Laurent polynomial mirrors for quiver flag zero loci

Abstract

The classification of Fano varieties is an important open question, motivated in part by the MMP. Smooth Fano varieties have been classified up to dimension three: one interesting feature of this classification is that they can all be described as certain subvarieties in GIT quotients; in particular, they are all either toric complete intersections (subvarieties of toric varieties) or quiver flag zero loci (subvarieties of quiver flag varieties). There is a program to use mirror symmetry to classify Fano varieties in higher dimensions. Fano varieties are expected to correspond to certain Laurent polynomials under mirror symmetry; given such a Fano toric complete intersections, one can produce a Laurent polynomial via the Hori-Vafa mirror. In this paper, we give a method to find Laurent polynomial mirrors to Fano quiver flag zero loci in Y -shaped quiver flag varieties. To do this, we generalise the Gelfand-Cetlin degeneration of type A flag varieties to Fano Y -shaped quiver flag varieties, and give a new description of these degenerations as canonical toric quiver varieties. We find conjectural mirrors to 99 four dimensional Fano quiver flag zero loci, and check them up to 20 terms of the period sequence.

GLSMs for exotic Grassmannians

“…In this section we will briefly discuss mirrors to these nonabelian GLSMs, following the nonabelian mirror ansatz discussed in [10]. (It should be noted that other notions of mirrors exist, with different UV presentations but apparently equivalent IR physics, see [6][7][8][9]28,29]. )…”

Unwinding Toric Degenerations and Mirror Symmetry for Grassmannians

“…A quiver moduli space is defined by assigning a vector space to each vertex of a quiver and choosing a stability condition [15]. If we assign a one-dimensional vector space to each vertex of the ladder quiver and choose a Fano stability condition, then we obtain a quiver moduli space that coincides with 𝑋 𝑃 𝑛,𝑟 : See [14] for details.…”

Unwinding toric degenerations and mirror symmetry for Grassmannians

“…A quiver moduli space is defined by assigning a vector space to each vertex of a quiver, and choosing a stability condition [14]. If we assign a one-dimensional vector space to each vertex of the ladder quiver and choose a Fano stability condition, then we obtain a quiver moduli space that coincides with X Pn,r : see [13] for details.…”

“…Then for any other coefficient c, the weight of c under the G-action can be expressed as a ratio of the d v . Whatever these ratios are, they give a collection of polynomial equations between the c i , which we call (13) P 1 .…”