×

zbMATH — the first resource for mathematics

GLSM realizations of maps and intersections of Grassmannians and Pfaffians. (English) Zbl 1390.81310
Summary: In this paper we give gauged linear sigma model (GLSM) realizations of a number of geometries not previously presented in GLSMs. We begin by describing GLSM realizations of maps including Veronese and Segre embeddings, which can be applied to give GLSMs explicitly describing non-complete intersection constructions such as the intersection of one hypersurface with the image under some map of another. We also discuss GLSMs for intersections of Grassmannians and Pfaffians with one another, and with their images under various maps, which sometimes form exotic constructions of Calabi-Yaus, as well as GLSMs for other exotic Calabi-Yau constructions of Kanazawa. Much of this paper focuses on a specific set of examples of GLSMs for intersections of Grassmannians \(G(2,N)\) with themselves after a linear rotation, including the Calabi-Yau case \(N=5\). One phase of the GLSM realizes an intersection of two Grassmannians, the other phase realizes an intersection of two Pfaffians. The GLSM has two nonabelian factors in its gauge group, and we consider dualities in those factors. In both the original GLSM and a double-dual, one geometric phase is realized perturbatively (as the critical locus of a superpotential), and the other via quantum effects. Dualizing on a single gauge group factor yields a model in which each geometry is realized through a simultaneous combination of perturbative and quantum effects.

