[Paper Review] Intersection pairings in moduli spaces of vector bundles of arbitrary rank over a Riemann surface

Lisa C. Jeffrey, Frances Kirwan|arXiv (Cornell University)|Aug 23, 1996

Algebraic Geometry and Number Theory9 citations

TL;DR

This paper provides a mathematical proof of Witten's formulas for intersection pairings in the cohomology of the moduli space M(n,d) of stable vector bundles of rank n and degree d on a Riemann surface of genus g ≥ 2, with gcd(n,d)=1. Using these intersection formulas, it derives the Verlinde formula for the dimension of holomorphic sections of line bundles over M(n,d), establishing a rigorous foundation for a key result in mathematical physics.

ABSTRACT

We prove formulas (found by Witten in 1992 using physical methods) for intersection pairings in the cohomology of the moduli space M(n,d) of stable holomorphic vector bundles of rank n and degree d (assumed coprime) on a Riemann surface of genus g greater than or equal to 2. We also use these formulas for intersection numbers to obtain a proof of the Verlinde formula for the dimension of the space of holomorphic sections of a line bundle over M(n,d).

Motivation & Objective

• To provide a rigorous mathematical proof of Witten's physical formulas for intersection pairings in the cohomology of M(n,d).
• To establish a connection between intersection theory on moduli spaces and the Verlinde formula.
• To demonstrate that intersection numbers computed via the proposed formulas yield the correct dimension of holomorphic sections of line bundles over M(n,d).
• To extend the understanding of cohomological invariants in moduli spaces of stable vector bundles over Riemann surfaces.

Proposed method

• Utilizes algebraic geometry techniques to analyze the cohomology ring of the moduli space M(n,d) of stable holomorphic vector bundles.
• Applies the theory of characteristic classes and Mumford's tautological classes to compute intersection pairings.
• Employs the assumption that rank n and degree d are coprime to ensure the moduli space is smooth and irreducible.
• Uses the Chern character and Todd class to express intersection numbers in terms of topological invariants.
• Relies on the structure of the moduli space as a projective variety with a natural polarization.
• Derives the Verlinde formula by interpreting the intersection numbers as dimensions of spaces of holomorphic sections.

Experimental results

Research questions

• RQ1 How can Witten’s physical formulas for intersection pairings in M(n,d) be rigorously derived using mathematical methods?
• RQ2 What is the precise relationship between intersection numbers in the cohomology of M(n,d) and the dimension of spaces of holomorphic sections of line bundles?
• RQ3 Can the Verlinde formula be obtained as a consequence of intersection theory on the moduli space of stable vector bundles?
• RQ4 What role does the coprimality of n and d play in ensuring the validity of the intersection formulas and the Verlinde formula?

Key findings

• The paper successfully proves Witten’s formulas for intersection pairings in the cohomology of M(n,d) using purely mathematical techniques.
• The derived intersection numbers match the predictions of the Verlinde formula for the dimension of holomorphic sections of line bundles over M(n,d).
• The Verlinde formula is established as a consequence of the intersection theory on the moduli space, providing a geometric interpretation.
• The results hold under the condition that gcd(n,d)=1, ensuring the moduli space is smooth and irreducible.
• The method confirms the consistency of mathematical and physical approaches to the geometry of vector bundle moduli spaces.
• The cohomological structure of M(n,d) is shown to encode deep arithmetic and topological information through intersection pairings.