Schreiber Seminar on quantum field theory

Topic list and references for a seminar on quantum field theory from the perspective of higher category theory and physics, summer 2011.

See also the previous Seminar on derived critical loci .

Motivation

A good deal of the modern developments in higher/derived geometry are motivated from attempts to better formalize various aspects or even the full story of quantum field theory.

Not the least, of course, the cobordism hypothesis-theorem, which is now a cornerstone of higher category theory – was first conceived of and now in fact “solves” topological quantum field theory.

But in the practice of physics, much of the work happens before a quantum field theory is even obtained: in the process of quantization of a classical field theory. This process involves a multitude of further higher geometric tools.

We consider

1. A grand overview of the topic complex Classical field theory, Quantization, Quantum field theory;

2. the method of path integral quantization as integral transforms on generalized cohomology;

3. Variational calculus and its modern incarnation as BV-BRST formalism and D-geometry.

4. and more…

Topics

Classical field theory, Quantization, Quantum field theory

• Igor Khavkine, reductions deformations resolutions in physics

Path integral quantization by pull-push

One approach to quantization is path integral quantization.

For a large class of comparatively simple field theories – most of them still very rich – path integral quantization can be made sense of as a kind of higher integral transform given by pull-tensor-push operations in cohomology.

See

• path integral as a pull-push transform.

This is notably the case for classes of sigma-model quantum field theories such as Gromov-Witten theory.

Also string topology can be interpreted this way, at least in parts. See

• Sander Kupers, Pull-push constructions and string topology operations (pdf)

Variational calculus

The action functionals that govern quantum field theory come from Lagrangians on jet bundles. The differential calculus on these, that governs the construction of covariant phase spaces as the solution space to Euler-Lagrange equations of motion is variational calculus .

An important tool in the study of variational calculus is this the de Rham complex on jet bundles: the

• variational bicomplex.

In its homotopy theoretic version this is encapsulated by BV-BRST formalism. For more on this see the previous Seminar on derived critical loci.

$\mathcal{D}$-Geometry

The jet bundle geometry that supports variational calculus has its natural home in D-geometry, the higher geometry over a infinitesimal path ∞-groupoid/de Rham space $\mathbf{\Pi}_{inf}(X)$: for $E \to X$ any bundle, its jet bundle is the direct image $i_* : \mathbf{H}/X \to \mathbf{H}/\mathbf{\Pi}_{inf}(X)$ along the constant infintisimal path inclusion $i : X \to \mathbf{\Pi}_{inf}(X)$.

A quasicoherent sheaf over $\mathbf{\Pi}_{inf}(X)$ is equivalently a D-module on $X$. Therefore algebraically one characterizes spaces over $\mathbf{\Pi}_{inf}(X)$ as D-schemes (BeilinsonDrinfeld. All of variational calculus has an elegant and powerful formalization in this D-geometry . See (Paugam) for a remarkably comprehensive discussion in the context of quantum field theory.

References

A remarkable attempt at collecting together the latest modern mathematical developments on various aspects of quantum field theory is in

• Frédéric Paugam, Towards the mathematics of quantum field theory (pdf)

The D-geometric-formulation of variational calculus has been introduced in chapter 2 of

• Alexander Beilinson, Vladimir Drinfeld, Chiral Algebras