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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.
(…)