Homotopy Colimits and Global Observables in Abelian Gauge Theory

Fortschritte der Physik

2019

Self Cite

“…Because there are no non‐trivial principal

double-struckR

‐bundles, the groupoid of gauge fields (cf. Example ) on M is given by

\begin{matrix} true \\ right B G^{con} (M) = ({, \begin{matrix} Obj: & A \in {normalΩ}^{1} false(M false) \\ Mor: & A \overset{ε}{⟶} A + d ε \\ with ε \in C^{\infty} false(M false) \end{matrix}) . \end{matrix}

Following, one easily computes the nerve of this groupoid and, after applying the Dold‐Kan correspondence to the resulting simplicial vector space, obtains the chain complex

F = (\overset{(0)}{{normalΩ}^{1} false(M false)} \overset{normald}{⟵} \overset{(1)}{{normalΩ}^{0} false(M false)}),

where we indicated in round brackets the homological degrees and identified functions with 0‐forms

{normalΩ}^{0} false(M false) = C^{\infty} false(M false)

. As expected, this chain complex has only ‘stacky’ positive degrees and no ‘derived’ negative degrees.…”

Section: Higher Structures In Gauge Theorymentioning

confidence: 99%

“…Following, [51] one easily computes the nerve of this groupoid and, after applying the Dold-Kan correspondence to the resulting simplicial vector space, obtains the chain complex…”

Section: Derived Geometry Of Linear Gauge Fieldsmentioning

confidence: 99%

See 1 more Smart Citation

Higher Structures in Algebraic Quantum Field Theory

Fortschritte der Physik

2019

Self Cite

“…Another advantage of our synthetic approach to classical field theory is that it is a suitable starting point for generalizations to gauge theories. In particular, the groupoids of gauge field configurations appearing in our recently proposed homotopy theoretic approach to gauge theories [BSS15] can be easily promoted to groupoid objects in the Cahiers topos, i.e. "generalized smooth groupoids".…”

Section: Introductionmentioning

confidence: 99%

“…"generalized smooth groupoids". The relevant homotopy theoretical concepts used in [BSS15] generalize to such "generalized smooth groupoids" [JT91], while locally-convex Lie groupoids (which arise in the framework of [BFR12]) are not suitable for homotopy theory.…”

Section: Introductionmentioning

confidence: 99%

Poisson Algebras for Non-Linear Field Theories in the Cahiers Topos

2016

Ann. Henri Poincaré

Self Cite

We develop an approach to construct Poisson algebras for non-linear scalar field theories that is based on the Cahiers topos model for synthetic differential geometry. In this framework the solution space of the field equation carries a natural smooth structure and, following Zuckerman's ideas, we can endow it with a presymplectic current. We formulate the Hamiltonian vector field equation in this setting and show that it selects a family of observables which forms a Poisson algebra. Our approach provides a clean splitting between geometric and algebraic aspects of the construction of a Poisson algebra, which are sufficient to guarantee existence, and analytical aspects that are crucial to analyze its properties.

Abelian Duality on Globally Hyperbolic Spacetimes

Becker

et al. 2016

Commun. Math. Phys.

Self Cite

We study generalized electric/magnetic duality in Abelian gauge theory by combining techniques from locally covariant quantum field theory and Cheeger-Simons differential cohomology on the category of globally hyperbolic Lorentzian manifolds. Our approach generalizes previous treatments using the Hamiltonian formalism in a manifestly covariant way and without the assumption of compact Cauchy surfaces. We construct semi-classical configuration spaces and corresponding presymplectic Abelian groups of observables, which are quantized by the CCR-functor to the category of C * -algebras. We demonstrate explicitly how duality is implemented as a natural isomorphism between quantum field theories. We apply this formalism to develop a fully covariant quantum theory of self-dual fields.