The motivic generating-function conjecture for stable-bundle moduli spaces

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Let SS be the completed and suitably localized Grothendieck ring of motives over C\mathbb{C}, and let Γ\Gamma denote the set of characters α=(n,d)\alpha=(n,d). For each α∈Γ\alpha\in\Gamma, let RαR_\alpha be the motive of the stack of semistable bundles on XX with character α\alpha, and let AαA_\alpha be the motive of the moduli space M(α)\mathcal{M}(\alpha) of stable bundles. For \mat\mutemp∈Q\mat{\mutemp}\in\mathbb{Q}, define

R\mat\mutemp=1+∑\mat\mutemp(α)=\mat\mutempRαxα,A\mat\mutemp=∑\mat\mutemp(α)=\mat\mutempAαxα.R_\mat{\mutemp}=1+\sum_{\mat{\mutemp}(\alpha)=\mat{\mutemp}}R_\alpha x^\alpha, \qquad A_\mat{\mutemp}=\sum_{\mat{\mutemp}(\alpha)=\mat{\mutemp}}A_\alpha x^\alpha.

Let SLtw[[x1,x2]]S^{\rm tw}_{\mathbb{L}}[[x_1,x_2]] have multiplication xα∘xβ=L−⟨α,β⟩xα+βx^\alpha\circ x^\beta=\mathbb{L}^{-\langle\alpha,\beta\rangle}x^{\alpha+\beta}. Motivic generating-function conjecture. One has

R\mat\mutemp∘Exp⁡(A\mat\mutemp1−L)R_\mat{\mutemp}\circ\operatorname{Exp}\left(\frac{A_\mat{\mutemp}}{1-\mathbb{L}}\right)

in SLtw[[x1,x2]]S^{\rm tw}_{\mathbb{L}}[[x_1,x_2]]. This is the motivic analogue of the preceding virtual-Hodge-polynomial formula; the supplied text does not state whether the conjecture has been resolved.

References

Primary source

Sergey Mozgovoy, “Poincare polynomials of moduli spaces of stable bundles over curves”, arXiv:0711.0634 (2007).

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