sigma-model -- exposition of higher gauge theories as sigma-models

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#### Quantum field theory
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#### Phyiscs
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#### $\infty$-Chern-Simons theory
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> This is a sub-entry of [[sigma-model]]. See there for background and context.

***

#Contents#
* table of contents
{:toc}

## Exposition of higher gauge theories as $\sigma$-models

We discuss how [[gauge theories]] and their higher analogs are
naturally regarded as $\sigma$-models.

### Higher geometric target spaces
 {#HigherGeometricTargetSpaces}

The [[sigma-model -- exposition of classical sigma-models|classical sigma-models]] all have [[target space]]s that are [[smooth manifold]]s. However, we saw that from [[dimension]] $d \geq 2$ on, the [[background gauge field]]s on these target spaces are naturally no longer just [[principal bundle]]s [[connection on a bundle|with connection]]: instead they are smooth [[principal 2-bundle]]s, then smooth principal 3-bundles, etc. and eventually generally [[principal ∞-bundle]]s with [[connection on an ∞-bundle|∞-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 groupoid]]s. This has a fairly long tradition: the _[[proper groupoid|proper]] [[étale groupoid|étale]]_ [[Lie groupoid]]s are precisely _[[orbifold]]s_, 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-model]]s. The [[orientifold]] background gauge fields mentioned [before](http://ncatlab.org/nlab/show/sigma-model+--+exposition+of+classical+sigma-models#RelativisticString) involve in general a $\mathbb{Z}_2$-orbifold target space, for instance.

But once we pass to the higher geometry of [[Lie groupoid]]s 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

$$
  \mathbf{B}G := *//G
  \,;
$$

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 [[isomorphism]]s themselves -- and hence in particular the [[automorphism]]s -- 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 infinity-groupoid|smooth groupoid]]s $\Sigma \to \mathbf{B}G$ (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

$$
  SmoothGrpd(\Sigma, \mathbf{B}G) \simeq G Bund(\Sigma) \in Grpd
  \,,
$$

whereas the [[geometric realization]] $B G \simeq \vert \mathbf{B}G\vert$ only sees the equivalence classes:
$$
  [\Sigma, B G] \simeq \pi_0 G Bund(\Sigma) \in Set
  \,.
$$

This indicates that ([[nonabelian cohomology|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 on a bundle|connection]]_. There is no Lie groupoid that that would similarly represent such connections as a target space object. But there is a _[[smooth infinity-groupoid|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 space]]s and [[smooth function]]s between them. Among all smooth groupoids, Lie groupoids -- and generally [[diffeological space|diffeological groupoid]]s -- are singled out as being the _[[concrete sheaf|concrete]]_ objects. While it is useful to know if a given smooth groupoid is [[concrete sheaf|concrete]] or even [[Lie groupoid|Lie]], it is in any case a fact that all of [[higher geometry|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 $\mathfrak{g}$-[[Lie algebra valued 1-form|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

$$
  SmoothGrpd(\Sigma , \mathbf{B}G_{conn})
   \simeq
  G Bund_{conn}(\Sigma)
  \,,
$$

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]]:

$$
  SmoothGrpd(\Sigma, \mathbf{B}G_{conn})
  \simeq
   Conf_{Yang-Mills}(\Sigma)
  \simeq
   Conf_{Chern-Simons}(\Sigma)
  \,.
$$

Notice that this configuration space is now itself a groupoid: morphisms are [[gauge transformation]]s. In fact, it is naturally itself a smooth groupoid (when we read the [[hom-object]] here as an [[internal hom]] in [[Smooth∞Grpd|SmoothGrpd]]). In the traditional physics literature these Lie groupoidal [[configuration space]]s of fields are best known in terms of their [[infinitesimal object|infinitesimal]] approximation $Lie(Conf(\Sigma)) \in LieAld$, which are _[[Lie algebroid]]s_, and these in turn are best known in terms of their [[function algebras on ∞-stacks|function algebras]], called the _[[Chevalley-Eilenberg algebra]]s_ $CE(Lie(Conf(\Sigma)))$: this [[dg-algebra]] is in physics called the [[BRST complex]]. Its degree-1 generators, the [[cotangent vector|cotangents]] to the [[morphism]]s of $Conf(\Sigma)$, are called the _ghost fields_ of gauge theory.

Of course we already saw secretly groupoidal configuration spaces in the [above](ExpositionClassical) list of examples of $\sigma$-models of relativistic [[branes]]. We said that their configuration spaces $C^\infty(\Sigma,X)//Diff(\Sigma)$ were [[quotient]]s; but really they are to be taken as higher categorical quotients, known as _[[homotopy colimit|homotopy quotients]]_ or _[[2-colimit|weak quotients]]_ : they are the [[action groupoid]]s 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-groupoid]]s, target [[3-groupoid]]s, ... target [[n-groupoid]]s  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 ∞-groupoid]]s as target spaces for $\sigma$-models, the questions is if there are useful [[background gauge field]]s 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 [[infinity-Chern-Weil theory|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:

$$
  \hat \mathbf{c} : \mathbf{B}G_{conn} \to \mathbf{B}^n U(1)_{conn}
  \,.
$$

This assignment generalizes the classical [[Chern-Weil homomorphism]]: we may speak of the _[[∞-Chern-Weil homomorphism]]_ . The first example [below](#SigmaCS) 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 field]]s given by the [[∞-Chern-Weil homomorphism]] this way as [[schreiber:∞-Chern-Simons theories]].

The second example [below](#SigmaDW) 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 field]]s given by the [[∞-Chern-Weil homomorphism]] this way as [[schreiber:∞-Dijkgraaf-Witten theories]].

### Chern-Simons theory as a $\sigma$-model
 {#SigmaCS}

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](#HigherGeometricTargetSpaces).

An ordinary (as opposed to higher) _[[gauge theory]]_ is a [[quantum field theory]] whose field configurations on a manifold $\Sigma$ are [[connection on a bundle|connections]] on $G$-[[principal bundle]]s 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 invariance|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]]

$$
  \exp(i S(-)) :  G Bund_{conn}(\Sigma) \to U(1)
$$

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:

$$
  \exp(i S_{YM}(-)) : (P, \nabla) \mapsto \exp(i \int_\Sigma
     \frac{1}{e^2} \langle F_\nabla \wedge \star F_\nabla\rangle
     +
     i \theta \langle F_\nabla \wedge  F_\nabla
   \rangle)
  \,.
$$

Here

* $P$ is any $G$-[[principal bundle]] and $\nabla$ a [[connection on a bundle|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

$$
  \exp(i S_{tYM}) : (P, \nabla) \mapsto \exp(i \int_\Sigma \langle F_\nabla \wedge F_\nabla \rangle)
$$

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](HigherGeometricTargetSpaces)),  there is no way that the above action functional is induced directly from the transgression of the [[higher parallel transport|higher holonomy]] of a [[circle n-bundle with connection]] on this target space.  This is because, at least for [[semisimple Lie group]]s $G$, 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 parallel transport|higher holonomy]] of $\nabla$ over $\partial_\Sigma$:

$$
  \exp(i \int_\Sigma \phi^* F_\nabla)
  =
  hol(\nabla, \gamma|_{\partial \Sigma})
  \,.
$$

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

$$
  \frac{1}{2}\hat \mathbf{p}_1 :
  \mathbf{B}G_{conn}
   \to
  \mathbf{B}^3 U(1)_{conn}
$$

from the smooth moduli stack of $G$-bundles with connection to the smooth moduli 3-groupoid of [[circle n-bundle with connection|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 [[differential string structure|smooth first fractional Pontryagin class]]

$$
  \frac{1}{2}\mathbf{p}_1 : \mathbf{B}G \to \mathbf{B}^3 U(1)
$$

which in turn is a smooth refinement of the fractional [[Pontryagin class]]

$$
  \frac{1}{2} p_1 : B G \to B^3 U(1) \simeq K(\mathbb{Z}, 4)
$$

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 group]]s 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]]

$$
  SmoothGrpd(\partial \Sigma, \mathbf{B}G_{conn})
  \simeq
  \Omega^1(\partial \Sigma, \mathfrak{g})//C^{\infty}(\partial \Sigma,G)
$$

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 form|Chern-Simons 3-form]]_

$$
  C = cs(A) = \langle A \wedge F_A \rangle + \frac{1}{6}\langle A \wedge [A \wedge A]\rangle
  \,.
$$

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

$$
  \exp(i S(-))_{\Sigma_3} : A \mapsto
  \exp(i \int_{\Sigma_3} cs(A))
  \,.
$$

The [[quantum field theory]] defined by this action functional is known as _[[Chern-Simons theory]]_ .

### AKSZ theory as a higher Chern-Simons $\sigma$-model
 {#AKSZSigmaModel}

#### Summary

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 theory|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](http://ncatlab.org/schreiber/show/infinity-Chern-Simons+theory+--+examples#ASKZTheory).

