The n-Category Café

Thanks! I’m honoured.

Maybe I should mention that today (Friday) I will have no time for digesting even the smallest bit, because we’re having our Scottish Category Theory Seminar. But in a couple of days my digestive capacity should be back to normal.

The Dijkgraaf-Witten theory entry in the nLab is missing the “gentle exposition” paragraph, although the header is already there - maybe, as a side effect of this discussion, we could fill that in. And maybe I can help with some particularly elementary questions.

Dijkgraaf-Witten theory in dimension n is the topological sigma-model quantum field theory…

I think I’ll let that pass on for now, John explained “sigma-model” over here, and the other terms will become clearer during the discussion, I hope.

…whose target space is the classifying space of a discrete group…

Since I don’t handle these concepts too often, I always need to refresh my memory, which is possible using the nLab and Wikipedia:

• A discrete group is a group with the trivial discrete topology, where all subsets are both open and closed,

• for a discrete group the classifying space is a pathconnected topological space X such that the fundamental group of X is isomorphic to G and the higher homotopy groups are trivial.

…and whose background gauge field is a circle n-bundle with connection on BG…

Here $G$ is our discrete group and $BG$ is its classifying space, but what is the “background gauge field”? And I’m also not sure that I know what a “circle n-bundle” is…

(From my viewpoint the nLab entry for “background gauge field” throws in more unfamiliar terms than it explains :-)

Exposition

We give a leisurely exposition of the idea of σ-models, aimed at readers with a background in category theory but trying to assume no other prerequisites.

What is called an $n$-dimensional σ-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 σ-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 is encoded by geometric data in a nice way: it describes physical configurations 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

6. Examples

See the comments with these titles further below.

Quantum field theory

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

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

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;

  • 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 .

• $\mathcal{C}$ 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;

• 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.

Back in the QVEST thread, John wrote:

A $\sigma$-model is a theory of physics where you have a manifold $M$ standing for ‘spacetime’, another manifold $X$, and a function

$$F: M \to X$$

which is called a ‘field’. Then there will be some differential equations that $F$ needs to satisfy.

Here are some unordered thoughts.

• Purely mathematically, this sounds almost like “a $\sigma$-model is a map of manifolds”.
• But only “almost”, because of that bit at the end about satisfying some differential equations.
• I realize that I don’t know what it means for a map between manifolds to satisfy a differential equation. I know that’s basic stuff and I’m not asking for an explanation; I’m sure I could look it up. But there we are.
• Although John more or less just said “a $\sigma$-model is a map of manifolds”, he also said that the domain was meant to represent spacetime. This reminds me of Lawvere’s definition of generalized element. A generalized element of an object $X$, of type $M$, is a map $M \to X$. One’s first reaction might be “but why do we need another word for map/morphism/arrow?” But it is actually useful terminology, because of the change of emphasis. The $M$ here is typically supposed to be some archetypal shape (such as a point or a circle), and the elements in $X$ of type $M$ are then the “figures in $X$ of shape $M$” (e.g. points or loops in $X$). That way of thinking seems like it might fit with $M$ being spacetime.

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

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

  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

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

    where in the middle we have a natural transformation;

  • composition of morphism is by forming 2-pullbacks:

    $$(\hat K_2 \circ \hat K_1) = \hat K_1 \prod_{K_2} \hat K_2 \,.$$

Let $\hat \Sigma : \Sigma_1 \to \Sigma_2$ be a [[cobordism]] and

$$\exp(i S(-))_{\Sigma} = \left( \array{ && Conf_{\hat \Sigma} \\ & {}^{\mathllap{(-)|_{in}}}\swarrow && \searrow^{\mathrlap{(-)|_{out}}} \\ Conf_{\Sigma_1} &&\swArrow_{\exp(i S(-)_{\hat \Sigma})}&& Conf_{\Sigma_2} \\ & {}_{V_{\Sigma_1}}\searrow && \swarrow_{\mathrlap{V_{\Sigma_2}}} \\ && \mathcal{C} } \right)$$

the value of a classical field theory on $\hat \Sigma$.

