:
 [1] Hull, C.M.; Townsend, P.K., Unity of superstring dualities, Nucl. phys. B, 438, 109-137, (1995), [hep-th/9410167] · Zbl 1052.83532 [2] Ceresole, A.; D’Auria, R.; Ferrara, S.; Van Proyen, A., Duality transformations in supersymmetric Yang-Mills theories coupled to supergravity, Nucl. phys. B, 444, 92, (1995), [hep-th/9502072] · Zbl 0990.81736 [3] Witten, E., String theory dynamics in various dimensions, Nucl. phys. B, 443, 85, (1995), [hepth/9503124] · Zbl 0990.81663 [4] Harvey, J.A.; Strominger, A., The heterotic string is a soliton, Nucl. phys. B, 449, 535, (1995), [hep-th/9504047] · Zbl 1076.81555 [5] Aspinwall, P.S., Enhanced gauge symmetries and K3 surfaces, Phys. lett. B, 357, 329, (1995), [hepth/9507012] [6] E. Witten, Some comments on string dynamics, Proc. Strings ’95 [hep-th/9507121]. · Zbl 1003.81535 [7] D. Kutasov, Orbifolds and solitons [hep-th/9512145]. [8] Kachru, S.; Vafa, C., Exact results for N = 2 compactifications of heterotic strings, Nucl. phys. B, 450, 69, (1995), [hep-th/9505105] · Zbl 0957.14509 [9] Ferrara, S.; Harvey, J.A.; Strominger, A.; Vafa, C., Second-quantized mirror symmetry, Phys. lett. B, 361, 59, (1995), [hep-th/9505162] · Zbl 0899.32012 [10] Kachru, S.; Klemm, A.; Lerche, W.; Mayr, P.; Vafa, C., Non-perturbative results on the point particle limit of N=2 heterotic string compactifications, Nucl. phys. B, 459, 537, (1996), [hep-th/9508155] · Zbl 1003.81524 [11] Aspinwall, P.S., An N = 2 dual pair and a phase transition, Nucl. phys. B, 460, 57, (1996), [hepth/9510142] · Zbl 1003.81525 [12] Bershadsky, M.; Sadov, V.; Vafa, C., D-strings on D-manifolds, Nucl. phys. B, 463, 398, (1996), [hep-th/9510225] · Zbl 1004.81535 [13] Aspinwall, P.S., Enhanced gauge symmetries and Calabi-Yau threefolds, Phys. lett. B, 371, 231, (1996), [hep-th/9511171] [14] Strominger, A., Massless black holes and conifolds in string theory, Nucl. phys. B, 451, 97, (1995), [hep-th/9504090] · Zbl 0925.83071 [15] Greene, B.R.; Morrison, D.R.; Strominger, A., Black hole condensation and the unification of string vacua, Nucl. phys. B, 451, 109, (1995), [hep-th/9504145] · Zbl 0908.53041 [16] Y. Hayakawa, Degeneration of Calabi-Yau manifold with Weil-Petersson metric [alg-geom/9507016]. [17] Reid, M., Canonical 3-folds, (), 273-310 [18] P. Berglund, S. Katz and A. Klemm, Extremal transitions between dual N = 2 string models I hepth/9605154]. · Zbl 0925.14012 [19] Ooguri, H.; Vafa, C., Two-dimensional black hole and singularities of CY manifolds, Nucl. phys. B, 463, 55, (1996), [hep-th/9511164] · Zbl 1003.83511 [20] Bershadsky, M.; Sadov, V.; Vafa, C., D-branes and topological field theories, Nucl. phys. B, 463, 420, (1996), [hep-th/9511222] · Zbl 1004.81560 [21] Aspinwall, P.S.; Morrison, D.R., U-duality and integral structures, Phys. lett. B, 355, 141, (1995), [hep-th/9505025] [22] Morrison, D.R., Mirror symmetry and the type II string, (), 146, [hep-th/9512016] · Zbl 0957.81596 [23] Becker, K.; Becker, M.; Strominger, A., Fivebranes, membranes and