Conformal blocks from vertex algebras and their
connections on ℳg,n

Damiolini, Chiara; Gibney, Angela; Tarasca, Nicola

doi:10.2140/gt.2021.25.2235

Cited by 9 publications

(17 citation statements)

References 60 publications

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“…al. [8] extended Zhu's results to to prove finite dimensionality of H ch 0 (X, A V ) for an arbitrary smooth curve X, where A V is the chiral algebra associated to the C 2 -cofinite vertex algebra V . It would be interesting to know if finite dimensionality of HK • (A) and HP • (R V ) sufice to guarantee finite dimensionality of higher chiral homology in arbitrary genera.…”

Section: Convergencementioning

confidence: 95%

“…To generalize this work to higher genera a natural approach is to use the techniques of [8]. As mentioned in the introduction, it would be quite interesting to know if the finiteness conditions found in genus 1 in this article suffice to guarantee finite dimensionality at arbitrary genus.…”

Section: Convergencementioning

confidence: 99%

“…Regarding chiral homology in higher genus. Recently, it was proved in a series of papers by Damiolini, Gibney and Tarasca [8,9] that the degree 0 chiral homology of a C 2 cofinite vertex algebra V is finite dimensional in arbitrary curves, not only genus 1. Their approach is to describe the chiral homology on genus g by understanding the structure in the boundary of the moduli space, thus reducing to lower genera.…”

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confidence: 99%

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The First Chiral Homology Group

Ekeren¹,

Heluani²

2021

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We study the first chiral homology group of elliptic curves with coefficients in vacuum insertions of a conformal vertex algebra V . We find finiteness conditions on V guaranteeing that these homologies are finite dimensional, generalizing the C2-cofinite, or quasi-lisse condition in the degree 0 case. We determine explicitly the flat connections that these homologies acquire under smooth variation of the elliptic curve, as insertions of the conformal vector and the Weierstrass ζ function. We construct linear functionals associated to self-extensions of V -modules and prove their convergence under said finiteness conditions. These linear functionals turn out to be degree 1 analogs of the n-point functions in the degree 0 case. As a corollary we prove the vanishing of the first chiral homology group of an elliptic curve with values in several rational vertex algebras, including affine sl2 at non-negative integral level, the (2, 2k + 1)-minimal models and arbitrary simple affine vertex algebras at level 1. Of independent interest, we prove a Fourier space version of the Borcherds formula.

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Section: Convergencementioning

confidence: 95%

Section: Convergencementioning

confidence: 99%

mentioning

confidence: 99%

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The First Chiral Homology Group

Ekeren¹,

Heluani²

2021

Preprint

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“…Bundles of coinvariants from vertex algebras have much in common with their classical counterparts. For instance, both support a projectively flat logarithmic connection [TUY,DGT1] and satisfy factorization [TUY,DGT2], a property that makes recursive arguments about ranks and Chern classes possible.…”

mentioning

confidence: 99%

“…As in [MOP + 2], the explicit computation of the Atiyah algebra giving rise to the projectively flat logarithmic connection allows one to determine the recursion. As the Atiyah algebra in the case of bundles of coinvariants from vertex algebras was determined in [DGT1], one is able to extend the reconstruction of the CohFTs of coinvariants from affine Lie algebras in [MOP + 2] to the general case of vertex algebras.…”

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confidence: 99%