# Schedule of the Workshop: Algebraic Quantum Field Theory and Local Symmetries

## Wednesday, September 26

9:30 - 10:30 |
Klaus Fredenhagen: Gauge symmetry in perturbative algebraic quantum field theory |

10:30 - 11:00 |
Coffee break |

11:00 - 12:00 |
Walter van Suijlekom: Quantization of gauge fields, graph polynomials, and ghost cycles |

12:00 - 15:00 |
Lunch break |

15:00 - 16:00 |
Chris Fewster: Endomorphisms and automorphisms of locally covariant quantum field theories |

16:00 |
Coffee |

## Thursday, September 27

9:30 - 10:30 |
Roberto Longo: How to add a boundary condition in Conformal Field Theory |

10:30 - 11:00 |
Coffee break |

11:00 - 12:00 |
Gandalf Lechner: KMS states of deformed quantum field theories |

12:00 - 15:00 |
Lunch break |

15:00 - 16:00 |
Thomas Hack: On the consistency of quantum supergravity |

16:00 |
Coffee |

19:30 |
Social dinner |

## Friday, September 28

9:00 - 10:00 |
Claudio Dappiaggi: New insights in the quantization of Maxwell's equations on curved backgrounds |

10:00 - 10:30 |
Coffee break |

10:30 - 11:30 |
Alexander Schenkel: Quantum field theory on affine bundles |

## Abstracts:

Klaus Fredenhagen: Gauge symmetry in perturbative algebraic quantum field theory

In gauge theories, the structure of the algebra of observables is typically too complicated for a direct treatment. Therefore one introduces auxiliary quantities which allow a treatment by standard methods. The crucial question is whether the obtained structure depends on the choices one has performed. The formalism of Batalin and Vilkovisky admits in principle a discussion of this question in terms of cohomology. But up to recently, this formalism could be made rigorous only for theories where spacetime is replaced by a finite set. The reason for this restriction was the essential use of the path integral. It will be shown that an adapted version of the Batalin-Vilkovisky formalism can be rigorously obtained in the framework of perturbative algebraic quantum field theory.

Walter van Suijlekom: Quantization of gauge fields, graph polynomials, and ghost cycles

We review quantization of gauge fields using algebraic properties of 3-regular graphs. We derive the Feynman integrand at n loops for a non-abelian gauge theory quantized in a covariant gauge from scalar integrands assigned by the help of graph polynomials to 3-regular graphs. This implies effectively a covariant quantization without ghosts, where all the relevant signs of the ghost sector are incorporated in a double complex furnished by a newly defined cycle cohomology and by graph cohomology.

Chris Fewster: Endomorphisms and automorphisms of locally covariant quantum field theories

In the framework of locally covariant quantum field theory, a theory is described as a functor from a category of spacetimes to a category of *-algebras. In this talk, I will describe how the global gauge group of such a theory can be identified as the automorphism group of thedefining functor. This is intended as the first step towards an implementation of the theory of superselection sectors at the functorial level. Various developments will be described. For instance, the gauge group may be given a topology and multiplets of fields may be identified at the functorial level. It will be shown that locally covariant theories obeying standard assumptions in Minkowski space, including energy compactness, have no proper endomorphisms (i.e., all endomorphisms are automorphisms) and that the global gauge group is compact under these circumstances. Finally, I will describe how the endomorphisms and automorphisms of a locally covariant theory may, in principle, be classified in any single spacetime. As an example, the endomorphisms and automorphisms of a system of finitely many free scalar fields

are completely classified.

The talk is based on arXiv:1201.3295; I will aim to discuss more of the technical detail than in other talks I have recently given on the subject.

Thomas Hack: On the consistency of quantum supergravity

We discuss the quantization of arbitrary supergravity theories on general curved Bosonic backgrounds. In this context, two potential obstacles have to be overcome: 1) as supergravity is a gauge theory, the equations of motions are not hyperbolic a priori, i.e. a causal propagation of the degrees of freedom is not manifest. 2) due to the Fermionic nature of the gauge fields in supergravity a nontrivial "unitarity problem" appears which has to be overcome in order to obtain a quantum theory on a Hilbert space. We address both issues by introducing a novel "causal" gauge fixing and by proving that the unitarity problem can be solved; our resulting formulation of the quantum theory is manifestly gauge-invariant. We discuss only the lowest order in perturbation theory, as the above-mentioned obstacles appear already at this level.

Roberto Longo: How to add a boundary condition in Conformal Field Theory

Given a conformal QFT local net of von Neumann algebras B2 on the two-dimensional Minkowski space-time based on the same completely rational chiral net A on the left/right light-ray, we show how to consistently add a boundary to B2: we provide a procedure to construct a Boundary CFT net B of von Neumann algebras on the half-plane x > 0, associated with A, and locally

isomorphic to B2. All such locally isomorphic Boundary CFT nets arise in this way. There are only finitely many locally isomorphic Boundary CFT nets and we get them all together. In essence, we show how to directly redefine the C* representation of the restriction of B2 to the half-plane by means of subfactors and local conformal nets of von Neumann algebras on the circle.

Gandalf Lechner: KMS states of deformed quantum field theories

Recently new quantum field theoretic models have been constructed by deformations of free field models in their vacuum representations. The field operators of these models satisfy certain residual (wedge-) locality properties which are closely connected to the spectrum condition of the undeformed theory. This talk is about a situation where the spectrum condition fails, namely the situation of a thermal equilibrium state at finite temperature. It is discussed how to compute KMS functionals of the deformed field algebra. The main problem consists in analyzing which of these functionals are positive, and some partial results in this direction will be presented.

Claudio Dappiaggi: New insights in the quantization of Maxwell's equations on

curved backgrounds

We show that the standard quantization scheme for field theories on curved backgrounds, when applied to Maxwell's equations, does not abide to the principle of general local covariance. We propose an approach to bypass this obstruction, which relies on the possibility of reading electromagnetism as a U(1) gauge theory and, thus, on the existence of an underlying natural principal bundle structure.

Alexander Schenkel: Quantum field theory on affine bundles

In quantum field theory on curved spacetimes the field configuration space is typically modeled in terms of the space of sections of a vector bundle over a globally hyperbolic spacetime. While this is certainly appropriate for scalar, spinor and vector fields, the Maxwell field (described in terms of principal connections) is different since the underlying bundle is an a this talk we study the formulation of noninteracting quantum field theories on general affine bundles. Using an affine generalization of the concept of linear hyperbolic operators we give a categorical formulation of such theories. We prove that this leads to a locally covariant quantum field theory, in particular the causality and time-slice axioms hold true. We compare this affine theory to linear quantum field theories on vector bundles and highlight nontrivial features.