G{LSM}s for gerbes (and other toric stacks)

Abstract

In this paper, we will discuss gauged linear sigma model descriptions of toric stacks. Toric stacks have a simple description in terms of (symplectic, GIT) C × quotients of homogeneous coordinates, in exactly the same form as toric varieties. We describe the physics of the gauged linear sigma models that formally coincide with the mathematical description of toric stacks and check that physical predictions of those gauged linear sigma models exactly match the corresponding stacks. We also see in examples that when a given toric stack has multiple presentations in a form accessible as a gauged linear sigma model, that the IR physics of those different presentations matches, so that the IR physics is presentation-independent, making it reasonable to associate CFTs to stacks, not just presentations of stacks. We discuss mirror symmetry for stacks, using MorrisonPlesser-Hori-Vafa techniques to compute mirrors explicitly, and also find a natural generalization of Batyrev's mirror conjecture. In the process of studying mirror symmetry, we find some new abstract CFTs, involving fields valued in roots of unity. e-print archive: http://lanl.arXiv.org/abs/hep-th/0502053 TONY PANTEV AND ERIC SHARPE

GLSMs for partial flag manifolds

“…Giving p charge m rather than charge 1 means that we are 'overgauging', gauging m rotations rather than a single rotation. This distinction is nonperturbatively meaningful, as discussed in [2], and the resulting low-energy theory is a sigma model on a Z m gerbe over the flag manifold, with characteristic class (q 1 mod m, q 2 mod m, · · · , q n mod m) 2.6.3 Grassmannian bundle over P…”

“…For example, strings on stacks [2,5,12,13,14] are described concretely by universality classes of gauged nonlinear sigma models, where the gauge group need be neither finite nor effectivelyacting. Because several naive consistency checks fail, a great deal of effort was expended in especially [2,13] to verify that universality classes are independent of presentation. Also, presentation-dependence issues arise in the application of derived categories to physics [14], though there the presentations involve open strings, not gauged sigma models.…”

“…Their primary original use was for describing toric varieties and complete intersections therein. More recently [2] they have been used to describe toric stacks and complete intersections therein.…”

Some recent advances in gauged linear sigma models

“…So far we have primarily focused on discrete gauge theories, but we can also understand the same effect in presentations as U(1) gauge theories which are spontaneously broken to a finite subgroiup. This is discussed in [57,58]. Briefly, on a compact worldsheet, to uniquely define the charged matter, one must specify which bundle the charged matter couples to, and this unambiguously distinguishes nonminimal charges from minimal charges.…”

“…Perhaps the most important subtlety is that, away from the locus {det A = 0} where some of the φ i become massless, the only massless fields have nonminimal charges. Due to subtleties in two-dimensional quantum field theories [16,57,58,59], if the only massless fields have charges ±2, then physics sees a double cover, and so the correct interpretation of the r ≪ 0 limit is as a nonlinear sigma model on a branched double cover of CP 1 , branched along the locus {det A = 0}. Since A is a 4 × 4 matrix, det A is a degree four polynomial in p's, so the branch locus is degree four.…”