A note on Virasoro constriants for products

1. Problem Statement

The paper addresses Problem 1.1, a fundamental question in Gromov-Witten (GW) theory regarding the Virasoro Conjecture.

• Context: The Virasoro conjecture posits that the total descendant potential $D_W$ of a smooth projective variety $W$ satisfies an infinite set of differential equations $L_m^W D_W = 0$ for $m \geq -1$, where $L_m^W$ are Virasoro operators.
• The Question: Does the Virasoro conjecture hold for a product variety $X \times Y$ if and only if it holds for both factors $X$ and $Y$ individually?
• Current Status: While the conjecture is proven for nonsingular curves and for varieties with semi-simple quantum cohomology (via Givental's approach), the general case for product varieties has seen little progress. The paper aims to resolve this specifically when one factor has semi-simple quantum cohomology.

2. Methodology

The author employs Givental's quantization formalism and the Givental-Teleman classification of Cohomological Field Theories (CohFTs). The methodology proceeds through the following steps:

A. Framework Setup

• CohFTs: The GW theory of a variety is treated as a CohFT. The paper utilizes the axioms of CohFTs (invariance, gluing, and forgetful maps) to define potentials.
• R-Matrix Action: The paper relies on the action of the loop group (specifically, the R-matrix action) on CohFTs. If a CohFT $\Omega$ is semi-simple, the Givental-Teleman theorem states it can be uniquely reconstructed from its topological part $\omega$ (degree 0 classes) via an R-matrix $R(z)$:
  $$\Omega = R \cdot \omega$$
• Product Formula: The author utilizes the known product formula for GW invariants, which states that the CohFT of a product $X \times Y$ is the tensor product of the CohFTs of the factors (with appropriate shifts).

B. Derivation of the Product Formula

1. Assumption: Assume the quantum cohomology of $Y$ is semi-simple at a specific point. This allows the application of the Givental-Teleman classification to $Y$.
2. Decomposition: The CohFT of $Y$ is expressed as $R_Y \cdot \omega_Y$, where $R_Y$ is the unique symplectic R-matrix.
3. Tensor Product Analysis: The author analyzes the CohFT of the product $X \times Y$ under the shift $s \otimes 1 + 1 \otimes t$. By applying the product formula and the axioms of CohFTs to the stable graph expansion, the author derives that the GW theory of $X \times Y$ is related to the GW theory of $X$ by the action of a specific loop group element:
   $$R = \text{Id} \otimes R_Y \in \text{End}(H^*(X) \otimes H^*(Y))[[z]]$$
4. Potential Relation: This leads to a formula for the total descendant potential $D_{X \times Y}$ in terms of $D_X$ and the action of $R$:
   $$D_{X \times Y} = \widehat{(S_{X \times Y})^{-1}} \widehat{R} \prod_i \widehat{S_X} D_X$$
   (Where $\widehat{S}$ and $\widehat{R}$ denote quantized operators).

C. Virasoro Operator Analysis

The core technical step involves analyzing how the Virasoro operators $L_m$ behave under this transformation.

• The Virasoro operators are defined via the quantization of differential operators involving the Hodge grading operator $\mu$ and the multiplication by the first Chern class $\rho$.
• The author proves that the operators involved in the product formula (specifically the conjugation by $S$ and $R$) preserve the grading.
• By conjugating the Virasoro operator $L_m^{X \times Y}$ across the transformation operators, the author shows it transforms into:
  $$L_m^{X \times Y} \longrightarrow L_m^{X} \otimes \text{Id} + \text{Id} \otimes L_m^{N \text{ points}}$$
  where $N = \text{rank}(H^*(Y))$.

3. Key Contributions

• Partial Resolution of Problem 1.1: The paper establishes that the Virasoro constraints for $X \times Y$ hold if and only if they hold for $X$, provided $Y$ has semi-simple quantum cohomology.
• Explicit Formula for Product Potentials: It provides an explicit formula (Equation 3.8) relating the descendant potential of a product space to the potential of one factor and the R-matrix action of the semi-simple factor.
• Extension to Orbifolds: The author notes in Remark 3.5 that the argument extends naturally to orbifold Gromov-Witten theory, as the necessary components (Givental-Teleman classification, product formulas, and Virasoro constraints) exist in that context.

4. Main Results

Theorem 1.2:

Suppose the quantum cohomology of $Y$ is semi-simple at some point. Then Virasoro constraints for $X \times Y$ hold if and only if Virasoro constraints hold for $X$.

Implication:
Since the Virasoro conjecture is known to hold for a point (or a collection of points, which corresponds to the semi-simple case of $Y$), the theorem implies that if $Y$ is semi-simple, the validity of the conjecture for the product depends entirely on the validity for $X$. If $X$ also satisfies the conjecture, then $X \times Y$ does.

5. Significance

• Bridging Approaches: The paper successfully bridges the "semi-simple" approach (Givental-Teleman) with the "product" structure of GW theory. Previously, these were often treated as separate domains.
• Momentum for Open Problems: By solving the product case under the semi-simplicity assumption, the paper creates a framework to potentially tackle the general case of Problem 1.1, where neither factor is necessarily semi-simple.
• Technical Rigor: The proof relies on the precise covariance properties of the loop group action on the Virasoro operators, demonstrating that the "twist" introduced by the semi-simple factor $Y$ does not disrupt the Virasoro constraints of the other factor $X$.

In summary, Tseng's paper provides a rigorous proof that the Virasoro conjecture behaves well under products when one factor is semi-simple, reducing the problem for the product to the problem for the non-semi-simple factor.