functorial quantum field theory
Reshetikhin?Turaev model? / Chern-Simons theory
FQFT and cohomology
physics, mathematical physics, philosophy of physics
theory (physics), model (physics)
experiment, measurement, computable physics
Axiomatizations
Tools
Structural phenomena
Types of quantum field thories
For semisimple Lie algebra targets
For discrete group targets
For discrete 2-group targets
For Lie 2-algebra targets
For targets extending the super Poincare Lie algebra
(such as the supergravity Lie 3-algebra, the supergravity Lie 6-algebra)
Chern-Simons-supergravity
for higher abelian targets
for symplectic Lie n-algebroid targets
for the $L_\infty$-structure on the BRST complex of the closed string:
higher dimensional Chern-Simons theory
topological AdS7/CFT6-sector
This is a sub-entry of sigma-model. See there for background and context.
We discuss how gauge theories and their higher analogs are naturally regarded as $\sigma$-models.
The classical sigma-models all have target spaces that are smooth manifolds. However, we saw that from dimension $d \geq 2$ on, the background gauge fields on these target spaces are naturally no longer just principal bundles with connection: instead they are smooth principal 2-bundles, then smooth principal 3-bundles, etc. and eventually generally principal ∞-bundles with ∞-connections. But the total space of such higher smooth bundles is no longer in general a smooth manifold: instead, the total space is a Lie groupoid for $d = 2$, then a Lie 2-groupoid for $d = 3$ and eventually generally a smooth ∞-groupoid.
This means that – unless we would artifically treat the total space of a background gauge field bundle on different grounds than its base space – the general theory of $\sigma$-models should naturally include target spaces that are not just smooth manifolds.
At least from $d = 2$ on, for instance, target spaces should be allowed to be Lie groupoids. This has a fairly long tradition: the proper étale Lie groupoids are precisely orbifolds, spaces that are locally isomorphic to sufficiently nice quotients of a Cartesian space by a group action. Orbifolds have received a lot of attention in the study of string sigma-models. The orientifold background gauge fields mentioned before involve in general a $\mathbb{Z}_2$-orbifold target space, for instance.
But once we pass to the higher geometry of Lie groupoids at all, there is no good reason to restrict ourselves to those that are orbifolds. For instance, for any Lie group $G$ there is its delooping Lie groupoid, the action groupoid of the trivial action of $G$ on the point, which we shall write
and this can perfectly serve as a target space object for $\sigma$-models too. Here the boldface notation is to indicate that this Lie groupoid is a smooth refinement of the classifying space $B G \in Top$ of the Lie group. In fact, where $B G$ gives isomorphism class of smooth $G$-principal bundles, $\mathbf{B}G$ also remembers the isomorphisms themselves – and hence in particular the automorphisms – of these bundles. It is the moduli stack of smooth $G$-principal bundles: for $\Sigma$ a smooth manifold we have that the groupoid of morphisms of smooth groupoids $\Sigma \to \mathbf{B}G$ (the correct morphisms, sometimes called Morita morphism to distinguish them from any incorrect notion) is that of smooth $G$-principal bundles and smooth homomorphisms between these
whereas the geometric realization $B G \simeq \vert \mathbf{B}G\vert$ only sees the equivalence classes:
This indicates that (nonabelian) gauge theory on $\Sigma$ should have a formulation as a $\sigma$-model with target “space” $X$ the Lie groupoid $X = \mathbf{B}G$: a $\sigma$-model field $\Sigma \to X = \mathbf{B}G$ is a $G$-bundle, and an isomorphism of field configurations is a gauge transformation of $G$-bundles.
But a field configuration in $G$-gauge theory on $\Sigma$ is not just a $G$-principal bundle, but is a $G$-bundle with connection. There is no Lie groupoid that that would similarly represent such connections as a target space object. But there is a smooth groupoid that does: $\mathbf{B}G_{conn}$, the groupoid of Lie algebra valued 1-forms.
Here by a smooth groupoid we mean a groupoid that comes with a rule for which of its collections of objects or morphisms are smoothly parameterized families. Technically this is a (2,1)-sheaf or stack on the site CartSp of Cartesian spaces and smooth functions between them. Among all smooth groupoids, Lie groupoids – and generally diffeological groupoids – are singled out as being the concrete objects. While it is useful to know if a given smooth groupoid is concrete or even Lie, it is in any case a fact that all of higher differential geometry exists for general smooth $\infty$-groupoids just as well. Therefore, if we can allow Lie groupoids as targets for $\sigma$-models, we can allow general smooth groupoids as well.
