nLab
higher category theory and physics

This entry lists and discusses applications and occurences of higher category theory in physics.

well, this entry is hugely imperfect at the moment, at best a sketch of a sketch. But some nice entry of this topic should eventually be available.

General considerations

Urs: I inserted the following quick rough sketch just as a catalyst for further development/criticism/revision, without really taking or even having the time to do justice to this topic. I don’t have the time and energy right now to run with this ball, but I thought I’d give it a kick anyway.

At this time, a systematic and comprehensive $n$-categorical formulation of fundamental physics, that was expected in one form or other for many years, is gradually beginning to take concrete shape in structures revolving around the concept of extended quantum field theory.

Given the current evidence there is some reason to expect that eventually higher category theory in general and (∞,n)-category theory in particular will find its role as the natural language for quantum field theory and all that entails in much the same way as, for instance, in the past

• differential geometry turned out to be the natural language for Maxwell theory and Einstein gravity;

• symplectic geometry turned out to be the natural language for Hamiltonian mechanics;

• etc.

For instance just recently with the cobordism hypothesis taking concrete shape we are seeing

• geometric ∞-function theory in derived stack (∞,1)-toposes as a natural language for (at least topological) sigma-model quantum field theory.

also… blah … blah

• AQFT/factorization algebra …

.. blah…

If we take the presently available evidence of the picture that is emerging here, then it appears that all higher category theoretical phenomena in physics that have been identified already in the past may be fit into and organized according to the following pattern:

• 0th quantization: classical physics: quantization is quantization of “action functionals”. Action functionals in fundamental physics tend to arise in the form of holonomies of higher differential cocycles. For instance the Chern-Simons action functional is the 3-dimensional holonomy of a $U(1)$ 2-gerbe, the Wess-Zumino model term the 2-dimensional holonomy of a 1-gerbe, and the gauge coupling term of the ordinary charged particle (an electron, or a quark) the ordinary holonomy of a bundle with connection. So 0th quantized physics is about parallel transport $n$-functors in differential cohomology $\nabla$ that send “classical trajectories” to the “phases” assigned to them by the “gauge coupling”. So 0th quantization is controled by $n$-functors of the form

  • classical action: target space $\to$ phases

    $\Pi (X)\to \mathrm{\infty }\mathrm{Mod}$

• 1st quantization: quantum mechanics: it has become clear that the quantum field theory that described the propagation of a fundamental $(n-1)$-brane (or ”$n$-particle”) is encoded in an $n$-functor from an (infinity,n)-category of cobordisms to some suitable codomain. This sends “worldvolumes” to “correlators”. So first quantization is controled by $n$-functors of the form

  • quantum propagation: cobordisms/Feynman diagrams $\to$ correlators

    $${\mathrm{Bord}}_{(\mathrm{\infty },n)}\to 𝒞$$Bord_{(\infty,n)} \to \mathcal{C}

• 2nd quantization: quantum field theory: quantum mechanical propagation functors as above that arise by quantizing classical data, i.e. that arise as sigma-models typically compute the amplitude assigned to as a cobordism a sum of phases over classical trajectories. This pattern continues: in second quantization one constructs ”$S$-matrix elements” by summing amplitudes over cobordisms/Feynman diagrams. This is by far the least developed bit, but it seems we want to say that the $n$-functor relevant for second quantized quantum field theory here is of the kind

  • scattering amplitude: reaction channel $\to$ $S$-matrix element

    $$\Pi (\mathrm{Bord})\to C$$\Pi(Bord) \to C

We will now list $n$-categorical physics ordered according to this rough pattern of successive quantization.

