[Paper Review] BPS invariants of semi-stable sheaves on rational surfaces

Jan Manschot|arXiv (Cornell University)|Sep 22, 2011

Algebraic Geometry and Number Theory33 references4 citations

TL;DR

This paper develops a systematic method to compute BPS invariants—refined supersymmetric indices—for semi-stable sheaves of rank ≤3 on rational surfaces, particularly Hirzebruch surfaces $\Sigma_\ell$ and $\mathbb{P}^2$. It introduces extended Harder-Narasimhan filtrations and a generalized blow-up formula to handle cases where standard invariants vanish or the $\gcd(c_1\cdot H, r) = 1$ condition fails, yielding explicit product formula generating functions and confirming agreement with known results under $\gcd(r,c_1)=1$.

ABSTRACT

BPS invariants are computed, capturing topological invariants of moduli spaces of semi-stable sheaves on rational surfaces. For a suitable stability condition, it is proposed that the generating function of BPS invariants of a Hirzebruch surface takes the form of a product formula. BPS invariants for other stability conditions and other rational surfaces are obtained using Harder-Narasimhan filtrations and the blow-up formula. Explicit expressions are given for rank <4 sheaves on a Hirzebruch surface or the projective plane. The applied techniques can be applied iteratively to compute invariants for higher rank.

Motivation & Objective

• To compute BPS invariants for semi-stable sheaves of rank ≤3 on Hirzebruch surfaces $\Sigma_\ell$ and $\mathbb{P}^2$, especially when standard invariants vanish due to stability conditions.
• To extend the applicability of generating functions for BPS invariants beyond the $\gcd(c_1\cdot H, r) = 1$ constraint that limits the original blow-up formula.
• To develop a recursive method using extended Harder-Narasimhan filtrations to compute invariants across walls of marginal stability.
• To provide explicit expressions for virtual Poincaré functions and Betti numbers of moduli spaces, verified against known results under $\gcd(r,c_1)=1$.

Proposed method

• Introduces extended HN filtrations with non-strict stability ordering $p_J(E_i,n) \succeq p_J(E_{i+1},n)$, enabling systematic wall-crossing computations.
• Proposes Eq. (4.2) as a generating function for virtual Poincaré functions of sheaves on $\Sigma_\ell$ with semi-stable restriction to fiber $f$, generalizing to virtual Hodge functions via Eq. (4.9).
• Applies wall-crossing via subtraction of generating functions corresponding to extended HN filtrations (Eq. (5.8)), analogous to techniques in vector bundles on curves.
• Uses the generalized blow-up formula to relate invariants on $\tilde{\mathbb{P}}^2$ to those on $\mathbb{P}^2$, extending beyond the $\gcd(c_1\cdot H, r)=1$ restriction.
• Transforms $\mu$-stable invariants to Gieseker invariants using the invariant (5.8), enabling recursive computation of BPS invariants for higher rank.
• Employs modular forms and theta functions (e.g., $h_{r,0}(z,\tau)$) to express generating functions, with final Betti numbers extracted after subtraction of $\frac{1}{3}h_{1,0}(3z,3\tau)$.

Experimental results

Research questions

• RQ1 How can BPS invariants be computed for semi-stable sheaves on rational surfaces when the standard blow-up formula fails due to $\gcd(c_1\cdot H, r) \neq 1$?
• RQ2 What is the role of extended Harder-Narasimhan filtrations in computing BPS invariants across walls of marginal stability?
• RQ3 How do the generating functions for BPS invariants on $\Sigma_\ell$ relate to those on $\mathbb{P}^2$ via the blow-up formula?
• RQ4 To what extent do the computed Betti numbers of moduli spaces agree with Euler numbers from toric localization when $\gcd(r,c_1) > 1$?
• RQ5 Can the method be generalized to compute BPS invariants for higher rank sheaves beyond rank 3?

Key findings

• The generating function for BPS invariants on $\Sigma_\ell$ with a suitable polarization is given by a product formula (Eq. (4.2)) for rank $r \geq 1$, valid even when $c_1 \cdot f \not\equiv 0 \mod r$.
• For $r=3$, $c_1=0$, and $3 \leq c_2 \leq 6$, the Betti numbers $b_n$ of moduli spaces on $\mathbb{P}^2$ are explicitly computed and listed in Table 1, with $\chi = 18, 216, 1512, 8109$ for $c_2 = 3,4,5,6$ respectively.
• The first non-vanishing Betti number $b_0 = 1$ for all $c_2 \geq 3$, and $b_2 = 1$ for $c_2=3$, increasing with $c_2$, consistent with expected complex dimension $\dim_{\mathbb{C}}\mathcal{M} = 2c_2 - 3$.
• The method correctly reproduces known results under $\gcd(r,c_1)=1$, confirming agreement with Refs. [26, 28, 32, 22] for $\mu$-stable loci.
• When $\gcd(r,c_1) > 1$, the computed BPS invariants differ from Euler numbers of $\mu$-stable loci, suggesting a non-trivial difference between BPS invariants and Euler characteristics in the semi-stable case.
• The recursive procedure based on extended HN filtrations and wall-crossing allows iterative computation of BPS invariants for higher rank, generalizing the rank 3 results.