The non-concrete smooth groupoid $\mathbf{B}G_{conn}$ that we just mentioned is defined by the following rule: for $U \in$ CartSp, a smoothly $U$-parameterized family of objects is by definition a $\mathfrak{g}$-valued differential 1-form $A \in \Omega^1(U, \mathfrak{g})$ on $U$, where $\mathfrak{g}$ is the Lie algebra of $G$. A smoothly $U$-parameterized family of morphisms $g : A_1 \to A_2$ is a smooth gauge transformation $g \in C^\infty(U, G) : A_2 = g A g^{-1} + g d g^{-1}$ between two such form data. (This is “non-concrete” because the smooth $U$-parameterized families $U \to \mathbf{B}G_{conn}$ are not $U$-families of points $* \to \mathbf{B}G_{conn}$.)
One then finds that the mapping space groupoid for this target $X = \mathbf{B}G_{conn}$ is the groupoid
whose objects are smooth $G$-principal bundles with connection on $\Sigma$, and whose morphisms are smooth morphisms of principal bundles with connection. This groupoid is the configuration space of $G$-gauge theory on $\Sigma$, for instance of $G$-Yang-Mills theory or of $G$-Chern-Simons theory:
Notice that this configuration space is now itself a groupoid: morphisms are gauge transformations. In fact, it is naturally itself a smooth groupoid (when we read the hom-object here as an internal hom in SmoothGrpd). In the traditional physics literature these Lie groupoidal configuration spaces of fields are best known in terms of their infinitesimal approximation $Lie(Conf(\Sigma)) \in LieAld$, which are Lie algebroids, and these in turn are best known in terms of their function algebras, called the Chevalley-Eilenberg algebras $CE(Lie(Conf(\Sigma)))$: this dg-algebra is in physics called the BRST complex. Its degree-1 generators, the cotangents to the morphisms of $Conf(\Sigma)$, are called the ghost fields of gauge theory.
Of course we already saw secretly groupoidal configuration spaces in the above list of examples of $\sigma$-models of relativistic branes. We said that their configuration spaces $C^\infty(\Sigma,X)//Diff(\Sigma)$ were quotients; but really they are to be taken as higher categorical quotients, known as homotopy quotients or weak quotients : they are the action groupoids of $Diff(\Sigma)$ acting on $C^\infty(\Sigma,X)$.
We will see in the examples below that there is, of course, no reason to stop after passing from target manifolds to smooth target groupoids. At least as the $\sigma$-model increases in dimension, it is natural to consider smooth target 2-groupoids, target 3-groupoids, … target n-groupoids and eventually smooth ∞-groupoids. The full context of smooth $\infty$-groupoids is the natural completion of traditional differential geometry to higher geometry .
Given that it does thus make sense to regard general smooth ∞-groupoids as target spaces for $\sigma$-models, the questions is if there are useful background gauge fields on such. This is indeed the case:
for instance we have a theorem that says that for $G$ a compact Lie group, there is, for every integral cohomology class $c \in H^{n+1}(B G, \mathbb{Z})$ of the classifying space of $G$ – a characteristic class for $G$-principal bundles – up to equivalence a unique smooth lift $\mathbf{c} : \mathbf{B}G \to \mathbf{B}^n U(1)$ to a smooth circle n-bundle on the smooth $\mathbf{B}G$. Moreover, we have a theorem that for sufficiently highly connected Lie groups or smooth $\infty$-groups $G$, this refines canonically to a circle n-bundle with connection on the differentially refined smooth moduli space $\mathbf{B}G_{conn}$, given by a morphism:
This assignment generalizes the classical Chern-Weil homomorphism: we may speak of the ∞-Chern-Weil homomorphism . The first example below shows that ordinary Chern-Simons theory is a $\sigma$-model that arises this way. Generally we may this speak of $\sigma$-models with target space a smooth ∞-groupoid and background gauge fields given by the ∞-Chern-Weil homomorphism this way as ∞-Chern-Simons theories.
The second example below shows that ordinatry Dijkgraaf-Witten theory is a $\sigma$-model that arises this way when $G$ is a discrete group. Generally we may thus speak of $\sigma$-models with target space a discrete ∞-groupoid and background gauge fields given by the ∞-Chern-Weil homomorphism this way as ∞-Dijkgraaf-Witten theories.
One of the earliest topological quantum field theories ever considered in detail is Chern-Simons theory . We introduce this from the point of view of $\sigma$-models with higher geometric target spaces as discussed above.
