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We propose a conjectural semiorthogonal decomposition for the derivedcategory of the moduli space of stable rank 2 bundles with fixed determinant ofodd degree, independently formulated by Narasimhan. We discuss some evidencefor, and furthermore propose semiorthogonal decompositions with additionalstructure.  We also discuss two other decompositions. One is a decomposition of thismoduli space in the Grothendieck ring of varieties, which relates to variousknown motivic decompositions. The other is the critical value decomposition ofa candidate mirror Landau-Ginzburg model given by graph potentials, which inturn is related under mirror symmetry to Munoz's decomposition of quantumcohomology. This corresponds to an orthogonal decomposition of the Fukayacategory. We will explain how these decompositions can be seen as evidence forthe conjectural semiorthogonal decomposition.

Decompositions of moduli spaces of vector bundles and graph potentials