nLab
sigma-model -- exposition of a general abstract formulation

This is a sub-entry of sigma-model. See there for further background and context.

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Exposition of a general abstract formulation

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

1. these arise from a simpler kind of field theory – called a classical field theory – by a process called quantization

2. 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:

1. Quantum field theory

2. Classical field theory

3. Quantization

4. Classical sigma-models

5. Quantum sigma-models

We draw from (FHLT, section 3).

Quantum field theory

Definition

For our purposes here, a quantum field theory of dimension $n$ is a symmetric monoidal functor

$$Z:{\mathrm{Bord}}_n^S\to 𝒞,$$Z : Bord_n^S \to \mathcal{C} \,,

where

• ${\mathrm{Bord}}_n^S$ 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

    $${\Sigma }_1\otimes {\Sigma }_2:={\Sigma }_1\coprod {\Sigma }_2;$$\Sigma_1 \otimes \Sigma_2 := \Sigma_1 \coprod \Sigma_2 \,;

• $𝒞$ is some symmetric monoidal category.

We think of data as follows:

• ${\mathrm{Bord}}_n^S$ is a model for being and becoming in physics (following Bill Lawvere’s terminology): the objects of ${\mathrm{Bord}}_n^S$ 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 $𝒞$ is the category of n-vector spaces among which the spaces of states of the quantum theory can be picked;

• the morphism $Z(\hat{\Sigma }):Z({\Sigma }_{\mathrm{in}})\to Z({\Sigma }_{\mathrm{out}})$ that $\Sigma$ assigns to any cobordism $\hat{\Sigma }$ with incoming boundary ${\Sigma }_{\mathrm{in}}$ and outgoing boundary ${\Sigma }_{\mathrm{out}}$ is the propagator? along $\hat{\Sigma }$: it maps every state $\psi \in Z({\Sigma }_{\mathrm{in}})$ of the system over ${\Sigma }_{\mathrm{in}}$ to the state $Z(\Psi )\in {\Sigma }_{\mathrm{out}}$ 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

$$F({\Sigma }_1\coprod {\Sigma }_2)\simeq Z({\Sigma }_1)\otimes Z({\Sigma }_2).$$F(\Sigma_1 \coprod \Sigma_2) \simeq Z(\Sigma_1) \otimes Z(\Sigma_2) \,.

Moreover, for $\hat{\Sigma }$ a closed cobordism, hence a morphism $\varnothing \stackrel{\hat{\Sigma }}{\to }\varnothing$ from the empty manifold to itself, we have that

• $Z(\varnothing )=1$ is the tensor unit of $𝒞$;

• $Z(\hat{\Sigma })\in \mathrm{End}(𝟙)$ is an endomorphism of this tensor unit, a number as seen internal to $𝒞$ – 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.

Examples

• A simple but archetypical example is this: let $S:=\mathrm{Riem}$ be Riemannian structure. Then the category ${\mathrm{Bord}}_1^{\mathrm{Riem}}$ of 1-dimensional cobordisms equipped with Riemannian structure is generated (as a symmetric monoidal category) from intervals

  $$•\stackrel{t}{\to }•$$\bullet \stackrel{t}{\to} \bullet

  equipped with a length $t\in {ℝ}_{+}$. Composition is given by addition of lengths

  $$(•\stackrel{t_1}{\to }•\stackrel{t_2}{\to })=(•\stackrel{t_1+t_2}{\to }•).$$(\bullet \stackrel{t_1}{\to} \bullet \stackrel{t_2}{\to}) = (\bullet \stackrel{t_1 + t_2}{\to} \bullet) \,.

  Therefore a 1-dimensional Euclidean quantum field theory

  $$Z:{\mathrm{Bord}}_1^{\mathrm{Riem}}\to \mathrm{Vect}$$Z : Bord_1^{Riem} \to Vect

  is specified by

  • a vector space $\mathscr{H}$ (“of states”) assigned to the point;

  • for each $t\in {ℝ}_{+}$ a linear endomorphism

    $$U(t):\mathscr{H}\to \mathscr{H}$$U(t) : \mathcal{H} \to \mathcal{H}

    such that

    $$U(t_1+t_2)=U(t_2)\circ U(t_1).$$U(t_1 + t_2) = U(t_2) \circ U(t_1) \,.

  This is just a system of quantum mechanics. If we demand that $Z$ respects the smooth structure on the space of morphisms in ${\mathrm{Bord}}_1^{\mathrm{Riem}}$ then there will be a linear map $iH:\mathscr{H}\to \mathscr{H}$ such that

  $$U(t)=\exp (iHt).$$U(t) = \exp(i H t) \,.

  This $H$ is called the Hamilton operator of the system.

