[Paper Review] Permutation-equivariant quantum K-theory III. Lefschetz' formula on $\overline{M}_{0,n}/S_n$ and adelic characterization

Alexander Givental|arXiv (Cornell University)|Aug 27, 2015

Algebraic Geometry and Number Theory8 references3 citations

TL;DR

This paper establishes a complete characterization of the big J-function in permutation-equivariant quantum K-theory for the point target space $X = \text{pt}$ using Lefschetz' fixed point formula on the moduli space $\overline{M}_{0,n+1}/S_n$. It derives a recursion via equivariant cohomology and proves that the J-function's range is swept by subspaces generated by the small J-function, providing a foundational adelic characterization applicable to general target spaces.

ABSTRACT

We continue our study of the genus-$0$ permutation-equivariant quantum K-theory of the target $X=pt$, and completely determine the "big J-function" of this theory. The computation is based on the application of Lefschetz' fixed point formula to the action of $S_n$ on $\overline{M}_{0,n+1}$. It is an instance of the general "adelic characterization" (which we state at the end with reference to arXiv:1106.3136) of quantum K-theory for any target $X$ in terms of quantum cohomology theory. Yet, some simplifications of non-conceptual nature occur in this example, making it a lucid illustration to the general theory.

Motivation & Objective

• To fully determine the big J-function in genus-0 permutation-equivariant quantum K-theory for the point target space $X = \text{pt}$.
• To apply Lefschetz' fixed point formula to the $S_n$-action on $\overline{M}_{0,n+1}$ to derive a recursive structure for correlators.
• To establish a general adelic characterization of genus-0 quantum K-theory for arbitrary target spaces $X$, extending prior results.
• To clarify the role of Adams operations and $\lambda$-algebra structures in the construction of the J-function and its range.
• To show that the J-function's range lies within a union of subspaces generated by the small J-function, enabling analytic continuation beyond the infinitesimal neighborhood of $1 - q$.

Proposed method

• Use of Lefschetz' fixed point formula on $\overline{M}_{0,n+1}$ with $S_n$-action to express equivariant traces of cohomology classes as holomorphic Euler characteristics on fixed-point loci.
• Application of the Quantum Hirzebruch-Riemann-Roch theorem to relate virtual and ordinary Euler characteristics in the case $X = \text{pt}$.
• Decomposition of the loop space $\mathcal{K} = \mathcal{K}_+ \oplus \mathcal{K}_-$ into positive and negative parts, with $\mathcal{K}_+$ consisting of Laurent polynomials and $\mathcal{K}_-$ of rational functions regular at $q=0$, to define the J-function as a graph $\mathcal{K}_+ \to \mathcal{K}_-$.
• Derivation of a recursion for the J-function by analyzing the action of permutations $h \in S_n$ near the $n+1$-st marked point, leading to a system of equations in the $\Lambda$-valued formal loop space.
• Use of the formal implicit function theorem to show that the projection of the J-function's range to $\mathcal{K}_+$ is surjective in a neighborhood of $1 - q$, ensuring full description of the graph.
• Generalization to arbitrary $X$ via fake holomorphic Euler characteristics and Kawasaki's Riemann-Roch formula on $X_{g,n,d}/S_n$, leading to an adelic characterization involving $m$-th roots of unity and $\Psi^m$-actions on operator series.

Experimental results

Research questions

• RQ1 How can the big J-function in permutation-equivariant quantum K-theory for $X = \text{pt}$ be completely determined using equivariant cohomological techniques?
• RQ2 What is the precise structure of the J-function’s range in the loop space $\mathcal{K}$, and how does it relate to the small J-function and $\Psi^k$-operations?
• RQ3 How does Lefschetz’ fixed point formula on $\overline{M}_{0,n+1}/S_n$ yield a recursion for the correlators in the J-function?
• RQ4 What is the general adelic characterization of genus-0 permutation-equivariant quantum K-theory for arbitrary target spaces $X$?
• RQ5 How do Adams operations $\Psi^m$ and the $\lambda$-algebra structure on $\Lambda$ interact with the $S_n$-equivariant geometry to define the J-function's values at roots of unity?

Key findings

• The range of the big J-function $\mathcal{J}_{\text{pt}}$ in the loop space $\mathcal{K}$ is swept by the family of subspaces $\bigcup_{\nu \in \Lambda_+} (1 - q) \exp\left(\sum_{k > 0} \frac{\Psi^k(\nu)}{k(1 - q^k)} \right) \mathcal{K}_+$, providing a complete description of the J-function's image.
• The J-function is characterized as the graph of a formal function $\mathcal{K}_+ \to \mathcal{K}_-$ defined in an infinitesimal neighborhood of $1 - q$, with the projection to $\mathcal{K}_+$ being surjective due to the formal implicit function theorem.
• For $X = \text{pt}$, the fake and ordinary quantum K-theory coincide because $\overline{M}_{0,n}$ are manifolds, allowing the use of standard holomorphic Euler characteristics.
• The general adelic characterization for arbitrary $X$ involves conditions at all primitive $m$-th roots of unity $\zeta$, where $\mathcal{J}_X(\mathbf{t})_{(\zeta)}(q^{1/m}/\zeta)$ lies in a specific $\Psi^m$-twisted image of the positive loop space $\widehat{\mathcal{K}}^X_+$.
• The operator series $S_\tau(q)^{-1}$, whose columns are partial derivatives of the small J-function, governs the tangent spaces to the J-function's range, and its $\Psi^m$-action is defined via the $\lambda$-structure on $\Lambda$ and $\Psi^m(q) = q^m$.
• The result confirms that the J-function's range is contained in the union of subspaces generated by the small J-function, even though the union is larger than the range, providing a natural analytic continuation of the function beyond the formal neighborhood of $1 - q$.