Last week I uploaded a newish paper to the arXiv: Decompositions of moduli spaces of vector bundles and graph potentials. Together with Sergey Galkin and Swarnava Mukhopadhyay we are in the process of splitting up the rather long preprint Graph potentials and moduli spaces of rank two bundles on a curve into 3 pieces:

We have replaced the original arXiv submission with the part concerning decompositions, because this is the part most cited so far, so that the arXiv id people have been using stays the same.

There are new results in this piece, and there will be in the other pieces, so stay tuned for more updates!

In this paper we propose a conjecture, which nowadays is referred to as the BGMN conjecture (where N stands for Narasimhan), describing a semiorthogonal decomposition for $\mathbf{D}^{\mathrm{b}}(\mathrm{M}_C(2,\mathcal{L}))$ where $C$ is a curve of genus $g\geq 2$ and $\deg\mathcal{L}$ is odd (so that this becomes a smooth projective Fano variety of dimension $3g-3$). Namely it is natural to expect a semiorthogonal decomposition $\mathbf{D}^{\mathrm{b}}(\mathrm{M}_C(2,\mathcal{L})) = \langle \mathbf{D}^{\mathrm{b}}(k), \mathbf{D}^{\mathrm{b}}(C), \mathbf{D}^{\mathrm{b}}(\operatorname{Sym}^2C), \ldots, \mathbf{D}^{\mathrm{b}}(\operatorname{Sym}^{g-2}C), \mathbf{D}^{\mathrm{b}}(\operatorname{Sym}^{g-1}C), \mathbf{D}^{\mathrm{b}}(\operatorname{Sym}^{g-2}C), \ldots, \mathbf{D}^{\mathrm{b}}(\operatorname{Sym}^2C), \mathbf{D}^{\mathrm{b}}(C), \mathbf{D}^{\mathrm{b}}(k) \rangle$

The main purpose of this paper is to give various pieces of evidence for it:

• graph potentials (introduced for completely different purposes involving quantum periods, discussed in the other 2 parts of the original preprint) are potential building blocks for a (so far conjectural) Landau–Ginzburg model mirror to these moduli spaces, and we show how the eigenspace decomposition of quantum cohomology is mirrored precisely by the decomposition of the critical locus for the potentials;
• the Grothendieck ring of categories is a linearisation of semiorthogonal decompositions, and receives a motivic realisation morphism from the Grothendieck ring of varieties: we exhibit a novel identity in the latter which (up to torsion) coincides with the identity defined by the conjectural semiorthogonal decomposition.