  (We are glossing here over some technical fine print in the definition of ${\mathrm{Bord}}_1^{\mathrm{Riem}}$. Done right we have that $\mathscr{H}$ may indeed be an infinite-dimensional vector space. See (1,1)-dimensional Euclidean field theories and K-theory)

Classical field 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

• a symmetric monoidal functor

  $$\exp (iS(-)):{\mathrm{Bord}}_n^S\to \mathrm{Span}(\mathrm{Grpd},𝒞),$$\exp(i S(-)) : Bord_n^S \to Span(Grpd, \mathcal{C}) \,,

  where

  • ${\mathrm{Bord}}_n^S$ is the same category of cobordisms as before;

  • $\mathrm{Span}(\mathrm{Grpd},𝒞)$ is the category of spans of groupoids over $𝒞$:

    • objects are groupoids $K$ equipped with functors $\varphi :K\to 𝒞$;

    • morphisms $(K_1,{\varphi }_1)\to (K_2,{\varphi }_2)$ are diagrams

      $$\begin{array}{ccc}& & \hat{K}\\ & \swarrow & & \searrow \\ K_1& & ⇙& & K_2\\ & \searrow & & \swarrow \\ & & 𝒞\end{array},$$\array{ && \hat K \\ & \swarrow && \searrow \\ K_1 &&\swArrow&& K_2 \\ & \searrow && \swarrow \\ && \mathcal{C} } \,,

      where in the middle we have a natural transformation;

    • composition of morphism is by forming 2-pullbacks:

      $$({\hat{K}}_2\circ {\hat{K}}_1)={\hat{K}}_1\prod _{K_2}{\hat{K}}_2.$$(\hat K_2 \circ \hat K_1) = \hat K_1 \prod_{K_2} \hat K_2 \,.

Let $\hat{\Sigma }:{\Sigma }_1\to {\Sigma }_2$ be a cobordism and

$$\exp (iS(-){)}_{\Sigma }=\left(\begin{array}{ccc}& & {\mathrm{Conf}}_{\hat{\Sigma }}\\ & {}^{(-){\mid }_{\mathrm{in}}}\swarrow & & {\searrow }^{(-){\mid }_{\mathrm{out}}}\\ {\mathrm{Conf}}_{{\Sigma }_1}& & {⇙}_{\exp (iS(-{)}_{\hat{\Sigma }})}& & {\mathrm{Conf}}_{{\Sigma }_2}\\ & {}_{V_{{\Sigma }_1}}\searrow & & {\swarrow }_{V_{{\Sigma }_2}}\\ & & 𝒞\end{array}\right)$$\exp(i S(-))_{\Sigma} = \left( \array{ && Conf_{\hat \Sigma} \\ & {}^{\mathllap{(-)|_{in}}}\swarrow && \searrow^{\mathrlap{(-)|_{out}}} \\ Conf_{\Sigma_1} &&\swArrow_{\exp(i S(-)_{\hat \Sigma})}&& Conf_{\Sigma_2} \\ & {}_{V_{\Sigma_1}}\searrow && \swarrow_{\mathrlap{V_{\Sigma_2}}} \\ && \mathcal{C} } \right)

the value of a classical field theory on $\hat{\Sigma }$. We interpret this data as follows:

• ${\mathrm{Conf}}_{{\Sigma }_1}$ 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 ${\mathrm{Conf}}_{{\Sigma }_2}$.

  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.

• ${\mathrm{Conf}}_{\hat{\Sigma }}$ is similarly the groupoid of field configurations on the whole cobordism, $\hat{\Sigma }$. If we think of an object in ${\mathrm{Conf}}_{\hat{\Sigma }}$ of a way of a brane of shape ${\Sigma }_1$ sitting in some target space, then an object in ${\mathrm{Conf}}_{\widehat{\mathrm{Sigma}}}$ is a trajectory of that brane in that target space, along which it evolves from shape ${\Sigma }_1$ to shape ${\Sigma }_2$.

• $V_{{\Sigma }_i}:{\mathrm{Conf}}_{{\Sigma }_i}\to 𝒞$ 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 $c\in {\mathrm{Conf}}_{{\Sigma }_1}$ may be thought of as an internal state of the brane of shape ${\Sigma }_1$ sitting in target space.

• $\exp (iS(-){)}_{\hat{\Sigma }}$ is the action functional that defines the classical field theory: the component

  $$\exp (iS(\gamma ){)}_{\hat{\Sigma }}:V_{\gamma {\mid }_{\mathrm{in}}}\to V_{\gamma {\mid }_{\mathrm{out}}}$$\exp(i S(\gamma))_{\hat \Sigma} : V_{\gamma|_{in}} \to V_{\gamma|_{out}}

of this natural transformation on a trajectory $\gamma \in {\mathrm{Conf}}_{\hat{\Sigma }}$ going from a configuration $\gamma {\mid }_{\mathrm{in}}$ to a configuration $\gamma {\mid }_{\mathrm{out}}$ is a morphism in $𝒞$ that maps the internal states of the ingoing configuration $\gamma {\mid }_{{\Sigma }_1}$ to the internal states of the outgoing configuration $\gamma {\mid }_{{\Sigma }_2}$. 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 $\mathrm{Span}(\mathrm{Grpd},𝒞)$. For more on this see classical field theory as quantum field theory?.

Quantization

We assume now that $𝒞$ has colimits and in fact biproducts.

