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geometry of physics -- prequantum gauge theory and gravity

Contents

this entry is going to contain one chapter of geometry of physics


Contents

Prequantum gauge theory and gravity

In the previous chapters we have set up prequantum field theory and classical field theory in generality. Here we discuss examples of such field theories in more detail.

Contents
  1. Model layer

    We introduce a list of important examples of field theories in fairly tradtional terms.

  2. Semantics layer

    We study the above physical systems with the tools of of cohesive (∞,1)-topos-theory as developed in the previous semantics-layers.

  3. Syntax layer

Model layer

Examples
  1. 1d Chern-Simons theory

  2. Nonabelian charged particle and Wilson loops

  3. 2d Chern-Simons theory

  4. 3d Chern-Simons theory

  5. (4k+3)d U(1)-Chern-Simons theory

  6. 7d Chern-Simons theory

  7. ∞-Chern-Simons theory

1d Chern-Simons theory

2d Chern-Simons theory

Nonabelian charged particle and Wilson loops

The prequantum field theory which describes the gauge interaction of a single nonabelian charged particle – a Wilson loop – turns out to be equivalent to what in mathematics is called the orbit method. We discuss here the traditional formulation of these matters. Below in Semantics layer – Nonabelian charged particle and Wilson loops we then show how all this is naturally understood from a certain extended Lagrangian which is induced by a regular coadjoint orbit.

A useful review of the following is also in (Beasley, section 4).

The group and its Lie algebra

Throughout, let GG be a semisimple compact Lie group. For some considerations below we furthermore assume it to be simply connected.

Write 𝔤\mathfrak{g} for its Lie algebra. Its canonical (up to scale) binary invariant polynomial we write

,:𝔤𝔤. \langle -,-\rangle : \mathfrak{g} \otimes \mathfrak{g} \to \mathbb{R} \,.

Since this is non-degenerate, we may equivalently think of this as an isomorphism

𝔤𝔤 * \mathfrak{g} \simeq \mathfrak{g}^*

that identifies the vector space underlying the Lie algebra with its dual vector space 𝔤 *\mathfrak{g}^*.

The coadjoint orbit and the coset space/ flag manifold

We discuss the coadjoint orbits of GG and their relation to the coset space/flag manifolds of GG.

Write

  1. TGT \hookrightarrow G inclusion of the maximal torus of GG.

1 𝔱𝔤\mathfrak{t} \hookrightarrow \mathfrak{g} the corresponding Cartan subalgebra

In all of the following we consider an element λ,𝔤 *\langle\lambda,-\rangle \in \mathfrak{g}^*.

Definition

For λ,𝔤 *\langle\lambda,-\rangle \in \mathfrak{g}^* write

𝒪 λ𝔤 * \mathcal{O}_\lambda \hookrightarrow \mathfrak{g}^*

for its coadjoint orbit

𝒪 λ={Ad g *(λ,)𝔤 *|gG}. \mathcal{O}_{\lambda} = \{ Ad_g^*(\langle\lambda,-\rangle) \in \mathfrak{g}^* | g \in G \} \,.

Write G λGG_\lambda \hookrightarrow G for the stabilizer subgroup of λ,\langle \lambda,-\rangle under the coadjoint action.

Proposition

There is an equivalence

G/G λ𝒪 λ G/G_\lambda \stackrel{\simeq}{\to} \mathcal{O}_\lambda

given by

gG λAd g *λ,. g G_\lambda \mapsto Ad_g^* \langle\lambda,-\rangle \,.
Definition

An element λ,𝔤 *\langle\lambda,-\rangle \in \mathfrak{g}^* is regular if its coadjoint action stabilizer subgroup coincides with the maximal torus: G λTG_\lambda \simeq T.

Example

For generic values of λ\lambda it is regular. The element in 𝔤 *\mathfrak{g}^* farthest from regularity is λ=0\lambda = 0 for which G λ=GG_\lambda = G instead.

The symplectic form

We describe a canonical symplectic form on the coadjoint orbit/coset 𝒪 λG/G λ\mathcal{O}_\lambda \simeq G/G_\lambda.

Write θΩ 1(G,𝔤)\theta \in \Omega^1(G, \mathfrak{g}) for the Maurer-Cartan form on GG.

Definition

Write

Θ λ:=λ,θΩ 1(G) \Theta_\lambda := \langle \lambda, \theta \rangle \in \Omega^1(G)

for the 1-form obtained by pairing the value of the Maurer-Cartan form at each point with the gixed element λ𝔤 *\lambda \in \mathfrak{g}^*.

Write

ν λ:=d dRΘ λ \nu_\lambda := d_{dR} \Theta_\lambda

for its de Rham differential.

Proposition

The 2-form ν λ\nu_\lambda from def.

  1. satisfies

    ν λ=12λ,[θθ]. \nu_\lambda = \frac{1}{2}\langle \lambda, [\theta\wedge \theta]\rangle \,.
  2. it descends to a closed GG-invariant 2-form on the coset space, to be denoted by the same symbol

    ν λΩ cl 2(G/G λ) G. \nu_\lambda \in \Omega^2_{cl}(G/G_\lambda)^G \,.
  3. this is non-degenerate and hence defines a symplectic form on G/G λG/G_\lambda.