#### Definition

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 action|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|pseudo]]-)[[Riemannian manifold]]
(for instance our [[spacetime]]) and where the background cocycle
is a [[circle bundle]] with [[connection on a bundle|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 [[connection on an infinity-bundle|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 [[space]]s
to be [[dg-geometry|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-geometry|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]]

$$
  \int_\Sigma \mathrm{ev}^* \omega
  :=
  p_! \mathrm{ev}^* \omega
$$

by [[integral transform|pull-push]] through the canonical
[[correspondence]]

$$
  \mathrm{Maps}(\Sigma,X)
  \stackrel{p}{\leftarrow}
  \mathrm{Maps}(\Sigma,X) \times \Sigma
  \stackrel{ev}{\to}
   X
  \,,
$$

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](#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](#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.

+-- {: .num_defn #TheAKSZAction}
###### Definition

For $(X,\omega)$ a [[symplectic Lie n-algebroid|symplectic dg-manifold]] of grade $n$, $\Sigma$ a smooth compact manifold of dimension $(n+1)$
and $k \in \mathbb{R}$, the
**AKSZ action functional**

$$
  S_{\mathrm{AKSZ},k} :
   \mathrm{SmoothGrMfd}(\mathfrak{T}\Sigma, X)
  \to
  \mathbb{R}
$$

(where $\mathfrak{T}\Sigma$ is the shifted tangent bundle)

is

$$
  S_{\mathrm{AKSZ},k}
   :
  \phi
  \mapsto
  \int_\Sigma
  \left(
    \frac{1}{2}\omega_{ab} \phi^a \wedge d_{\mathrm{dR}}\phi^b
    +
    k
    \,
    \pi(\phi\wedge \cdots \wedge \phi)
  \right)
  \,,
$$

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](#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](#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.

### Dijkgraaf-Witten theory as a $\sigma$-model
 {#SigmaDW}

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](http://ncatlab.org/schreiber/show/infinity-Chern-Simons+theory+--+examples#DiscreteTargets)

### The Yetter model as a $\sigma$-model

(...)

### $\infty$-Dijkgraaf-Witten theory

(...)

### $\infty$-Chern-Simons theory

See [[schreiber:∞-Chern-Simons theory]].

## References

The [[AKSZ sigma-model]] is discussed in

* M. Alexandrov, [[Maxim Kontsevich|M. Kontsevich]], [[Albert Schwarz|A. Schwarz]], O. Zaboronsky, _The geometry of the master equation and topological quantum field theory_, Int. J. Modern Phys. A 12(7):1405--1429, 1997 ([arXiv:hep-th/9502010](http://arxiv.org/abs/hep-th/9502010))
 {#AKSZ}

* [[Dmitry Roytenberg]], _AKSZ-BV Formalism and Courant Algebroid-induced Topological Field Theories_ Lett.Math.Phys.79:143-159,2007 ([arXiv](http://arxiv.org/abs/hep-th/0608150)).
 {#Roytenberg}

General $\infty$-Chern-Simons theory is discussed in

* [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], _[[schreiber:infinity-Chern-Simons theory]]_


sigma-model -- exposition of quantum sigma-models

+-- {: .rightHandSide}
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###Context###
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Phyiscs
+--{: .hide}
[[!include physicscontents]]
=--
#### $\infty$-Chern-Simons theory
+--{: .hide}
[[!include infinity-Chern-Simons theory - contents]]
=--
=--
=--

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

***

#Contents#
* table of contents
{:toc}

## Exposition of quantum sigma-models

Above we have discussed some standard [classical sigma-models](#ExpositionClassical) and [higher gauge theories as sigma-models](#GaugeTheoryAsSigmaModel), also mostly classically. Here we talk about the [[quantization]] of these models (or some of them) to [[QFT]]s: _quantum $\sigma$-models_ .

### Particle on the line

for the moment see the discussion at

* [[path integral as a pull-push transform]]

### Topological string on a smooth manifold and string topology operations
 {#StringTopology}

Chas and Sullivan famously noticed, that the [[homology group]]s of the [[free loop space]] $L X$ of a [[compact space|compact]] [[orientation|oriented]] [[smooth manifold]] $X$ are equipped with an interesting paring operation

$$
  H_\bullet(L X) \otimes H_\bullet(L X) \to H_{\bullet - dim X}(L X)
$$

that generalizes the [[Goldman bracket]] on $H_0(L X) \simeq F \pi_1(X)$. Since this operation is induced from concatenating loops, they called it the _string product_ . Its study has come to be known as [[string topology]], now a branch of [[differential topology]].

It was soon realized that there indeed ought to be a relation to [[string]] physics: there ought to be a 2-dimensional [[quantum field theory]] associated with $X$, as follows:

for $\Sigma$ a 2-dimensional [[surface]] with incoming and outgoing [[boundary]] components $\partial_{in} \Sigma \stackrel{in}{\to} \Sigma \stackrel{out}{\leftarrow} \partial_{out} \Sigma$, the "space of [[state]]s" of the theory ought to be given by the [[homology group]]s $H_\bullet(X^{\partial_{in} \Sigma})$ and $H_\bullet(X^{\partial_{out} \Sigma})$, and the [[path integral as a pull-push transform]] along $\Sigma$ ought to be given by push-forward and dual [[fiber integration]]

$$
  (X^{out})_* \circ (X^{in})^! : H_\bullet(X^{\partial_{in} \Sigma})
   \to
  H_{\bullet- dim X}(X^{\partial_{out} \Sigma})
$$

induced by the [[mapping space]] [[span]]

$$
  \array{
     && X^{\Sigma}
     \\
     & {}^{\mathllap{X^{in}}}\swarrow && \searrow^{\mathrlap{X^{out}}}
     \\
    X^{\partial_{in} \Sigma} &&&& X^{\partial_{out} \Sigma}
  }
  \,.
$$

For $\Sigma \simeq S^1 \vee S^1$ the 3-holed sphere with two incoming and one outgoing circle, this would describe an operation on string states induced by the merging of two closed strings to a single one

$$
  H_\bullet(X^{S^1 \coprod S^1})
 \simeq
  H_\bullet(L X) \times H_\bullet(L X)
  \to
  H_\bullet(L X)
$$

and this ought to be Chas-Sullivan string product operation.

That this is indeed the case was finally demonstrated by [[Ralph Cohen]] and [[Veronique Godin]]. (See [[string topology]] for all references.) While the idea is rather simple, the concrete realization, especially when taking open strings into account, is fairly technical (see for instance [this MO discussion](http://mathoverflow.net/questions/66977/pull-push-in-godins-hcft-for-string-topology)).

But there should be more to it: one expects that these operations on [[homology group]]s are just a shadow of a refined construction on [[chain complex]]es (for instance [[singular homology|singular chains]]): while Godin's construction gives an _[[HQFT]]_ -- a [[quantum field theory]]  that depends only on the [[homology]] of the [[moduli space]]s of the relevant [[cobordism]]s -- one expect that this is the homology of a genuine [[extended TQFT]] (which in this dimension is widely but somewhat unfortunately known under the term "[[TCFT]]"). Remarks on how that might be obtained have been made in print by Costello and Lurie.

In the context of our discussion of $\sigma$-models, we would want to refine this even one further step and ask: is the string-topology TCFT of a manifold (given that it exists) formally a $\sigma$-model with [[target space]] that manifold, and using some suitable [[background gauge field]]?

Given that the string topology [[TCFT]] itself has not been fully identified yet, we cannot expect a complete answer to this at the moment, but we will discuss some crucial ingredients that are available.

Notably we can first ignore the dynamics of the system, just consider the kinematics and ask the simple question: _which quantum $\sigma$-models on $X$ have [[(∞,n)-vector space]]s of [[states]] whose [[decategorification]] are graded [[homology group]]s $H_\bullet(X^{\partial \Sigma})$ of mapping spaces of $X$?_

The answer to this question is more more transparent after we formulate the question in more generality: as observed by Cohen and Godin in their _[[A Polarized View of String Topology]]_ , we may assume without restriction that the [[homology group]]s here are with respect to the [[generalized homology]] with coefficients in any [[E-∞ ring|commutative ∞-ring]] $K$ (as long as this is an "$\infty$-field" and as long as $X$ is $K$-[[orientation in generalized cohomology|oriented]]):

$$
  H_\bullet(X^{\partial_{in} \Sigma})
   :=
  H_\bullet(X^{\partial_{in}} \Sigma, K)
  \,.
$$

Most every statement about ordinary commutative rings has its analog for
[[E-∞ ring|commutative ∞-rings]], and so we can just follow our nose:

An _[[(∞,1)-vector space]]_ over $K$ is an $K$-[[module spectrum]] and we have an [[(∞,1)-category]] $K$[[Mod]] of such $\infty$-vector spaces. Notice that these are a [[categorification]] of the ordinary notion of [[vector space]] only in the "$r$"-direction of the lattice of [[(n,r)-category|(r,n)-categories]]. A genuine [[(∞,n)-vector space]] over $K$ -- as appears in the description of general $n$-dimensional $\sigma$-models -- is instead an object of $(\cdots ((K Mod) Mod) \cdots ) Mod $. In particular the $(\infty,2)$-category of $(\infty,2)$-vector spaces over $K$ is
something like $(K Mod) Mod$.