We interpret this data as follows:

• $Conf_{\Sigma_1}$ 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

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

  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.

Quantization

We assume now that $\mathcal{C}$ has colimits and in fact bilimits (colimits coinciding with limits).

Then for every functor $\phi : K \to \mathcal{C}$ the colimit

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

exists, and this construction extends to a functor

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

We call this the path integral functor.

For

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

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

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

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

It acts

• on objects by sending

  $$\begin{aligned} \Sigma_{in} & \mapsto (V_{\Sigma_{in}} : Conf_{\Sigma_{in}} \to \mathcal{C}) \\ & \mapsto \mathcal{H}_{\Sigma_{in}} := \int^K V_{\Sigma_{in}} \end{aligned}$$

  the vector bundle on the configuration space over some boundary $\Sigma_{in}$ 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

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

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

Classical sigma-models

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 cohesive 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

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

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

  • …where the association is via a representation $$\rho : \mathbf{B}^{n} U(1) \to (n+1) Vect$$ 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 $\exp(i S(-))_{\hat \Sigma}$ 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

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

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

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

being the evaluation of $[\gamma^* \alpha]$ 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.

This is a case of Descartes before the horse

What is called an n-dimensional σ-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 σ-models is that

even the toc lists quantum before classical!

Above John was trying to explain the idea of typical classical $\sigma$-models; concretely, without general abstraction.

Of course this is the traditional theory that is descibed in the introductory physics textbooks. My original idea of this thread here had been different: the strategy was to exploit the fact that one can speak of the concept of $\sigma$-model in an entirely different way – a general abstract way – a way that does not require knowledge of the concept of differential equations but just of basic category theory; and thus get abstract category theorists into the boat. But for the sake of completeness I should also provide some of the traditional discussion. So I’ll do that now.

As before I’ll post this in bite-sized bits in the comments below.

Exposition of classical sigma-models

1. The Newtonian particle

2. The relativistic particle

3. The relativistic (n-1)-brane

4. The relativistic string

The Newtonian particle

With hindsight, the earliest $\sigma$-model ever considered was also the very origin of the science of physics:

In order to describe the motion of matter particles in space, Isaac Newton wrote down a differential equation with the famous symbols

$$\vec F = m \vec a \,.$$

More in detail, this is meant to describe the following situation:

• write $X := \mathbb{R}^3$ for the Cartesian space of dimension 3; think of this as a crude model for physical space;

• write $\Sigma := \mathbb{R}$ for the Cartesian space of dimension 1; think of this as the abstract trajectory of a point particle;

• write $\gamma : \Sigma \to X$ for any smooth function; think of this as an actual trajectory of a point particle in $X$;

  write furthermore

  • $\vec v := \dot \gamma \in Hom(T \Sigma, T X)$ for the derivative of $\gamma$; think of this as the velocity of the particle; and

  • $\vec a := \ddot \gamma$ for the second derivative, the acceleration of the particle.

We call then the collection of all smooth functions

$$Conf := C^\infty(\Sigma, X)$$

the configuration space of a physical model of a point particle propagating on $X$.

In order to define the model – the model of some physical situation –

• pick a vector field $\vec F \in \Gamma(T X)$ on $X$.

  Think of this as expressing at each point a force acting on the particle with trajectory.

  For instance $\vec F := q \vec E$ could be an electric field influencing the propagation of an electrically charged particle of charge $q$.

In modern language we may say:

• $\Sigma$ is the worldline of the particle;

• $X$ is the target space (expressing the fact that it is the codomain of a function $\gamma : \Sigma \to X$)

• $\vec F$ is the background gauge field ;

and the collection $(\Sigma, X, \vec F)$ of all three is a $\sigma$-model .

