A gerbe for the elliptic gamma function

Felder, Giovanni; Henriques, André; Rossi, Carlo A.; Zhu, Chenchang

doi:10.1215/s0012-7094-08-14111-0

Cited by 11 publications

(13 citation statements)

References 22 publications

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“…Remark 1. If we have any group action of Γ on W , similar results hold as long as we pass to the stack [W/Γ] as proven in Theorem A.6 of [14]. In particular, if H i (W, ρ −1 S)) = 0 for all i > 0 then…”

Section: Cohomology and Alternating Classesmentioning

confidence: 66%

“…One option is to study meromorphic sections of a given gerbe. This direction has been pursued in a different context in the paper of Felder, Henriques, Rossi, and Zhu [14] and we hope that in our context those kind of relationships exist as well.…”

Section: Future Directionsmentioning

confidence: 97%

“…We define the moduli stack for topologicaly trivial gerbes as [H 2 (X, O)/H 2 (X, Z)] and define the universal (Poincaré) gerbe on the product of a torus and the moduli stack, proving that the moduli stack is fine. In the second appendix and comments on future work, we call on some tools from and explain some relations with a recent preprint of Polishchuk [20] and comment on similarities with work of Felder, Henriques, Rossi, and Zhu [14].…”

Section: Introductionmentioning

confidence: 99%

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Gerbes and the holomorphic Brauer group of complex tori

Ben-Bassat¹

2012

J. Noncommut. Geom.

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Contents 1. Introduction 1 2. Conventions 2 3. Acknowledgments 2 4. Cohomology and Alternating Classes 3 5. Recollection of Some Facts About Line Bundles 5 6. Topological Classes of Gerbes 8 7. The Appell-Humbert Theorem for Gerbes 12 8. A Groupoid in Sheaves of Sets 16 9. The Interaction With the Group Structure 17 10. The Universal Gerbe 23 11. Future Directions 29 12. Appendix 1 29 13. Appendix 2 30 References 34

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Section: Cohomology and Alternating Classesmentioning

confidence: 66%

Section: Future Directionsmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Gerbes and the holomorphic Brauer group of complex tori

Ben-Bassat¹

2012

J. Noncommut. Geom.

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“…In fact, it is worth noting that the very same cubic polynomials also appear in the modular transformation properties of the elliptic Gamma functions. These objects are not sections of a line bundle over an elliptic curve, rather they are sections of a gerbe on the universal triptic curve [112], and as such they enjoy SL(3, Z) Z 3 modular properties [113] rather than SL(2, Z) Z 2 as Theta functions, which do appear as 1-loop determinants of honest torus partition functions (7.15). Therefore, the D 2 × T 2 partition functions can be thought as defined on the torus of holonomies only in this generalized sense.…”

Section: Anomalies and Modularitymentioning

confidence: 99%

Localization of 4d $$ \mathcal{N} $$ = 1 theories on 𝔻2× 𝕋2

Longhi

Nieri

Pittelli

2019

J. High Energ. Phys.

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We consider 4d N = 1 gauge theories with R-symmetry on a hemisphere times a torus. We apply localization techniques to evaluate the exact partition function through a cohomological reformulation of the supersymmetry transformations. Our results represent the natural elliptic lifts of the lower dimensional analogs as well as a field theoretic derivation of the conjectured 4d holomorphic blocks, from which partition functions of compact spaces with diverse topology can be recovered through gluing. We also analyze the different boundary conditions which can naturally be imposed on the chiral multiplets, which turn out to be either Dirichlet or Robin-like. We show that different boundary conditions are related to each other by coupling the bulk to 3d N = 1 degrees of freedom on the boundary three-torus, for which we derive explicit 1-loop determinants.

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“…O × -gerbes on M can be tensored over Pic M giving the structure of a group to the set of equivalence classes which agrees with the cup product on cohomology classes. Some material on holomorphic gerbes both old and recent can be found in [5], [8], [4], [11], [9], [6], [2], [3], [14], [16] and [10]. In this paper, we will refer to O × -gerbes simply as grebes.…”

Section: Introductionmentioning

confidence: 99%

Equivariant gerbes on complex tori

Ben-Bassat¹

2013

Journal of Geometry and Physics

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We explore a new direction in representation theory which comes from holomorphic gerbes on complex tori. The analogue of the theta group of a holomorphic line bundle on a (compact) complex torus is developed for gerbes in place of line bundles. The theta group of symmetries of the gerbe has the structure of a Picard groupoid. We calculate it explicitly as a central extension of the group of symmetries of the gerbe by the Picard groupoid of the underlying complex torus. We discuss obstruction to equivariance and give an example of a group of symmetries of a gerbe with respect to which the gerbe cannot be equivariant. We calculate the obstructions to invariant gerbes for some group of translations of a torus to be equivariant. We survey various types of representations of the group of symmetries of a gerbe on the stack of sheaves of modules on the gerbe and the associated abelian category of sheaves on the gerbe (twisted sheaves).

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A gerbe for the elliptic gamma function

Cited by 11 publications

References 22 publications

Gerbes and the holomorphic Brauer group of complex tori

Gerbes and the holomorphic Brauer group of complex tori

Localization of 4d $$ \mathcal{N} $$ = 1 theories on 𝔻2× 𝕋2

Equivariant gerbes on complex tori

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