This means that an [[(∞,1)-vector bundle]] with flat [[connection on an ∞-bundle|∞-connection]] over some [[manifold]] $X$ is equivalently encoded by an [[(∞,1)-functor]]

$$
  \alpha : \Pi X \to K Mod
$$

out of the [[fundamental ∞-groupoid]] of $X$. This assigns to each point of $X$ a $K$-module -- the [[fiber]] of the [[(∞,1)-vector bundle]] thus encoded -- to each path in $X$ an [[equivalence in an (∞,1)-category|equivalence]] between the fibers over its endpoints, and so on: this is the [[higher parallel transport]] of a flat $\infty$-connection. We can also think of this as an [[∞-representation]] of $\Pi X$ on $K$-modules, also called a _[[representation up to homotopy]]_ . For instance if $X$ is the classifying space $X = B G$ of a [[discrete ∞-group]], then flat [[(∞,1)-vector bundle]]s on $X$ are precisely [[∞-representation]]s of $G$.

There is an evident [[full sub-(∞,1)-category]]

$$
  K Line \hookrightarrow K Mod
$$

of 1-dimensional $(\infty,1)$-vector spaces:  $K$-lines -- $K$-modules that are equivalent to $K$ itself regarded as a $K$-module.

An $(\infty,1)$-vector bundle $\nabla : \Pi(X) \to K Mod$ that factors through this inclusion is a _$K$-line $\infty$-bundle_ .

One finds, as for the case of ordinary 1-vector spaces, that

$$
  K Line \simeq B GL_1(K) \simeq B Aut(K)
$$

is the [[delooping]] of the [[automorphism ∞-group]] of $K$. This means that $K$-line $\infty$-bundles are equivalently $GL_1(K)$-[[principal ∞-bundle]]s. We can think of the inclusion

$$
  \rho : B GL_1(K) \stackrel{\simeq}{\to} K Line \hookrightarrow K Mod
$$

as being the canonical linear [[∞-representation]] of $GL_1(K)$; and for $g : \Pi X \to B GL_1(K)$ a $GL_1(K)$-[[principal ∞-bundle]] of the $(\infty,1)$-vector bundle

$$
  \Pi X \stackrel{g}{\to} B GL_1(K) \stackrel{\rho}{\to} K Mod
$$

as the corresponding [[associated ∞-bundle]].

Therefore it makes sense to consider $\sigma$-models with [[target space]] $X$ and [[background gauge field]] given by an $K$-line $\infty$-bundle.

For instance for $K = K U$ the [[K-theory spectrum]], there is a canonical morphism $B^2 U(1) \to B GL_1(KU)$ and hence to every [[circle n-bundle|circle 2-bundle]] $\alpha : \Pi X \to B^2 U(1)$ is associated the corresponding $K U$-line $\infty$-bundle

$$
 \Pi X \stackrel{\alpha}{\to} B^2 U(1) \stackrel{}{\to}
   B GL_1(K U) \stackrel{\rho}{\to} K U Mod
  \,.
$$

Or for $K = tmf$ the [[tmf]] spectrum, there is a canonical morphism $B^3 U(1) \to B GL_1(tmf)$ and hence to every [[circle n-bundle|circle 3-bundle]] $\alpha : \Pi X \to B^3 U(1)$ is associated the corresponding $tmf$-line $\infty$-bundle

$$
 \Pi X \stackrel{\alpha}{\to} B^3 U(1) \stackrel{}{\to}
   B GL_1(tmf) \stackrel{\rho}{\to} tmf Mod
  \,.
$$

This was amplified by Ando, Blumberg, Gepner, Hopkins, and Rezk (see the reference at [[(∞,1)-vector bundle]]),  who notice much of the theory of [[generalized (Eilenberg-Steenrod) cohomology|K-(co)homology]] -- including notably its [[Thom spectrum]] theory and its [[twisted cohomology]] -- is neatly captured by simple statements about such $A$-line $(\infty,1)$-bundles. For instance the notion of [[orientation in generalized cohomology]] simply boils down to the notion of trivialization of such $K$-line $\infty$-bundles:

a vector bundle $E \to X$ is _$K$-oriented_ precisely if the corresponding [[Thom space]]-bundle -- which is a [[sphere spectrum]]-line $\infty$-bundle $V: \Pi(X) \to S Line$ is such that the canonically associated $K$-line bundle is trivializable:

$$
  (V is K-orientable)
    \Leftrightarrow
  ( (\Pi(X) \stackrel{V}{\to} S Line \to K Line)
    \simeq
    const_K  )
  \,.
$$

For our discussion here this means that Cohen-Godin's finding that the string topology HQFT exists for $K$ such that $X$ is $K$-orientable meaks that the $K$-line $\infty$-bundle background field that we are to consider in this context are to be trivializable.

Recall that if we interpret an $(\infty,2)$-vector bundle as a [[background gauge field]] for a $\sigma$-model, then for $\Sigma$ any 2-dimensional [[cobordism]] the corresponding [[(∞,1)-vector space]] of states assigned to, say, the incoming boundary $\partial_{in} \Sigma$ is defined to be the $\infty$-vector space of sections of the [[transgression]] of this $(\infty,2)$-vector bundle to an [[(∞,1)-vector bundle]] the [[mapping space]] $X^{\partial_{in} \Sigma}$. The transgression of a trivial bundle is again the trivial bundle. And the $\infty$-vector space of (co)sections is, in the discrete case,  as we had discussed [above](ExpositionQuantization), the [[(∞,1)-colimit]]

$$
  \Gamma_{\partial_{in} \Sigma}
    :=
  \lim_\to(\Pi X^{\partial_{in} \Sigma} \to S Line \to  K Mod )
  \,.
$$

This one can compute. By triviality of the bundle, Ando-Blumberg-Gepner-Hopkins-Rezk observe that this is the $K$-homology spectrum of $X^{\partial_{in} \Sigma}$

$$
  \cdots \simeq (\Sigma^\infty X^{\partial_{in} \Sigma}) \wedge K
  \in K Mod
 \,.
$$

(Here the main point is that for the bundle _not_ being trivial the result encodes the corresponding _[[twisted cohomology]]_ , but for our purposes at the moment we want the oriented/trivializable case.)

This is hence the $\infty$-vector space of states over $\partial_{in} \Sigma$ assigned by a $\sigma$-model with [[background gauge field]] a $K$-line $\infty$-bundle over a $K$-oriented [[target space]].

Notice that for $K = H \mathbb{Z}$ the [[Eilenberg-MacLane spectrum]] for the [[integer]]s, we have an equivalence

$$
  H \mathbb{Z} Mod \simeq Ch_\bullet
$$

and in fact

$$
  H \mathbb{Z} Alg \simeq dgAlg_{\mathbb{Z}}
$$

betwee the [[(∞,1)-category]] of $H \mathbb{Z}$-[[module spectra]]/$H \mathbb{Z}$-[[algebra spectra]] (see there for details on this equivalence) and the $(\infty,1)$-category [[presentable (infinity,1)-category|presented]] by the [[model structure on chain complexes]]/[[model structure on dg-algebras|on dg-algebras]]. Under this equivalence the above [[module spectrum]]-space of states over the circle ought to be identified with the ordinary integral homology chain complex

$$
  (\Sigma^\infty X^{S^1}) \wedge K
  \sim
  C_\bullet(X^{S^1})
  \,.
$$

The "[[decategorification]]" of this $\infty$-vector space of states is precisely the tower of homology groups of $X$:

$$
  \pi_\bullet(\Gamma_{\partial \Sigma}) = H_\bullet(X^{\partial_{in} \Sigma}, K)
  \,.
$$

### The topological $A$-model string and Gromov-Witten theory
 {#GWTheory}

The quantum [[string]]-[[sigma model]] whose [[target space]] is a [[symplectic manifold]] $(X, \omega)$ -- often called the [[A-model]] -- is given by the

* [[Gromov-Witten theory]]

of $X$. For the moment see there for more details.