Then for every functor $\varphi :K\to 𝒞$ the colimit

$${\int }^K\varphi \in 𝒞$$\int^{K} \phi \in \mathcal{C}

exists, and (using the existence of biproducts) this construction extends to a functor

$$\int :\mathrm{Span}(\mathrm{Grpd},𝒞)\to 𝒞.$$\int : Span(Grpd, \mathcal{C}) \to \mathcal{C} \,.

We call this the path integral functor.

For

$$\exp (iS(-)):{\mathrm{Bord}}_n^S\to \mathrm{Span}(\mathrm{Grpd},𝒞)$$\exp(i S(-)) : Bord_n^S \to Span(Grpd, \mathcal{C})

a classical field theory, we get this way a quantum field theory by forming the composite functor

$$Z:=\int \circ \exp (iS(-)):{\mathrm{Bord}}_n^S\stackrel{\exp (iS(-))}{\to }\mathrm{Span}(\mathrm{Grpd},𝒞)\stackrel{\int }{\to }𝒞.$$Z := \int \circ \exp(i S(-)) : Bord_n^S \stackrel{\exp(i S(-))}{\to} Span(Grpd, \mathcal{C}) \stackrel{\int}{\to} \mathcal{C} \,.

This $Z$ we call the quantization of $\exp (iS(-))$.

It acts

• on objects by sending

  $$\begin{array}{rl}{\Sigma }_{\mathrm{in}}& \mapsto (V_{{\Sigma }_{\mathrm{in}}}:{\mathrm{Conf}}_{{\Sigma }_{\mathrm{in}}}\to 𝒞)\\ & \mapsto {\mathscr{H}}_{{\Sigma }_{\mathrm{in}}}:={\int }^K{V}_{{\Sigma }_{\mathrm{in}}}\end{array}$$\begin{aligned} \Sigma_{in} & \mapsto (V_{\Sigma_{in}} : Conf_{\Sigma_{in}} \to \mathcal{C}) \\ & \mapsto \mathcal{H}_{\Sigma_{in}} := \int^K V_{\Sigma_{in}} \end{aligned}

  the vector bundle on the configuration space over some boundary ${\Sigma }_{\mathrm{in}}$ of worldvolume to its space ${\mathscr{H}}_{{\Sigma }_{\mathrm{in}}}$ of gauge invariant sections. In typical situations this ${\mathscr{H}}_{{\Sigma }_{\mathrm{in}}}$ is the famous Hilbert space of states in quantum mechanics, only that here it is allowed to be any object in $𝒞$;

• on morphisms by sending a natural transformation

  $$\begin{array}{rl}\hat{\Sigma }& \mapsto (\exp (iS(-){)}_{\hat{\Sigma }}:\gamma \mapsto V_{\gamma {\mid }_{\mathrm{in}}}\to V_{\gamma {\mid }_{\mathrm{out}}})\\ & \mapsto ({\int }^K\exp (iS(-){)}_{\hat{\Sigma }}:{\mathscr{H}}_{{\Sigma }_1}\to {\mathscr{H}}_{{\Sigma }_2})\end{array}$$\begin{aligned} \hat \Sigma & \mapsto (\exp(i S(-))_{\hat \Sigma} : \gamma \mapsto V_{\gamma|_{in}} \to V_{\gamma|_{out}}) \\ & \mapsto (\int^K \exp(i S(-))_{\hat \Sigma} : \mathcal{H}_{\Sigma_1} \to \mathcal{H}_{\Sigma_2} ) \end{aligned}

  to the integral transform that it defines, weighted by the groupoid cardinality of ${\mathrm{Conf}}_{\hat{\Sigma }}$ : the path integral .

Classical $\sigma$-models

A classical $\sigma$-model is a classical field theory such that

• the configuration spaces ${\mathrm{Conf}}_{\Sigma }$ are mapping spaces $H(\Sigma ,X)$ in some suitable category – some higher topos in fact – , for $X$ some fixed object of that category called target space ;

• the bundles $V_{\Sigma }:{\mathrm{Conf}}_{\Sigma }\to 𝒞$ “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

    $$\alpha :X\to B^{n}U(1)$$\alpha : X \to \mathbf{B}^{n} U(1)

    into the circle n-group in $H$; equipped with an n-connection $\nabla$…

  • …where the association is via a representation

    $$\rho :B^{n+1}U(1)\to (n+1)\mathrm{Vect}$$\rho : \mathbf{B}^{n+1} U(1) \to (n+1) Vect

    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 $\exp (iS(-){)}_{\hat{\Sigma }}$ 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

$$\alpha :X\to B^{n}U(1).$$\alpha : X \to \mathbf{B}^{n} U(1) \,.

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

$$\exp (iS(\gamma ){)}_{\hat{\Sigma }}={\int }_{\hat{\Sigma }}[\alpha ]$$\exp(i S(\gamma))_{\hat \Sigma} = \int_{\hat \Sigma} [\alpha]

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.

Quantum $\sigma$-models

(…)

References

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