Given this data, the space of solutions to the original differential equation

$$P := \{ \gamma \in Conf | \vec F = m \vec a \}$$

is called the covariant phase space of the model. The configurations in $P \subset Conf$ have the interpretation of being those potential configurations, that describe actual trajectories of particles observed in nature.

The relativistic particle

The Newtonian particle propagating on $\mathbb{R}^3$, discussed above, is a special and limiting case of a particle propagating on a 4-dimensional pseudo-Riemannian manifold: spacetime. For historical reasons (the same that led to the theory of gravity being called a theory of relativity ) this is called the relativistic particle .

The $\sigma$-model describing the relativistic particle is the following.

• Target space is a pseudo-Riemannian manifold $(X,g)$, thought of as spacetime.

• Parameter space $\Sigma = \mathbb{R}$ is the real line, thought of as the abstract worldline of the particle.

• The background gauge field is given by a circle bundle with connection, which for the moment we shall assume to be topologically trivial and hence be equivalently given by a differential 1-form $A \in \Omega^1(X)$. Its curvature exterior derivative $F := d A$ is the field strength of an electromagnetic field on $X$.

• The configuration space is the quotient

  $$Conf = C^\infty(\mathbb{R}, X)//Diff(\mathbb{R}) \,,$$

  of smooth functions $\Sigma \to X$ by diffeomorphisms $\Sigma \stackrel{\simeq}{\to} \Sigma$. Each point in cofiguration space is a trajectory of a particle in spacetime.

• The covariant phase space – the subspace of the configuration space of those configurations that satisfy the equations of motion – is defined to be

  $$P = \{ [\gamma] \in Conf |\;\; m g(\nabla_{\dot \gamma} \dot \gamma/ {\vert \dot \gamma}\vert,-) = q F(\dot \gamma, -) \;\; \} \subset Conf \,,$$

  where

  • $\nabla$ denotes the covariant derivative of the Levi-Civita connection of the background metric $g$ on the tangent bundle $T X$ of $X$.

  • the equation is taken to hold in each cotangent space $T^*_{\gamma(\tau)} X$ for each $\tau \in \mathbb{R}$.

To see what this means physically, consider some special cases. First regard the case that the background field strenght vanishes, $F = 0$. Then the equations of motion reduce to

$$\nabla_{\dot \gamma} \dot \gamma = 0 \,.$$

This says that the trajectory $\gamma$ exhibits parallel transport of its tangent vectors with respect to the Levi-Civita connection of the background metric. These curves are precisely the geodesics of the background geometry. This models motion under the force exerted by the field of gravity on our particle.

In the even more special case that $X$ is Minkowski spacetime, where we may find a global coordinate chart $(\mathbb{R}^4, \eta) \simeq (X,g)$, this are exactly the straight lines in $\mathbb{R}^4$. Given any such, there is precisely one representative in the diffeomorphism class for which $\mathbb{R} \stackrel{\gamma}{\to} \mathbb{R}^4 \stackrel{x^0}{\to} \mathbb{R}$ is the identity, hence for which the worldline parameter coincides precisely with the chosen global time coordinate $t := x^0$ on $\mathbb{R}^4$. For those the equations of motions are again those of the free Newtonian particle $\vec a = 0$.

Remaining in the case that $X$ is Minkowski space but allowing now a nontrivial background field, notice that we may write the 2-form $F$ always as

$$F = \vec E_i \cdot d x^i \wedge d t + \vec B^i \epsilon_{i j k} d x^j \wedge d x^j \,,$$

where $\vec E \in \mathbb{R}^3$ is the electric field strength vector and $\vec B \in \mathbb{R}^3$ the magnetic field strength vector. The spatial part of the above equations of motion are in this case again as for a Newtonian particle

$$m \vec a = q \vec E + q \vec v \times \vec B \,,$$

where in the second term we have the cross product of vectors in $\mathbb{R}^3$. The force on the right is the Lorentz force exerted by an electromagnetic field on a charged particle.

