functorial quantum field theory
Reshetikhin?Turaev model? / Chern-Simons theory
FQFT and cohomology
physics, mathematical physics, philosophy of physics
theory (physics), model (physics)
experiment, measurement, computable physics
Axiomatizations
Tools
Structural phenomena
Types of quantum field thories
For semisimple Lie algebra targets
For discrete group targets
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For targets extending the super Poincare Lie algebra
(such as the supergravity Lie 3-algebra, the supergravity Lie 6-algebra)
Chern-Simons-supergravity
for higher abelian targets
for symplectic Lie n-algebroid targets
for the $L_\infty$-structure on the BRST complex of the closed string:
higher dimensional Chern-Simons theory
topological AdS7/CFT6-sector
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 $\sigma$-models, aimed at readers with a background in category theory but trying to assume no other prerequisites.
What is called an $n$-dimensional $\sigma$-model is first of all an instance of an $n$-dimensional quantum field theory (to be explained). The distinctive feature of those quantum field theories that are $\sigma$-models is that
these arise from a simpler kind of field theory – called a classical field theory – by a process called quantization
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).
For our purposes here, a quantum field theory of dimension $n$ is a symmetric monoidal functor
where
$Bord_n^S$ is a category of n-dimensional cobordisms that are equipped with some structure $S$;
its objects are $(n-1)$-dimensional topological manifolds;
its morphisms are isomorphism classes of $n$-dimensional cobordisms between such manifolds
and all manifolds are equipped with $S$-structure . For instance $S$ could be Riemannian structure . Then we would call $Z$ a Euclidean quantum field theory (confusingly). If $S$ is “no structure” then we call $Z$ a topological quantum field theory.
the monoidal category-structure is given by disjoint union of manifolds
$\mathcal{C}$ is some symmetric monoidal category.
We think of data as follows:
$Bord_n^S$ is a model for being and becoming in physics (following Bill Lawvere’s terminology): the objects of $Bord_n^S$ are archetypes of physical spaces that are and the morphisms are physical spaces that evolve ;
the object $Z(\Sigma)$ that $Z$ assigns to any $(n-1)$-manifold $\Sigma$ is to be thought of as the space of all possible states over the space $\Sigma$ of a the physical system to be modeled;
so $\mathcal{C}$ is the category of n-vector spaces among which the spaces of states of the quantum theory can be picked;
the morphism $Z(\hat \Sigma) : Z(\Sigma_{in}) \to Z(\Sigma_{out})$ that $\Sigma$ assigns to any cobordism $\hat \Sigma$ with incoming boundary $\Sigma_{in}$ and outgoing boundary $\Sigma_{out}$ is the propagator along $\hat \Sigma$: it maps every state $\psi \in Z(\Sigma_{in})$ of the system over $\Sigma_{in}$ to the state $Z(\Psi) \in \Sigma_{out}$ that is the result of the evolution of $\psi$ along $\hat \Sigma$ by the dynamics of the system. Or conversely: the action of $Z$ encodes what this dynamics is supposed to be.
Notice that since $Z$ is required to be a symmetric monoidal functor it sends disjoint unions of manifolds to tensor products
Moreover, for $\hat \Sigma$ a closed cobordism, hence a morphism $\emptyset \stackrel{\hat \Sigma}{\to} \emptyset$ from the empty manifold to itself, we have that
$Z(\emptyset) = \mathbf{1}$ is the tensor unit of $\mathcal{C}$;
$Z(\hat \Sigma) \in End(\mathbb{1})$ is an endomorphism of this tensor unit, a number as seen internal to $\mathcal{C}$ – this is the invariant associated to $\hat \Sigma$ by $Z$, called the partition function of $Z$ over $\hat \Sigma$. We can think of $Z$ 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 $S := Riem$ be Riemannian structure. Then the category $Bord_1^{Riem}$ of 1-dimensional cobordisms equipped with Riemannian structure is generated (as a symmetric monoidal category) from intervals
equipped with a length $t \in \mathbb{R}_+$. Composition is given by addition of lengths
Therefore a 1-dimensional Euclidean quantum field theory
is specified by
a vector space $\mathcal{H}$ (“of states”) assigned to the point;
for each $t \in \mathbb{R}_+$ a linear endomorphism
such that
This is just a system of quantum mechanics. If we demand that $Z$ respects the smooth structure on the space of morphisms in $Bord_1^{Riem}$ then there will be a linear map $i H : \mathcal{H} \to \mathcal{H}$ such that
This $H$ is called the Hamilton operator of the system.
