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This is a subentry of sigma-model. See there for background and context.
We survey, starting from the very basics, classical field theory aspects of $\sigma$-models that describe dynamics of particles, strings and branes on geometric target spaces.
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
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 model for physics 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 a 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 (strictly speaking this is the covariant derivative with respect to the trivial connection on the (canonically trivialized) tangent bundle on $\mathbb{R}^3$, see below for the fully fledged discussion).
We call then the collection of all smooth functions
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 $\vec E$ 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
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.
Notice that in the case of vanishing force field $\vec F = 0$, the equations of motion of the Newtonian particle
may be read as characterizing precisely the geodesics in $\mathbb{R}^3$ regarded as a Riemannian manifold using the canonical metric. This is a special and limiting case of the relativistic particle discussed below.
A cautionary note is in order. While the Newtonian particle may serve as an introductory example for motivating the concept of $\sigma$-models, it in general lacks some of the nice properties that later on we shall take to be characteristic of $\sigma$-models. Mainly this is due to the fact that the Newtonian particle is but a limiting approximation to the relativistic particle to which we turn next.
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 smooth 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 (or rather action groupoid)
of smooth functions $\Sigma \to X$ by diffeomorphisms $\Sigma \stackrel{\simeq}{\to} \Sigma$. Each object in configuration space is a trajectory $\gamma : \Sigma \to X$ of a particle in spacetime, each morphism/equivalence $\gamma_1 \stackrel{\simeq}{\to} \gamma_2$ : a gauge transformation.
The covariant phase space – the subspace of the configuration space of those configurations that satisfy the equations of motion – is defined to be
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, consider some special cases. First regard the case that the background field strength vanishes, $F = 0$. Then the equations of motion reduce to
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 $(\mathbb{R}^4, \eta) \simeq (X,g)$, these 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 these 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
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
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
Therefore a trajectory that solves the equations of motion and whose tangent vector is timelike or spacelike or lightlike, 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
happen to be the local critical points of the functional
given by
where on the left we have the integral of the volume form of the pullback $\gamma^* g \in Sym^2 T^* \Sigma$ 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 called the gauge coupling action.
Typically one characterizes $\sigma$-models in terms of such action functionals, so that the covariant phase space is then 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 circle bundle with connection $\nabla$: if we pass to the exponentiated action functional
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 $\Sigma = S^1$)
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
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 $d \mu_{Wien}$
(at least after replacing the kinetic Nambu-Goto action by the classically equivalent Polyakov action) 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 (see differential cohomology and differential cohomology in a cohesive topos), this suggests that there should be a general abstract theory of $\sigma$-models. Aspects of this are discussed below.
It is hard not to consider the following generalization of the relativistic particle $\sigma$-model, that we discussed above:
notice that nothing in the structure of the relativistic particle’s action functional (1) relies on the dimension of $\Sigma$ being $1$. Instead, it is just the degree-1 case of the following family of types of classical $\sigma$-models, that make sense for all $n \in \mathbb{N}$:
let $\Sigma$ be of dimension $n$;
let target space be a pseudo-Riemannian manifold $(X,g)$ as before;
let the background gauge field be given by a smooth differential n-form $A \in \Omega^n(X)$.
let configuration space be the weak quotient
let then finally the action functional be given by
This is the same formula as for the relativistic particle as before, only that now the differential forms are taken to be of degree $n$ and integrals to be over $n$-dimensional spaces.
Moreover, for each $n \in \mathbb{N}$ there is an analog of the generalization
to the generalization
Given any circle n-bundle with connection $\nabla$ and closed $\Sigma$ of dimension $n$, there is a higher holonomy functional
that extends the functional $A \mapsto \exp(i \int_\Sigma \gamma^* A)$.
Therefore, generally, we may take for $n \in \mathbb{N}$
the background gauge field on $X$ to be a circle n-bundle with connection $\nabla$;
the exponentiated action functional to be
For $n = 3$ such a $\sigma$-model describes an analog of a relativistic particle which is not point-like, but 2-dimensional (with 3-dimensional trajectory) hence which reminds one of a membrane. Inspired by this term, the general case has come to be known as the relativistic $(n-1)$-brane.
The case $n = 2$ is called the relativistic string, which we consider in more detail below. This has received a lot of attention (in string theory) not just because it is the next simplest in an infinite hierarchy of cases, but also because its quantum theory turns out to have various interesting features that seem to make it special. Moreover, many of the $(n-1)$-branes for other $n$ re-appear in one way or other in the study of the string (as its boundary D-branes in all dimensions $0 \leq n \leq 10$, as its “strongly coupled” version: the M-theory membrane, or as its electric-magnetic dual: the NS5-brane). If nothing else, the seemingly innocent step from $n = 1$ to $n = 2$ in the $\sigma$-model shows that there is a rich pattern of higher dimensional ($\sigma$-model) quantum field theories that are all interrelated in intricate ways.
Another important special case for the general discussion of $\sigma$-models is the case of the membrane, $n = 3$, for which the background gauge field is a Chern-Simons circle 3-bundle for some $G$-principal bundle on $X$, for $G$ some suitable Lie group. In this case the gauge-coupling Lagrangian of the $\sigma$-model is, locally, the Chern-Simons form $CS(\nabla_\mathfrak{g})$ of a $G$-connection $\nabla_{\mathfrak{g}}$, hence the action functional is (locally) the Chern-Simons functional
Below we will see that when $\sigma$-models are considered internal to a suitable cohesive (∞,1)-topos, then there are universal $\sigma$-models of this Chern-Simons type, whose target space is no longer a smooth manifold, but a smooth ∞-groupoid incarnation of a classifying space $B G$.
