Benchmarks

56 MIGUEL MOREIRA

Remark 9.2. We point out that one can also define analogues of these operators in the quasi-parabolic descendent algebras $\backslash mathbb\{D\}\_\{\backslash alpha,\backslash bullet\}^\{\backslash mathrm\{qpar\}\}$ ; recall that this is a supercommutative algebras generated by symbols $\backslash mathrm\{ch\}\_k(\backslash gamma)$ for $\backslash gamma\backslash in\; H^*(C)$ and $\backslash mathrm\{ch\}\_k(e\_j)$ for $0\backslash le\; j\backslash le\; l+1$ , modulo certain simple relations; alternatively, it is generated by $\backslash mathrm\{ch\}\_k^H(\backslash gamma),\; \backslash mathrm\{ch\}\_k^H(e\_j)$ . One defines $\backslash mathsf\{L\}\_n^\{\backslash mathrm\{qpar\}\}:\backslash mathbb\{D\}\_\{\backslash alpha,\backslash bullet\}^\{\backslash mathrm\{qpar\}\}\backslash rightarrow\backslash mathbb\{D\}\_\{\backslash alpha,\backslash bullet\}^\{\backslash mathrm\{qpar\}\}$ by $\backslash mathsf\{L\}\_n^\{\backslash mathrm\{qpar\}\}=\backslash mathsf\{R\}\_n^\{\backslash mathrm\{qpar\}\}+\backslash mathsf\{T\}\_n^\{\backslash mathrm\{qpar\}\}$ where $\backslash mathsf\{R\}\_n^\{\backslash mathrm\{qpar\}\}$ is defined exactly in the same way as in the parabolic case, and $\backslash mathsf\{T\}\_n$ is

$$\backslash mathsf\{T\}\_n^\{\backslash mathrm\{qpar\}\}=\backslash sum\_\{a+b=n\}a!b!\backslash left((1-g)\backslash mathrm\{ch\}\_a(\backslash mathrm\{pt\})\backslash mathrm\{ch\}\_b(\backslash mathrm\{pt\})+\backslash sum\_\{j=0\}^l\backslash mathrm\{ch\}\_a(e\_j)\backslash mathrm\{ch\}\_b(e\_\{j+1\}-e\_j)\backslash right).$$

In particular, the Virasoro constraints do not depend on a choice of weights within a fixed chamber.

The main result of this section are the Virasoro constraints for moduli spaces of parabolic bundles:

Theorem 9.3. Let $\backslash alpha=(r,d,f\_\backslash bullet,c)\backslash in\; C(I)$ . For every $D\backslash in\backslash mathbb\{D\}^\{\backslash mathrm\{par\}\}$ we have

$$\backslash int\_\{[M\_\backslash alpha]\}\backslash mathsf\{L\}\_\{\backslash mathrm\{wt\}\_0\}(D)=0.$$

These generalize the Virasoro constraints for moduli spaces of stable bundles shown in [BLM]. For the proof, we will need compatibility statements between the Virasoro constraints and the structures that we have discussed earlier in the paper: wall-crossing, flag bundles, and Hecke operators. We develop these compatibilities first, and then give the proof in Section 9.5.

9.1. Primary states and wall-crossing compatibility.

Compatibility with wall-crossing is already the key ingredient in [BLM]. To state it, let us introduce the Lie subalgebra of primary states.

Definition 9.4. The space of primary states of weight $i$ on $\backslash mathbf\{V\}\_\{\backslash mathrm\{tr\}\}^\{\backslash mathrm\{par\}\}$ is

$$P\_i=\backslash left\backslash \{v\backslash in\backslash mathbf\{V\}\_\{\backslash mathrm\{tr\}\}^\{\backslash mathrm\{par\}\}:L\_0(v)=iv\backslash text\{\; and\; \}L\_n(v)=0\backslash text\{\; for\; \}n>0\backslash right\backslash \}.$$

The space of primary states on the Lie algebra $\backslash tilde\{\backslash mathbf\{V\}\}\_\{\backslash mathrm\{tr\}\}^\{\backslash mathrm\{par\}\}$ is

$$\backslash tilde\{P\}\_0=P\_1/T(P\_0)\backslash subseteq\backslash tilde\{\backslash mathbf\{V\}\}\_\{\backslash mathrm\{tr\}\}^\{\backslash mathrm\{par\}\}.$$

By [LM, Corollary 5.7], a class $u\backslash in\backslash tilde\{\backslash mathbf\{V\}\}\_\{\backslash mathrm\{tr\}\}^\{\backslash mathrm\{par\}\}$ (of non-trivial topological type) is a primary state if and only if $L\_\{\backslash mathrm\{wt\}\_0\}(u)=0$ , where

$$L\_\{\backslash mathrm\{wt\}\_0\}=\backslash sum\_\{n\backslash ge-1\}\backslash frac\{(-1)^n\}\{(n+1)!\}L\_\{-1\}^\{n+1\}\backslash circ\; L\_n:\backslash tilde\{\backslash mathbf\{V\}\}\_\{\backslash mathrm\{tr\}\}^\{\backslash mathrm\{par\}\}\backslash rightarrow\backslash mathbf\{V\}\_\{\backslash mathrm\{tr\}\}^\{\backslash mathrm\{par\}\}$$

is the dual of $L\_\{\backslash mathrm\{wt\}\_0\}$ . ¹⁷ In particular, the moduli space $M\_\backslash alpha$ satisfies the Virasoro constraints (i.e. Theorem 9.3 holds) if and only if $[M\_\backslash alpha]\backslash in\backslash tilde\{P\}\_0$ is a primary state.

We recall the reader that $\backslash mathbf\{V\}\_\{\backslash mathrm\{tr\}\}^\{\backslash mathrm\{par\}\}$ is a lattice vertex algebra (cf. Proposition 3.1). When the symmetrized Euler pairing $\backslash chi\_\{\backslash mathrm\{tr\}\}^\{\backslash mathrm\{sym\}\}$ is non-degenerate, this vertex algebra

¹⁷ When $\backslash mathbf\{V\}\_\{\backslash mathrm\{tr\}\}^\{\backslash mathrm\{par\}\}$ admits a conformal element $\backslash omega$ , $L\_\{\backslash mathrm\{wt\}\_0\}$ is a canonical lift of the operator $[-,\backslash omega]$ , see [BLM, Proposition 3.13, Lemma 3.15].