(We are glossing here over some technical fine print in the definition of $Bord_1^{Riem}$. Done right we have that $\mathcal{H}$ may indeed be an infinite-dimensional vector space. See (1,1)-dimensional Euclidean field theories and K-theory)
A special class of examples of $n$-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 $n$ is
where
$Bord_n^S$ is the same category of cobordisms as before;
$Span(Grpd, \mathcal{C})$ is the category of spans of groupoids over $\mathcal{C}$:
objects are groupoids $K$ equipped with functors $\phi : K \to \mathcal{C}$;
morphisms $(K_1, \phi_1) \to (K_2, \phi_2)$ are diagrams
where in the middle we have a natural transformation;
composition of morphism is by forming 2-pullbacks:
Let $\hat \Sigma : \Sigma_1 \to \Sigma_2$ be a cobordism and
the value of a classical field theory on $\hat \Sigma$. We interpret this data as follows:
$Conf_{\Sigma_1}$ is the configuration space of a classical field theory over $\Sigma_1$: objects are “field configurations” on $\Sigma_1$ and morphisms are gauge transformations between these. Similarly for $Conf_{\Sigma_2}$.
Here a “physical field” can be something like the electromagnetic field. But it can also be something very different. For the special case of $\sigma$-models that we are eventually getting at, a “field configuration” here will instead be a way of an particle of shape $\Sigma_1$ sitting in some target space.
$Conf_{\hat \Sigma}$ is similarly the groupoid of field configurations on the whole cobordism, $\hat \Sigma$. If we think of an object in $Conf_{\hat \Sigma}$ of a way of a brane of shape $\Sigma_1$ sitting in some target space, then an object in $Conf_{\hat Sigma}$ is a trajectory of that brane in that target space, along which it evolves from shape $\Sigma_1$ to shape $\Sigma_2$.
$V_{\Sigma_i} : Conf_{\Sigma_i} \to \mathcal{C}$ is the classifying map of a kind of vector bundle over configuration space: a state $\psi \in Z(\Sigma_1)$ 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 $V_{c}$ of $V_{\Sigma_1}$ over a configuration $c \in Conf_{\Sigma_1}$ may be thought of as an internal state of the brane of shape $\Sigma_1$ sitting in target space.
$\exp(i S(-))_{\hat \Sigma}$ is the action functional that defines the classical field theory: the component
of this natural transformation on a trajectory $\gamma \in Conf_{\hat \Sigma}$ going from a configuration $\gamma|_{in}$ to a configuration $\gamma|_{out}$ is a morphism in $\mathcal{C}$ that maps the internal states of the ingoing configuration $\gamma|_{\Sigma_1}$ to the internal states of the outgoing configuration $\gamma|_{\Sigma_2}$. 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 $Span(Grpd, \mathcal{C})$. For more on this see classical field theory as quantum field theory?.
We assume now that $\mathcal{C}$ has colimits and in fact biproducts.
Then for every functor $\phi : K \to \mathcal{C}$ the colimit
exists, and (using the existence of biproducts) this construction extends to a functor
We call this the path integral functor.
For
a classical field theory, we get this way a quantum field theory by forming the composite functor
This $Z$ we call the quantization of $\exp(i S(-))$.
It acts
on objects by sending
the vector bundle on the configuration space over some boundary $\Sigma_{in}$ of worldvolume to its space $\mathcal{H}_{\Sigma_{in}}$ of gauge invariant sections. In typical situations this $\mathcal{H}_{\Sigma_{in}}$ is the famous Hilbert space of states in quantum mechanics, only that here it is allowed to be any object in $\mathcal{C}$;
on morphisms by sending a natural transformation
to the integral transform that it defines, weighted by the groupoid cardinality of $Conf_{\hat \Sigma}$ : the path integral .
A classical $\sigma$-model is a classical field theory such that
the configuration spaces $Conf_{\Sigma}$ are mapping spaces $\mathbf{H}(\Sigma,X)$ in some suitable category – some higher topos in fact – , for $X$ some fixed object of that category called target space ;
the bundles $V_{\Sigma} : Conf_{\Sigma} \to \mathcal{C}$ “of internal states” over these mapping spaces are
the transgression to these mapping spaces…
…of an associated higher bundle…
…asscociated to a circle n-bundle with connection on target space…
…encoded by a classifying morphism
into the circle n-group in $\mathbf{H}$; equipped with an n-connection $\nabla$…
…where the association is via a representation
on n-vector spaces, which is usually taken to be the canonical 1-dimensional one.
One calls $(\alpha,\nabla)$ the background gauge field of the $\sigma$-model.
The action functionals $\exp(i S(-))_{\hat \Sigma}$ are given by the higher parallel transport of $\nabla$ over $\hat \Sigma$.
So an $n$-dimensional $\sigma$-model is a classical field theory that is represented, in a sense, by a circle n-bundle with connection on some target space.
More specifically and more simply, in cases where $X$ is just a discrete ∞-groupoid – the case of sigma-models of Dijkgraaf-Witten type, every principal ∞-bundle on $X$ is necessarily flat, hence the background gauge field is given just by the morphism
Then for $\hat \Sigma$ a closed $n$-dimensional manifold, the action functional of the sigma-model on $\Sigma$ on a field configuration $\gamma : \hat \Sigma \to X$ has the value
being the evaluation of $[\alpha]$ regarded as a class in ordinary cohomology $H^n(\hat \Sigma, U(1))$ evaluated on the fundamental class of $X$.
One says that $[\alpha]$ is the Lagrangian of the theory.
(…)
Created on August 3, 2011 at 16:17:44. See the history of this page for a list of all contributions to it.