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



Quantum field theory


physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes

theory (physics), model (physics)

experiment, measurement, computable physics

\infty-Chern-Simons theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory




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


Exposition of a general abstract formulation

We give a leisurely exposition of a general abstract formulation σ\sigma-models, aimed at readers with a background in category theory but trying to assume no other prerequisites.

What is called an nn-dimensional σ\sigma-model is first of all an instance of an nn-dimensional quantum field theory (to be explained). The distinctive feature of those quantum field theories that are σ\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


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

Z:Bord n S𝒞, Z : Bord_n^S \to \mathcal{C} \,,


We think of data as follows:

  • Bord n SBord_n^S is a model for being and becoming in physics (following Bill Lawvere‘s terminology): the objects of Bord n SBord_n^S are archetypes of physical spaces that are and the morphisms are physical spaces that evolve ;

  • the object Z(Σ)Z(\Sigma) that ZZ assigns to any (n1)(n-1)-manifold Σ\Sigma is to be thought of as the space of all possible states over the space Σ\Sigma of a the physical system to be modeled;

  • so 𝒞\mathcal{C} is the category of n-vector spaces among which the spaces of states of the quantum theory can be picked;

  • the morphism Z(Σ^):Z(Σ in)Z(Σ out)Z(\hat \Sigma) : Z(\Sigma_{in}) \to Z(\Sigma_{out}) that Σ\Sigma assigns to any cobordism Σ^\hat \Sigma with incoming boundary Σ in\Sigma_{in} and outgoing boundary Σ out\Sigma_{out} is the propagator along Σ^\hat \Sigma: it maps every state ψZ(Σ in)\psi \in Z(\Sigma_{in}) of the system over Σ in\Sigma_{in} to the state Z(Ψ)Σ outZ(\Psi) \in \Sigma_{out} that is the result of the evolution of ψ\psi along Σ^\hat \Sigma by the dynamics of the system. Or conversely: the action of ZZ encodes what this dynamics is supposed to be.

Notice that since ZZ is required to be a symmetric monoidal functor it sends disjoint unions of manifolds to tensor products

F(Σ 1Σ 2)Z(Σ 1)Z(Σ 2). F(\Sigma_1 \coprod \Sigma_2) \simeq Z(\Sigma_1) \otimes Z(\Sigma_2) \,.

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

  • Z()=1Z(\emptyset) = \mathbf{1} is the tensor unit of 𝒞\mathcal{C};

  • Z(Σ^)End(𝟙)Z(\hat \Sigma) \in End(\mathbb{1}) is an endomorphism of this tensor unit, a number as seen internal to 𝒞\mathcal{C} – this is the invariant associated to Σ^\hat \Sigma by ZZ, called the partition function of ZZ over Σ^\hat \Sigma. We can think of ZZ as being a rule for computing such invariants by building them up from smaller pieces. This is the locaity of quantum field theory.


  • A simple but archetypical example is this: let S:=RiemS := Riem be Riemannian structure. Then the category Bord 1 RiemBord_1^{Riem} of 1-dimensional cobordisms equipped with Riemannian structure is generated (as a symmetric monoidal category) from intervals

    t \bullet \stackrel{t}{\to} \bullet

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

    (t 1t 2)=(t 1+t 2). (\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:Bord 1 RiemVect Z : Bord_1^{Riem} \to Vect

    is specified by

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

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

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

      such that

      U(t 1+t 2)=U(t 2)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 ZZ respects the smooth structure on the space of morphisms in Bord 1 RiemBord_1^{Riem} then there will be a linear map iH:i H : \mathcal{H} \to \mathcal{H} such that

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

    This HH is called the Hamilton operator of the system.

    (We are glossing here over some technical fine print in the definition of Bord 1 RiemBord_1^{Riem}. Done right we have that \mathcal{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 nn-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 nn is

  • a symmetric monoidal functor

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


    • Bord n SBord_n^S is the same category of cobordisms as before;

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

      • objects are groupoids KK equipped with functors ϕ:K𝒞\phi : K \to \mathcal{C};

      • morphisms (K 1,ϕ 1)(K 2,ϕ 2)(K_1, \phi_1) \to (K_2, \phi_2) are diagrams

        K^ K 1 K 2 𝒞, \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:

        (K^ 2K^ 1)=K^ 1 K 2K^ 2. (\hat K_2 \circ \hat K_1) = \hat K_1 \prod_{K_2} \hat K_2 \,.