Classical and Quantum Physics

0th quantized

• the objects of interest in physics: space and quantity: motivation for sheaves, cohomology and higher stacks

• target spaces (spacetimes) and configuration spaces: space , generalized smooth space, ∞-stack/derived stack, structured generalized space

• background gauge fields – forces:

  • integral description: higher gauge theory: the description of background fields: higher bundles with connection

  • differential description: rational homotopy theory – Lie ∞-algebroid

• tools for handling $\mathrm{\infty }$-categorical action functionals

  • BV theory

  • AKSZ theory

1st quantized

• FQFT

  • cobordism hypothesis

  • (∞,n)-category of cobordisms

• $\mathrm{\infty }$-sigma-models

  • geometric infinity-function theory

• AQFT: sheaves of operator algebra, nonabelian cohomology

• factorization algebra

The path integral

The path integral is becoming expected to have a formalization in terms of higher category theory as a suitable “push-forward of a classical theory to a point”.

• a proposal for how to think of this is in Topological Quantum Field Theories from Compact Lie Groups

linear $(\mathrm{\infty }\mathrm{,1})$-categorical QFT

the following a stub obtained from copy-and-pasting material that David Ben-Zvi mentioned here

Kontsevich’s 1994 ICM article, is one of the seminal papers of the 1990s. This paper invented the categorical side of mirror symmetry (homological mirror symmetry), discovered D-branes (before physicists realized their role — and directly inspiring many physicists) and the fact that they naturally form dg-categories and $A_{\mathrm{\infty }}$-category, and thus led to a deluge of papers involving category theory and higher category theory in close relation with mathematical physics.

In particular the Moore–Segal paper should be seen in the light of this development. On a similar note, roughly contemporary with Moore–Segal (which is arXiv:math/0609042 though developed earlier) are the works of Costello (arXiv:math/0412149) and Kontsevich–Soibelman (arXiv:math/0606241 — some of the results were lectured on in various places by Kontsevich in 2003 and in particular helped inspire Costello) proving a much stronger result, which is the TCFT (or equivalently differential graded or $A_{\mathrm{\infty }}$-) version of the classification of open-closed 2d TQFTs. These papers were directly motivated by homological mirror symmetry and topological? string theory, and have greatly influenced work in areas such as string topology? which you mention and the cobordism hypothesis (Hopkins-Lurie started from Costello’s paper and abstracted the argument, before the general argument in n-dimensions came around).

2nd quantized

• Hopf-algebraic renormalization?

Further topics

• Discrete Differential Geometry
• Lattice Gauge Theory

Urs: not sure which $n$-categorical aspect of these two subjects we might want to discuss. As soon as I know, I would try to move it into the above pattern, but for the moment I am not sure.

…

References

A historical introduction to some aspects of $n$-categories in physics can be found here:

• John Baez and Aaron Lauda, A prehistory of $n$-categorical physics (pdf), to appear in Deep Beauty: Mathematical Innovation and the Search for an Underlying Intelligibility of the Quantum World, ed. Hans Halvorson.

For blog discussion of this paper see:

• A prehistory of $n$-categorical physics, (n-Category Cafe)

• A prehistory of $n$-categorical physics II, (n-Category Cafe)

Jim Stasheff is also writing a historical introduction to cohomological physics:

• Jim Stasheff, A Survey of Cohomological Physics

The following book-to-be aims to give picture of the present state of the art of describing the general abstract nonsense structure of the universe, as far as fundamental physics is concerned

• Hisham Sati, Urs Schreiber, Branislav Jurco (eds.) Mathematical Foundations of Quantum Field and Perturbative String Theory

On the quantization procedure

First sketches of the idea that path integral quantization may have a formalization in terms of higher category theory appeared in

• Dan Freed, Higher Algebraic Structures and Quantization (arXiv:hep-th/9212115)

and its companing (coming? companion?) paper

• Dan Freed, …

More formal aspects along these lines appear in

• Simon Willerton The twisted Drinfeld double of a finite group via gerbes and finite groupoids (arXiv:math/0503266)

An indication of a full formalization of what that may mean for discrete QFTs such as Dijkgraaf-Witten theory is in

• Dan Freed, Mike Hopkins, Jacob Lurie, Constantin Teleman Topological Quantum Field Theories from Compact Lie Groups

based on work in

• Jacob Lurie, On the Classification of Topological Field Theories