[Paper Review] GLSM's, gerbes, and Kuznetsov's homological projective duality

Eric Sharpe|arXiv (Cornell University)|Apr 29, 2010

Black Holes and Theoretical Physics28 references4 citations

TL;DR

This paper demonstrates that certain gauged linear sigma models (GLSMs) for complete intersections of quadrics exhibit Landau-Ginzburg points with a Z₂ gerbe structure, leading to a geometric interpretation via branched double covers. Using the decomposition conjecture, it shows these phases are not birational but instead related by Kuznetsov’s homological projective duality, and that the GLSMs physically realize noncommutative resolutions of singular branched covers through matrix factorizations and sheaves of Clifford algebras.

ABSTRACT

In this short note we give an overview of recent work on string propagation on stacks and applications to gauged linear sigma models. We begin by outlining noneffective orbifolds (orbifolds in which a subgroup acts trivially) and related phenomena in two-dimensional gauge theories, which realize string propagation on gerbes. We then discuss the `decomposition conjecture,' equating conformal field theories of strings on gerbes and strings on disjoint unions of spaces. Finally, we apply these ideas to gauged linear sigma models for complete intersections of quadrics, and use the decomposition conjecture to show that the Landau-Ginzburg points of those models have a geometric interpretation in terms of a (sometimes noncommutative resolution of a) branched double cover, realized via nonperturbative effects, rather than as the vanishing locus of a superpotential. These examples violate old unproven lore on GLSM's (e.g., that geometric phases must be related by birational transformations), and we conclude by observing that in these examples (and conjecturing more generally in GLSM's), the phases are instead related by Kuznetsov's `homological projective duality.'

Motivation & Objective

• To understand string propagation on gerbes and noneffective orbifolds in 2D gauge theories.
• To investigate the physical realization of noncommutative resolutions in GLSMs via nonperturbative effects.
• To challenge the long-standing belief that geometric phases in GLSMs are always birational.
• To establish a physical realization of Kuznetsov’s homological projective duality in GLSMs.
• To show that Landau-Ginzburg points in certain GLSMs correspond to branched double covers via the decomposition conjecture.

Proposed method

• Use of noneffective orbifolds, where a subgroup K ⊂ G acts trivially on X, to model gerbes as stacks [X/G].
• Application of the decomposition conjecture, equating CFTs on gerbes with those on disjoint unions of spaces.
• Analysis of one-loop partition functions in noneffective orbifolds to show physical distinctions from effective orbifolds.
• Identification of Z₂ gerbe structures in GLSMs for complete intersections of quadrics.
• Use of matrix factorizations and sheaves of Clifford algebras over P³ to describe D-branes at the Landau-Ginzburg point.
• Application of Kuznetsov’s homological projective duality to relate non-birational geometric phases in GLSMs.

Experimental results

Research questions

• RQ1 How do noneffective group actions in 2D gauge theories lead to physical distinctions from effective orbifolds?
• RQ2 Can the decomposition conjecture be used to give a geometric interpretation of Landau-Ginzburg points in GLSMs?
• RQ3 Why do some GLSMs violate the standard lore that geometric phases are birational?
• RQ4 How is Kuznetsov’s homological projective duality realized physically in GLSMs?
• RQ5 What is the role of noncommutative resolutions in GLSMs with singular branched double covers?

Key findings

• The Landau-Ginzburg point of the GLSM for P⁷[2,2,2,2] is physically realized as a branched double cover of P³, not as the vanishing locus of a superpotential.
• The branched double cover is singular, but the GLSM physics behaves as if it were smooth, indicating a noncommutative resolution.
• The noncommutative resolution is defined by sheaves of B-modules over P³, where B is the even part of the Clifford algebra associated with the universal quadric.
• D-branes at the Landau-Ginzburg point are described by matrix factorizations, which match the sheaves of B-modules, confirming the noncommutative resolution.
• The two geometric phases of the GLSM are not birational, contradicting established lore in the GLSM community.
• The phases are related by Kuznetsov’s homological projective duality, and the paper conjectures this holds universally in GLSMs.