zbMATH — the first resource for mathematics

Nonabelian 2D gauge theories for determinantal Calabi-Yau varieties. (English) Zbl 1397.81378
Summary: The two-dimensional supersymmetric gauged linear sigma model (GLSM) with abelian gauge groups and matter fields has provided many insights into string theory on Calabi-Yau manifolds of a certain type: complete intersections in toric varieties. In this paper, we consider two GLSM constructions with nonabelian gauge groups and charged matter whose infrared CFTs correspond to string propagation on determinantal Calabi-Yau varieties, furnishing another broad class of Calabi-Yau geometries in addition to complete intersections. We show that these two models – which we refer to as the PAX and the PAXY model – are dual descriptions of the same low-energy physics. Using GLSM techniques, we determine the quantum Kähler moduli space of these varieties and find no disagreement with existing results in the literature.

81T60 Supersymmetric field theories in quantum mechanics
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
Full Text: DOI arXiv
[1] Witten, E., Phases of N = 2 theories in two-dimensions, Nucl. Phys., B 403, 159, (1993)
[2] Morrison, DR; Plesser, MR, Summing the instantons: quantum cohomology and mirror symmetry in toric varieties, Nucl. Phys., B 440, 279, (1995)
[3] J.-P. Serre, Sur les modules projectifs, Algèbre et théorie des nombres (Séminaire Dubreil)14, Secrétariat mathématique, Paris (1960-1961).
[4] Buchsbaum, DA; Eisenbud, D., Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension 3, Am. J. Math., 99, 447, (1977)
[5] Okonek, C., Notes on varieties of codimension 3 in P\^{}{N}, Manuscripta Math., 84, 421, (1994)
[6] Walter, CH, Pfaffian subschemes, J. Algebraic Geom, 5, 671, (1996)
[7] Tonoli, F., Construction of Calabi-Yau 3-folds in P\^{}{6}, J. Algebraic Geom., 13, 209, (2004)
[8] Kustin, A.; Miller, M., Structure theory for a class of grade four Gorenstein ideals, Trans. Amer. Math. Soc., 270, 287, (1982)
[9] Gulliksen, TH; Negard, OG, Un complexe résolvant pour certains idéaux déterminantiels, C. R. Acad. Sci. Paris Sér. A-B, 274, a16, (1972)
[10] J. Harris, Algebraic geometry, Graduate Texts in Mathematics133, Springer-Verlag, New York, U.S.A. (1992).
[11] Hori, K.; Tong, D., Aspects of non-abelian gauge dynamics in two-dimensional N =(2,2) theories, JHEP, 05, 079, (2007)
[12] E.A. Rødland, The Pfaffian Calabi-Yau, its Mirror and their link to the Grassmannian G(2\(,\) 7), Compositio Math.122 (2000) [math/9801092].
[13] E. Witten, The Verlinde algebra and the cohomology of the Grassmannian, in S.-T. Yau ed., Geometry, Topology, & Physics for Raoul Bott (Conference Proceedings and Lecture Notes in Geometry and Topology), vol. IV, Int. Press, Cambridge, MA, U.S.A. (1995), pg. 357-422 [hep-th/9312104] [INSPIRE].
[14] W. Lerche, P. Mayr and J. Walcher, A New kind of McKay correspondence from nonAbelian gauge theories, hep-th/0103114 [INSPIRE].
[15] Donagi, R.; Sharpe, E., GLSM’s for partial flag manifolds, J. Geom. Phys., 58, 1662, (2008)
[16] K. Hori, Duality In Two-Dimensional (2\(,\) 2) Supersymmetric Non-Abelian Gauge Theories, arXiv:1104.2853 [INSPIRE].
[17] K. Hori and J. Knapp, Phases of non-abelian gauged linear sigma models, lectures given at Workshop on Noncommutative Algebraic Geometry and D-branes, Simons Center for Geometry and Physics, 12-16 December 2011.
[18] M.-A. Bertin, Examples of Calabi-Yau 3-folds of P\^{}{7}with ρ = 1, math/0701511.
[19] G. Kapustka and M. Kapustka, A cascade of determinantal Calabi-Yau threefolds, arXiv:0802.3669.
[20] S. Hosono and H. Takagi, Mirror symmetry and projective geometry of Reye congruences I, arXiv:1101.2746 [INSPIRE].