### Quantum abelian Chern-Simons
 {#QuantumAbelianCS}

For the moment see

* [[Topological Quantum Field Theories from Compact Lie Groups]].


sigma-model -- exposition of second quantization of sigma-models

+-- {: .rightHandSide}
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###Context###
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Phyiscs
+--{: .hide}
[[!include physicscontents]]
=--
#### $\infty$-Chern-Simons theory
+--{: .hide}
[[!include infinity-Chern-Simons theory - contents]]
=--
=--
=--

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

***

#Contents#
* table of contents
{:toc}

## Exposition of second-quantization of sigma-models

Low dimensional $\sigma$-models that describe the dynamics of [[particle]]s --  or generally [[brane]]s -- propagating in a [[target space]] $X$ subject to the forces exerted by a [[background field]], such as discussed [above](#ExpositionClassical), are just the first ingredient in a description of the quantum physics _on [[target space]] $X$_ : one is interested in describing a [[quantum field theory]] (or possibly some higher analog, such as a _[[string field theory]]_ ) on $X$, such that the original branes described by the $\sigma$-model are _quanta_ of the fields on $X$. The $\sigma$-model quantum field theory on the [[worldvolumes]] $\Sigma$ is supposed to induce, in turn, another quantum field theory, or something similar, but now on $X$. This process -- or its idea -- goes by the name _[[second quantization]]_ . We shall try to indicate below in which sense this should indeed be a direct iteration of the ("first") quantization of the original $\sigma$-model itself, as described [above](ExpositionQuantization). But we will (have to) be more vague and schematic than before.

Assume now that we started with a classical $\sigma$-model and have quantized it to obtain the [[FQFT|functorial QFT]]

$$
  Z : Bord_n^S \to Vect
  \,,
$$

which we take here to be 1-categorical (un-[[extended quantum field theory|extended]]), not to overburden the discussion. Assume also for simplicity that the corresponding [[target space]] is of the form $X = Y \times \mathbb{R}$, where the second factor is the time-axis.

Using this, we define a space of [[state]]s of the _second quantization_  to be the _Fock space_

$$
  \begin{aligned}
    Sym^\bullet(\oplus_{[\Sigma_{n-1}]} Z(\Sigma_{n-1}) )
    & :=
    Sym^\bullet( \mathcal{V} )
    \\
    & \simeq
    \mathbf{1}
    \oplus
    \mathcal{V}
    \oplus
    (\mathcal{V} \otimes \mathcal{V})_{sym}
    \oplus
    (\mathcal{V} \otimes \mathcal{V} \otimes \mathcal{V})_{sym}
    \cdots
  \end{aligned}
  \,,
$$

the free commutative [[tensor algebra]] over the vector space $\mathcal{V}$ of [[state]]s of the given single [[particle]], [[string]] or, generally, [[brane]]. This has a contribution for each equivalence class $[\Sigma_{n-1}]$ of connected shapes $\Sigma_{n-1}$ of this brane. For instance for $n = 1$ there is just the point $\Sigma_0 = *$, but for the [[string]] with $n = 2$ there is one contribution from the closed string $\Sigma_1 = S^1$ and one contribution from the open string $\Sigma_1 = [0,1]$; and in turn several contributions of this type if the open string carries boundary [[D-brane]] labels. (In the presence of [[fermion]]s the vector spaces appearing here are [[super vector space]]s and the Fock space construction is the corresponding [[super algebra]] version.)

This space of states is that of a quantum theory whose _classical_ field configurations are given by _many_ states of the original quantum $\sigma$-model. We therefore say that the original brane of shape $\Sigma_{n-1}$ is a _quantum_ or a _single excitation_ of the field theory that it induces on target space $X$: where the original $\sigma$-model describes the dynamics of a single brane on $X$, its second quantization describes the dynamics of many and interacting copies of that brane.

Of course $Sym^\bullet(-) : Vect \to Vect$ is a [[functor]]. This is traditionally stated as the notorious slogan "second quantization is a functor", usually meant to state a contrast with first quantization. Notice however, firstly, that the ("first") quantization of $\sigma$-models -- at least for the version discussed [above](#ExpositionQuantization) -- also is a functor (what is not functorial is the quantization of a random [[Poisson manifold]], but $\sigma$-models carry much more structure than that), and, secondly, that the $Sym^\bullet$-construction is at best half of what second quantization is about: the other half is the description of the _interaction_ of the many branes:

for $X = Y \times (-t,t)$ a piece of target space, the would-be second quantized theory on $X$ should assign a morphism

$$
  S : Sym^\bullet(\mathcal{V}) \to Sym^\bullet(\mathcal{V})
$$

between the space of states of the incoming [[space]] $Y$, to the outgoing space $Y$, here taken to be the same. Usually one assumes here $t = \infty$ and calls $S$ the _[[S-matrix]]_ of the theory, where "S" is for _scattering_ : for $(v^{in}_1 \otimes \dots v^{in}_r) \in Sym^r \mathcal{V}$ the state of $r$ incoming brane quanta and $(v^{out}_1 \otimes \dots v^{out}_s) \in Sym^s \mathcal{V}$ a state of $s$ outgoing brane quanta, the [[matrix]] element

$$
  \left( \left(v^{out}_1 \otimes \dots v^{out}_s\right) , S \left(v^{in}_1 \otimes \dots v^{in}_r\right) \right)
$$

(using an [[inner product space]]-structure on $\mathcal{V}$) is the [[probability amplitude]] for $r$-many branes in the given states propagating through $X$, interacting among each other, thereby transmuting into other states -- hence _scattering off each other_ --  and eventually emerging again in the state $(v^{out}_1 \otimes \dots v^{out}_s)$.

The idea is that this scattering happens in all possible ways that it can happen, with each way weighted by the probability amplitude for it to happen as seen by the quantum $\sigma$-model $Z$: the S-matrix is taken to be given by an expression like

$$
  S
  :=
  \int_{\Sigma_n \in Bord^S_n}
  Z(\Sigma)
  \,\,
  d\mu(\Sigma)
  \,,
$$

and called the brane _perturbation series_ . For instance the _[[Feynman perturbation series]]_ for $n = 1$, or the _[[string perturbation series]]_ for $n = 2$: we sum over all possible $S$-structured [[cobordism]]s $\Sigma$ -- each representing an interaction "channel" for branes of shape $\partial_{in} \Sigma$ to scatter into shape $\partial_{out} \Sigma$ -- the linear maps $S(\Sigma) : \mathcal{V}_{\partial_{in} \Sigma} \to \mathcal{V}_{\partial_{out} \Sigma}$ weighted by some [[measure]] $d\mu(\Sigma)$, and all regarded as giving one single endomorphism on $Sym^\bullet(\mathcal{V})$ in the evident way.

Except for simple cases, plenty of technical subtleties may have to be taken care of here in order to get anything close to being well defined. For instance there is an [[integral]] over some suitably compactified [[moduli space]] of $S$-structured cobordisms involved. For sufficiently simple but still nontrivial $\sigma$-models this has been made fully precise (for instance for $n = 1$ in standard QFT perturbation theory with [[renormalization]] or for $n = 2$ in the example of [[Gromov-Witten theory]], discussed [above](GWTheory)), and intuition and motivation is drawn from these cases, but in full generality it remains an open problem to fully realize this idea.

This [[S-matrix]]-construction provides at least the rudiments of a quantum field theory on target space $X$ obtained by second quantization of a $\sigma$-model describing brane dynamics on $X$.

In application to phenomenological physics we think of $X$ here as our [[spacetime]]. Everything that is measured in particle accelerator experiments is explained with such a construction for $n = 1$. Everything that perturbative [[string theory]] hypothesizes is a refinement of this theory relevant at energies not visible in current accelerator experiments is described with such a construction for $n = 2$. A little bit of investigation has gone into exploring the $n = 3$-case. In principle one could investigate this further for $n \geq 4$.  but so far the jump in complexity given by the step from $n = 1$ to $n = 2$ has kept mankind busy enough. There is already an intricate interrelation of quantum field theories showing up at this level. For instance the second quantization of the 2-dimensional [A-model string](#GWTheory) $\sigma$-model (see there) has be shown to be [[Chern-Simons theory]], which we have seen, [above](#SigmaCS), may itself be understood as a 3-dimensional $\sigma$-model. The second quantization of the non-topological string $\sigma$-model is more complicated but follows this pattern. It contains even [[Yang-Mills theory]] and [[gravity]], and this is what has driven much of the interest in this structure.

Given the evident importance of the "brane perturbation series" or "second quantization" of $\sigma$-models it would be desireable to have a more general abstract and systematic description of it, in the spirit of the [above discussion](#ExpositionQuantization) of general abstract first quantization. Here is an observation that might be suggestive:

we had amplified that the input data for a classical ("0-quantized") $\sigma$-model is a [[background field]] on a target space $X$. At least in nice cases this background field is entirely encoded in its [[higher parallel transport]] and [[holonomy]]-assignment, which is a map

$$
  tra_\nabla : Bord_n(X) \to n Vect
$$

from $n$-dimensional bordisms _in_ $X$ to [[n-vector space]]s (we had mostly discussed an equivalent characteristic morphism $\nabla : X \to \mathbf{B}^n U(1)_{conn}$ and should eventually discuss in more detail how both perspectives are related...).

