[Paper Review] K-Theoretic $I$-function of $V//_{ heta} \mathbf{G}$ and Application

Yaoxiong Wen|arXiv (Cornell University)|May 30, 2019

Nonlinear Waves and Solitons16 references4 citations

TL;DR

This paper computes the K-theoretic $I$-function with level structures for GIT quotients $V//_{\theta}\mathbf{G}$ using abelian and non-abelian correspondences, generalizing Givental-Lee's Toda operator results to nontrivial level structures in quantum K-theory. It establishes a finite-difference difference equation for the $\mathcal{J}$-function on complete flag varieties, showing that the $K_{\mathbf{T}}(X)$-valued series is an eigenstate of a generalized Toda operator with level-dependent coefficients.

ABSTRACT

In this paper, we compute K-theoretic $I$-function with level structure (defined by quasi-map theory) of GIT-quotient of a vector space via abelian and non-abelian correspondence. As a consequence, we generalize Givental-Lee's result to find the analogous "Toda operators" for the $I$-function with nontrivial level structures in the case of complete flag variety.

Motivation & Objective

• To extend Givental-Lee's Toda operator framework to quantum K-theory with nontrivial level structures in the context of complete flag varieties.
• To establish a K-theoretic abelian/non-abelian correspondence for $I$-functions incorporating level structures.
• To derive difference equations satisfied by the $\mathcal{J}$-function under level structures, generalizing the finite-difference structure of the original Toda system.
• To verify that the $K_{\mathbf{T}}(X)$-valued $\tilde{\mathcal{J}}$-function with level structures is an eigenstate of a generalized Toda operator.

Proposed method

• Uses quasi-map theory and $\mathbb{C}^{*}$-equivariant localization to compute the $I$-function as a sum over virtual classes on moduli spaces of quasimaps.
• Applies abelian/non-abelian correspondence techniques to relate the $I$-function of $V//\mathbf{G}$ to that of $V//\mathbf{T}$, incorporating level structures via determinant line bundles.
• Introduces level structures via $\mathcal{D}^{R,l}$, defined as $\operatorname{det}^{-l_i} R^\bullet \pi_* (\mathfrak{P} \times_{\mathbf{G}} \mathbb{C}_{\theta_i})$, which modifies the $I$-function by inserting $p_i^{l_i(d_i-1)} q^{l_i d_i(d_i-1)/2}$ factors.
• Derives the generalized Toda operator $\widetilde{H}^{\theta_i,l_i}_{Q,q}$ by modifying the original Givental-Lee operator to include $q^{l_i Q_i \partial_{Q_i}}$ terms in the $i$-th term.
• Employs the mirror theorem for partial flag varieties to show that the small $I$-function with level zero equals the small $\mathcal{J}$-function, enabling the extension to non-zero levels.
• Uses Plücker embedding to express line bundles $l_i$ in terms of $p_i$, allowing explicit comparison with known results from Givental and Lee.

Experimental results

Research questions

• RQ1 How can the K-theoretic $I$-function with level structures be computed for GIT quotients $V//_{\theta}\mathbf{G}$ using abelian/non-abelian correspondence?
• RQ2 Does the generalized Toda operator structure persist when level structures are introduced in the $I$-function for complete flag varieties?
• RQ3 Can the $\mathcal{J}$-function with nontrivial level structures be shown to satisfy a finite-difference difference equation analogous to the original Toda system?
• RQ4 What is the precise form of the modified Toda operator $\widetilde{H}^{\theta_i,l_i}_{Q,q}$ that governs the $\tilde{\mathcal{J}}$-function with level structures?
• RQ5 How do level structures modify the $I$-function via determinant line bundles and virtual classes on quasimap moduli spaces?

Key findings

• The K-theoretic abelian/non-abelian correspondence is generalized to include level structures, yielding a formula involving $w\left[\prod_{\alpha}\frac{\prod_{k=-\infty}^{\tilde{\beta}(\alpha)}(1-L^{\vee}_\alpha q^k)}{\prod_{k=-\infty}^{0}(1-L^{\vee}_\alpha q^k)} I_{\tilde{\beta}}^{V//\mathbf{T},R,l}\right]$.
• For partial flag varieties, the small $I$-function with level zero coincides with the small $\mathcal{J}$-function, confirming a mirror theorem in this case.
• The $K_{\mathbf{T}}(X)$-valued $\tilde{\mathcal{J}}$-function with nontrivial level structures is an eigenstate of the generalized Toda operator $\widetilde{H}^{\theta_i,l_i}_{Q,q}$ with eigenvalue $\Lambda_0^{-1} + \cdots + \Lambda_r^{-1}$.
• The modified Toda operator includes level-dependent terms: $q^{Q_{i+1}\partial_{Q_{i+1}} - Q_i\partial_{Q_i}}(1 - Q_i \circ q^{l_i Q_i \partial_{Q_i}})$, replacing the original $q^{Q_i\partial_{Q_i}}(1 - Q_i)$ term.
• The $I$-function with level $l_i$ is modified by inserting $p_i^{l_i(d_i-1)} q^{l_i d_i(d_i-1)/2}$, reflecting the contribution of the level structure to the quantum invariants.
• Explicit computation on $Fl_3(\mathbb{C}^4)$ confirms agreement with Givental and Lee's results when level structures are set to zero, validating the framework.