[21] Seiberg, N., Electric-magnetic duality in supersymmetric nonabelian gauge theories, Nucl. Phys., B 435, 129, (1995)
[22] Horrocks, G.; Mumford, D., A rank 2 vector bundle on P\^{}{4} with 15,000 symmetries, Topology, 12, 63, (1973)
[23] Schoen, C., On the geometry of a special determinantal hypersurface associated to the Mumford-Horrocks vector bundle, J. Reine Angew. Math., 364, 85, (1986)
[24] M. Gross and S. Popescu, Calabi-Yau threefolds and moduli of Abelian surfaces. 1., math/0001089 [INSPIRE].
[25] W. Fulton, Intersection theory, Ergeb. Math. Grenzgeb.2 Springer-Verlag, Berlin, Germany (1984).
[26] A. Kanazawa, Pfaffian Calabi-Yau Threefolds and Mirror Symmetry, arXiv:1006.0223.
[27] E.N. Tjøtta, Quantum cohomology of a Pfaffian Calabi-Yau variety: verifying mirror symmetry predictions, math/9906119.
[28] Wall, CTC, Classification problems in differential topology. V. on certain 6-manifolds, Invent. Math., 1, 355, (1966)
[29] D. Berenstein and M.R. Douglas, Seiberg duality for quiver gauge theories, hep-th/0207027 [INSPIRE].
[30] Burch, L., On ideals of finite homological dimension in local rings, Proc. Cambridge Philos. Soc., 64, 941, (1968)
[31] W. Decker and S. Popescu, On surfaces inP\^{}{4}and 3-folds inP\^{}{5}, in Vector bundles in algebraic geometry, N.J. Hitchin, P. E.Newstead and W.M. Oxbury eds., London Math. Soc. Lecture Note Ser. volume 208, Cambridge University Press, Cambridge U.K (1995) [alg-geom/9402006].
[32] J.R. Sendra, F. Winkler and S. Pérez-D´ıaz, Rational algebraic curves: A computer algebra approach, Algorithms and Computation in Mathematics22, Springer, Berlin (2008).
[33] D. Hilbert, Theory of algebraic invariants (Translated from the German by R.C. Laubenbacher), Cambridge University Press, Cambridge, U.K. (1993).
[34] Witten, E., Constraints on supersymmetry breaking, Nucl. Phys., B 202, 253, (1982)
[35] P.M.H. Wilson, Flops, Type III contractions and Gromov-Witten invariants on Calabi-Yau threefolds, in New trends in algebraic geometry , K. Hulek, M. Reid, C. Peters and F. Catanese eds., London Math. Soc. Lecture Note Ser. volume 264, Cambridge University Press, Cambridge, U.K. (1999) [alg-geom/9707008].
[36] Căldăraru, A.; Distler, J.; Hellerman, S.; Pantev, T.; Sharpe, E., Non-birational twisted derived equivalences in abelian glsms, Commun. Math. Phys., 294, 605, (2010)
[37] Batyrev, VV, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Algebraic Geom., 3, 493, (1994)
[38] L.A. Borisov, Towards the mirror symmetry for Calabi-Yau complete intersections in Gorenstein toric Fano varieties [alg-geom/9310001].
[39] V.V. Batyrev and L.A. Borisov, On Calabi-Yau complete intersections in toric varieties, alg-geom/9412017 [INSPIRE].
[40] Batyrev, V.; Ciocan-Fontanine, I.; Kim, B.; Straten, D., Conifold transitions and mirror symmetry for Calabi-Yau complete intersections in Grassmannians, Nucl. Phys , B 514, 640, (1998)
[41] Batyrev, VV; Ciocan-Fontanine, I.; Kim, B.; Straten, D., Mirror symmetry and toric degenerations of partial flag manifolds, Acta Math., 184, 1, (2000)
[42] J. Böehm, Mirror symmetry and tropical geometry, arXiv:0708.4402.
[43] J. Böehm, A framework for tropical mirror symmetry, arXiv:1103.2673.
[44] Morrison, DR; Plesser, MR, Towards mirror symmetry as duality for two-dimensional abelian gauge theories, Nucl. Phys. Proc. Suppl., 46, 177, (1996)
[45] K. Hori and C. Vafa, Mirror symmetry, hep-th/0002222 [INSPIRE].
[46] K. Hori et al., Mirror symmetry, Clay Mathematics Monographs1, American Mathematical Society, Providence, RI, U.S.A. (2003).
[47] S. Hosono and Y. Konishi, Higher genus Gromov-Witten invariants of the Grassmannian and the Pfaffian Calabi-Yau threefolds, Adv. Theor. Math. Phys.13 (2009) [arXiv:0704.2928] [INSPIRE].
[48] Shimizu, M.; Suzuki, H., Open mirror symmetry for Pfaffian Calabi-Yau 3-folds, JHEP, 03, 083, (2011)
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.