The prequantum bundle

We discuss the geometric prequantization of the symplectic manifold given by the coadjoint orbit 𝒪 λ\mathcal{O}_\lambda equipped with its symplectic form ν λ\nu_\lambda of def. .

Assume now that GG is simply connected.

Proposition

The weight lattice Γ wt𝔱 *𝔱\Gamma_{wt} \subset \mathfrak{t}^* \simeq \mathfrak{t} of the Lie group GG is isomorphic to the group of group characters

Γ wtHom LieGrp(G,U(1)) \Gamma_{wt} \stackrel{\simeq}{\to} Hom_{LieGrp}(G,U(1))

where the identification takes α,𝔱 *\langle \alpha , -\rangle \in \mathfrak{t}^* to ρ α:TU(1)\rho_\alpha : T \to U(1) given on t=exp(ξ)t = \exp(\xi) for ξ𝔱\xi \in \mathfrak{t} by

ρ α:exp(ξ)exp(iα,ξ). \rho_\alpha : \exp(\xi) \mapsto \exp(i \langle \alpha, \xi\rangle) \,.
Proposition

The symplectic form ν λΩ cl 2(G/T)\nu_\lambda \in \Omega^2_{cl}(G/T) of prop. is integral precisely if λ,\langle \lambda, - \rangle is in the weight lattice.

The Hamiltonian GG-action / coadjoint moment map

The group GG canonically acts on the coset space G/G λG/G_{\lambda} (by multiplication from the left). We discuss a lift of this action to a Hamiltonian action with respect to the symplectic manifold structure (G/T,ν λ)(G/T, \nu_\lambda) of prop. , equivalently a momentum map exhibiting this Hamiltonian action.

Wilson loops and 1d Chern-Simons σ\sigma-models with target the coadjoint orbit

Above (…) we discussed how an irreducible unitary representation of GG is encoded by the prequantization of a coadjoint orbit (𝒪 λ,ν λ)(\mathcal{O}_\lambda, \nu_\lambda). Here we discuss how to express Wilson loops/holonomy of GG-principal connections in this representation as the path integral of a topological particle charged under this background field, whose action functional is that of a 1-dimensional Chern-Simons theory.

Let A| S 1Ω 1(S 1,𝔤)A|_{S^1} \in \Omega^1(S^1, \mathfrak{g}) be a Lie algebra valued 1-form on the circle, equivalently a GG-principal connection on the circle.

For

ρ:GAut(V) \rho : G \to Aut(V)

a representation of GG, write

W S 1 R(A):=hol S 1 R(A):=Tr R(tra S 1(A)) W_{S^1}^R(A) := hol^R_{S^1}(A) := Tr_R( tra_{S^1}(A) )

for the holonomy of AA around the circle in this representation, which is the trace of its parallel transport around the circle (for any basepoint). If one thinks of AA as a background gauge field then this is alse called a Wilson loop.

Definition

Let the action functional

exp(iCS λ() A):[S 1,G/T]U(1) \exp(i CS_\lambda(-)^A) \;\colon\; [S^1, G/T] \to U(1)

be given by sending gT:S 1G/Tg T : S^1 \to G/T represented by g:S 1Gg : S^1 \to G to

exp(i S 1λ,A g), \exp(i \int_{S^1} \langle \lambda, A^g\rangle ) \,,

where

A g:=Ad g(A)+g *θ A^g := Ad_g(A) + g^* \theta

is the gauge transformation of AA under gg.

Proposition

The Wilson loop of AA over S 1S^1 in the unitarry irreducible representation RR is proportional to the path integral of the 1-dimensional sigma-model with

  1. target space the coadjoint orbit 𝒪 λG/T\mathcal{O}_\lambda \simeq G/T for λ,\langle \lambda, - \rangle the weight corresponding to RR under the Borel-Weil-Bott theorem

  2. action functional the functional of def. :

W S 1 R(A) [S 1,𝒪 λ]D(gT)exp(i S 1λ,A g). W_{S^1}^R(A) \propto \int_{[S^1, \mathcal{O}_\lambda]} D(g T) \exp(i \int_{S^1} \langle \lambda, A^g\rangle) \,.

See for instance (Beasley, (4.55)).

Remark

Notice that since 𝒪 λ\mathcal{O}_\lambda is a manifold of finite dimension, the path integral for a point particle with this target space can be and has been defined rigorously, see at path integral.

3d Chern-Simons theory

(4k+3)(4k+3)d U(1)U(1)-Chern-Simons theory

7d Chern-Simons theory

\infty-Chern-Simons theory

String field theory

Gauge fields and gravity – Einstein-Maxwell-Yang-Mills theory
Kaluza-Klein compactification
Standard model of particle physics
Standard model of cosmology

Semantic Layer

an exposition and survey is in (FSS 13).

