### BRST Quantization, Symplectic Reduction, and Syzygies I

I want to talk a bit about quantizing gauge theories with syzygies (I’ll explain what I mean by this). I’ll probably take a few posts to do this. First an overview.

A gauge theory, roughly speaking, is a QFT whose physical configurations form a category with non-trivial morphisms. Take BF theory for example. This has the action functional

,

where and are a 1-form and 2-form, respectively. This theory has the gauge symmetries

.

These describe an isomorphism between the configurations and . There are even 2-morphisms. If we take

,

then we get an equivalent gauge transformation. This is an example of a syzygy–a relation between relations.

Put another way, our configuration space, is actually a 2-groupoid. Its fundamental 2-group with basepoint at a particular configuration is presented by the crossed module (of sheaves on a suitable site)

where the target map is the de Rham differential into the first summand. The BV-BRST formalism gives us a method to use this presentation to quantize BF theory.

What does it mean to quantize a gauge theory? Let’s recall what we need to do geometrically and then (in later posts) translate this into algebra. First we form the unreduced phase space of the theory. What this means is that we need to come up with some canonically conjugate coordinates. If we pick a coordinate , we can use the Lagrangian , which in this case is to pick a particularly nice conjugate momentum : . In this case, we see that and are a natural choice of conjugate variables. We define a Poisson bracket so

.

This uniquely defines a symplectic structure on . Now we must restrict to the submanifold cut out by the equations of motion. We can describe as the homology of the complex

.

This is the sheafy version of D’Alembert’s principle. Here, denotes -linear derivations, ie. vector fields, and is a one-form describing the variation of the action along some vector field. Roughly speaking, what this says is that the classical solutions are the configurations where the action is stationary.

Note that , called the Noether ideal, consists of those vector fields which generate symmetries of the action **off shell**, ie. away from where the action is stationary. This includes gauge transformations as well as ordinary symmetries. Resolution of this ideal sheaf will concern us quite a bit in quantizing this theory.

In this case, one finds (try it!) that the equations cutting out are

as long as the spacetime is without boundary. Now we want to identify physically equivalent states, ie. our reduced phase space should be . This is just the sum of the first and second homology of our spacetime, which until now I have not mentioned, but should be some 4-manifold . This space is not even dimensional in general, but it is if we insist for some 3-manifold . It seems like this is the only case where it is possible to quantize this theory in this method, but it is really the one we’re interested in for constructing BF theory as a TQFT. More on why this is the case in later posts. It has to do with Anton Kapustin and others’ program of extending TQFTs by Kaluza-Klein compactifications. This will include a discussion of symplectic reduction next time.

Once we have a good phase space, one thing we can do is geometric quantization. We can cook up a line bundle with connection whose curvature gives the symplectic form. This is called the prequantum line bundle. Then we pick a polarization–some kind of involutive distribution on this line bundle such as a complex structure, and ask for the Hilbert space completion of the space of polarized (think holomorphic) sections of this line bundle. That’s our Hilbert space, and there are ways to take functions on the phase space and produce Hermitian operators on this Hilbert space. Voila! Quantization.