×

zbMATH — the first resource for mathematics

Towards mirror symmetry as duality for two-dimensional abelian gauge theories. (English) Zbl 0957.81656
Summary: Superconformal sigma models with Calabi-Yau target spaces described as complete intersection subvarieties in toric varieties can be obtained as the low-energy limit of certain abelian gauge theories in two dimensions. We formulate mirror symmetry for this class of Calabi-Yau spaces as a duality in the abelian gauge theory, giving the explicit mapping relating the two Lagrangians. The duality relates inequivalent theories which lead to isomorphic theories in the low-energy limit. This formulation suggests that mirror symmetry could be derived using abelian duality. The application of duality in this context is complicated by the presence of nontrivial dynamics and the absence of a global symmetry. We propose a way to overcome these obstacles, leading to a more symmetric Lagrangian. The argument, however, fails to produce a derivation of the conjecture.

MSC:
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
32G81 Applications of deformations of analytic structures to the sciences
32J81 Applications of compact analytic spaces to the sciences
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Dixon, L.J., Some world-sheet properties of superstring compactifications, on orbifolds and otherwise, (), 67-126, Singapore, New Jersey, Hong Kong
[2] Lerche, W.; Vafa, C.; Warner, N.P., Chiral rings in N=2 superconformal theories, Nucl. phys., B324, 427-474, (1989)
[3] Candelas, P.; Lynker, M.; Schimmrigk, R., Calabi-Yau manifolds in weighted ℙ_{4}, Nucl. phys., B341, 383-402, (1990) · Zbl 0962.14029
[4] Greene, B.R.; Plesser, M.R., Duality in Calabi-Yau moduli space, Nucl. phys., B338, 15-37, (1990)
[5] Batyrev, V.V., Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. algebraic geom., 3, 493-535, (1994) · Zbl 0829.14023
[6] Borisov, L., Towards the mirror symmetry for Calabi-Yau complete intersections in toric Fano varieties, (1993), University of Michigan, Preprint
[7] Batyrev, V.V.; Borisov, L.A., Dual cones and mirror symmetry for generalized Calabi-Yau manifolds, To appear in “Essays on Mirror Manifolds II” · Zbl 0927.14019
[8] Witten, E., Phases of N=2 theories in two dimensions, Nucl. phys., B403, 159-222, (1993) · Zbl 0910.14020
[9] Silverstein, E.; Witten, E., Global U(1) R-symmetry and conformal invariance of (0,2) models, Phys. lett., 328B, 307-311, (1994)
[10] Morrison, D.R.; Plesser, M.R., Summing the instantons: quantum cohomology and mirror symmetry in toric varieties, Nucl. phys., B440, 279-354, (1995) · Zbl 0908.14014
[11] Candelas, P.; de la Ossa, X.; Katz, S., Mirror symmetry for Calabi-Yau hypersurfaces in weighted ℙ_{4} and extensions of Landau-Ginzburg theory, Nucl. phys., B450, 267-356, (1995) · Zbl 0896.14023
[12] Berglund, P.; Hübsch, T., A generalized construction of mirror manifolds, Nucl. phys., B393, 377-391, (1993) · Zbl 1245.14039
[13] Aspinwall, P.S.; Greene, B.R., On the geometric interpretation of N=2 superconformal theories, Nucl. phys., B437, 205-230, (1995) · Zbl 1052.32502
[14] Aspinwall, P.S.; Greene, B.R.; Morrison, D.R., Calabi-Yau moduli space, mirror manifolds and spacetime topology change in string theory, Nucl. phys., B416, 414-480, (1994) · Zbl 0899.32006
[15] Aspinwall, P.S.; Greene, B.R.; Morrison, D.R., The monomial-divisor mirror map, Internat. math. res. notices, 319-337, (1993) · Zbl 0798.14030
[16] Batyrev, V.V., Quantum cohomology rings of toric manifolds, (), 9-34, of Astérisque · Zbl 0806.14041
[17] S. Katz and D. R. Morrison, The Multinomial-Divisor Mirror Map, In preparation.
[18] Seiberg, N.; Witten, E.; Seiberg, N.; Witten, E., Electric-magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory, Nucl. phys., Nucl. phys., B430, 485-486, (1994), Erratum · Zbl 0996.81511
[19] Seiberg, N.; Witten, E., Monopoles, duality and chiral symmetry breaking in N=2 supersymmetric QCD, Nucl. phys., B431, 484-550, (1994) · Zbl 1020.81911
[20] Seiberg, N., Electric-magnetic duality in supersymmetric nonabelian gauge theories, Nucl. phys., B435, 129-146, (1995) · Zbl 1020.81912
[21] Gates, S.J.; Hull, C.M.; Roček, M., Twisted multiplets and new supersymmetric nonlinear sigma models, Nucl. phys., B248, 157-186, (1984)
[22] Giveon, A., Target space duality and stringy black holes, Modern phys. lett., A6, 2843-2854, (1991) · Zbl 1020.81842
[23] Roček, M.; Verlinde, E., Duality, quotients, and currents, Nucl. phys., B373, 630-646, (1992)
[24] Giveon, A.; Witten, E., Mirror symmetry as a gauge symmetry, Phys. lett., 332B, 44-50, (1994)
[25] Berglund, P.; Henningson, M., Landauginzhurg orbifolds, mirror symmetry and the elliptic genus, Nucl. phys., B433, 311-332, (1995) · Zbl 0899.58068
[26] Giveon, A.; Roček, M., Introduction to duality, To appear in “Essays on Mirror Manifolds II” · Zbl 1129.81341
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.