[Paper Review] A Cyclic Operad in the Category of Artin Stacks and Gravitational Correlators

Ivan Kausz|ArXiv.org|Feb 25, 2008

Homotopy and Cohomology in Algebraic Topology3 citations

TL;DR

This paper constructs a cyclic operad in the category of Artin stacks using a new moduli stack $\widetilde{\mathcal{M}}_{0,2}$, which serves as a substitute for the non-existent moduli space of stable two-pointed genus-zero curves. By passing to homology, the authors obtain a linear cyclic operad that conjecturally governs the cohomology of smooth projective varieties and gravitational quantum cohomology, with explicit verification of known gravitational correlator relations via algebraic identities in the homology ring.

ABSTRACT

We define an Artin stack which may be considered as a substitute for the non-existing (or empty) moduli space of stable two-pointed curves of genus zero. We show that this Artin stack can be viewed as the first term of a cyclic operad in the category of stacks. Applying the homology functor we obtain a linear cyclic operad. We formulate conjectures which assert that cohomology of a smooth projective variety has the structure of an algebra over this homology operad and that gravitational quantum cohomology can naturally expressed in terms of this algebra. As a test for these conjectures we show how certain well-known relations between gravitational correlators can be deduced from them.

Motivation & Objective

• To resolve the absence of a well-defined moduli space for stable two-pointed genus-zero curves by constructing a substitute stack $\widetilde{\mathcal{M}}_{0,2}$.
• To extend the standard operad $A_*(\overline{M}_{0,n+1})$ to include $n=1$ by defining a cyclic operad structure on stacks $\widetilde{\mathcal{M}}_{0,n+1}$ for $n \geq 1$.
• To conjecture that the cohomology of a smooth projective variety carries an algebraic structure over the homology of this extended operad.
• To show that gravitational correlators can be naturally expressed in terms of this structure, using the new classes $\tilde{\psi}_i$ on the extended stacks.

Proposed method

• Define $\mathscr{W}$ as a zero-dimensional Artin stack parameterizing modifications of a fixed one-pointed curve.
• Construct $\widetilde{\mathcal{M}}_{0,2} := \mathbf{B}\mathbb{G}_m \times \mathscr{W}$ and $\widetilde{\mathcal{M}}_{0,n+1} := \overline{M}_{0,n+1} \times \mathscr{W}^{n+1}$ for $n \geq 2$, forming a cyclic operad in the category of Artin stacks.
• Prove that $\widetilde{\mathcal{M}}_{0,2}$ has a semigroup structure via the clutching of 'zollstocks'—two-pointed prestable genus-zero curves with linear dual graphs and marked points on extremal components.
• Pass to homology to obtain the linear cyclic operad $(A_*(\widetilde{\mathcal{M}}_{0,n+1}))_{n \geq 1}$, with $A_*(\widetilde{\mathcal{M}}_{0,2}) \cong R[t_0]$ and $A_*(\widetilde{\mathcal{M}}_{0,n+1}) \cong A_*(\overline{M}_{0,n+1}) \otimes R^{\otimes(n+1)}$ for $n \geq 2$, where $R$ is the intersection ring of $\mathscr{W}$.
• Define a non-commutative, non-unital product $\odot$ on $R[t_0]$ induced by the semigroup structure, and show it admits a basis of elements $t_0^{k_1} \odot \cdots \odot t_0^{k_r}$.
• Construct natural morphisms $\widetilde{\operatorname{st}}: \overline{M}_{g,n}(V,\beta) \to \widetilde{\mathcal{M}}_{g,n}$, lifting the usual stabilization map, and define $\tilde{\psi}_i$ classes as pullbacks of tautological classes on $\widetilde{\mathcal{M}}_{g,n}$.

Experimental results

Research questions

• RQ1 Can a well-defined substitute for the non-existent moduli space of stable two-pointed genus-zero curves be constructed in the category of Artin stacks?
• RQ2 Does the stack $\widetilde{\mathcal{M}}_{0,2}$ admit a natural semigroup structure compatible with a cyclic operad structure on $\widetilde{\mathcal{M}}_{0,n+1}$ for $n \geq 1$?
• RQ3 Is the cohomology of a smooth projective variety naturally an algebra over the homology operad $A_*(\widetilde{\mathcal{M}}_{0,n+1})$?
• RQ4 Can gravitational correlators be systematically expressed using the new $\tilde{\psi}_i$ classes and the extended operad structure?

Key findings

• The stack $\widetilde{\mathcal{M}}_{0,2}$ is isomorphic to $\mathbf{B}\mathbb{G}_m \times \mathscr{W}$, where $\mathscr{W}$ is a zero-dimensional Artin stack parameterizing modifications of a fixed one-pointed curve.
• The homology ring $A_*(\widetilde{\mathcal{M}}_{0,2})$ is isomorphic to $R[t_0]$, where $R$ is the intersection ring of $\mathscr{W}$, and carries a non-commutative, non-unital product $\odot$ induced by the semigroup structure of $\widetilde{\mathcal{M}}_{0,2}$.
• The operad $(\widetilde{\mathcal{M}}_{0,n+1})_{n \geq 1}$ is a cyclic operad in the category of Artin stacks, with $A_*(\widetilde{\mathcal{M}}_{0,n+1}) \cong A_*(\overline{M}_{0,n+1}) \otimes R^{\otimes(n+1)}$ for $n \geq 2$.
• The conjecture is supported by showing that Theorem 1.2 of [KM] follows from a simple identity in $R[t_0]$, and that an extension of this theorem to the unstable range also follows from the conjectures.
• The morphism $\widetilde{\operatorname{st}}: \overline{M}_{g,n}(V,\beta) \to \widetilde{\mathcal{M}}_{g,n}$ lifts the usual stabilization map, and the $\tilde{\psi}_i$ classes are pullbacks of tautological classes on $\widetilde{\mathcal{M}}_{g,n}$, enabling a potential formulation of gravitational quantum cohomology in terms of the extended operad.