[Paper Review] Intersection theory on moduli spaces of holomorphic bundles of arbitrary rank on a Riemann surface

Lisa C. Jeffrey, Frances Kirwan|ArXiv.org|Aug 23, 1996

Algebraic Geometry and Number Theory23 references4 citations

TL;DR

This paper provides a rigorous mathematical proof of Witten's formulas for intersection numbers on the moduli space of stable holomorphic vector bundles of arbitrary rank on a Riemann surface of genus g ≥ 2. Using equivariant cohomology and localization techniques, the authors derive explicit formulas for intersection pairings in the cohomology ring of M(n,d), and use them to establish the Verlinde formula for the dimension of spaces of holomorphic sections of line bundles over M(n,d).

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 mathematically rigorous derivation of intersection numbers on the moduli space M(n,d) of stable holomorphic vector bundles of rank n and degree d on a Riemann surface of genus g ≥ 2.
• To prove Witten's conjectured formulas for intersection pairings in the cohomology of M(n,d), originally derived using physical methods.
• To use the derived intersection formulas to give a complete proof of the Verlinde formula for the dimension of the space of holomorphic sections of a line bundle over M(n,d).
• To correct an error in the original version of the paper concerning the structure of the moduli space and its equivariant cohomology.
• To establish a bridge between mathematical physics and algebraic geometry by rigorously validating a key physical prediction in the context of stable vector bundles.

Proposed method

• Employing equivariant cohomology techniques on the space of holomorphic bundles to analyze the action of a maximal torus on the moduli space M(n,d).
• Applying the Atiyah-Bott localization formula to compute intersection numbers by restricting to fixed points of the torus action.
• Using the Kirwan surjection to relate the cohomology of M(n,d) to the equivariant cohomology of the space of connections on the Riemann surface.
• Computing the Chern classes of tautological bundles and their intersections via residue computations in equivariant cohomology.
• Deriving explicit formulas for the intersection numbers of products of Chern classes of universal bundles on M(n,d).
• Verifying that the derived formulas match Witten's physical predictions and satisfy the required consistency conditions for the Verlinde formula.

Experimental results

Research questions

• RQ1 What are the precise formulas for intersection pairings in the cohomology ring of the moduli space M(n,d) of stable holomorphic vector bundles of rank n and degree d on a Riemann surface of genus g ≥ 2?
• RQ2 How can Witten’s physical derivation of these intersection numbers be rigorously reconstructed using algebraic and differential geometry?
• RQ3 Can the derived intersection formulas be used to prove the Verlinde formula for the dimension of the space of holomorphic sections of a line bundle over M(n,d)?
• RQ4 What is the correct structure of the equivariant cohomology of the space of connections on a Riemann surface, and how does it relate to the cohomology of M(n,d)?
• RQ5 How do the intersection numbers behave under the action of the mapping class group, and what symmetries are encoded in the formulas?

Key findings

• The authors derive explicit formulas for intersection pairings in the cohomology ring of M(n,d) for arbitrary rank n and degree d, with g ≥ 2, confirming Witten’s conjectures.
• The intersection numbers are computed via localization in equivariant cohomology, yielding rational functions in the equivariant parameters that encode the topological invariants of M(n,d).
• The derived formulas are shown to be consistent with the Verlinde formula, providing a complete mathematical proof of the dimension of the space of holomorphic sections of a line bundle over M(n,d).
• An error in the original version concerning the structure of the Kirwan map and the equivariant cohomology of the frame bundle is corrected in the revised version.
• The intersection numbers are expressed as residues of rational functions, and their invariance under the action of the mapping class group is established.
• The final formulas for the Verlinde dimension match the known physical predictions and are valid for all coprime pairs (n,d) and genus g ≥ 2.