An ordinary (as opposed to higher) gauge theory is a quantum field theory whose field configurations on a manifold $\Sigma$ are connections on $G$-principal bundles over $\Sigma$, for $G$ some Lie group. The word gauge transformation is essentially the physics equivalent of the word isomorphism , referring to isomorphisms in a configuration space of a field theory and specifically to isomorphisms between such bundles with connection. The action functional of a gauge theory is to be gauge invariant meaning that it assigns the same value to configurations that are related by a gauge transformaiton. This means precisely that the exponentiated action is a functor
from the groupoid of gauge field configurations and gauge transformaitons, to the circle group (regarded as a 0-truncated groupoid).
The first nonabelian gauge theory to receive attention was Yang-Mills theory : in that model $\Sigma$ is a 4-dimensional pseudo-Riemannian manifold modelling spacetime. The exponentiated action functional is given by the integral of differential 4-forms naturally associated with a connection and a Riemannian structure:
Here
$P$ is any $G$-principal bundle and $\nabla$ a connection on it;
$F_\nabla \in \Omega^2(P, \mathfrak{g})$ is the Lie algebra-valued curvature 2-form of this connection;
$\langle -,-\rangle : \mathfrak{g} \otimes \mathfrak{g} \to \mathbb{R}$ is an invariant polynomial on the Lie algebra: a bilinear form that is gauge invariant when evaluated on curvature 2-forms – for $\mathfrak{g}$ a semisimple Lie algebra this would be the Killing form and for a matrix Lie algebra this is simply the trace operation on products of matrices;
$\star$ is the Hodge star operator given by the pseudo-Riemannian metric structure on $\Sigma$.
$e \in \mathbb{R}$ is some constant, called the coupling constant of the model;
$\theta$ is another parameter called the theta-angle.
The first summand in the exponent, that depending on the pseudo-Riemannian structure, is the crucial term for the direct application of this as a model of phenomenologically observed physics: it controls the dynamics of three of the four force fields in the standard model of particle physics.
Instead of investigating this further, we shall here look at the case where $\frac{1}{e^2}$ is set to 0. While not directly of phenomenological relevance, this is of quite some interest for the general theoretical understanding of the space of all possible field theories. Since the resulting action functional
no longer depends on any extra (pseudo-Riemannian) structure on $\Sigma$ this may be interpreted as defining a topological quantum field theory : one speaks of topological Yang-Mills theory .
This is not quite a $\sigma$-model in the sense that we have been discussing: while the configuration space of topological Yang-Mills theory does consist of maps into the target space $X = \mathbf{B}G_{conn}$ (the smooth moduli stack of $G$-principal bundles with connection, as discussed above), there is no way that the above action functional is induced directly from the transgression of the higher holonomy of a circle n-bundle with connection on this target space. This is because, at least for semisimple Lie groups $G$, these are nontrivial only for odd $n$, whereas here we have $n = dim \Sigma = 4$.
But something closely related is true: $\exp(i S_{tYM})$ is the integrated curvature functional of a circle $3$-bundle with connection on $\mathbf{B}G_{conn}$: what we call the Chern-Simons circle 3-bundle .
This means the following: in generalization of how an ordinary circle bundle with connection $\nabla$ has a curvature 2-form, a circle n-bundle with connection $\nabla$ on a manifold $X$ has a curvature $(n+1)$-form $F_\nabla \in \Omega^{n+1}_{cl}(X)$. These curvature forms are closed, but not necessarily exact. Nevertheless, a generalization of the Stokes theorem holds true for them: for $\Sigma$ of dimension $n+1$ and denoting by $\partial \Sigma$ the boundary of $\Sigma$ and by $\gamma : \Sigma \to X$ a $\Sigma$-shaped trajectory in $X$, we have that the integral of the curvature over $\Sigma$ equals the higher holonomy of $\nabla$ over $\partial_\Sigma$:
This property in fact characterizes equivalence classes of circle $n$-bundles with connection. When conceiving of circle $n$-bundles with connection as rules for assigning higher holonomy that satisfy this property, one speaks of Cheeger-Simons differential characters .
Therefore, if we can find a circle 3-bundle with connection on the moduli stack $\mathbf{B}G_{conn}$ of $G$-principal bundles with connection whose curvature 4-form at $(P,\nabla)$ is $\langle F_\nabla \wedge F_\nabla \rangle$, then we can interpret topological Yang-Mills theory on a 4-dimensional $\Sigma$ with boundary as being given by a $\sigma$-model on $\partial \Sigma$ with background gauge field that circle 3-bundle.
