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Let $G$ be a semisimple algebraic group and $\Sigma$ a smooth proper curve. Then $\text{Bun}_G(\Sigma)$ comes equipped with a line bundle $\mathcal{L}$ which generates the torsion free part of $\text{Pic}\text{Bun}_G(\Sigma)$ (e.g. in type A or C it's the determinant bundle or in type B or D the Pfaffian bundle). Then the Verlinde formula is an explicit formula for $$H^0(\text{Bun}_G(\Sigma),\mathcal{L}^{\otimes k}),$$ intimately related to fusion products.

There are lots of nice proofs of the Verlinde formula published in the 1990s, e.g. Beauville's Conformal blocks, fusion rules and the Verlinde formula. However, they are all quite algebraic which makes it hard (at least for me) to understand what's going on. Given how much better the geometric side has been understood in recent decades (e.g. fusion and the BD Grassmannian), is there written up anywhere a slightly cleaner/more geometric proof of the Verlinde formula?

Edit: I'm mainly curious about whether there is a "more geometric" version of Beaville's proof, but am very happy to see other methods also.

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    $\begingroup$ P.S. There is a chance that Falting's 1994 paper A proof for the Verlinde formula just answers this, but I can't access a copy online. $\endgroup$
    – Pulcinella
    Commented Apr 7, 2021 at 9:48
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    $\begingroup$ I don't know what "modern" or "geometric" mean to you, but I would say the proof of Betram-Szenes (type A) is quite geometric. The main idea is that because the Verlinde dimensions are deformation-invariant and the moduli space of curves is irreducible, it suffices to prove it for a hyperelliptic curve, and then use the connection between moduli of bundles on hyperelliptic curves and Fano schemes of intersections of quadrics. The proof by Thaddeus (using stable pairs) I also geometric. $\endgroup$
    – ssx
    Commented Apr 7, 2021 at 10:26
  • $\begingroup$ The proof in Beauville's paper assumes the factorisation rules: the main point there is to show why this implies the formula actually conjectured/claimed by Verlinde. $\endgroup$
    – naf
    Commented Apr 9, 2021 at 12:40
  • $\begingroup$ Would you consider the approach of Alekseev-Meinreinken-Woodward as geometric (and modern)? arxiv.org/pdf/math/0005047.pdf -- though it is for compact groups $\endgroup$
    – DamienC
    Commented Apr 9, 2021 at 21:35

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This proof is geometric and from 2020, so it might satisfy the two requirements of the OP: A finite dimensional proof of Verlinde Formula.

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