The process of ("first") quantization of this $\sigma$-model involves _some_ kind of extension of this functor through the projection to abstract cobordisms

$$
  \array{
    Bord_n^S(X) &\stackrel{tra_\nabla}{\to}& Vect
    \\
    \downarrow & \nearrow_{Z_\nabla}
    \\
    Bord_n^S
  }
$$

given by the [[path integral]]

$$
  Z_\nabla :=  \int_{\gamma \in Bord^S_n} tra_\nabla(\gamma) \;\; d \mu(\gamma)
$$

over all trajectories $\gamma : \Sigma \to X$ of given shape $\Sigma$. We saw that second quantization reads in this $Z_\nabla$, in turn, and from the perturbation series

$$
  S := \int_{\Sigma \in Bord^S_n} Z(\Sigma) \; d \mu(\Sigma)
$$

produces a target space quantum field theory. Hence apparently there is a pattern of iterated path integrals, the first over morphisms in $Bord_n^S(X)$, the second over morphisms in $Bord_n^S$:

* classical (0-quantized) $\sigma$-model ([[higher parallel transport]] of [[background gauge field]]):

  $$
    tra_\nabla : Bord_n^S(X) \to n Vect
    \,;
  $$

* quantum (1st quantized) $\sigma$-model ([[FQFT]]):

  $$
    Z_\nabla = \int_{\gamma \in Bord^S_n(X)} tra_\nabla(\gamma) \,
      d\mu(\gamma) : Bord_n^S \to n Vect
    \,;
  $$

* second quantized model ([[S-matrix]]):

  $$
    S = \int_{\Sigma \in Bord^S_n} Z_\nabla(\Sigma)\, d\mu(\Sigma)
    \,.
  $$

Remember that these formulas are to be taken with a grain of salt. Quite some additional effort is in general needed to make them well-defined. Already in the well-understood case $n = 1$ the [[path integral]] in the expression for $Z_\nabla$ needs attention, then the single terms in the expression for $S$ may still need [[renormalization]] and after all that the sum still may need [[resummation]], at least for models richer than for instance the $\infty$-Dijkgraaf-Witten theories, for which all integrals reduce to finite sums.


sigma-orientation

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###Context###
#### Cohomology
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[[!include cohomology - contents]]
=--
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A $\sigma$-orientation is an [[orientation in generalized cohomology]] for [[elliptic cohomology]]. A universal such orientation is a  _[[string orientation of tmf]]_. See there for more details.

## References

* [[Matthew Ando]], [[Mike Hopkins]], [[Neil Strickland]], _Elliptic spectra, the Witten genus, and the theorem of the cube_, Inventiones Mathematicae, 146 (2001) 595–687

* [[Matthew Ando]], _The sigma orientation for analytic circle-equivariant elliptic cohomology_, Geom. Topol. 7 (2003) 91-153 ([arXiv:math/0201092](http://arxiv.org/abs/math/0201092))


sigma-pretopos

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#### Category theory
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[[!include category theory - contents]]
=--
#### Topos Theory
+--{: .hide}
[[!include topos theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[categorification]] of the notion of [[sigma-frame|$\sigma$-frame]].

## Definition ##

A __$\sigma$-pretopos__ is a [[pretopos]] $\mathcal{E}$ where all [[countable]] [[unions]] of [[subobjects]] (exist and) are [[pullback-stable]].

### Girard-esque axioms ###

A __$\sigma$-pretopos__ is

* a [[category]]

* with a [[generating set]],

* with all [[finite limits]],

* with all [[countable]] [[coproducts]], which are [[disjoint]], and [[pullback-stable]],

* where all [[congruences]] have [[effective quotient|effective]] [[quotient objects]], which are also [[pullback-stable]].

## Examples

* Any [[pretopos#infinitary|infinitary pretopos]] is a $\sigma$-topos.

* The category of [[countable sets]] (for instance, [[subquotients]] of the [[natural numbers]] $\mathbb{N}$) is a $\sigma$-pretopos.

* The [[category of sets]] in the [[predicative mathematics|predicative]] [[set theory]] $CZF_{Exp}$ from ([Simpson-Streicher 2012](#SimpsonStreicher12)) is a $\sigma$-pretopos.

## See also ##

* [[pretopos]]

* [[Grothendieck topos]]

* [[sigma-frame]]

* [[geometric category]]

## References ##

The concept is defined and discussed around Lemma A.1.4.19 in

* [[Peter Johnstone]], [[Sketches of an Elephant]] -- A Topos Theory Compendium**, Oxford University Press 2002. Volume 1 ([ISBN:9780198534259](https://global.oup.com/academic/product/sketches-of-an-elephant-9780198534259)), Volume 2 ([ISBN:9780198515982](https://global.oup.com/academic/product/sketches-of-an-elephant-9780198515982))

A predicative approach to foundations using $\sigma$-pretoposes is given in

* {#SimpsonStreicher12} [[Alex Simpson]] and [[Thomas Streicher]], _Constructive toposes with countable sums as models of constructive set theory_, Annals of Pure and Applied Logic **163** (2012), 1419–1436.

Simplicial homotopy internal to a $\sigma$-pretopos is studied in:

* {#Low14} [[Zhen Lin Low]], _Internal and local homotopy theory_, [arXiv:1404.7788](https://arxiv.org/abs/1404.7788) (2014).

[[!redirects sigma-pretopoi]]


sigma-topological space

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### Context
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
=--
=--

\tableofcontents

## Idea

A **$\sigma$-topological space** is a set $X$ equipped with a **$\sigma$-topology**: a collection $O(X)$ of subsets of $X$, called [[open sets]], which is closed under [[finite]] [[intersections]] and [[countable]] [[unions]].

The [[open sets]] $O(X)$ of a **$\sigma$-topological space** form a [[sigma-frame|$\sigma$-frame]].

An example of a $\sigma$-topological space is a [[sigma-algebra|$\sigma$-algebra]], whose collection $O(X)$ is also closed under [[complements]] and countable intersections. Another example of a $\sigma$-topological space is a [[zero-set structure]].

## Properties

\begin{theorem}
Let $X = \prod_{n \to \alpha} X_n$ be an internal set, and let $\mathcal{F}_X$ be the collection of all subsets of $X$ that can be expressed as the union of at most countably many internal subsets of $X$. Then $(X, \mathcal{F}_X)$ is a [[countably compact]] [[T_1-topological space|$T_1$]] $\sigma$-topological space.
\end{theorem}

## Related concepts

* [[topological space]]
* [[sigma-frame]]
* [[sigma-locale]]
* [[sigma-algebra]]
* [[zero-set structure]]
* [[countably compact space]]
* [[first countable space]]

## References

* Fedor Petrov, *"countable" topology*, MathOverflow ([web](https://mathoverflow.net/questions/173255/countable-topology))

* [[Vitaly Bergelson]], [[Terence Tao]], *Multiple recurrence in quasirandom groups*. ([arXiv:1211.6372](https://arxiv.org/abs/1211.6372))

[[!redirects sigma-topology]]
[[!redirects sigma-topologies]]

[[!redirects sigma-topological space]]
[[!redirects sigma-topological spaces]]


sign function

# Contents
* table of contents
{: toc}

## Idea

The _sign function_ on the [[real numbers]].

## Definition

$$
  \array{
    \mathbb{R} &\overset{sgn}{\longrightarrow}& \mathbb{R}
    \\
    x &\mapsto& \left\{ \array{ 1 &\vert& x \gt 0 \\ 0 &\vert& x = 0 \\ -1 &\vert& x \lt 0 }\right.
  }
$$

Some definitions will modify the value at $0$, usually to make it $1$, $-1$, or undefined.  In many applications, the sign function is essentially treated as a [[measurable function]] on the real line with [[Lebesgue measure]], and then these are all essentially the same since they are all [[almost equal]].    But some applications require the function to be [[left-continuous function|left-]] or [[right-continuous function|right-continuous]], and then the value of $0$ must be chosen appropriately, while the handy formula $x/{|x|}$ naturally makes the value at $0$ undefined.

# Related concepts

* [[positive real number]]

* [[negative real number]]

* [[real absolute value]]

* [[step function]]

## References

See also

* Wikipedia, _[Sign function](https://en.wikipedia.org/wiki/Sign_function)_

[[!redirects sign]]
[[!redirects sign function]]
[[!redirects sgn]]


sign problem in lattice QCD

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=--
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#Contents#
* table of contents
{:toc}

## Idea

In the context of [[nuclear physics]], by the *sign problem* in [[lattice quantum chromodynamics]] is is meant the issue that

* for computational effectivity the [[partition function]] needs to be written as a (discretized) [[path integral]] of the exponentiated [[Euclidean field theory|Euclidean]] [[action functional]] against a [[probability density]] (the "quark matrix"),

but

* in many situations, notably at finite [[chemical potential]], the would-be [[probability]] [[measure]] turns out not to be positive definite, and in fact typically turns out to be [[complex number|complex]],

* which means that various numerical evaluation methods fail.

