1,354,741 research outputs found
A classification of equivariant gerbe connections
No full textLet G be a compact Lie group acting on a smooth manifold M. In this paper, we consider Meinrenken's G-equivariant bundle gerbe connections on M as objects in a 2-groupoid. We prove this 2-category is equivalent to the 2-groupoid of gerbe connections on the differential quotient stack associated to M, and isomorphism classes of G-equivariant gerbe connections are classified by degree 3 differential equivariant cohomology. Finally, we consider the existence and uniqueness of conjugation-equivariant gerbe connections on compact semisimple Lie groups.<br
The Nori fundamental gerbe of a fibered category
No full textWe extend Nori's theory of the fundamental group scheme to a theory of the fundamental gerbe, which applies to schemes, algebraic stacks, and more general fibered categories, even in absence of a rational point. We give a tannakian interpretation of the fundamental gerbe in terms of essentially finite bundles, extending Nori's correspondence for complete varieties with a rational point. We also show how our formalism allows a natural formulation of Grothendieck's section conjecture in arbitrary characteristic
A gerbe-like construction in gauge theory
No full textIn 2022 Baraglia and Konno showed the following: for a smooth family of a
homotopy surface , if the tangent
bundle along the fibers admits a spin structure, then
also admits a spin structure, where
is the vector bundle consisting of self-dual
harmonic 2-forms. In this paper, we show that admits a canonical spin structure. The proof is
carried out by canonically constructing a lifting -gerbe for the spin
structure on using the families Seiberg--Witten
equations, starting from a lifting -gerbe for the spin structure on .Comment: 71 page
Z2 lattice Gerbe Theory
No full textDiscretized formulations of 2-form Abelian and non-Abelian gauge fields on d-dimensional hypercubiclattices have been discussed in the past by various authors and most recently by Lipstein and Reid-Edwards [J. High Energy Phys. 09 (2014) 034]. In this paper we recall that the Hamiltonian of a Z2 variant ofsuch theories is one of the family of generalized Ising models originally considered by Wegner. For such“Z2 lattice Gerbe theories” general arguments can be used to show that a phase transition for Wilsonsurfaces will occur for d > 3 between volume and area scaling behavior. In 3d the model is equivalentunder duality to an infinite coupling model and no transition is seen, whereas in 4d the model is dual to the 4d Ising model and displays a continuous transition. In 5d the Z 2 lattice Gerbe theory is self-dual in the presence of an external field and in 6d it is self-dual in zero external field
A gerbe for the elliptic gamma function
No full textThe identities for elliptic gamma functions discovered by Felder and Varchenko [8] are generalized to an infinite set of identities for elliptic gamma functions associated to pairs of planes in 3-dimensional space. The language of stacks and gerbes gives a natural framework for a systematic description of these identities and their domain of validity. A triptic curve is the quotient of the complex plane by a subgroup of rank three. (It is a stack.) Our identities can be summarized by saying that elliptic gamma functions form a meromorphic section of a hermitian holomorphic abelian gerbe over the universal oriented triptic curv
Transgression of the index gerbe
No full textThe holonomy of an unitary line bundle with connection over some base space B is a U(1)-valued function on the loop space LB. In a parallel manner, the holonomy of a gerbe with connection on B is a line bundle with connection over LB. Given a family of graded Dirac operators on B and some additional geometric data one can define the determinant line bundle with Quillen metric and Bismut-Freed connection. According to Witten, Bismut-Freed the holonomy of this determinant bundle can be expressed in terms of an adiabatic limit of eta invariants of an associated family of Dirac operators over LB. Recently, for a family of ungraded Dirac operators on B J. Lott constructed an index gerbe with connection., In the present paper we show, in analogy to the holonomy formula for the determinant bundle, that the holonomy of the index gerbe coincides with an adiabatic limit of determinant bundles of the associated family of Dirac operators over LB
The basic bundle gerbe on unitary groups
No full textWe consider the construction of the basic bundle gerbe on SU(n) introduced by Meinrenken and show that it extends to a range of groups with unitary actions on a Hilbert space including U(n) and Up(H), the Banach Lie group of unitaries differing from the identity by an element of a Schatten ideal. In all these cases we give an explicit connection and curving on the basic bundle gerbe and calculate the real Dixmier–Douady class. Extensive use is made of the holomorphic functional calculus for operators on a Hilbert space
Catalogue (annonce) des éditions Delormel : “Pour paraître prochainement : Gerbe de fleurs”
No full textAnnonce des éditions Delormel : “Pour paraître prochainement : Gerbe de fleurs, mélodie”, poésie de Maurice Boukay, musique de René de Buxeuil. Au verso d’une partition de “La chanson des yeux clos” (des mêmes auteurs, 1916). Le titre "Gerbe de fleurs" (édition Delormel) est inconnu du catalogue de la BNF ; aucune attestation de l’existence d’une édition graphique (partition, revue) ou phonographique de ce titre n’a été trouvée. Datation par © du titre au recto
The nori fundamental gerbe of a fibered category. arXiv:1204.1260
No full textAbstract We extend Nori's theory of the fundamental group scheme to a theory of the fundamental gerbe, which applies to schemes, algebraic stacks, and more general fibered categories, even in the absence of a rational point. We give a Tannakian interpretation of the fundamental gerbe in terms of essentially finite bundles, extending Nori's correspondence for complete varieties with a rational point. We also show how our formalism allows a natural formulation of Grothendieck's Section Conjecture in arbitrary characteristic
The 2-Hilbert space of a prequantum bundle gerbe
No full textWe construct a prequantum 2-Hilbert space for any line bundle gerbe whose Dixmier–Douady class is torsion. Analogously to usual prequantization, this 2-Hilbert space has the category of sections of the line bundle gerbe as its underlying 2-vector space. These sections are obtained as certain morphism categories in Waldorf’s version of the 2-category of line bundle gerbes. We show that these morphism categories carry a monoidal structure under which they are semisimple and abelian. We introduce a dual functor on the sections, which yields a closed structure on the morphisms between bundle gerbes and turns the category of sections into a 2-Hilbert space. We discuss how these 2-Hilbert spaces fit various expectations from higher prequantization. We then extend the transgression functor to the full 2-category of bundle gerbes and demonstrate its compatibility with the additional structures introduced. We discuss various aspects of Kostant–Souriau prequantization in this setting, including its dimensional reduction to ordinary prequantization.</p
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