Sigma Models and Phase Transitions for Complete Intersections

Abstract

We study a one-parameter family of gauged linear sigma models (GLSMs) naturally associated to a complete intersection in weighted projective space. In the positive phase of the family we recover Gromov-Witten theory of the complete intersection, while in the negative phase we obtain a Landau-Ginzburg-type theory. Focusing on the negative phase, we develop foundational properties which allow us to state and prove a genus-zero comparison theorem that generalizes the multiple log-canonical correspondence and should be viewed as analogous to quantum Serre duality in the positive phase. Using this comparison result, along with the crepant transformation conjecture and quantum Serre duality, we prove a genus-zero correspondence between the GLSMs which arise at the two phases, thereby generalizing the Landau-Ginzburg/Calabi-Yau correspondence to complete intersections.The right-hand vertical arrow is quantum Serre duality (QSD) and the quantum Lefschetz hyperplane principle, developed by Coates-Givental [14] and Tseng [26]. It is an identification of a T-equivariant extension of the Gromov-Witten theory of Z to the T-equivariant Gromov-Witten theory of X + . The top of the diagram is the crepant transformation conjecture (CTC), proved in this setting by Coates-Iritani-Jiang [15].The equivalence in the left-hand vertical arrow is new and is the technical heart of our paper. It generalizes the multiple log-canonical (MLK) correspondence of [23] and serves as the analogue in the negative phase of quantum Serre duality:

Towards a mirror theorem for GLSMs

“…This recovers the hybrid-model I-function computed by Clader-Ross in [16,17]. Their I-function appears, for instance, as I X − ,W in [16, Section 7.4], in the proof of Lemma 7.6.…”

“…These GLSMs arise when one starts with a GLSM representing a complete intersection as in the previous section, but chooses a new stability condition θ − . These were computed by Clader-Ross in [16,17]. More precisely, let G = C * act on V = Spec C[x 1 , .…”

“…Despite the dramatic recent progress in defining GLSM invariants, with the exception of affine phases, geometric phases, and a class of Picard-rank one hybrid model phases [16], there have been very few computations of these new invariants. In this paper we develop a method for computing genus-zero invariants for a large class of GLSMs.…”

Mirror Symmetry Constructions

“…The dotted horizontal arrow is a special case of a conjecture known as the Landau-Ginzburg/Calabi-Yau (LG/CY) correspondence, which relates the various phases of a GLSM to one another. The conjecture has been proven in many instances; see [10] [11] [1] [15], among many other references, for more information.…”

“…Given the discussion of the previous paragraphs and the fact that W → W T as t → ∞, the diagram (15) can be re-expressed as:…”