[Paper Review] Positivity of equivariant Gromov-Witten invariants

Dave Anderson, Linda Chen|arXiv (Cornell University)|Oct 26, 2011

Geometric and Algebraic Topology4 references3 citations

TL;DR

This paper establishes the positivity of equivariant Gromov-Witten invariants for projective homogeneous spaces $G/P$ by showing they lie in the semiring $\mathbb{N}[\alpha_1,\ldots,\alpha_n]$ of polynomials with nonnegative integer coefficients in the positive roots. Using transversality and geometric intersection theory on equivariant moduli spaces, the authors represent these invariants as degrees of effective zero-cycles, proving that generalized EQLR coefficients in the equivariant quantum cohomology ring are Graham-positive, offering a new proof of Mihalcea's result and extending it to multiple-point invariants.

ABSTRACT

We show that the equivariant Gromov-Witten invariants of a projective homogeneous space G/P exhibit Graham-positivity: when expressed as polynomials in the positive roots, they have nonnegative coefficients.

Motivation & Objective

• To establish the Graham-positivity of multiple-point equivariant Gromov-Witten invariants in the equivariant quantum cohomology of $G/P$.
• To provide a geometric interpretation of these invariants as degrees of effective zero-cycles via transversality arguments.
• To generalize Mihalcea’s positivity result for EQLR coefficients to higher-point invariants using a new geometric approach.
• To derive a Giambelli formula for $QH_T^*(SL_n/P)$ using dimension estimates from the main proof.
• To show that the associativity relations in the equivariant quantum cohomology ring are subtraction-free, reinforcing the positivity structure.

Proposed method

• Represent the equivariant Gromov-Witten invariants as pushforwards of cup products of equivariant Schubert classes via the equivariant pushforward $\pi^T_*$ on the moduli space $\overline{M}_{0,r+1}(X,\mathbf{d})$.
• Use approximation spaces $\mathcal{Z} = \mathbb{E} \times^T Z$ for $T$-varieties to compute equivariant cohomology with finite-dimensional models.
• Apply transversality theorems to show that intersections of equivariant Schubert varieties with evaluation loci yield Cohen-Macaulay, pure-dimensional cycles.
• Define the cycle $V_\gamma = (\mathrm{ev}_1^T)^{-1}(\gamma_1\mathcal{Y}(v_1)) \cap \cdots \cap (\mathrm{ev}_r^T)^{-1}(\gamma_r\mathcal{Y}(v_r)) \cap \mathcal{Z}_J$ as an effective zero-cycle whose degree gives the invariant coefficient.
• Establish dimension counts for $V_\gamma$ and its boundary $\partial V_\gamma$ to determine when the cycle is zero-dimensional and thus has a well-defined degree.
• Use the fiber product identity $[V_\gamma] = (\mathrm{ev}^T)^* y(v_1) \cdots (\mathrm{ev}^T)^* y(v_r) \cdot (\mathrm{ev}_{r+1}^T)^* x(w) \cdot [\overline{\mathcal{M}}_J]$ to equate the cycle class to the equivariant cohomology class.

Experimental results

Research questions

• RQ1 Are the equivariant Gromov-Witten invariants $I^T_{\mathbf{d}}(y(v_1)\cdots y(v_r) \cdot x(w))$ Graham-positive for $r > 2$?
• RQ2 Can the coefficients of these invariants be interpreted geometrically as degrees of effective cycles?
• RQ3 Does the associativity of the equivariant quantum cohomology ring imply that higher-point EQLR coefficients are also Graham-positive?
• RQ4 Can dimension estimates from the moduli space geometry lead to a Giambelli formula for $QH_T^*(SL_n/P)$?
• RQ5 Is there a geometric proof of positivity that avoids the algebraic methods used by Mihalcea?

Key findings

• The equivariant Gromov-Witten invariants $I^T_{\mathbf{d}}(y(v_1)\cdots y(v_r) \cdot x(w))$ lie in $\mathbb{N}[\alpha_1,\ldots,\alpha_n]$, proving their Graham-positivity for any $r \geq 2$.
• The coefficients $c_J$ in the expansion of the invariant as $\sum c_J \alpha^J$ are equal to the degree of the effective zero-cycle $V_\gamma$, which is a nonnegative integer.
• When the dimension of the intersection $V_\gamma$ is zero, it consists of finitely many points in the open moduli space $\mathcal{M}$, and its degree is well-defined and nonnegative.
• The proof establishes that the generalized EQLR coefficients $c_{v_1,\ldots,v_r}^{w,\mathbf{d}}$ in the equivariant quantum cohomology ring are Graham-positive, via induction and the $r=2$ case.
• The associativity relations in $QH_T^*X$ are subtraction-free, confirming that the structure constants are nonnegative polynomials in the positive roots.
• The dimension estimates from the proof are used to derive a Giambelli formula for $QH_T^*(SL_n/P)$ in a subsequent work [AC].