Quantum $K$-invariants via Quot schemes I

1. Problem Statement

The paper addresses the computation and structural understanding of Quantum K-theory for Grassmannians, specifically $Gr(r, N)$. Quantum K-theory, introduced by Givental and Lee, generalizes Gromov–Witten theory by defining invariants as Euler characteristics of K-theory classes on moduli spaces of stable maps.

The primary challenges in this field include:

• Computational Complexity: Calculating quantum K-invariants (structure constants of the quantum K-ring) traditionally relies on the Kontsevich moduli space of stable maps, $M_{0,n}(X, d)$, which can be geometrically intricate.
• Lack of Explicit Formulas: While the quantum cohomology of Grassmannians is well-understood via Quot schemes, the K-theoretic analogues lacked a similarly robust computational framework, particularly for higher genera.
• Structural Understanding: Proving fundamental properties like the finiteness of structure constants (polynomiality in the Novikov variable $q$) and symmetries (e.g., $S_3$-symmetry) often requires complex geometric arguments.

The authors aim to establish a new framework using Quot schemes to compute these invariants, prove they form a Topological Quantum Field Theory (TQFT), and derive explicit presentations for the quantum K-ring.

2. Methodology

The authors employ a multi-faceted approach combining algebraic geometry, representation theory, and wall-crossing techniques:

• Quot Scheme Compactification: Instead of using the Kontsevich moduli space of stable maps, the authors utilize the Grothendieck Quot scheme, $Quot_d(C, N, r)$, which parametrizes short exact sequences $0 \to S \to \mathcal{O}_C^{\oplus N} \to Q \to 0$. This space provides a simpler compactification for morphisms from a curve $C$ to the Grassmannian.
• Virtual Structure Sheaves: Since Quot schemes over curves of genus $g > 0$ are not necessarily smooth, the authors utilize the virtual fundamental class and virtual structure sheaf ($O^{vir}$) constructed by Marian and Oprea. They define virtual Euler characteristics $\chi^{vir}$ to compute invariants.
• Stacky Evaluation Maps: A key technical innovation is the use of "stacky" evaluation maps ($\tilde{ev}_p$) from the Quot scheme to the quotient stack $X = [M_{r \times N} / GL_r(\mathbb{C})]$. This allows the definition of K-theoretic invariants even when the target is not a smooth manifold in the classical sense, bridging the gap between the Quot scheme and the representation ring $R(GL_r(\mathbb{C}))$.
• Wall-Crossing Formulas: To relate Quot scheme invariants to the standard quantum K-invariants (defined via stable maps), the authors apply two sequences of wall-crossing:
  1. $\epsilon$-wall-crossing: Relating stable maps ($\epsilon = \infty$) to $\epsilon$-stable quasimaps ($\epsilon = 0^+$).
  2. Light-to-Heavy wall-crossing: Relating quasimaps to Quot scheme invariants.
     They prove that for up to 3 marked points, these invariants coincide.
• Degeneration Formulas: The authors adapt degeneration techniques (Li, Liu, etc.) to the K-theoretic setting. They prove that virtual invariants satisfy gluing formulas that allow the decomposition of higher-genus invariants into lower-genus ones, establishing the TQFT structure.
• Vanishing Theorems: A critical component is a vanishing result (from their companion paper [SZ]) stating that the Euler characteristics of certain Schur bundles over high-degree Quot schemes vanish. This is proven via equivariant torus localization.

3. Key Contributions and Results

A. TQFT Structure

The authors establish that the virtual Euler characteristics of sheaves over Quot schemes of curves of any genus $g$ form a $(1+1)$-dimensional Topological Quantum Field Theory (TQFT) valued in the power series ring $\mathbb{Z}[[q]]$.

• Gluing Formula: They prove a gluing formula (Theorem 1.5) where invariants of a genus $g$ curve are obtained by summing products of invariants of genus $g_1$ and $g_2$ curves over a basis of the K-group.
• Orthogonality: They establish an orthogonality relation for the genus-zero pairing, showing that the "quantized dual" basis is orthogonal to the Schubert basis under the Quot scheme pairing.

B. Equivalence of Invariants

Theorem 1.13 is a central result: It proves that the 3-pointed genus-zero K-theoretic stable map invariants of the Grassmannian coincide exactly with the K-theoretic invariants defined via the Quot scheme.

• This equivalence allows the authors to replace the complex stable map moduli space with the more tractable Quot scheme for computing quantum K-products.

C. Structure Constants and Symmetries

Using the Quot scheme formulation, the authors derive a new formula for the structure constants $N^\nu_{\lambda, \mu}$ of the quantum K-ring:
$$N^\nu_{\lambda, \mu} = \sum_{d \geq 0} q^d \chi^{vir}(Quot_d(P^1, N, r), \tilde{O}_\lambda \cdot \tilde{O}_\mu \cdot \tilde{O}^*_\nu)$$

• $S_3$-Symmetry: This formula immediately reveals the $S_3$-symmetry ($N^\nu_{\lambda, \mu} = N^{\mu^*}_{\lambda, \nu^*}$), a property that was previously difficult to see from the stable map definition.
• Finiteness: They prove that the structure constants are polynomials in $q$ of degree at most $\min(r, N-r)$, providing a geometric explanation for the finiteness of the quantum K-ring via the vanishing of high-degree Euler characteristics.

D. Quantum Reduction Map and Ring Presentation

The authors introduce a Quantum Reduction Map $\kappa: R(GL_r(\mathbb{C})) \otimes \mathbb{Z}[[q]] \to QK(Gr(r, N))$.

• This map is a surjective ring homomorphism.
• It allows the reconstruction of the quantum K-ring from the representation ring of $GL_r(\mathbb{C})$.
• Using this map, they provide an independent proof of the Quantum K-Whitney presentation for the Grassmannian, deriving the relations:
  $$\wedge^y S \bullet \wedge^y Q = (1+y)^N - \frac{q}{1-q} y^k (\wedge^y S - 1) \bullet \det Q$$
  where $S$ and $Q$ are the universal subbundle and quotient bundle.

4. Significance

• New Computational Paradigm: The paper shifts the computational paradigm for Quantum K-theory of Grassmannians from stable maps to Quot schemes. This is significant because Quot schemes often admit simpler geometric descriptions and torus localization techniques.
• Unification of TQFTs: It demonstrates that the K-theoretic invariants of Grassmannians fit into a TQFT framework analogous to (but distinct from) the cohomological Verlinde TQFT. This provides a powerful structural tool for calculating invariants of arbitrary genus.
• Explicit Formulas: The approach offers a pathway to explicit formulas for quantum K-invariants (further explored in Part II of their work), similar to the Vafa-Intriligator formula in cohomology.
• Generalizability: The methodology (using Quot schemes, wall-crossing, and degeneration) is designed to be extendable to other target spaces, such as partial flag manifolds and isotropic Grassmannians, and potentially to general GIT quotients.

In summary, Sinha and Zhang provide a rigorous geometric foundation for Quantum K-theory of Grassmannians, proving that Quot schemes serve as a complete and computationally superior alternative to stable maps for defining and calculating these invariants, while revealing deep structural symmetries and ring presentations.