### Course listing

Graduate Seminar on Advanced Geometry (S4D3)

Hauptseminar Geometrie (S2D1)

University of Bonn, Winter semester 2019/20

Instructors: Christian Blohmann, Lory Kadiyan

Time/venue: Wednesday 14:15-16:00, Max Planck Institute for Mathematics, seminar room

#### Seminar description

From the perspective of physics, the notion of factorization algebra can be seen as a framework to formulate classical and (perturbative) quantum field theory, which allows to extend the deformation quantizaton from classical mechanics to field theories. From the perspective of mathematics, a factorization algebra is a cosheaf on a manifold with values in vector spaces (or differential complexes) with an additional factorization property, which defines a homology theory and, therefore, produces topological invariants. This is a math seminar, so we will focus on the mathematical side and use the physics background as motivation and source of examples.

#### Prerequisites

A background in basic differential geometry (sheaves on manifolds, de Rham cohomology) and homological algebra is assumed. Some basic knowledge of homotopical algebra will be helpful. A background in Physics is not necessary, but may be helpful for motivation (quantum mechanics, classical field theory, quantum field theory).

#### Seminar organization

The seminar consists of short talks (ca. 30-45 minutes) by the participants on well-defined parts of the seminar material (e.g. a presentation of an important result with proof or an introduction of a new concept) which are framed and connected by short introductory talks by the seminar organizers.

**Registration**

If you are interested in participating you can send an email to blohmann@mpim-bonn.mpg.de and/or sign up at the organizational meeting on **October 1**. In addition, you will have to register officially via Basis. If you have a preference for one or several of the talks, please let me know by email.

#### Course credit

In order to obtain credit you will have to participate actively. This means that you will have to a) come to all seminar meetings and b) give one of the short talks listed below. The seminar is officially registered for both, Bachelor studies (S2D1) and Master studies (S4D3).

#### Seminar plan

**Prefactorization algebras**(23 Oct 2019), Chair:*Christian Blohmann*

Motivation from physics and mathematics; definition, basic properties, associative algebras as example, morphisms; relation to operads, to precosheaves, and to multicategories**Basic examples: Associative algebras and bimodules**(30 Oct 2019), Chair:*Lory Kadiyan*

Associative algebras are locally constant factorization algebras on $\mathbb{R}$; bimodules as factorization algebras**Basic example: Quantum mechanics**(6 Nov 2019), Chair:*Christian Blohmann*

The apparatus of quantum mechanics; quantum mechanics as factorization algebra over $\mathbb{R}$.**Basic example: The universal enveloping algebra of a Lie algebra**(13 Nov 2019), Chair:*Lory Kadiyan*

The universal envelopping algebra: universal property and construction; Lie algebra homology; the universal enveloping algebra as factorization algebra**Background on generalized differential structures**(20 Nov 2019), Chair:*Christian Blohmann*

Convenient vector spaces; differntiable vector spaces; differentiable prefactorization algebras; tensor product; sections of vector bundles; algebra of observables**The factorization envelope**(27 Nov 2019), Chair:*Lory Kadiyan*

Differential graded (dg) Lie algebras; sheaves of dg Lie algebras; local dg Lie algebras; the factorization envelope; the twisted envelope**The derived critical locus**(11 Dec 2019), Chair:*Christian Blohmann*

Gaußian integrals; divergence operator; multi-vector fields; divergence complex; definition of derived locus; relation to Chevalley-Eilenberg chain complex; relation to left derived tensor product**Free field theories**(18 Dec 2019), Chair:*Christian Blohmann*

Free scalar theory; derived critical locus; the algebra of classical observables; the 1-dimensional case; Poisson structure**1-dimensional free field theories and canonical quantization**(8 Jan 2020), Chair:*Christian Blohmann*

The Weyl algebra; quantum observables of a free field theory; the hamiltonian; recovering the Weyl algebra by factorization homology**Abelian Chern-Simons theory**(15 Jan 2020), Chair:*Christian Blohmann*

Chern-Simons action; classical and quantum observables; the quantum observables on a split spacetime; Chern-Simons theory on a cobordism; relation to knot theory**Factorization algebras**(22 Jan 2020), Chair:*Christian Blohmann*

Cosheaves; coverage, sites, examples for sites; Weiss covers; the descent condition; Hochschild homology as factorization homology**Computing factorization homology**(29 Jan 2020), Chair:*Christian Blohmann*

Spectral sequences; factorization homology of enveloping algebras, of 1-dimensional free scalar field theory, of Kac-Moody algebras

Remarks: The dates of the session are subject to changes and not strict since some topics take more time than an entire meeting and some less. Due to the dies academicus there will be no seminar on Dec 4, 2019.

#### Talks

Talks marked with * are more difficult.

1: The definition in explicit terms, basic properties, morphisms (Secs. 3.1.1 and 3.1.4 in [1]), **Sebastian Meyer**

2a: Proof that associative algebras are locally constant factorization algebras on $\mathbb{R}$ (Secs. 3.1.1 and 3.2 in [1]), **Stefano Ronchi**

2b: Bimodules and factorization algebras (Sec. 3.3.1 in [1]), **Federica Bertolotti**

3: Quantum mechanics as factorization algebra over $\mathbb{R}$ (Sec. 3.3.2 in [1]), **Milan Kroemer**

4a: The Lie algebra homology complex (App. A.4 in [1]), **Yannick Burchart**

4b: The enveloping algebra as factorization algebra (Prop. 3.4.1 in [1])

5*: Proof of Lem. 3.5.15 in [1], **Lennart Ronge**

6a: Twisted factorization envelope (Sec. 3.6.3 in [1])

6b: The Kac-Moody factorization algebra (Examples on p. 78 of [1]), **Jan Nöller**

7a: The divergence complex of a measure (Sec. 4.1.1 in [1]), **Matthias Hippold**

7b*: Derived tensor products, **Jakob Kraasch**

8: Proof of Lem. 4.2.4 in [1] computing the cohomology ring of classical observables on the line, **Sid Maibach**

9*: Proof of Prop. 4.3.3 in [1] recovering the Weyl algebra as factorization homology, **Leonard Hofmann**

10a: Proof of Prop. 4.5.2 in [1] computing the algebra of space observables

10b*: Relation of Chern-Simons theory with knot theory (Sec. 4.5.4 in [1]), **Annika Tarnowsky**

11*: Proof of Thm. 6.4.2 in [1] relating Hochschild homology to factorization homology on a circle

12a: Proof of Prop. 8.1.1 in [1] (enveloping algebras)

12b: Proof of Prop. 8.1.2 in [1] (Kac-Moody algebras)

#### Literature

**Main reference**

[1] Kevin Costello, Owen Gwilliam, *Factorization Algebras in Quantum Field Theory*, Vol. 1, Cambridge UP, 2017

**Additional literature**

[2] Owen Gwilliam, *Factorization algebras and free field theories*, PhD thesis, Northwestern University 2012

[3] Kevin Costello, Owen Gwilliam, *Factorization Algebras in Quantum Field Theory*, Vol. 2 (28 April 2016), manuscript, available online at https://people.math.umass.edu/~gwilliam/vol2may8.pdf, last accessed on 9/30/2019

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