Fourier duality for quantum cohomology central charges (K-theory generalization)
Establish a Fourier duality for central charges associated with K-theory classes, proving that for E ∈ K^0_T(X) and τ ∈ H^2_T(X), the equivariant central charge Π_X^{eq}(E;τ) is Fourier dual to Π_Y(κ(E);σ_Y(τ−λ⋅t)) for a suitable GIT quotient Y=X//T and mirror map σ_Y, where κ denotes the K-theoretic Kirwan map.
References
The quantum volume equals, up to a simple factor, the quantum cohomology central charge of the structure sheaf $O$ in . We can generalize the above conjecture to central charges associated with elements of the $K$-group of vector bundles. We define
\Pi_X(E; \tau) := \int_X J_X(\tau, -z) \cup z{n-\frac{\deg}{2} z{c_1(X)} _X (2\pi i){\frac{\deg}{2} ch(E)
for $E$ in the topological $K$-group $K0(X)$. We can define the equivariant version $\Pi{\rm eq}_X(E;\tau)$ associated with equivariant classes $E \in K0_T(X)$ and $\tau\in H2_T(X)$ similarly. Then we conjecture the Fourier duality between
\Pi{\rm eq}_X(E;\tau) \quad \underset{\rm FT}{\longleftrightarrow} \quad \Pi_Y(\kappa(E); \sigma_Y(\tau-\lambda \cdot t))
where $Y$ is a GIT quotient of $X$ and $\kappa_Y(E)\in K0(Y)$ is the image of the $K$-theoretic Kirwan map.