[Paper Review] Mirror Symmetry and Discriminants

Paul S. Aspinwall, M. Ronen Plesser+1 more|arXiv (Cornell University)|Feb 15, 2017

Topological and Geometric Data Analysis22 references3 citations

TL;DR

This paper establishes a precise correspondence between the discriminant of singular N=(2,2) superconformal field theories and the GKZ A-determinant in the context of noncompact toric Calabi–Yau varieties via gauged linear σ-model (GLSM) techniques. It shows that multiplicities in the GKZ determinant arise naturally from the Higgs branch of the GLSM and are encoded in the derived category of massless D-branes, with monodromy around discriminant components described by spherical functors, providing a categorification of the GKZ determinant.

ABSTRACT

We analyze the locus, together with multiplicities, of "bad" conformal field theories in the compactified moduli space of N=(2,2) superconformal field theories in the context of the generalization of the Batyrev mirror construction using the gauged linear sigma-model. We find this discriminant of singular theories is described beautifully by the GKZ "A-determinant" but only if we use a noncompact toric Calabi-Yau variety on the A-model side and logarithmic coordinates on the B-model side. The two are related by "local" mirror symmetry. The corresponding statement for the compact case requires changing multiplicities in the GKZ determinant. We then describe a natural structure for monodromies around components of this discriminant in terms of spherical functors. This can be considered a categorification of the GKZ A-determinant. Each component of the discriminant is naturally associated with a category of massless D-branes.

Motivation & Objective

• To understand the origin of multiplicities in the GKZ A-determinant within the framework of N=(2,2) superconformal field theories.
• To establish a match between the discriminant of singular CFTs and the GKZ A-determinant using noncompact mirror symmetry.
• To interpret the multiplicities in the GKZ determinant as ranks of topological K0-theory and algebraic K0-theory via GLSM Higgs branches.
• To categorify the GKZ A-determinant by associating each component of the discriminant with a category of massless D-branes and describing monodromy via spherical functors.
• To clarify the role of logarithmic coordinates on the B-model side and their necessity for matching the GKZ determinant in the noncompact setting.

Proposed method

• Use of the gauged linear σ-model (GLSM) to analyze the moduli space of N=(2,2) SCFTs and decompose it into Higgs and Coulomb branches.
• Application of local mirror symmetry to relate the A-model on a noncompact toric Calabi–Yau to the B-model on a compact Calabi–Yau with logarithmic coordinates.
• Computation of the GKZ A-determinant as a product of monomials in parameters, with exponents encoding multiplicities.
• Identification of the rank of the algebraic K0-theory group with the number of objects in an exceptional collection of the massless D-brane category.
• Construction of monodromy functors via spherical twists on derived categories of D-branes, linked to generic monodromy around discriminant components.
• Use of toric geometry to equate Chow ring and cohomology ring, ensuring agreement between topological and algebraic K0-theory ranks.

Experimental results

Research questions

• RQ1 How do the multiplicities in the GKZ A-determinant arise from the structure of the gauged linear σ-model?
• RQ2 Why is noncompact mirror symmetry necessary to match the GKZ A-determinant with the discriminant of singular CFTs?
• RQ3 What is the categorical interpretation of the discriminant components in terms of massless D-branes?
• RQ4 How is monodromy around a discriminant component encoded in the derived category of D-branes?
• RQ5 What is the role of logarithmic coordinates on the B-model side in achieving the match with the GKZ determinant?

Key findings

• The discriminant of singular N=(2,2) SCFTs is precisely described by the GKZ A-determinant when using noncompact toric Calabi–Yau varieties on the A-model side and logarithmic coordinates on the B-model side.
• The multiplicities in the GKZ A-determinant correspond to the rank of the algebraic K0-theory group, which equals the number of objects in an exceptional collection of the massless D-brane category.
• Monodromy around a generic component of the discriminant is realized as a spherical twist functor acting on the derived category of massless D-branes.
• The derived category of the massless D-branes for a wall or hybrid phase is generated by an exceptional collection, and its structure matches the topology of the corresponding toric variety.
• For the compact case, the match between GKZ multiplicities and GLSM requires modifying the GKZ determinant, as the K0-theory structure becomes more complex and infinite-dimensional in K3-like categories.
• The GLSM provides a physical interpretation of the GKZ determinant's anomalous degree through the Witten index and orbifold Euler characteristic of the Higgs theory.