The sign problem is one of the reasons that [[lattice QCD]] fails to be a fully satisfactory construction of [[non-perturbative quantum field theory|non-perturbative]] [[QCD]] even for practical purposes. For instance, much of the [[phase diagram]] of [[QCD]] remains unknown (e.g. [de Forcrand 13](#deForcrand13) [CQF21](#skyrmion#ChenQiuFukushima21)), since the sign problem prevents lattice QCD computations at "almost all points" in phase space ([Hsu-Reeb 08](#HsuReeb08)).

## References

Exposition:

* {#deForcrand13} Philippe de Forcrand, *The sign problem in Lattice QCD*, ETH 2013 ([pdf](https://www.int.washington.edu/talks/WorkShops/int_13_2a/People/deForcrand_P/deForcrand.pdf))

Detailed review:

* Keitaro Nagata, *Finite-density lattice QCD and sign problem: current status and open problems* ([arXiv:2108.12423](https://arxiv.org/abs/2108.12423))

Original articles:

* E. Y. Loh, Jr., J. E. Gubernatis, R. T. Scalettar, S. R. White, D. J. Scalapino, and R. L. Sugar, *Sign problem in the numerical simulation of many-electron systems* Phys. Rev. B **41** 9301 (1990) ([doi:10.1103/PhysRevB.41.9301](https://doi.org/10.1103/PhysRevB.41.9301))

* {#HsuReeb08} Stephen D.H. Hsu, David Reeb, *On the sign problem in dense QCD*, Int. J. Mod. Phys. A 25, pp. 53-67 (2010) ([arXiv:0808.2987](https://arxiv.org/abs/0808.2987))

* V. A. Goy, V. Bornyakov, D. Boyda, A. Molochkov, A. Nakamura, A. Nikolaev, V. Zakharov, *Sign problem in finite density lattice QCD*, Prog Theor Exp Phys (2017) 2017 (3): 031D01 ([arXiv:1611.08093](https://arxiv.org/abs/1611.08093))

See also:

* Wikipedia, _[Numerical sign problem](https://en.wikipedia.org/wiki/Numerical_sign_problem)_

* Xu Zhang, Gaopei Pan, Xiao Yan Xu, Zi Yang Meng, *Sign Problem Finds Its Bounds* ([arXiv:2112.06139](https://arxiv.org/abs/2112.06139))

Proposal using [[quantum computation]]:

* Zohreh Davoudi, Niklas Mueller, Connor Powers, *Toward Quantum Computing Phase Diagrams of Gauge Theories with Thermal Pure Quantum States* [[arXiv:2208.13112](https://arxiv.org/abs/2208.13112)]

[[!redirects sign problems in lattice QCD]]

[[!redirects sign problem]]
[[!redirects sign problems]]


sign relation

The material previously on this page is now located at the [Online Encyclopedia of Integer Sequences Wiki](https://oeis.org/wiki/Sign_relation).

[[!redirects semeiotic]]
[[!redirects semeiotics]]
[[!redirects sign relations]]


sign rules in homological superalgebra -- table

**[[signs in supergeometry|sign rule]] for [[differential graded-commutative superalgebras]]**

 (different but [equivalent](chain+complex+in+super+vector+spaces#EquivalenceTwoSymmetricMonoidalStructuresOnChSuperVect))

|   | $\phantom{A}$[Deligne's convention](signs+in+supergeometry#TheSignRuleFromInternalization)$\phantom{A}$ | $\phantom{A}$[Bernstein's convention](signs+in+supergeometry#SuperOddConvention)$\phantom{A}$ |
|---|------------------------------------|----------------|
| $\phantom{A}$$ \alpha_i \cdot  \alpha_j = $$\phantom{A}$ | $\phantom{A}$$(-1)^{ (n_i \cdot n_j + \sigma_i \cdot \sigma_j) }  \alpha_j \cdot \alpha_i$$\phantom{A}$ | $\phantom{A}$$ (-1)^{ (n_i + \sigma_i) \cdot (n_j + \sigma_j) } \alpha_j \cdot \alpha_i$$\phantom{A}$ |
| $\phantom{A}$common in$\phantom{A}$
 $\phantom{A}$discussion of$\phantom{A}$ | $\phantom{A}$[[nLab:supergravity]]$\phantom{A}$ | $\phantom{A}$[[nLab:AKSZ sigma-models]]$\phantom{A}$ |
| $\phantom{A}$representative$\phantom{A}$
 $\phantom{A}$references$\phantom{A}$ | $\phantom{A}$[Bonora et. al 87](signs+in+supergeometry#BonoraBregolaLechnerPastiTonin87),$\phantom{A}$
$\phantom{A}$[Castellani-D'Auria-Fré 91](signs+in+supergeometry#CastellaniDAuriaFre91),$\phantom{A}$
 $\phantom{A}$[Deligne-Freed 99](signs+in+supergeometry#DeligneFreed99)$\phantom{A}$ | $\phantom{A}$[AKSZ 95](AKSZ+sigma-model#AKSZ),$\phantom{A}$
$\phantom{A}$[Carchedi-Roytenberg 12](signs+in+supergeometry#CarchediRoytenberg12)$\phantom{A}$ |


signal

\tableofcontents

\section{Idea}

A _signal_ is, roughly speaking, the use of a particular physical entity, typically a [[waveform]] of a certain kind, to convey information about another. Signals fall into two broad categories: analogue and [[digital signal|digital]].

In both the analogue and digital cases, the physical conversion from information $I$ about an entity $E$ to a signal $S_E(I)$ sent through a certain medium conveying $I$ is carried out in the final instance by a [[transducer]]. The waveform $S_E(I)$ itself which the transducer sends out is typically constructed by [[waveform modulation|modulating]] a waveform of a standard form, or more than one such waveform, according to an algorithm to which $I$ is an input.

\section{Related concepts}

* [[signal processing]]

* [[noise]], [[white noise]], [[quantum noise]]

* [[error correction code]], [[quantum error correction code]]

\section{References}

See also:

* Wikipedia, *[Signal](https://en.wikipedia.org/wiki/Signal)*

[[!redirects signals]]


signal processing

#Contents#
* table of contents
{:toc}

## Idea

[Wikipedia](https://en.wikipedia.org/wiki/Signal_processing) defines signal processing as

> the analysis, synthesis, and modification of [[signal|signals]], which are broadly defined as functions conveying "information about the behavior or attributes of some phenomenon"

## Related concepts

* [[sampling theory]]

* [[time series]]

* [[information geometry]]

* [[noise]], [[white noise]]

* [[error correction]], [[quantum error correction]]

## References

* H. Akaike, _Information theory and an extension of the maximum likelihood principle_, in hoc. 2nd Int. Symp. Inform.392
Theory, suppl. Problems of Control and Inform. Theory, 1973,
* M. Wax, T. Kailath, _Detection of signals by information theoretic criteria_, IEEE Transactions on Acoustics, Speech, and Signal Processing __33__:2, 1985 [doi](https://doi.org/10.1109/TASSP.1985.1164557)

category: applications, probability


signature

The term **signature** refers to a number of quite different notions in mathematics, including but not limited to:

* [[signature of a permutation]],

* [[signature of a quadratic form]]

* [[signature of a metric]]

* [[signature (in logic)]]

* [[signature genus]]

* etc....

[[!redirects signature]]
[[!redirects signatures]]
[[!redirects signature (disambiguation)]]

category: disambiguation


signature (in logic)

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# Contents#
* table of contents
{: toc}

## Idea

In [[logic]], a **signature** is a collection of data which prescribes the 'alphabet' of non-logical symbols of a logical [[theory]], saying which operations and [[predicates]] are taken to be primitive.

The signature of a structure is the union of the collection of constant symbols, function symbols, and  relation symbols in the language of said structure.

## Definition

### Types

Formally, a **signature** $\Sigma$ consists of

* A set $S$ whose elements are called **sorts** or **[[types]]** of $\Sigma$,

* A set $Rel(\Sigma)$ whose elements are called **relation symbols**, equipped with a function $ar: Rel(\Sigma) \to S^*$ to the [[free monoid]] on $S$ which prescribes an [[arity]] for each relation symbol,

* A set $Func(\Sigma)$ whose elements are called **function symbols**, equipped with a function
$$\langle dom, cod \rangle: Func(\Sigma) \to S^* \times S$$
which prescribes an arity or type for each function symbol. A function symbol $f$ may be written as $dom(f) \to cod(f)$, or as $d_0(f) \to d_1(f)$, to indicate its arity.