non-perturbative string theory, Nucl. phys. B, 456, 130, (1995), [hep-th/9507158] · Zbl 0925.81161 [24] Witten, E., Phases of N = 2 theories in two dimensions, Nucl. phys. B, 403, 159, (1993), [hepth/9301042] · Zbl 0910.14020 [25] Aspinwall, P.S.; Greene, B.R.; Morrison, D.R., Calabi-Yau moduli space, mirror manifolds and spacetime topology change in string theory, Nucl. phys. B, 416, 414, (1994), [hep-th/9309097] · Zbl 0899.32006 [26] Witten, E., Topological quantum field theory, Commun. math. phys., 117, 353, (1988) · Zbl 0656.53078 [27] Duff, M., Strong/weak coupling duality from the dual string, Nucl. phys. B, 442, 47, (1995), [hepth/9501030] · Zbl 0990.81661 [28] Seiberg, N., Supersymmetry and non-perturbative beta functions, Phys. lett. B, 206, 75, (1988) [29] Strominger, A., Special geometry, Commun. math. phys., 133, 163, (1990) · Zbl 0716.53068 [30] Seiberg, N.; Witten, E., Electromagnetic duality, monopole condensation and confinement in N = 2 supersymmetric Yang-Mills theory, Nucl. phys. B, 426, 19, (1994), [hep-th/9407087] · Zbl 0996.81510 [31] Vafa, C., A stringy test of the fate of the conifold, Nucl. phys. B, 447, 252, (1995), [hep-th/9505023] · Zbl 1009.81542 [32] Witten, E., Bound states of strings and p-branes, Nucl. phys. B, 460, 335, (1996), [hep-th/9510135] · Zbl 1003.81527 [33] A. Strominger, Open p-branes [hep-th/95120591. [34] Greene, B.R.; Shapere, A.; Vafa, C.; Yau, S.-T., Stringy cosmic strings, Nucl. phys. B, 337, 1, (1990) · Zbl 0744.53045 [35] P.S. Aspinwall and D.R. Morrison, String theory on K3 surfaces, in: Essays on Mirror Manifolds II, B.R. Greene and S.-T. Yau, eds., to appear [hep-th/9404151]. · Zbl 0931.14020 [36] Callan, C.; Harvey, J.; Strominger, A., Supersymmetric string solitons, (), 208, [hep-th/9112030] [37] Gimon, E.G.; Polchinski, J., Consistency conditions for orientifolds and D-manifolds, Phys. rev. D, 54, 1667, (1996), [hep-th/9601038] [38] Kodaira, K.; Kodaira, K., On compact analytic surfaces III, Ann. math., Ann. math., 78, 1, (1963) · Zbl 0171.19601 [39] Batyrev, V.V., Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in tonic varieties, J. alg. geom., 3, 493, (1994), [alg-geom/9310003] · Zbl 0829.14023 [40] Candelas, P.; de la Ossa, X.; Font, A.; Katz, S.; Morrison, D.R., Mirror symmetry for two parameter models (1), Nucl. phys. B, 416, 481, (1994), [hep-th/9308083] · Zbl 0899.14017 [41] C. Gómez, R. Hernández and E. López, S-duality and the Calabi-Yau interpretation of the N = 4 to N = 2 flow [hep-th/9512017]. [42] Gel’fand, I.M.; Zelevinskil, A.V.; Kapranov, M.M.; Gel’fand, I.M.; Zelevinskil, A.V.; Kapranov, M.M., Hypergeometric functions and toral (tonic) manifolds, Func. anal. appl., Funk. anal. pril., 23, 94, (1989), [Russian] · Zbl 0721.33006 [43] Aspinwall, P.S.; Greene, B.R.; Morrison, D.R., Measuring small distances in N = 2 sigma models, Nucl. phys. B, 420, 184, (1994), [hep-th/9311042] · Zbl 0990.81689 [44] Candelas, P.; de la Ossa, X.C.; Green, P.S.; Parkes, L., A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory, Nucl. phys. B, 