Bundle Gerbes Applied to Quantum Field Theory

Journal of Geometry and Physics, 2007

The infinitesimal symmetries of a fully decomposed non-Abelian gerbe can be generated in terms of a nilpotent BRST operator, which is here constructed. The appearing fields find a natural interpretation in terms of the universal gerbe, a generalisation of the universal bundle. We comment on the construction of observables in the arising Topological Quantum Field Theory. It is also shown how the BRST operator and the trace part of a suitably truncated set of fields on the non-Abelian gerbe reduce directly to the coboundary operator and the pertinent cochains of the underlyingČech-de Rham complex.

1999

We consider the canonical quantization of fermions on an odd dimensional manifold with boundary, with respect to a family of elliptic hermitean boundary conditions for the Dirac hamiltonian. We show that there is a topological obstruction to a smooth quantization as a function of the boundary conditions. The obstruction is given in terms of a gerbe and its Dixmier-Douady class is evaluated. : 81T50, 58B25, 19K56 Key words: Gerbe, Hamiltonian quantization, Dirac boundary value problems 0. INTRODUCTION In this paper we study the Hamiltonian quantization of massless fermions on a compact odd-dimensional manifold X with boundary Y = ∂X. Field theories on manifolds with boundary arise in several situations including gravitation on odd dimensional anti-de Sitter spacetimes [W]

Communications in Mathematical Physics, 2000

In ref.

2016

A pedagogical but concise overview of fiber bundles and their connections is provided, in the context of gauge theories in physics. The emphasis is on defining and visualizing concepts and relationships between them, as well as listing common confusions, alternative notations and jargon, and relevant facts and theorems. Special attention is given to detailed figures and geometric viewpoints, some of which would seem to be novel to the literature. Topics are avoided which are well covered in textbooks, such as historical motivations, proofs and derivations, and tools for practical calculations. The present paper is best read in conjunction with the similar paper on Riemannian geometry cited herein.

Communications in Mathematical Physics, 2005

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 H4(BG,) for a compact semisimple 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 H4(BG,) to H3(G,). The notion of a multiplicative bundle gerbe accounts geometrically for the subtleties in this correspondence for non-simply connected Lie groups. The implications for Wess-Zumino-Witten models are also discussed.

Communications in Mathematical Physics, 2001

Using properties of the determinant line bundle for a family of elliptic boundary value problems, we explain how the Fock space functor defines an axiomatic quantum field theory which formally models the Fermionic path integral. The 'sewing axiom' of the theory arises as an algebraic pasting law for the determinant of the Dirac operator. We show how representations of the boundary gauge group fit into this description and that this leads to a Fock functor description of certain gauge anomalies.

arXiv (Cornell University), 2004

Journal of Physics A: Mathematical and Theoretical, 2007

We construct a U(1) gerbe with a connection over a finite-dimensional, classical phase space P. The connection is given by a triple of forms A, B, H: a potential 1-form A, a Neveu-Schwarz potential 2-form B, and a field-strength 3-form H = dB. All three of them are defined exclusively in terms of elements already present in P, the only external input being Planck's constant . U(1) gauge transformations acting on the triple A, B, H are also defined, parametrised either by a 0-form or by a 1-form. While H remains gauge invariant in all cases, quantumness vs. classicality appears as a choice of 0-form gauge for the 1-form A. The fact that [H]/2πi is an integral class in de Rham cohomology is related with the discretisation of symplectic area on P. This is an equivalent, coordinate-free reexpression of Heisenberg's uncertainty principle. A choice of 1-form gauge for the 2-form B relates our construction with generalised complex structures on classical phase space. Altogether this allows one to interpret the quantum mechanics corresponding to P as an Abelian gauge theory.

This is the second in a series of papers discussing, in the framework of gerbe theory, canonical and geometric aspects of the two-dimensional non-linear sigma model in the presence of conformal defects in the world-sheet. Employing the formal tools worked out in the first paper of the series, 1101.1126 [hep-th], a thorough analysis of rigid symmetries of the sigma model is carried out, with emphasis on algebraic structures on generalised tangent bundles over the target space of the theory and over its state space that give rise to a realisation of the symmetry algebra on states. The analysis leads to a proposal for a novel differential-algebraic construct extending the original definition of the (gerbe-twisted) Courant algebroid on the generalised tangent bundles over the target space in a manner co-determined by the structure of the 2-category of abelian bundle gerbes with connection over it. The construct admits a neat interpretation in terms of a relative Cartan calculus associated with the hierarchy of manifolds that compose the target space of the multi-phase sigma model. The paper also discusses at length the gauge anomaly for the rigid symmetries, derived and quantified cohomologically in a previous work of Gawȩdzki, Waldorf and the author. The ensuing reinterpretation of the small gauge anomaly in terms of the twisted relative Courant algebroid modelling the Poisson algebra of Noether charges of the symmetries is elucidated through an equivalence between a category built from data of the gauged sigma model and that of principal bundles over the world-sheet with a structural action groupoid based on the target space. Finally, the large gauge anomaly is identified with the obstruction to the existence of topological defect networks implementing the action of the gauge group of the gauged sigma model and those giving a local trivialisation of a gauge bundle of an arbitrary topology over the world-sheet.

An explicit model of fiber bundle with local fibers being distinct copies of isotopic space is introduced. The local isotopic spaces are endowed with frames which are used as local isotopic ones. The field local of isotopic frames are considered as gauge field itself while the form of gauge connections is derived from it. The field equation for that of local frames is found. It is shown that Yang-Mills equation follows from it, but variety of solutions of the new equation is highly reduced such that no ambiguities (Yang-Wu and vacuum ones) arise. It is shown that Lagrangian for the field gives non-zero trace for the stress-energy tensor and zero value for the field of plane wave. New solutions for the fields of punctual source and spherical wave are found.