The majority of (but far from all) mathematical concepts are described by means of a **single-sorted** signature, where $S$ consists of a single element $X$. Examples where multisorted signatures are used is the theory of [[categories]] (with an [[object]] sort and a [[morphism]] sort) and [[quiver|(multi)directed graphs]] (with a vertex sort and an edge sort). But in the single-sorted or one-sorted case, the free monoid on the one-sort set is isomorphic to $\mathbb{N}$, and the arity of a relation is a number $n$. Hence we speak of an $n$-ary relation (unary, binary, etc.). Similarly, the arity of a function symbol or operation $f$ is the number $n$ which indexes $dom(f)$ (hence we speak of unary operations, binary operations, etc.) In either the single-sorted or multisorted case, a 0-ary operation (where the domain empty) is usually called a **constant**.

A **relational** signature is where $Func(\Sigma)$ is empty, and an **equational** or **algebraic** signature is where $Rel(\Sigma)$ is empty. (In the latter case, the only relation symbol is the equality symbol, but this is typically considered a logical symbol.) Occasionally one allows a relational signature to have constants.

### Models

A ([[set theory|set-theoretic]]) **model** or **interpretation** $M$ of a signature consists of functions which assign

* To each sort $s$ a [[set]] $M(s)$,

* To each relation symbol $R$ of arity $\langle s_1, \ldots, s_n \rangle$ a [[subset]] $M(R) \hookrightarrow M(s_1) \times \ldots \times M(s_n)$,

* To each function symbol $f$ of type $\langle s_1, \ldots, s_n \rangle \to s$ a [[function]] $M(f): M(s_1) \times \ldots \times M(s_n) \to M(s)$. Such a function is called an **operation** (that interprets $f$). In the case of constants of type $s$, the empty product on the left is taken to be a terminal or 1-element set $1$, whose element needs no name but is often given a generic name like '$*$'.

More generally a signature may be interpreted in other [[category|categories]] $C$ besides [[Set]]; for this it is reasonable to suppose that $C$ is at least a [[regular category]]. See also

* [[categorical semantics]].

### Theories

In terms of a signature one may formulate [[proposition]]s, [[sequents]] and then [[theories]]. See there for more details.

## Examples

An important point to bear in mind is that _essentially_ the same theories (where 'same' means that the categories of models are equivalent or even isomorphic) may have very different signatures. For example,

* The theory of [[group]]s is usually described as a single-sorted theory whose signature has one binary operation (called 'multiplication') $m$, one constant $e$ (called 'identity'), and one unary operation (called 'inversion') $i$. But it can also be described using a signature with just one (binary) operation $m$; there one adds in extra axioms which posit the existence of an identity and of inverse elements. Or, it can be described using a signature with a single (binary) operation $d$ (for 'division'). All these are examples of equational signatures.

* A [[Boolean ring]] is usually described as a commutative ring with identity in which multiplication is idempotent, hence the theory of Boolean rings is usually presented using the signature normally reserved for rings (with identity). A [[Boolean algebra]] may be described using a variety of signatures, for example non-equationally, involving a binary relation $\leq$ and function symbols $\wedge$, $\neg$, $\top$.

* The theory ZFC is usually described using a single-sorted relational signature with one binary relation $\in$. However, other approaches are possible: one can also describe ZFC with a relational symbol $\subseteq$ together with a unary function symbol $sing$ (to be interpreted as taking a 'set' $S$, i.e., an element of a ZFC structure, to a singleton 'set' $\{S\}$).

## Remarks

The multiplicity of ways in which one can describe essentially the same class of objects is a phenomenon which in higher category theory is often referred to as [[bias]]. Often one desires to use an 'unbiased' approach which includes different signatures under one roof and on the same footing. In the case of algebraic (equational) theories, this can be done using the device of [[clone]]s or [[Lawvere theory|Lawvere theories]], which does not single out certain non-logical operations as 'privileged'.

There are several reasons though why one might prefer to retain some bias, for example:

* Tradition, familiarity, thus ease of reference. For example, a group _is_ a set equipped with a binary $m$, an unary $i$, and a constant $e$, rather than a set equipped with a single division operation, or a finite product-preserving functor for that matter.

* Mathematical intuition. That is, different signatures may invoke different intuitions or attitudes toward a subject; for example, "Boolean rings" may invoke a more algebraic or algebro-geometric attitude (as in [[Stone duality]]) whereas "Boolean algebras" may invoke a more logical attitude (propositional logic).

* Differences in logical strength. That is, first-order logic should be thought of as coming in layers, ranging from equational logic up to [[pretopos]] logic, and different signatures may need differing levels of logical strength for them to be interpreted as intended. Equational logic may be preferred on occasion because it is a very weak logic, whereas relational signatures often require at least the strength of regular categories for them to be interpreted correctly.

## The language of a signature

Each signature $\Sigma$ generates a language whose elements are first-order predicates: expressions formed by using an alphabet of logical symbols together with $\Sigma$, considered as an alphabet of non-logical symbols, according to certain rules. For an account of the traditional logical syntax, see [Wikipedia](https://en.wikipedia.org/wiki/First-order_logic#Syntax).

First-order languages can also be structured categorically through the notion of [[hyperdoctrine]]. The base of the hyperdoctrine is a [[cartesian monoidal category]] $\mathbf{Term}(\Sigma)$ consisting of sorts and terms, or rather, the objects are formal (cartesian) products of sorts, written as finite lists, and the morphisms are tuples of terms generated by the function symbols of the signature. A detailed categorical description of $\mathbf{Term}(\Sigma)$ may be found [here](https://ncatlab.org/nlab/show/free%20cartesian%20category#free_cartesian_category_on_a_signature).

Given the base $\mathbf{Term}(\Sigma)$, the language itself is structured as a certain Boolean hyperdoctrine, i.e., a functor

$$Lang(\Sigma): \mathbf{Term}(\Sigma)^{op} \to Bool$$

where for each morphism $f$ in the base, the morphism $f^\ast = Lang(\Sigma)(f)$ has a left adjoint $\exists_f$ and a right adjoint $\forall_f$, subject to a Beck-Chevalley condition for each pullback square in the base. To describe it with more traditional logical terminology: the functor takes each object (i.e., product of sorts) $w = s_1 s_2 \ldots s_n$ to the Boolean algebra $Pred(w)$ of predicates of arity $w$ that built up from the formal relations of the signature using first-order logic operations. The sorts $s_i$ appearing in the arity are matched bijectively with the free variables of the predicate, and are the types of those variables. Then, the functor takes each morphism $f: v \to w$, given by an $n$-tuple of terms $\tau_i: v \to s_i$, to the Boolean algebra map $f^\ast: Pred(w) \to Pred(v)$ obtained by substituting, in predicates of arity $w$, the term $\tau_i$ for the free variable $x_i$ that is matched with $s_i$, resulting in a predicate of arity $v$.

A more precise categorical description of $Lang(\Sigma)$, in terms of a universal property, is as follows. Let $S^\ast$ denote the underlying set of free monoid generated by the set of sorts $S$; regard $S^\ast$ as a discrete category. The formal relations of the signature give a functor

$$i: S^\ast \to Set$$

sending each product $w = s_1 \ldots s_n$ to the set of relations $R \in Rel(\Sigma)$ that have arity $w$. Let $j: S^\ast \to \mathbf{Term}(\Sigma)$ be the obvious inclusion functor (the identity on objects). Define an *interpretation* of $\Sigma$ to be a pair consisting of a Boolean hyperdoctrine $M: \mathbf{Term}(\Sigma)^{op} \to Bool$ together with a natural transformation $\phi: i \to U M j$:

$$\array{
S^\ast & \stackrel{i}{\to} & Set \\
\mathllap{j} \downarrow & \Downarrow \mathrlap{\phi} & \uparrow \mathrlap{U} \\
\mathbf{Term}(\Sigma)^{op} & \underset{M}{\to} & Bool
}$$

where $U: Bool \to Set$ is the underlying set functor. There is an obvious notion of morphism between interpretations $(M, \phi) \to (M', \phi')$, given by a natural transformation $M \Rightarrow M'$ compatible with the data $\phi, \phi'$.

+-- {: .num_defn}
###### Definition
The *language* $Lang(\Sigma)$ is the initial object in the category of interpretations of $\Sigma$.
=--

## References

* Categorical Logic and Type Theory, Bart Jacobs

[[!redirects signatures in logic]]
[[!redirects signature in logic]]

[[!redirects relation symbol]]
[[!redirects function symbol]]
[[!redirects relation symbols]]
[[!redirects function symbols]]

[[!redirects signature of a structure]]
[[!redirects signatures of structures]]


signature genus

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#### Manifolds and cobordisms
+--{: .hide}
[[!include manifolds and cobordisms - contents]]
=--
=--
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#Contents#
* table of contents
{:toc}

## Idea

The _signature_ of an [[orientation|oriented]] [[manifold]] $X$ of [[dimension]] $4 k$, $k \in \mathbb{N}$ is the [[signature of a quadratic form|signature of the quadratic form]] which is the [[intersection pairing]] in [[integral cohomology]]

$$
  H^{2k}(X,\mathbb{Z}) \times
  H^{2k}(X,\mathbb{Z})
  \stackrel{-\cup-}{\to}
  H^{4k}(X, \mathbb{Z})
  \stackrel{\langle -,X\rangle}{\to}
  \mathbb{Z}
$$

If the signature of a manifold with dimension not divisible by 4 is taken to be 0.

