Summing the instantons: Quantum cohomology and mirror symmetry in toric varieties

Abstract

We use the gauged linear sigma model introduced by Witten to calculate instanton expansions for correlation functions in topological sigma models with target space a toric variety V or a Calabi-Yau hypersurface M ⊂ V . In the linear model the instanton moduli spaces are relatively simple objects and the correlators are explicitly computable; moreover, the instantons can be summed, leading to explicit solutions for both kinds of models. In the case of smooth V , our results reproduce and clarify an algebraic solution of the V model due to Batyrev. In addition, we find an algebraic relation determining the solution for M in terms of that for V . Finally, we propose a modification of the linear model which computes instanton expansions about any limiting point in the moduli space. In the smooth case this leads to a (second) algebraic solution of the M model. We use this description to prove some conjectures about mirror symmetry, including the previously conjectured "monomial-divisor mirror map" of Aspinwall, Greene, and Morrison. 12/94

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

“…Integrating out the Y i yields an effective superpotential for Σ identical to that obtained by one-loop calculations in [3]. If instead we integrate out Σ, then in principle we are left with an effective Landau-Ginzburg theory whose B model correlators should match the A model correlation functions of the G k…”

“…In [3], predictions for quantum cohomology of toric varieties were derived by computing one-loop corrections to effective actions in gauged linear sigma models. However, the authors of [3] were aware [8] of the physical distinction between gauged linear sigma models with minimal and nonminimal charges, and although they did not understand the mathematical interpretation, were careful to write down results also valid for gauged linear sigma models with nonminimal charges. Thus, on the face of it, the quantum cohomology calculations in [3] should also apply equally well to toric stacks, not just toric varieties.…”

“…However, the authors of [3] were aware [8] of the physical distinction between gauged linear sigma models with minimal and nonminimal charges, and although they did not understand the mathematical interpretation, were careful to write down results also valid for gauged linear sigma models with nonminimal charges. Thus, on the face of it, the quantum cohomology calculations in [3] should also apply equally well to toric stacks, not just toric varieties. Indeed, we have checked that statement in previous sections by comparing several calculations of quantum cohomology for Z k gerbes on projective spaces, some of those calculations coming from [3].…”

“…Integrating out the Y 's returns an effective superpotential for the Σ's identical to that calculated from one-loop effects in [3]. Integrating out the Σ's gives the constraints 6…”

Large N Duality, Lens Spaces and the Chern-Simons Matrix Model

“…Setting m = 0 in [21] gives the above geometry. {Q ∈ Z p+3 | i Q i v i = 0} [19]. After rearranging the columns, these are given by…”