MSC:
81T13 Yang-Mills and other gauge theories in quantum field theory
81T60 Supersymmetric field theories in quantum mechanics
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Jockers, H.; Kumar, V.; Lapan, JM; Morrison, DR; Romo, M., Two-sphere partition functions and Gromov-Witten invariants, Commun. Math. Phys., 325, 1139, (2014) · Zbl 1301.81253
[2] Benini, F.; Eager, R.; Hori, K.; Tachikawa, Y., elliptic genera of 2\(d\)\( \mathcal{N} \) = 2 gauge theories, Commun. Math. Phys., 333, 1241, (2015) · Zbl 1321.81059
[3] Honda, D.; Okuda, T., exact results for boundaries and domain walls in 2d supersymmetric theories, JHEP, 09, 140, (2015) · Zbl 1388.81218
[4] K. Hori and M. Romo, Exact results in two-dimensional (2\(,\) 2) supersymmetric gauge theories with boundary, arXiv:1308.2438 [INSPIRE].
[5] Closset, C.; Cremonesi, S.; Park, DS, The equivariant A-twist and gauged linear σ-models on the two-sphere, JHEP, 06, 076, (2015) · Zbl 1388.81713
[6] Hori, K.; Tong, D., aspects of non-abelian gauge dynamics in two-dimensional\( \mathcal{N} \) = (2\(,\) 2) theories, JHEP, 05, 079, (2007)
[7] Caldararu, A.; etal., Non-birational twisted derived equivalences in abelian glsms, Commun. Math. Phys., 294, 605, (2010) · Zbl 1231.14035
[8] Hori, K., duality in two-dimensional (2\(,\) 2) supersymmetric non-abelian gauge theories, JHEP, 10, 121, (2013) · Zbl 1342.81635
[9] Jockers, H.; etal., nonabelian 2D gauge theories for determinantal Calabi-Yau varieties, JHEP, 11, 166, (2012) · Zbl 1397.81378
[10] Donagi, R.; Sharpe, E., GLSM’s for partial flag manifolds, J. Geom. Phys., 58, 1662, (2008) · Zbl 1218.81091
[11] Hori, K.; Knapp, J., Linear σ-models with strongly coupled phases — one parameter models, JHEP, 11, 070, (2013)
[12] K. Hori and J. Knapp, A pair of Calabi-Yau manifolds from a two parameter non-Abelian gauged linear σ-model, arXiv:1612.06214 [INSPIRE].
[13] Gerhardus, A.; Jockers, H., Dual pairs of gauged linear σ-models and derived equivalences of Calabi-Yau threefolds, J. Geom. Phys., 114, 223, (2017) · Zbl 1359.81136
[14] Gerhardus, A.; Jockers, H., Quantum periods of Calabi-Yau fourfolds, Nucl. Phys., B 913, 425, (2016) · Zbl 1349.81155
[15] Kanazawa, A., Pfaffian Calabi-Yau threefolds and mirror symmetry, Commun. Num. Theor. Phys., 6, 661, (2012) · Zbl 1274.14047
[16] L. Borisov, A. Căldăraru, A. Perry, Intersections of two Grassmannians in ℙ\^{}{9}, arXiv:1707.00534.
[17] J. Ottem and J. Rennemo, A counterexample to the birational Torelli problem for Calabi-Yau threefolds, arXiv:1706.09952. · Zbl 1393.14006
[18] P. Griffiths and J. Harris, Principles of algebraic geometry, John Wiley & Sons, New York U.S.A. (1978). · Zbl 0408.14001
[19] J. Harris, Algebraic geometry: a first course, Springer, Germany (1992). · Zbl 0779.14001
[20] Pantev, T.; Sharpe, E., GLSM’s for gerbes (and other toric stacks), Adv. Theor. Math. Phys., 10, 77, (2006) · Zbl 1119.14038
[21] T. Pantev and E. Sharpe, Notes on gauging noneffective group actions, hep-th/0502027 [INSPIRE].
[22] Hellerman, S.; Henriques, A.; Pantev, T.; Sharpe, E.; Ando, M., Cluster decomposition, T-duality and gerby CFT’s, Adv. Theor. Math. Phys., 11, 751, (2007) · Zbl 1156.81039
[23] Sharpe, E., Decomposition in diverse dimensions, Phys. Rev., D 90, (2014)
[24] E. Witten, The Verlinde algebra and the cohomology of the Grassmannian, hep-th/9312104 [INSPIRE]. · Zbl 0863.53054
[25] J. Weyman, Cohomology of vector bundles and syzygies, Cambridge University Press, Cambridge U.K. (2003). · Zbl 1075.13007
[26] C. van Enckevort and D. van Straten, Electronic data base of Calabi-Yau equations, http://www.mathematik.uni-mainz.de/CYequations/db/. · Zbl 1117.14043
[27] Distler, J.; Sharpe, E., Heterotic compactifications with principal bundles for general groups and general levels, Adv. Theor. Math. Phys., 14, 335, (2010) · Zbl 1213.81191
[28] E.A. Rødland, The Pfaffian Calabi-Yau, its mirror, and their link to the Grassmannian G(2\(,\) 7), Composito Math.122 (2000) 135 [math/9801092]. · Zbl 0974.14026
[29] Aspinwall, PS; Plesser, MR, Decompactifications and massless D-branes in hybrid models, JHEP, 07, 078, (2010) · Zbl 1290.81095
[30] L. Manivel, Double spinor Calabi-Yau varieties, arXiv:1709.07736.
[31] L. Pertusi, On the double EPW sextic associated to a Gushel-Mukai fourfold, arXiv:1709.02144.
[32] C. van Enckevort and D. van Straten, Monodromy calculations of fourth order equations of Calabi-Yau type, AMS/IP Stud. Adv. Math.38 (2006) 539 [math/0412539]. · Zbl 1117.14043
[33] F. Tonoli, Canonical surfaces in ℙ\^{}{5} and Calabi-Yau threefolds in ℙ\^{}{6}, Ph.D. Thesis, University of Padova, Padova, Italy (2000).
[34] Tonoli, F., construction of Calabi-Yau 3-folds in ℙ\^{}{6}, J. Alg. Geom., 13, 209, (2004) · Zbl 1060.14060
[35] Cynk, S.; Straten, D., Calabi-Yau conifold expansions, Fields Inst. Commun., 67, 499, (2018) · Zbl 1302.14033
[36] S. Cynk and D. van Straten, Periods of double octic Calabi-Yau manifolds, arXiv:1709.09751 [INSPIRE]. · Zbl 1302.14033
[37] S. Cynk and D. van Straten, Picard-Fuchs operators for octic arrangements I (The case of orphans), arXiv:1709.09752. · Zbl 1321.81059
[38] S. Galkin, Joins and Hadamard products, lecture given at Categorical and analytic invariants in algebraic geometry 1 , September 14-18, Stekov Mathematical Institute, Moscow, Russia (2015). · Zbl 1301.81253
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.