([Milnor-Stasheff 74, p. 224](#MilnorStasheff74))

This defines a [[genus]]: the **signature genus**.

## Properties

(...)

* $L_0 = 1$

* $L_1 = p_1 / 3$

* $L_2 = (7 p_2  - (p_1)^2) / 45$

## Related concepts

* [[Hirzebruch signature theorem]]

[[!include quadratic invariants - table]]

## References

* {#MilnorStasheff74} [[John Milnor]], [[James Stasheff]], p. 224 of _Characteristic classes_, Princeton University Press, 1974

See also

* Wikipedia, _[L genus and the Hirzebruch signature theorem](http://en.wikipedia.org/wiki/Hirzebruch_L-polynomial#L_genus_and_the_Hirzebruch_signature_theorem)_


signature of a metric

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#Contents#
* table of contents
{:toc}

## Idea

Given a [[pseudo-Riemannian manifold]] $(X, g)$ with [[metric tensor]] $g$, its _signature_ is the signature of $g$ as a [[quadratic form]] on any [[tangent space]] of $X$, in the sense of _[[signature of a quadratic form]]_.

If $(X,g)$ is regarded as a model for [[spacetime]], then this is also called _spacetime signature_.

Here the case of signature $(-,+, \cdots, +)$ corresponds to [[Lorentzian geometry]], being the basis of Einstein-[[gravity]] in the sense of "[[general relativity]]".

The case of signature $(+,+, \cdots, +)$ corresponds to [[Riemannian geometry]] (as opposed to [[pseudo-Riemannian geometry|pseudo-Riemannian]]). In a context of [[spacetime]] models this pertains for instance to the usual [[fibers]] in [[KK-compactifications]], or to all of spacetime after "[[Wick rotation]]" ([[Euclidean gravity]]).

Spacetime signatures of the form $(-,\cdots,-, +,\cdots, +)$ have been considered in the context of [[D=12 supergravity]] and [[D=14 supersymmetry]].

## Related concepts

* [[Lorentzian geometry]]

* [[Euclidean gravity]], [[Wick rotation]]

* [[D=12 supergravity]]

* [[D=14 supersymmetry]]

* [[relativistic field theory]], [[Euclidean field theory]]

[[!redirects spacetime signature]]


signature of a permutation

> This entry is about the _signature_ of a [[permutation]]. For other notions of [[signature]] see there.

#Contents#
* table of contents
{:toc}

## Definition

For $Aut(\{1, \cdots , n\}) \simeq S_n$ the [[symmetric group]] on $n \in \mathbb{N}$ elements, the _signature_ is the unique [[group]] [[homomorphism]]

$$
  sign : S_n \to \mathbb{Z}_2 = \{1, -1\}
$$

that sends each transposition $s_{i, i+1} : \{1, \cdots, n\} \to \{1, \cdots, n\}$, which interchanges the $i$th element with its neighbour and leaves the other elements fixed, to the nontrivial element $(-1) \in \mathbb{Z}_2$.

Permutations in the kernel of $sign$ are called _even_ permutations, and the rest are called _odd_ permutations.

+-- {: .un_lem}
###### Lemma
The signature is well-defined.
=--

+-- {: .proof}
###### Proof
One way of seeing this is invoking a standard [[group presentation]] of $S_n$ where generators $\sigma_i$ for $i = 1$ to $n-1$ (representing $s_{i, i+1}$) are subject to relations

$$\sigma_{i}^2 = 1, \qquad (\sigma_i \sigma_{i+1})^3 = 1, \qquad \sigma_i \sigma_j = \sigma_j \sigma_i \; ({|i-j|} \gt 1),$$

and checking that the sign applied to both sides of a relation equation gives the same result.

Another is by invoking a tautological representation of $S_n$ on a [[polynomial algebra]] $\mathbb{Z}[x_1, \ldots, x_n]$,

$$S_n \stackrel{\cong}{\to} Set_{core}(\{x_1, \ldots, x_n\}, \{x_1, \ldots, x_n\}) \to CRing_{core}(\mathbb{Z}[x_1, \ldots, x_n], \mathbb{Z}[x_1, \ldots, x_n])$$

(where [[core]] refers to the groupoid of invertible morphisms) and recognizing that for the special polynomial

$$D \coloneqq \prod_{i \lt j} (x_i - x_j)$$

we have, for each permutation $\tau$, either $\tau \cdot D = D$ or $\tau \cdot D = -D$. (The polynomial $\Delta \coloneqq D^2$, which is invariant under the action, is called the [[discriminant]].)
=--

## Computations

There are various means for computing the signature (also called **sign**) of a permutation.

The definition itself suggests one method: if we linearly [[total order|order]] the set $\{x_1, \ldots, x_n\}$ by $x_1 \lt \ldots \lt x_n$, then we can exhibit a permutation $\tau$ by a [[string diagram]] and simply count the number of crossings $I(\tau)$; then we have

$$sign(\tau) = (-1)^{I(\tau)}.$$

Each crossing corresponds to a pair of elements $x_i \lt x_j$ such that $\tau(x_i) \gt \tau(x_j)$, called an **inversion**.

Another method which does not depend on choosing a total order is to exhibit a permutation through its [cycle decomposition](http://ncatlab.org/nlab/show/permutation#via_cycle_decompositions_6). Each cycle of period $k$ contributes a sign $(-1)^{k-1}$, and the overall sign is the product of these contributions taken over all the cycles. Thus the signature is given by the parity of the number of cycles of even length.

This cycle description can actually be used to give an independent definition of the signature. It is manifestly well-defined and invariant on conjugacy classes. To check that it defines a homomorphism to $\{1, -1\}$, it suffices to check that multiplication by a transposition changes the parity of the number of even-length cycles by one. This is easy if we note that transposing two elements belonging to different cycles merges two cycles into one, whereas transposing two elements belonging to the same cycle splits one cycle into two.

## Related concepts

* [[Levi-Civita symbol]]

[[!redirects signatures of permutations]]


signature of a structure > history

see *[[signature (in logic)]]*


signed graph

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### Context
#### Graph theory
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[[!include graph theory - contents]]
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#Contents#
* table of contents
{:toc}

## Definition

\begin{definition}
A *signed graph* is an undirected [[graph]] (e.g., a [[simple graph]], which is a [[set]] $V$ of *[[vertices]]* together with a collection $E$ of two-element subsets of $V$ called *[[edges]]*) together with a function from $E$ to a two-element set of "signs", e.g., $\sigma: E \to \{+, -\}$.
not necessarily of edges.

One says that the sign of a cycle in a signed graph is the product of the signs of its edges.
\end{definition}

There are several notions of *[[homomorphisms]]* of signed graphs. They are all homomorphisms of the underlying graphs, by differ by how strictly they preserve the extra sign structure:

1. *sign preserving homomorphisms* preserve the signs of individual edges,

1. *switching homomorphisms* are only required to preserve the signs of cycles but not necessarily of individual edges.

\begin{definition}
Let $\binom{V}{2}$ denotes the set of two-element [[subsets]] of $V$. A signed *simple* graph is equivalently a set $V$ together with a [[partial function]] $\binom{V}{2} \to \{+, -\}$. The [[domain]] of this partial function is declared to be $E$, so the data of a signed graph can be represented as $V$ together with a [[span]] of type
$$
  \binom{V}{2}
    \hookleftarrow
  E
    \overset{\sigma}{\longrightarrow}
  \{+, -\}
  \,.
$$
\end{definition}

Signed graphs (and the basic theorems of their theory) seem to be rediscovered again and again. See [Zaslavsky 2018](#Zaslavsky18) for an annotated bibliography on signed graphs and their various guises.

\begin{remark}
Signed graphs are *not* *directed graphs*: signs on undirected edges are not the same as directional information. For example, if vertices represent people and edges represent being acquainted then the signs could indicate being allies or enemies.

If a [[linear order]] has been imposed on the vertex set, then the sign function can be interpreted as imparting directions to edges (say if $x \lt y$ in the order and an edge $e$ between $x$ and $y$ has sign $+$, then the direction is $x \to y$; if $e$ has sign $-$, then the direction is $y \to x$). However, such linear orders on vertices will not be preserved by morphisms of signed graphs, so this does not give a [[functor|functorial]] way to derive directed graphs from signed graphs.
\end{remark}

## References

* {#Zaslavsky18} Thomas Zaslavsky, *A Mathematical Bibliography of Signed and Gain Graphs and Allied Areas*, Electronic Journal of Combinatorics, Dynamic Surveys in Combinatorics, DS8 (2018) [[doi:10.37236/29](https://doi.org/10.37236/29)]

[[!redirects signed graphs]]