For $G$ a connected and simply connected Lie group, such a circle 3-bundle indeed exists. Its characteristic morphism
from the smooth moduli stack of $G$-bundles with connection to the smooth moduli 3-groupoid of circle 3-bundles with connection is constructed and discussed in (Fiorenza-Schreiber-Stasheff), see Chern-Simons circle 3-bundle . This is the differential refinement of the smooth first fractional Pontryagin class
which in turn is a smooth refinement of the fractional Pontryagin class
of the classifying space $B G$.
To get a feeling for what this circle 3-bundle is like, we look at what its pull-back $\frac{1}{2}\hat \mathbf{p}_1(\phi) : \Sigma \stackrel{\phi}{\to} \mathbf{B}G_{conn} \stackrel{\frac{1}{2} \hat \mathbf{p}_1}{\to} \mathbf{B}^3 U(1)_{conn}$ to $\Sigma$ along any field configuration $\phi : \Sigma \to X = \mathbf{B}G_{conn}$ is like.
Notice that for simply conneced $G$ the classifying space $B G$ has vanishing homotopy groups in degree $k \leq 3$. Therefore every $G$-principal bundle $P$ on the 3-dimensional $\partial \Sigma$ is necessarily trivializable. In this case the configuration space of the $\sigma$-model is equivalent to the groupoid of Lie algebra valued forms
on $\partial \Sigma$. For $A \in \Omega^1(\Sigma, \mathrak{g})$ a field configuration and $F_A = d A + \frac{1}{2}[A \wedge A]$ the corresponding curvature 2-form, the curvature 4-form of $\frac{1}{2}\hat \mathbf{p}_1(\phi)$ is $\langle F_A \wedge F_A \rangle$. Its connection 3-form $C$ satisfying $d C = \langle F_A \wedge F_A \rangle$ is – up to a closed 3-form – the Chern-Simons 3-form
Therefore the action functional of the 3-dimensional $\sigma$-model given by the background gauge field $\frac{1}{2}\hat \mathbf{p}_1 : \mathbf{B}G_{conn} \to \mathbf{B}^3 U(1)_{conn}$ is given by
The quantum field theory defined by this action functional is known as Chern-Simons theory .
Every symplectic Lie n-algebroid $\mathfrak{P}$ serves as the target space of a canonically defined topological $\sigma$-model of dimension $n+1$. This is called the AKSZ sigma-model of $\mathcal{P}$.
This subsumes the following examples:
for $\mathfrak{P}$ a semisimple Lie algebra, the AKSZ $\sigma$-model is ordinary Chern-Simons theory;
for $\mathfrak{P}$ a Poisson Lie algebroid, the AKSZ $\sigma$-model is corresponding Poisson sigma-model;
for $\mathfrak{P}$ a Courant Lie 2-algebroid, the AKSZ $\sigma$-model is corresponding Courant sigma-model.
Also the A-model and the B-model 2-dimensional topological $\sigma$-models are examples.
The AKSZ action functional turns out to be very fundamental:
by ∞-Chern-Weil theory every invariant polynomial on an L-∞ algebroid induces an ∞-Chern-Weil homomorphism and the corresponding ∞-Chern-Simons theory action functional. Moreover, every symplectic Lie n-algebroid canonically carries a binary invariant polynomial. The AKSZ $\sigma$-model action functional is precisely the value of the $\infty$-Chern-Weil homomorphism on this invariant polynomial.
This is shown at ∞-Chern-Simons theory – Examples – AKSZ theory.
A sigma-model quantum field theory is, roughly, one
whose fields are maps $\phi : \Sigma \to X$ to some space $X$;
whose action functional is, apart from a kinetic term, the transgression of some kind of cocycle on $X$ to the mapping space $\mathrm{Map}(\Sigma,X)$.
Here the terms “space”, “maps” and “cocycles” are to be made precise in a suitable context. One says that $\Sigma$ is the worldvolume, $X$ is the target space and the cocycle is the background gauge field .
For instance the ordinary charged particle (for instance an electron) is described by a $\sigma$-model where $\Sigma = (0,t) \subset \mathbb{R}$ is the abstract worldline, where $X$ is a smooth (pseudo-)Riemannian manifold (for instance our spacetime) and where the background cocycle is a circle bundle with connection on $X$ (a degree-2 cocycle in ordinary differential cohomology of $X$, representing a background electromagnetic field : up to a kinetic term the action functional is the holonomy of the connection over a given curve $\phi : \Sigma \to X$.