Let Σ^:Σ 1Σ 2\hat \Sigma : \Sigma_1 \to \Sigma_2 be a cobordism and

exp(iS()) Σ=( Conf Σ^ ()| in ()| out Conf Σ 1 exp(iS() Σ^) Conf Σ 2 V Σ 1 V Σ 2 𝒞) \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 Σ^\hat \Sigma. We interpret this data as follows:

  • Conf Σ 1Conf_{\Sigma_1} is the configuration space of a classical field theory over Σ 1\Sigma_1: objects are “field configurations” on Σ 1\Sigma_1 and morphisms are gauge transformations between these. Similarly for Conf Σ 2Conf_{\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 σ\sigma-models that we are eventually getting at, a “field configuration” here will instead be a way of an particle of shape Σ 1\Sigma_1 sitting in some target space.

  • Conf Σ^Conf_{\hat \Sigma} is similarly the groupoid of field configurations on the whole cobordism, Σ^\hat \Sigma. If we think of an object in Conf Σ^Conf_{\hat \Sigma} of a way of a brane of shape Σ 1\Sigma_1 sitting in some target space, then an object in Conf Sigma^Conf_{\hat Sigma} is a trajectory of that brane in that target space, along which it evolves from shape Σ 1\Sigma_1 to shape Σ 2\Sigma_2.

  • V Σ i:Conf Σ i𝒞V_{\Sigma_i} : Conf_{\Sigma_i} \to \mathcal{C} is the classifying map of a kind of vector bundle over configuration space: a state ψZ(Σ 1)\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 cV_{c} of V Σ 1V_{\Sigma_1} over a configuration cConf Σ 1c \in Conf_{\Sigma_1} may be thought of as an internal state of the brane of shape Σ 1\Sigma_1 sitting in target space.

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

    exp(iS(γ)) Σ^:V γ| inV γ| out \exp(i S(\gamma))_{\hat \Sigma} : V_{\gamma|_{in}} \to V_{\gamma|_{out}}

of this natural transformation on a trajectory γConf Σ^\gamma \in Conf_{\hat \Sigma} going from a configuration γ| in\gamma|_{in} to a configuration γ| out\gamma|_{out} is a morphism in 𝒞\mathcal{C} that maps the internal states of the ingoing configuration γ| Σ 1\gamma|_{\Sigma_1} to the internal states of the outgoing configuration γ| Σ 2\gamma|_{\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 Span(Grpd,𝒞)Span(Grpd, \mathcal{C}). For more on this see classical field theory as quantum field theory?.


We assume now that 𝒞\mathcal{C} has colimits and in fact biproducts.

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

Kϕ𝒞 \int^{K} \phi \in \mathcal{C}

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

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

We call this the path integral functor.


exp(iS()):Bord n SSpan(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:=exp(iS()):Bord n Sexp(iS())Span(Grpd,𝒞)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 ZZ we call the quantization of exp(iS())\exp(i S(-)).

It acts

  • on objects by sending

    Σ in (V Σ in:Conf Σ in𝒞) Σ in:= KV Σ in \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 Σ in\Sigma_{in} of worldvolume to its space Σ in\mathcal{H}_{\Sigma_{in}} of gauge invariant sections. In typical situations this Σ in\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 𝒞\mathcal{C};

  • on morphisms by sending a natural transformation

    Σ^ (exp(iS()) Σ^:γV γ| inV γ| out) ( Kexp(iS()) Σ^: Σ 1 Σ 2) \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 Σ^Conf_{\hat \Sigma} : the path integral .

Classical σ\sigma-models

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

So an nn-dimensional σ\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 XX is just a discrete ∞-groupoid – the case of sigma-models of Dijkgraaf-Witten type, every principal ∞-bundle on XX is necessarily flat, hence the background gauge field is given just by the morphism

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

Then for Σ^\hat \Sigma a closed nn-dimensional manifold, the action functional of the sigma-model on Σ\Sigma on a field configuration γ:Σ^X\gamma : \hat \Sigma \to X has the value

exp(iS(γ)) Σ^= Σ^[α] \exp(i S(\gamma))_{\hat \Sigma} = \int_{\hat \Sigma} [\alpha]

being the evaluation of [α][\alpha] regarded as a class in ordinary cohomology H n(Σ^,U(1))H^n(\hat \Sigma, U(1)) evaluated on the fundamental class of XX.

One says that [α][\alpha] is the Lagrangian of the theory.

Quantum σ\sigma-models



Last revised on November 11, 2023 at 08:55:58. See the history of this page for a list of all contributions to it.