The important case $n = 2$ of the general (n-1)-brane sigma-model that we considered above is called the string-$\sigma$-model. Even though this is just the first step after the relativistic particle, the theory of this $\sigma$-model is already considerably richer classically and all the more so after quantization. For the purposes of this exposition here we only briefly indicate the physical interpretation of the $\sigma$-model and then consider some qualitatively new higher gauge theory aspects, that appear in this dimension.
First notice that by the general reasoning of relativistic $(n-1)$-branes, the background gauge field is now given (if we assume for the moment a topological trivial class) by a 2-form, which is traditionally denoted $B \in \Omega^2(X)$ and called the B-field. Its 3-form curvature field strength is traditionally denoted $H := d B$.
The action functional of the string’s $\sigma$-model for a pseudo-Riemannian target space $(X,g)$ with background gauge field $B$ is
To gain insight into the physical meaning of this, consider the simple case that target space $(X,g)$ is Minkowski spacetime and that the worldsheet $\Sigma = \mathbb{R} \times S^1$ is the cylinder. With $(\tau,\sigma)$ the two canonical coordinates on $\Sigma$, we still write
for the derivative “along the trajectory” (along the $\mathbb{R}$-factor), but now we also have the derivative $\partial_\sigma \gamma$ which we may think of as being tangential to the string at any instant of its trajectory. A field configuration $\gamma : \Sigma \to X$ may be thought of as the trajectory of a circle propagating in $X$.
The critical trajectories $\gamma : \Sigma \to X$ are found to be those that satisfy the 2-dimensional wave equation
on the worldsheet. Comparison with the equation of motion of the relativistic particle shows that $H(\partial_\sigma \gamma, -,-)$ plays the role of an electromagnetic field strength 2-form. Hence the string behaves as if electric charge is spread out evenly along it.
For point particle limit configurations $\gamma$, where the string has vanishing extension in that $\partial_\sigma \gamma = 0$, the above equation reduces again to free motion
and for general $(X,g)$ to the corresponding geodesic motion.
Therefore close to these point particle configurations the string looks like a little oscillating loop whose dynamics is that of its “center of mass” point, but slightly modified by the energy in the oscillations and the way these interact with the background fields. After quantization of the $\sigma$-model, these oscillations have a discrete ( quantized!) set of possible frequencies, and indeed each of the oscillation modes makes the string appear in the point particle limit as one species or other of a relativistic particle. (For more on this see string theory.)
Next we have a look at aspects of higher gauge theory that appears in $n = 2$.
The above 2-form $B$ is in general just the local connection form of a circle 2-bundle with connection $\nabla$ on $X$, given (as its homotopy fiber) by a morphism of smooth ∞-groupoids $\alpha : X \to \mathbf{B}^2 U(1)$. Equivalently this is a $U(1)$-bundle gerbe with connection.
There is a canonical 1-dimensional 2-representation of the circle 2-group $\mathbf{B} U(1)$ on 2-vector spaces:
Hence the corresponding associated 2-bundle is classified by a morphism
One can consider the string $\sigma$-model for worldsheets with boundary. A careful analysis then shows that the consistent Dirichlet-type boundary conditions that can be added correspond, roughly, to certain subspaces of target space – called D-branes – that are equipped with a section $V : \mathbf{1} \to \rho(g)|_{D-brane}$ of the background gauge field 2-vector bundle restricted to the $D$-brane. Such a section is precisely a twisted vector bundle on the brane, where the twist is the class in integral cohomology $H^3(X, \mathbb{Z})$ of the background gauge field. More generally, these twisted bundles are cocycles in twisted K-theory and differential K-theory. Hence more differential cohomology appears on the target space for the string in the presence of string boundaries.
More generally, the structure 2-group of the background principal 2-bundle need not be $\mathbf{B}U(1)$, which is given by the crossed module $[U(1) \to 1]$. Instead, it can be the automorphism 2-group $AUT(U(1))$, which is given by the crossed module $(U(1) \to Aut(U(1)) \simeq \mathbb{Z}_2)$. An $AUT(U(1))$-principal 2-bundle on $X$ is equivalently a double cover of $X$, equipped with a circle 2-bundle that has a twisted equivariance under the $\mathbb{Z}_2$-action. Such a background gauge field structure is called a string orientifold background. This is a kind of higher structure that the relativistic particle alone cannot see.
More such higher structure appears as one passes to the supergeometry analogs of the $\sigma$-models that we have considered so far: the superstring. The presence of the additional fermion fields that this brings with it (both on target space as well as on the worldsheet) influences all the structures that we have considered so far. For instance, a phenonemon called a fermionic quantum anomaly forces the above background circle 2-bundle to become a twisted 2-bundle, where the twist is given by a fivebrane charge Chern-Simons circle 3-bundle. This is discussed in detail at differential string structure
These are the first examples of a general phenomenon: as $n$ increases, a background gauge $n$-bundle with connection may constitute considerably more structure then one might naively expect from a generalization of the ordinary notion of a connection. More examples of this phenomenon arise when we allow our target spaces to be general smooth ∞-groupoids, below.