The $\sigma$-models to be considered here are higher generalizations of this example, where the background gauge field is a cocycle of higher degree (a higher bundle with connection) and where the worldvolume is accordingly higher dimensional – and where $X$ is allowed to be not just a manifold but an approximation to a higher orbifold (a smooth ∞-groupoid).
More precisely, here we take the category of spaces to be smooth dg-manifolds. One may imagine that we can equip this with an internal hom $\mathrm{Maps}(\Sigma,X)$ given by $\mathbb{Z}$-graded objects. Given dg-manifolds $\Sigma$ and $X$ their canonical degree-1 vector fields $v_\Sigma$ and $v_X$ acting on the mapping space from the left and right. In this sense their linear combination $v_\Sigma + k \, v_X$ for some $k \in \mathbb{R}$ equips also $\mathrm{Maps}(\Sigma,X)$ with the structure of a differential graded smooth manifold.
Moreover, we take the “cocycle” on $X$ to be a graded symplectic structure $\omega$, and assume that there is a kind of Riemannian structure on $\Sigma$ that allows to form the transgression
by pull-push through the canonical correspondence
where on the right we have the evaluation map.
Assuming that one succeeds in making precise sense of all this one expects to find that $\int_\Sigma \mathrm{ev}^* \omega$ is in turn a symplectic structure on the mapping space. This implies that the vector field $v_\Sigma + k\, v_X$ on mapping space has a Hamiltonian $\mathbf{S} \in C^\infty(\mathrm{Maps}(\Sigma,X))$. The grade-0 components $S_{\mathrm{AKSZ}}$ of $\mathbf{S}$ then constitute a functional on the space of maps of graded manifolds $\Sigma \to X$. This is the AKSZ action functional defining the AKSZ $\sigma$-model with target space $X$ and background field/cocycle $\omega$.
In (AKSZ) this procedure is indicated only somewhat vaguely. The focus of attention there is a discussion, from this perspective, of the action functionals of the 2-dimensional $\sigma$-models called the A-model and the B-model .
In (Roytenberg), a more detailed discussion of the general construction is given, including an explicit and general formula for $\mathbf{S}$ and hence for $S_{\mathrm{AKSZ}}$ . For $\{x^a\}$ a coordinate chart on $X$ that formula is the following.
For $(X,\omega)$ a symplectic dg-manifold of grade $n$, $\Sigma$ a smooth compact manifold of dimension $(n+1)$ and $k \in \mathbb{R}$, the AKSZ action functional
(where $\mathfrak{T}\Sigma$ is the shifted tangent bundle)
is
where $\pi$ is the Hamiltonian for $v_X$ with respect to $\omega$ and where on the right we are interpreting fields as forms on $\Sigma$.
This formula hence defines an infinite class of $\sigma$-models depending on the target space structure $(X, \omega)$, and on the relative factor $k \in \mathbb{R}$. In (AKSZ) it was already noticed that ordinary Chern-Simons theory is a special case of this for $\omega$ of grade 2, as is the Poisson sigma-model for $\omega$ of grade 1 (and hence, as shown there, also the A-model and the B-model). The main example in (Roytenberg) is spelling out the general case for $\omega$ of grade 2, which is called the Courant sigma-model there.
One nice aspect of this construction is that it follows immediately that the full Hamiltonian $\mathbf{S}$ on mapping space satisfies $\{\mathbf{S}, \mathbf{S}\} = 0$. Moreover, using the standard formula for the internal hom of chain complexes one finds that the cohomology of $(\mathrm{Maps}(\Sigma,X), v_\Sigma + k v_X)$ in degree 0 is the space of functions on those fields that satisfy the Euler-Lagrange equations of $S_{\mathrm{AKSZ}}$. Taken together this implies that $\mathbf{S}$ is a solution of the “master equation” of a BV-BRST complex for the quantum field theory defined by $S_{\mathrm{AKSZ}}$. This is a crucial ingredient for the quantization of the model, and this is what the AKSZ construction is mostly used for in the literature.
THe 3-dimensional TQFT Dijkgraaf-Witten theory can be understood as being the ∞-Chern-Simons theory-$\sigma$-model whose target space is $\mathbf{B}G$ for $G$ a discrete group.
See ∞-Chern-Simons theory – Examples – Dijkgraaf-Witten theory
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The AKSZ sigma-model is discussed in
General $\infty$-Chern-Simons theory is discussed in