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

Surveys, textbooks and lecture notes

$\infty$-Chern-Simons theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory

Examples

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

Contents

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:

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

$Z:{\mathrm{Bord}}_{n}^{S}\to 𝒞\phantom{\rule{thinmathspace}{0ex}},$Z : Bord_n^S \to \mathcal{C} \,,

where

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 $Z\left(\Sigma \right)$ that $Z$ assigns to any $\left(n-1\right)$-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 $Z\left(\stackrel{^}{\Sigma }\right):Z\left({\Sigma }_{\mathrm{in}}\right)\to Z\left({\Sigma }_{\mathrm{out}}\right)$ that $\Sigma$ assigns to any cobordism $\stackrel{^}{\Sigma }$ with incoming boundary ${\Sigma }_{\mathrm{in}}$ and outgoing boundary ${\Sigma }_{\mathrm{out}}$ is the propagator? along $\stackrel{^}{\Sigma }$: it maps every state $\psi \in Z\left({\Sigma }_{\mathrm{in}}\right)$ of the system over ${\Sigma }_{\mathrm{in}}$ to the state $Z\left(\Psi \right)\in {\Sigma }_{\mathrm{out}}$ that is the result of the evolution of $\psi$ along $\stackrel{^}{\Sigma }$ 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

$F\left({\Sigma }_{1}\coprod {\Sigma }_{2}\right)\simeq Z\left({\Sigma }_{1}\right)\otimes Z\left({\Sigma }_{2}\right)\phantom{\rule{thinmathspace}{0ex}}.$F(\Sigma_1 \coprod \Sigma_2) \simeq Z(\Sigma_1) \otimes Z(\Sigma_2) \,.

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

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

• $Z\left(\stackrel{^}{\Sigma }\right)\in \mathrm{End}\left(𝟙\right)$ is an endomorphism of this tensor unit, a number as seen internal to $𝒞$ – this is the invariant associated to $\stackrel{^}{\Sigma }$ by $Z$, called the partition function of $Z$ over $\stackrel{^}{\Sigma }$. 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

$\left(•\stackrel{{t}_{1}}{\to }•\stackrel{{t}_{2}}{\to }\right)=\left(•\stackrel{{t}_{1}+{t}_{2}}{\to }•\right)\phantom{\rule{thinmathspace}{0ex}}.$(\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 $ℋ$ (“of states”) assigned to the point;

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

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

such that

$U\left({t}_{1}+{t}_{2}\right)=U\left({t}_{2}\right)\circ U\left({t}_{1}\right)\phantom{\rule{thinmathspace}{0ex}}.$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 $iH:ℋ\to ℋ$ such that

$U\left(t\right)=\mathrm{exp}\left(iHt\right)\phantom{\rule{thinmathspace}{0ex}}.$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 $ℋ$ 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

• $\mathrm{exp}\left(iS\left(-\right)\right):{\mathrm{Bord}}_{n}^{S}\to \mathrm{Span}\left(\mathrm{Grpd},𝒞\right)\phantom{\rule{thinmathspace}{0ex}},$\exp(i S(-)) : Bord_n^S \to Span(Grpd, \mathcal{C}) \,,

where

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

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

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

• morphisms $\left({K}_{1},{\varphi }_{1}\right)\to \left({K}_{2},{\varphi }_{2}\right)$ are diagrams

$\begin{array}{ccc}& & \stackrel{^}{K}\\ & ↙& & ↘\\ {K}_{1}& & ⇙& & {K}_{2}\\ & ↘& & ↙\\ & & 𝒞\end{array}\phantom{\rule{thinmathspace}{0ex}},$\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:

$\left({\stackrel{^}{K}}_{2}\circ {\stackrel{^}{K}}_{1}\right)={\stackrel{^}{K}}_{1}\prod _{{K}_{2}}{\stackrel{^}{K}}_{2}\phantom{\rule{thinmathspace}{0ex}}.$(\hat K_2 \circ \hat K_1) = \hat K_1 \prod_{K_2} \hat K_2 \,.

Let $\stackrel{^}{\Sigma }:{\Sigma }_{1}\to {\Sigma }_{2}$ be a cobordism and

$\mathrm{exp}\left(iS\left(-\right){\right)}_{\Sigma }=\left(\begin{array}{ccc}& & {\mathrm{Conf}}_{\stackrel{^}{\Sigma }}\\ & {}^{\left(-\right){\mid }_{\mathrm{in}}}↙& & {↘}^{\left(-\right){\mid }_{\mathrm{out}}}\\ {\mathrm{Conf}}_{{\Sigma }_{1}}& & {⇙}_{\mathrm{exp}\left(iS\left(-{\right)}_{\stackrel{^}{\Sigma }}\right)}& & {\mathrm{Conf}}_{{\Sigma }_{2}}\\ & {}_{{V}_{{\Sigma }_{1}}}↘& & {↙}_{{V}_{{\Sigma }_{2}}}\\ & & 𝒞\end{array}\right)$\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 $\stackrel{^}{\Sigma }$. 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.

