
## 2. Moduli

1. #### Problem 2.1.

a) Is there an explicit cycle $C_k \subset \overline{\mathcal{M}}_{g,1}(\mathbb{P}^1,d) \cap {\rm ev}_1^*(0)$ representing the class $\left[\overline{\mathcal{M}}_{g,1}(\mathbb{P}^1,d) \right]^{{\rm vir}}\cap {\rm ev}_1^*(0)\; \cdot \; \psi_1^k$ in $\overline{\mathcal{M}}_{g,1}(\mathbb{P}^1,d)$?

Partial answer (Okounkov, Pandharipande): if $k=\text{virdim} \,\overline{\mathcal{M}}_{g}(\mathbb{P}^1,d)$, then the answer is in terms of completed cycles that appear in the representation theory of symmetric groups.

b) Construct a virtual fundamental class of maps $f: (C,x) \rightarrow (\mathbb{P}^1,0)$ such that $df$ admits an $r$-th root locally at $x$. Show that this virtual fundamental class is equal to $\left[\overline{\mathcal{M}}_{g,1}(\mathbb{P}^1,d) \right]^{{\rm vir}}\cap {\rm ev}_1^*(0)\; \cdot \; \psi_1^k,$ ideally that it is represented by the cycle $C_k$.

Comforting evidence: in the smooth case, the local equation of $f$ is $f(z)=z^{k+1}\Longleftrightarrow df = (k+1)z^k dz$ has a $k$-th root.
• #### Problem 2.2.

Consider \begin{align*} (1) &\,\, \text{stab}_*([\overline{\mathcal{M}}^{\,\sim}_{g,n} (\mathbb{P}^1,a_i p_i)]^{\text{vir}});\\ (2) &\, \,\text{stab}_*(\text{Adm}_{g,n}(\mathbb{P}^1,a_ip_i)), \text{where Adm is the space of admissible covers;}\\ (3) &\, \,\sigma^*(0), \text{where}\,\, \sigma: \mathcal{M}_{g,n}^{ct}\rightarrow Pic^0_g \,\,\text{is given by}\,\,(C,p_1,..,p_n)\mapsto (C, \mathcal{O}(\Sigma a_ip_i)). \end{align*} It is known that (1)=(3) on $\mathcal{M}_{g,n}^{\rm ct}$.

a) Find an expression for (1) and (2) in the tautological ring of $\overline{\mathcal{M}}_{g,n}$.

b) Does (3) extend to $\overline{\mathcal{M}}_{g,n}$?

Cite this as: AimPL: Integrable systems in Gromov-Witten and symplectic field theory, available at http://aimpl.org/gwsymplectic.