GLSM realizations of maps and intersections of Grassmannians and Pfaffians

Abstract

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.

GLSMs, joins, and nonperturbatively-realized geometries

“…In passing, we observe that the structure above is part of the structure that describes G(2, 5) ∩ G(2, 5) in [11]. There, the intersection G(2, 5) ∩ G(2, 5) was described in terms of a…”

“…Thus, we begin in section 2 by giving an introduction to joins and their physical realization in some simple one-parameter GLSM examples. At some level this section of our paper is also a continuation of our efforts in [11] to give GLSM-based realizations of other constructions in algebraic geometry. In fact, as we shall see explicitly, one of the models discussed in [11] fits into the framework of joins.…”

“…At some level this section of our paper is also a continuation of our efforts in [11] to give GLSM-based realizations of other constructions in algebraic geometry. In fact, as we shall see explicitly, one of the models discussed in [11] fits into the framework of joins. The one-parameter example we discuss there all have multiple non-abelian factors in their gauge groups to which two-dimensional gauge dualities can be applied, so we explore the dual GLSMs and verify that the phases of the duals have the same geometric interpretation as in the original GLSM.…”

“…To begin, we shall describe the Join of G(2, 5) with itself that was also discussed in [9]. We will see that the result is related to the intersection G(2, 5) ∩ G(2, 5) described physically in [11], following [10]. (See also [16] for a related theory.…”

“…• We can smooth the singularities by deforming the superpotential, but when we do so, we recover a model that was extensively analyzed in [11], including how the phases of dual GLSMs are related to one another.…”