T-Duality, Jacobi Forms and Witten Gerbe Modules

Bundle gerbes for Chern-Simons and Wess-Zumino-Witten theories

We develop the theory of Chern-Simons bundle 2-gerbes and multiplicative bundle gerbes associated to any principal $G$-bundle with connection and a class in $H^4(BG, \ZZ)$ for a compact semi-simple Lie group $G$. The Chern-Simons bundle 2-gerbe realises differential geometrically the Cheeger-Simons invariant. We apply these notions to refine the Dijkgraaf-Witten correspondence between three dimensional Chern-Simons functionals and Wess-Zumino-Witten models associated to the group $G$. We do this by introducing a lifting to the level of bundle gerbes of the natural map from $H^4(BG, \ZZ)$ to $H…

A quantum Leray-Hirsch theorem for banded gerbes

For a gerbe $\Y$ over a smooth proper Deligne-Mumford stack $\B$ banded by a finite group $G$, we prove a structure result on the Gromov-Witten theory of $\Y$, expressing Gromov-Witten invariants of $\Y$ in terms of Gromov-Witten invariants of $\B$ twisted by various flat $U(1)$-gerbes on $\B$. This is interpreted as a Leray-Hirsch type of result for Gromov-Witten theory of gerbes.

A generalization of Abel's Theorem and the Abel--Jacobi map

We generalize Abel's classical theorem on linear equivalence of divisors on a Riemann surface. For every closed submanifold $M^d \subset X^n$ in a compact oriented Riemannian $n$--manifold, or more generally for any $d$--cycle $Z$ relative to a triangulation of $X$, we define a (simplicial) $(n-d-1)$--gerbe $Λ_{Z}$, the Abel gerbe determined by $Z$, whose vanishing as a Deligne cohomology class generalizes the notion of `linear equivalence to zero'. In this setting, Abel's theorem remains valid. Moreover we generalize the classical Inversion Theorem for the Abel--Jacobi map, thereby proving th…

Affine su(2) fusion rules from gerbe 2-isomorphisms

We give a geometric description of the fusion rules of the affine Lie algebra su(2)_k at a positive integer level k in terms of the k-th power of the basic gerbe over the Lie group SU(2). The gerbe can be trivialised over conjugacy classes corresponding to dominant weights of su(2)_k via a 1-isomorphism. The fusion-rule coefficients are related to the existence of a 2-isomorphism between pullbacks of these 1-isomorphisms to a submanifold of SU(2) x SU(2) determined by the corresponding three conjugacy classes. This construction is motivated by its application in the description of junctions of…

Gerbes, (twisted) K-theory, and the supersymmetric WZW model

The aim of this talk is to explain how symmetry breaking in a quantum field theory problem leads to a study of projective bundles, Dixmier-Douady classes, and associated gerbes. A gerbe manifests itself in different equivalent ways. Besides the cohomological description as a DD class, it can be defined in terms of a family of local line bundles or as a prolongation problem for an (infinite-dimensional) principal bundle, with the fiber consisting of (a subgroup of) projective unitaries in a Hilbert space. The prolongation aspect is directly related to the appearance of central extensions of (br…

gauge理论II：同调环面的情况

A gerbe-like construction in gauge theory II: the case of homology tori

本文研究同调环面的奇确定性，通过构造反线性Z/4作用，恢复Baraglia(2023)关于闭合spin4流形mod2Seiberg-Witten不变量的部分结果。