Generalized Theta Functions, Strange Duality, and Odd Orthogonal Bundles on Curves

Mukhopadhyay, Swarnava; Wentworth, Richard A.

doi:10.1007/s00220-019-03482-9

Cited by 8 publications

(4 citation statements)

References 58 publications

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“…In [40], Laszlo showed that the connection constructed by Hitchin and the one constructed by Tsuchiya-Ueno-Yamada [67] coincide under the natural identification between H 0 (M G (C), Det ⊗ℓ ) and V † 0 (C, g, ℓ). A similar result for twisted Spin groups was also proved by Mukhopadhyay-Wentworth [47]. The following questions are natural in the context of parabolic moduli spaces:…”

Section: Introductionsupporting

confidence: 55%

Geometrization of the TUY/WZW/KZ connection

Biswas¹,

Mukhopadhyay²,

Wentworth³

2021

Preprint

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Given a simple, simply connected, complex algebraic group G, a flat projective connection on the bundle of nonabelian theta functions on the moduli space of semistable parabolic G-bundles over any family of smooth projective curves with marked points was constructed by the authors in an earlier paper. Here, it is shown that the identification between the bundle of nonabelian theta functions and the bundle of WZNW conformal blocks is flat with respect to this connection and the one constructed by Tsuchiya-Ueno-Yamada. As an application, we give a geometric construction of the Knizhnik-Zamolodchikov connection on the trivial bundle over the configuration space of points in the projective line whose typical fiber is the space of invariants of tensor product of representations. BISWAS, MUKHOPADHYAY, AND WENTWORTH 7.1. The line bundle on the universal moduli stack 21 7.2. The relative uniformization and parabolic theta functions 22 8. Sugawara tensor and the parabolic Hitchin heat operator 22 8.1. Twisted D-modules via quasi-section of Drinfeld-Simpson 23 8.2. Projective heat operator from Sugawara 24 8.3. The parabolic duality pairing and the Hitchin symbol 25 8.4. Proof of the Main theorem (Theorem 1.1) 27 9. Proof of Theorem 6.3 28 9.1. Reduction to the SL r case 28 9.2. Reduction to the SL r with full flag 28 9.3. Reduction to abelian varieties 31 9.4. Abelianization and determinant of cohomology 33 10. Geometrization of the KZ equation on invariants 34 10.1. KZ connection 34 10.2. Invariants as global sections 34 10.3. Differential operators 35 References 36

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Section: Introductionsupporting

confidence: 55%

Geometrization of the TUY/WZW/KZ connection

Biswas¹,

Mukhopadhyay²,

Wentworth³

2021

Preprint

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“…An application of this fact is the following: Hitchin's construction [11] of a projectively flat connection on the space of generalized Spin(N ) theta functions works as well in the twisted case. This connection was used by the authors in [15] in their study of strange duality for odd orthogonal bundles.…”

Section: Resultsmentioning

confidence: 99%

Spectral Data for Spin Higgs Bundles

Mukhopadhyay

Wentworth

2019

International Mathematics Research Notices

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In this paper we determine the spectral data parametrizing Higgs bundles in a generic fiber of the Hitchin map for the case where the structure group is the special Clifford group with fixed Clifford norm. These are spin and "twisted" spin Higgs bundles. The method used relates variations in spectral data with respect to the Hecke transformations for orthogonal bundles introduced by Abe. The explicit description also recovers a result from the geometric Langlands program which states that the fibers of the Hitchin map are the dual abelian varieties to the corresponding fibers of the moduli spaces of projective orthogonal Higgs bundles (in the even case) and projective symplectic Higgs bundles (in the odd case).

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“…The conjecture was proved in type A by Belkale [Bel08], and Marian-Oprea [MO07]. Abe in [Abe08] proved strange duality in the symplectic setting conjectured by Beauville [Bea06] (see also [Bel12]), and has been studied for other cases [Muk16a,Muk16b,BP10,MW19]. In [DGT19c, Question 1] it was asked whether there are analogous geometric interpretations of dual spaces for vector spaces of conformal blocks defined by vertex operator algebras.…”

Section: Particular Examples: Computing Degreesmentioning

confidence: 99%

On global generation of vector bundles on the moduli space of curves from representations of vertex operator algebras

Damiolini¹,

Gibney²

2021

Preprint

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We consider global generation of sheaves of coinvariants on the moduli space of curves given by simple modules over certain vertex operator algebras, extending results for affine VOAs at integrable levels on stable pointed rational curves. Examples where global generation fails, and further evidence of positivity are given.When studying a projective variety, a natural objective is to characterize the maps admitted by it. From this perspective, vector bundles -locally free sheaves of finite rank -are important, as their sections may be regarded as (twisted) functions. Sections of globally generated bundles give rise to morphisms; and globally generated coherent sheaves determine rational maps which are regular on the locus where they are free.Sheaves of coinvariants V g (V, W • ) on M g,n , the moduli space of stable n-pointed curves of genus g, are given by admissible modules W i over a vertex operator algebra V [DGT19a]. If the vertex operator algebra (VOA for short) is C 2 -cofinite and of CFT-type (see §1.2 for definitions), these are coherent [DGT19a]. If also rational, simple modules form vector bundles [DGT19b]. Chern characters determine (rational) cohomological field theories if the vertex algebra is also self-contragredient, and in this case, Chern classes are in the tautological ring [DGT19c]. Well-known examples are given by representations of simple affine vertex operator algebras L (g) (derived from a simple Lie algebra g), the minimal series W -algebras (like the unitary discrete series Vir c ), the even lattice vertex algebras V L , and holomorphic vertex algebras (like the moonshine module V ). There are many others, for instance from VOAs obtained as tensor products, orbifold algebras, and through coset constructions, including parafermions K(g, ), and code VOAs ([DGT19b, §9], and [DGT19c, §6]).Affine vertex operator algebras and their coinvariants are well-understood. For g a simple Lie algebra, and ∈ C, = −h ∨ , where h ∨ is the dual Coxeter number, the simple affine vertex operator algebra L (g) is defined by a non-negatively graded vector space, with one dimensional degree zero component (so of CFTtype), and generated by its degree one component (isomorphic to g). For ∈ Z >0 , the sheaves V g (L (g), W • ) are vector bundles on M g,n [TUY89], which are globally generated on M 0,n [TUY89, Fak12].Here we initiate an investigation of global generation for other sheaves of coinvariants. Our main result is:Theorem 1. The sheaf of coinvariants V 0 (V, W • ) on M 0,n , defined by n simple admissible modules over a vertex operator algebra V of CFT-type, and generated in degree 1, is globally generated.By Corollary A, sheaves of coinvariants as in Theorem 1, are coherent on M 0,n , and by Corollary B, are locally free of finite rank on M 0,n . If V is also C 2 -cofinite and rational, then by Corollary C, they are globally generated vector bundles on M 0,n .By [DM06], for V of CFT-type, generated in degree 1, if V is also simple, rational, and self-contragredient, so that Theorem 1 and all three Coroll...

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Generalized Theta Functions, Strange Duality, and Odd Orthogonal Bundles on Curves

Cited by 8 publications

References 58 publications

Geometrization of the TUY/WZW/KZ connection

Geometrization of the TUY/WZW/KZ connection

Spectral Data for Spin Higgs Bundles

On global generation of vector bundles on the moduli space of curves from representations of vertex operator algebras

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