[Paper Review] General GLSM Invariants and Their Cohomological Field Theories

David Favero, Bumsig Kim|arXiv (Cornell University)|Jun 22, 2020

Homotopy and Cohomology in Algebraic Topology33 references4 citations

TL;DR

This paper constructs GLSM invariants for general stability conditions in both narrow and broad sectors by introducing a virtual fundamental class in the local cohomology of a twisted Hodge complex. Using Thom-Sullivan and Godement resolutions of matrix factorizations and a localized Chern character from the Atiyah class, the authors prove these invariants form a cohomological field theory, extending prior work in enumerative geometry and Landau-Ginzburg models.

ABSTRACT

We construct GLSM invariants for a general choice of stability in both the narrow and broad sector cases and prove they form a Cohomological Field Theory. This is obtained by forming the analogue of a virtual fundamental class which lives in the local cohomology of the twisted Hodge complex. This general construction comes from the use of two new ingredients. First, the use of the Thom-Sullivan and Godement resolutions applied to matrix factorizations are introduced to handle poorly behaved (non-separated) moduli spaces. Second, a localized Chern character map built from the Atiyah class of a matrix factorization is utilized to forgo the use of Hochschild homology.

Motivation & Objective

• To extend the construction of GLSM invariants to general stability conditions in both narrow and broad sectors.
• To define a virtual fundamental class in local cohomology of the twisted Hodge complex, avoiding reliance on Hochschild homology.
• To establish that these invariants satisfy the axioms of a cohomological field theory.
• To unify methods from matrix factorization, resolution theory, and localized Chern characters in a single framework.

Proposed method

• Utilizes the Thom-Sullivan and Godement resolutions to handle non-separated moduli spaces arising in GLSMs.
• Constructs a virtual factorization using cdga resolutions and defines homotopy classes for factorizations over non-separated stacks.
• Introduces a localized Chern character map derived from the Atiyah class of a matrix factorization, bypassing Hochschild homology.
• Applies the Atiyah class to define a trace map in the $C^{ty}$-setting, enabling integration over critical loci.
• Employs the twisted Hodge complex with support conditions to define a virtual class in local cohomology.
• Establishes independence of choices in evaluation maps and stability parameters via gluing and forgetful operations.

Experimental results

Research questions

• RQ1 Can GLSM invariants be constructed for arbitrary stability conditions in both narrow and broad sectors?
• RQ2 How can a virtual fundamental class be defined in the absence of separated moduli spaces?
• RQ3 Can the use of Hochschild homology be avoided in constructing GLSM invariants?
• RQ4 What role does the Atiyah class of a matrix factorization play in defining a localized Chern character?
• RQ5 Do the resulting invariants satisfy the full axiomatic structure of a cohomological field theory?

Key findings

• The GLSM invariants $\Omega_{g,r,d}$ form a cohomological field theory with unit under mild assumptions.
• The virtual fundamental class $[\mathfrak{U}]_{W}^{\mathrm{vir}}$ is constructed in the local cohomology of the twisted Hodge complex.
• The localized Chern character map from the Atiyah class replaces Hochschild homology in the construction.
• Independence of the invariants from choices of evaluation maps and stability parameters is proven via gluing and forgetful operations.
• The construction is valid for both convex hybrid models and broad sector invariants, generalizing previous results.
• The method applies to non-separated moduli stacks via Thom-Sullivan and Godement resolutions of matrix factorizations.