# A Framework for Geometric Field Theories and their Classification in Dimension One

@article{Ludewig2021AFF, title={A Framework for Geometric Field Theories and their Classification in Dimension One}, author={Matthias Ludewig and Augusto Stoffel}, journal={Symmetry, Integrability and Geometry: Methods and Applications}, year={2021} }

In this paper, we develop a general framework of geometric functorial field theories, meaning that all bordisms in question are endowed with geometric structures. We take particular care to establish a notion of smooth variation of such geometric structures, so that it makes sense to require the output of our field theory to depend smoothly on the input. We then test our framework on the case of 1-dimensional field theories (with or without orientation) over a manifold M. Here the expectation…

#### 7 Citations

A T ] 1 N ov 2 02 1 The geometric cobordism hypothesis

- 2021

We prove a generalization of the cobordism hypothesis of Baez–Dolan and Hopkins–Lurie for bordisms with arbitrary geometric structures, such as Riemannian metrics, complex and symplectic structures,…

Smooth functorial field theories from B-fields and D-branes

- Mathematics, PhysicsJournal of Homotopy and Related Structures
- 2021

In the Lagrangian approach to 2-dimensional sigma models, B-fields and D-branes contribute topological terms to the action of worldsheets of both open and closed strings. We show that these terms…

The geometric cobordism hypothesis

- Mathematics, Physics
- 2021

We prove a generalization of the cobordism hypothesis of Baez–Dolan and Hopkins–Lurie for bordisms with arbitrary geometric structures, such as Riemannian metrics, complex and symplectic structures,…

Connes fusion of spinors on loop space

- Mathematics, Physics
- 2020

The loop space of a string manifold supports an infinite-dimensional Fock space bundle, which is an analog of the spinor bundle on a spin manifold. This spinor bundle on loop space appears in the…

Extended field theories are local

- Mathematics, Physics
- 2020

We show that all extended functorial field theories, both topological and nontopological, are local. Precisely, we show that the functor that sends a target geometry S to the smooth…

Sheaves of Higher Categories on Generalised Spaces

- Mathematics, Physics
- 2020

We study the extension of higher presheaves on a category C to its free cocompletion Ĉ. Any pretopology on C induces a canonical pretopology of generalised coverings on Ĉ. We show that with respect…

Smooth 1-dimensional algebraic quantum field theories

- Physics, Mathematics
- 2020

This paper proposes a refinement of the usual concept of algebraic quantum field theories (AQFTs) to theories that are smooth in the sense that they assign to every smooth family of spacetimes a…

#### References

SHOWING 1-10 OF 23 REFERENCES

Smooth one-dimensional topological field theories are vector bundles with connection

- Mathematics
- 2015

We prove that smooth 1-dimensional topological field theories over a manifold are the same as vector bundles with connection. The main novelty is our definition of the smooth 1-dimensional bordism…

Topological quantum field theories

- Mathematics
- 1988

© Publications mathematiques de l’I.H.E.S., 1988, tous droits reserves. L’acces aux archives de la revue « Publications mathematiques de l’I.H.E.S. » (http://www.…

Topological quantum field theories, Inst

- Hautes Études Sci. Publ. Math
- 1988

Parallel Transport and Functors

- Mathematics
- 2009

Parallel transport of a connection in a smooth fibre bundle yields a functor from the path groupoid of the base manifold into a category that describes the fibres of the bundle. We characterize…

Grothendieck topologies, fibered categories and descent theory, in Fundamental Algebraic Geometry

- Math. Surveys Monogr
- 2005

Grothendieck topologies, fibered categories and descent theory, in Fundamental Algebraic Geometry, Math

- Surveys Monogr.,
- 2005

Gerbes and homotopy quantum field theories

- Mathematics
- 2004

For smooth finite dimensional manifolds, we characterise gerbes with connection as functors on a certain surface cobordism category. This allows us to relate gerbes with connection to Turaev's…