FQFT and cohomology
Types of quantum field thories
For semisimple Lie algebra targets
For discrete group targets
For discrete 2-group targets
For Lie 2-algebra targets
For targets extending the super Poincare Lie algebra
for higher abelian targets
for symplectic Lie n-algebroid targets
This is a sub-entry of sigma-model. See there for further background and context.
We give a leisurely exposition of a general abstract formulation -models, aimed at readers with a background in category theory but trying to assume no other prerequisites.
What is called an -dimensional -model is first of all an instance of an -dimensional quantum field theory (to be explained). The distinctive feature of those quantum field theories that are -models is that
moreover, this simpler kind of field theory encoded bygeometric data in a nice way: it describes physical configuration spaces that are mapping spaces into a geometric space equipped with some differential geometric structure.
We give expositions of these items step-by-step:
We draw from (FHLT, section 3).
and all manifolds are equipped with -structure . For instance could be Riemannian structure . Then we would call a Euclidean quantum field theory (confusingly). If is “no structure” then we call a topological quantum field theory.
is some symmetric monoidal category.
We think of data as follows:
the morphism that assigns to any cobordism with incoming boundary and outgoing boundary is the propagator along : it maps every state of the system over to the state that is the result of the evolution of along by the dynamics of the system. Or conversely: the action of encodes what this dynamics is supposed to be.
Notice that since is required to be a symmetric monoidal functor it sends disjoint unions of manifolds to tensor products
is the tensor unit of ;
is an endomorphism of this tensor unit, a number as seen internal to – this is the invariant associated to by , called the partition function of over . We can think of as being a rule for computing such invariants by building them up from smaller pieces. This is the locaity of quantum field theory.
A simple but archetypical example is this: let be Riemannian structure. Then the category of 1-dimensional cobordisms equipped with Riemannian structure is generated (as a symmetric monoidal category) from intervals
equipped with a length . Composition is given by addition of lengths
Therefore a 1-dimensional Euclidean quantum field theory
is specified by
This is called the Hamilton operator of the system.
(We are glossing here over some technical fine print in the definition of . Done right we have that may indeed be an infinite-dimensional vector space. See (1,1)-dimensional Euclidean field theories and K-theory)
A special class of examples of -dimensional quantum field theories, as discussed above, arise as deformations or averages of similar, but simpler structure: classical field theories . The process that constructs a quantum field theory out of a classical field theory is called quantization . This is discussed below. Here we describe what a classical field theory is. We shall inevitably oversimplify the situation such as to still count as a leisurely exposition. The kind of examples that the following discussion applies to strictly are field theories of Dijkgraaf-Witten type. But despite its simplicity, this case accurately reflects most of the general abstract properties of the general theory.
For our purposes here, a classical field theory of dimension is
Let be a cobordism and
the value of a classical field theory on . We interpret this data as follows:
Here a “physical field” can be something like the electromagnetic field. But it can also be something very different. For the special case of -models that we are eventually getting at, a “field configuration” here will instead be a way of an particle of shape sitting in some target space.
is similarly the groupoid of field configurations on the whole cobordism, . If we think of an object in of a way of a brane of shape sitting in some target space, then an object in is a trajectory of that brane in that target space, along which it evolves from shape to shape .
is the classifying map of a kind of vector bundle over configuration space: a state of the quantum field theory that will be associated to this classical field theory by quantization will be a section of this vector bundle. Such a section is to be thought of as a generalization of a probability distribution on the space of classical field configurations. The generalized elements of a fiber of over a configuration may be thought of as an internal state of the brane of shape sitting in target space.
is the action functional that defines the classical field theory: the component
of this natural transformation on a trajectory going from a configuration to a configuration is a morphism in that maps the internal states of the ingoing configuration to the internal states of the outgoing configuration . This evolution of internal states encodes the classical dynamics of the system.
Notice that this way a classical field theory is taken to be a special case of a quantum field theory, where the codomain of the symmetric monoidal functor is of the special form . For more on this see classical field theory as quantum field theory?.
We call this the path integral functor.
This we call the quantization of .
on objects by sending
the vector bundle on the configuration space over some boundary of worldvolume to its space of gauge invariant sections. In typical situations this is the famous Hilbert space of states in quantum mechanics, only that here it is allowed to be any object in ;
A classical -model is a classical field theory such that
the bundles “of internal states” over these mapping spaces are
…of an associated higher bundle
…encoded by a classifying morphism
on n-vector spaces, which is usually taken to be the canonical 1-dimensional one.
One calls the background gauge field of the -model.
More specifically and more simply, in cases where is just a discrete ∞-groupoid – the case of sigma-models of Dijkgraaf-Witten type, every principal ∞-bundle on is necessarily flat, hence the background gauge field is given just by the morphism
Then for a closed -dimensional manifold, the action functional of the sigma-model on on a field configuration has the value
One says that is the Lagrangian of the theory.