Notice that the equations of motion imply, generally, that the norm of $\dot \gamma$ is constant along the trajectory

$$\begin{aligned} \frac{d}{d \tau} g(\dot \gamma, \dot \gamma) &= 2 g(\nabla_{\dot \gamma} \dot \gamma, \dot \gamma) \\ & = 2 g(g^{-1}(\iota_{\dot \gamma} F, -), \dot \gamma) \\ & \propto F(\dot \gamma, \dot \gamma) \\ & = 0 \end{aligned} \,.$$

Therefore a trajectory that solves the equations of motion and whose tangent vector is timelike or spacelike , respectively at any instant is so throughout. In particular, no choice of gravitational and electromagnetic background field strength can accelerate a physical particle from being timelike to being light-like.

Experiments around the second half of the 19th and the beginning of the 20th century established that this covariant phase space correctly describes the dynamics of gravitationally and electromagnetically charged relativistic particles. But also formally this phase space is not a randomly chosen space; instead, it is the critical locus of a (mathematically) natural action functional .

The points in the covariant phase space

$$P := \{ [\gamma] \in Conf | m g(\nabla_{\dot \gamma} \dot \gamma / {\vert \dot \gamma}\vert,-) = q \iota_{\dot \gamma} F \} \subset Conf$$

happen to be the local critical points of the functional

$$S : Conf \to \mathbb{R}$$

given by

$$\begin{aligned} S([\gamma]) & := S_{kin}([\gamma]) + S_{gauge}([\gamma]) \\ & := m \int_\Sigma dvol(\gamma^\ast g) + q \int_\Sigma \gamma^\ast A \end{aligned} \,,$$

where on the left we have the integral of the volume form of the pullback $\gamma^\ast g \in Sym^2 T^\ast \Sigma$ of the metric on target space to the worldline.

This is called the action functional of the relativistic particle $\sigma$-model. The first summand is called the kinetic action , the second is calle the gauge coupling action.

Typically one characterizes $\sigma$-models in terms of such action functionals, so that the covariant phase space is given as their critical locus. This usually yields a simpler and deeper description of the model.

Notably the above action functional has an evident generalization to the case where the background electromagnetic field is not given by a globally defined 1-form, but more generally by a cicle bundle with connection $\nabla$: if we pass to the exponentiated action functional

$$\exp(i S(-)) : Conf \to U(1)$$

$$\exp(i S(\gamma)) = \exp(S_{kin}(\gamma)) \;\; \exp(i q \int_\Sigma \gamma^* A)$$

the second factor is precisely the holonomy of $\nabla$ over the worldline. Hence for general electromagnetic background gauge fields the action functional is (assuming for simplicity now closed curves with $\Sigma = S^1$)

$$\exp(i S(\gamma)) = \exp(S_{kin}(\gamma)) \;\; hol(\nabla, \gamma) \,.$$

This is the beginning of an important pattern: most $\sigma$-models are determined by a kind of higher gauge field $\nabla$ on target space (a cocycle in the differential cohomology of target space) and their dynamics is determined by an action functional that is the higher holonomy functional of this gauge field.

At the same time the kinetic action functional factor is usually to be understood as part of the measure on configuration space $Conf$. For the particle this has been made precise: the path integral

$$\int_{\gamma \in Conf} hol(\nabla,\gamma) \;\;\exp(i S_{kin}(\gamma)) [d \gamma] := \int_{\gamma \in Conf} hol(\nabla,\gamma) d \mu_{Wien}$$

can be interpreted as the integral with respect to the Wiener measure on path space (after Wick rotation, at least). The kinetic part of the action functional is then absorbed into the Wiener measure $d \mu_{Wien}$

$$\exp(i S_{kin}(\gamma)) [d\gamma] := d \mu_{Wien}$$

and the path integral is just the “expectation value” (after Wick rotation) of the holonomy, taken over all trajectories.

Since there is a good general abstract theory of higher gauge fields and their higher holonomies, this suggests that there should be a general abstract theory of $\sigma$-models.