• ${\mathrm{Conf}}_{\stackrel{^}{\Sigma }}$ is similarly the groupoid of field configurations on the whole cobordism, $\stackrel{^}{\Sigma }$. If we think of an object in ${\mathrm{Conf}}_{\stackrel{^}{\Sigma }}$ of a way of a brane of shape ${\Sigma }_{1}$ sitting in some target space, then an object in ${\mathrm{Conf}}_{\stackrel{^}{\mathrm{Sigma}}}$ 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 $\psi \in Z\left({\Sigma }_{1}\right)$ 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.

• $\mathrm{exp}\left(iS\left(-\right){\right)}_{\stackrel{^}{\Sigma }}$ is the action functional that defines the classical field theory: the component

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

of this natural transformation on a trajectory $\gamma \in {\mathrm{Conf}}_{\stackrel{^}{\Sigma }}$ 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 $\mathrm{Span}\left(\mathrm{Grpd},𝒞\right)$. 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

$\int :\mathrm{Span}\left(\mathrm{Grpd},𝒞\right)\to 𝒞\phantom{\rule{thinmathspace}{0ex}}.$\int : Span(Grpd, \mathcal{C}) \to \mathcal{C} \,.

We call this the path integral functor.

For

$\mathrm{exp}\left(iS\left(-\right)\right):{\mathrm{Bord}}_{n}^{S}\to \mathrm{Span}\left(\mathrm{Grpd},𝒞\right)$\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

$Z:=\int \circ \mathrm{exp}\left(iS\left(-\right)\right):{\mathrm{Bord}}_{n}^{S}\stackrel{\mathrm{exp}\left(iS\left(-\right)\right)}{\to }\mathrm{Span}\left(\mathrm{Grpd},𝒞\right)\stackrel{\int }{\to }𝒞\phantom{\rule{thinmathspace}{0ex}}.$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 $\mathrm{exp}\left(iS\left(-\right)\right)$.

It acts

• on objects by sending

$\begin{array}{rl}{\Sigma }_{\mathrm{in}}& ↦\left({V}_{{\Sigma }_{\mathrm{in}}}:{\mathrm{Conf}}_{{\Sigma }_{\mathrm{in}}}\to 𝒞\right)\\ & ↦{ℋ}_{{\Sigma }_{\mathrm{in}}}:={\int }^{K}{V}_{{\Sigma }_{\mathrm{in}}}\end{array}$\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 ${ℋ}_{{\Sigma }_{\mathrm{in}}}$ of gauge invariant sections. In typical situations this ${ℋ}_{{\Sigma }_{\mathrm{in}}}$ 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

$\begin{array}{rl}\stackrel{^}{\Sigma }& ↦\left(\mathrm{exp}\left(iS\left(-\right){\right)}_{\stackrel{^}{\Sigma }}:\gamma ↦{V}_{\gamma {\mid }_{\mathrm{in}}}\to {V}_{\gamma {\mid }_{\mathrm{out}}}\right)\\ & ↦\left({\int }^{K}\mathrm{exp}\left(iS\left(-\right){\right)}_{\stackrel{^}{\Sigma }}:{ℋ}_{{\Sigma }_{1}}\to {ℋ}_{{\Sigma }_{2}}\right)\end{array}$\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 ${\mathrm{Conf}}_{\stackrel{^}{\Sigma }}$ : 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 $H\left(\Sigma ,X\right)$ 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

One calls $\left(\alpha ,\nabla \right)$ the background gauge field of the $\sigma$-model.

• The action functionals $\mathrm{exp}\left(iS\left(-\right){\right)}_{\stackrel{^}{\Sigma }}$ are given by the higher parallel transport of $\nabla$ over $\stackrel{^}{\Sigma }$.

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

$\alpha :X\to {B}^{n}U\left(1\right)\phantom{\rule{thinmathspace}{0ex}}.$\alpha : X \to \mathbf{B}^{n} U(1) \,.

Then for $\stackrel{^}{\Sigma }$ a closed $n$-dimensional manifold, the action functional of the sigma-model on $\Sigma$ on a field configuration $\gamma :\stackrel{^}{\Sigma }\to X$ has the value

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

being the evaluation of $\left[\alpha \right]$ regarded as a class in ordinary cohomology ${H}^{n}\left(\stackrel{^}{\Sigma },U\left(1\right)\right)$ evaluated on the fundamental class of $X$.

One says that $\left[\alpha \right]$ is the Lagrangian of the theory.

(…)

References

Created on August 3, 2011 16:17:44 by Urs Schreiber (89.204.153.126)