359, 2l, (1991) [45] Aspinwall, P.S.; Morrison, D.R., Topological field theory and rational curves, Commun. math. phys. 151, 245, (1993), [hep-th/9110048] · Zbl 0776.53043 [46] Morrison, D.R., Beyond the Kähler cone, (), 361-376, [alg-geom/9407007] · Zbl 0847.32034 [47] Burns, D.; Rapoport, M., On the Torelli problem for Kählerian K3 surfaces, Ann. sci. école norm. sup., 8, 235, (1975) · Zbl 0324.14008 [48] Wilson, P.M.H., The Kähler cone on Calabi-Yau threefolds, Inv. math., 107, 561, (1992) · Zbl 0766.14035 [49] Morrison, D.R., Through the looking Glass, (), to appear · Zbl 0935.32020 [50] Namikawa, Y.; Steenbrink, J.H.M., Global smoothing of Calabi-Yau threefolds, Inv. math. 122, 403, (1995) · Zbl 0861.14036 [51] M. Gross, Deforming Calabi-Yau threefolds [alg-geom/9506022]. · Zbl 0935.14026 [52] M. Gross, Primitive Calabi-Yau threefolds [alg-geom/9512002]. [53] Clemens, C.H.; Griffiths, P.A., The intermediate Jacobian of the cubic threefold, Ann. math., 5, 281, (1972) · Zbl 0214.48302 [54] Bourbaki, N., Groupes et algèbres de Lie, (1968), Hermann Paris · Zbl 0186.33001 [55] Zerger, T.E., Contracting rational curves on smooth complex threefolds, () [56] Berglund, P.; Katz, S.; Klemm, A., Mirror symmetry and the moduli space for generic hypersurfaces in tonic varieties, Nucl. phys. B, 456, 153, (1995), [hep-th/9506091] · Zbl 0899.32007 [57] D.R. Morrison, Making enumerative predictions by means of mirror symmetry, Essays on Mirror Manifolds II, B.R. Greene and S.-T. Yau, eds., to appear [alg-geom/9504013]. · Zbl 0932.14021 [58] Bershadsky, M.; Cecotti, S.; Ooguri, H.; Vafa, C., Holomorphic anomalies in topological field theories, (), 279, [hep-th/9302103] · Zbl 0908.58074 [59] Hosono, S.; Klemm, A.; Theisen, S.; Yau, S.-T., Mirror symmetry, mirror map and applications to Calabi-Yau hypersurfaces, Common. math. phys., 167, 301, (1995), [hep-th/9308122] · Zbl 0814.53056 [60] Hosono, S.; Klemm, A.; Theisen, S.; Yau, S.-T., Mirror symmetry, mirror map and applications to complete intersection Calabi-Yau spaces, Nucl. phys. B, 433, 501, (1995), [hep-th/9406055] · Zbl 1020.32508 [61] Aspinwall, P.S.; Greene, B.R.; Morrison, D.R., The monomial-divisor mirror map, Int. math. res. notices, 319, (1993), [alg-geom/9309007] · Zbl 0798.14030 [62] Aspinwall, P.S.; Louis, J., On the ubiquity of K3 fibrations in string duality, Phys. lett. B, 369, 233, (1996), [hep-th/9510234] [63] Kapranov, M.M., A characterization of A-discriminantal hypersurfaces in terms of the logarithmic Gauss map, Math. ann., 290, 277, (1991) · Zbl 0714.14031 [64] Morrison, D.R.; Plesser, M.R., Summing the instantons: quantum cohomology and mirror symmetry in toric varieties, Nucl. phys. B, 440, 279, (1995), [hep-th/9412236] · Zbl 0908.14014 [65] Klemm, A.; Mayr, P., Strong coupling singularities and non-abelian gauge symmetries in N = 2 string theory, Nucl. phys. B, 469, 37, (1996), [hep-th/9601014] · Zbl 1002.81536
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