{}
  1. 1d Chern-Simons theory

  2. Nonabelian charged particle trajectories – Wilson loops

  3. 2d CS theory: WZW terms and Chan-Paton gauge fields

  4. 3d Chern-Simons theory with Wilson loops

  5. Chan-Paton gauge fields on D-branes

1d Chern-Simons theory

For some nn \in \mathbb{N} let

det:U(n)U(1) det \;\colon\; U(n) \to U(1)

be the Lie group homomorphism from the unitary group to the circle group which is given by sending a unitary matrix to its determinant.

Being a Lie group homomorphism, this induces a map of deloopings/moduli stacks

Bdet:BU(n)BU(1) \mathbf{B}det \;\colon\; \mathbf{B}U(n) \to \mathbf{B}U(1)

Under geometric realization of cohesive infinity-groupoids this is the universal first Chern class

|Bdet|c 1:BU(n)BU(1)K(,2). {\vert \mathbf{B}det\vert} \simeq c_1 \;\colon\; B U(n) \to B U(1) \simeq K(\mathbb{Z},2) \,.

Moreiver this has the evident differential refinement

Bdet^:BU(n) connBU(1) conn \widehat {\mathbf{B} det} \;\colon\; \mathbf{B} U(n)_{conn} \to \mathbf{B} U(1)_{conn}

given on Lie algebra valued 1-forms by taking the trace

tr:𝔲(n)𝔲(1). tr \;\colon\; \mathfrak{u}(n) \to \mathfrak{u}(1) \,.

So we get a 1d Chern-Simons theory with Bdet^\widehat{\mathbf{B}det} as its extended Lagrangian.

Nonabelian charged particle trajectories – Wilson loops

We consider now extended Lagrangians defined on fields as above in Nonabelian charged particle trajectories – Wilson loops. This provides a natural reformulation in higher geometry of the constructions in the orbit method as reviewed above in Model layer – Nonabelian charged particle.

{}
  1. Survey

  2. Definitions and constructions

  3. Nonabelian charged particle trajectories – Wilson loops

  4. 3d Chern-Simons theory with Wilson loops.

Survey

We discuss how for λ𝔤\lambda \in \mathfrak{g} a regular element, there is a canonical diagram of smooth moduli stacks of the form

𝒪 λ G/T θ Ω 1(,𝔤)//T λ, BU(1) conn J * BG conn c B 3U(1) conn, \array{ \mathcal{O}_\lambda &\stackrel{\simeq}{\to}& G/T &\stackrel{\mathbf{\theta}}{\to}& \Omega^1(-,\mathfrak{g})//T &\stackrel{\langle \lambda, - \rangle}{\to}& \mathbf{B} U(1)_{conn} \\ && \downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\mathbf{J}}} \\ && * &\stackrel{}{\to}& \mathbf{B}G_{conn} &\stackrel{\mathbf{c}}{\to}& \mathbf{B}^3 U(1)_{conn} } \,,

where

  1. J\mathbf{J} is the canonical 2-monomorphism;

  2. the left square is a homotopy pullback square, hence θ\mathbf{\theta} is the homotopy fiber of J\mathbf{J};

  3. the bottom map is the extended Lagrangian for GG-Chern-Simons theory, equivalently the universal Chern-Simons circle 3-bundle with connection;

  4. the top map denoted λ,\langle \lambda,- \rangle is an extended Lagrangian for a 1-dimensional Chern-Simons theory;

  5. the total top composite modulates a prequantum circle bundle which is a prequantization of the canonical symplectic manifold structure on the coadjoint orbit Ω λG/T\Omega_\lambda \simeq G/T.

Definitions and constructions

Write H=\mathbf{H} = Smooth∞Grpd for the cohesive (∞,1)-topos of smooth \infty-groupoids.

For the following, let λ,𝔤 *\langle \lambda, - \rangle \in \mathfrak{g}^* be a regular element, def. , so that the stabilizer subgroup is identified with a maximal torus: G λTG_\lambda \simeq T.

As usual, write

BG connΩ 1(,𝔤)//GH \mathbf{B}G_{conn} \simeq \Omega^1(-,\mathfrak{g})//G \in \mathbf{H}

for the moduli stack of GG-principal connections.

Definition

Write

J:=(Ω 1(,𝔤)//TΩ 1(,𝔤)//GBG conn)H Δ 1 \mathbf{J} := ( \Omega^1(-,\mathfrak{g})//T \to \Omega^1(-,\mathfrak{g})//G \simeq \mathbf{B}G_{conn} ) \in \mathbf{H}^{\Delta^1}

for the canonical map, as indicated.

Remark

The map J\mathbf{J} is the differential refinement of the delooping BTBG\mathbf{B}T \to \mathbf{B}G of the defining inclusion. By the general discussion at coset space we have a homotopy fiber sequence

𝒪 λG/T BT BG. \array{ \mathcal{O}_\lambda \simeq G/T &\to& \mathbf{B}T \\ && \downarrow \\ && \mathbf{B}G } \,.

By the discussion at ∞-action this exhibits the canonical action ρ\rho of GG on its coset space: it is the universal rho-associated bundle.

The following proposition says what happens to this statement under differential refinement

Proposition

The homotopy fiber of J\mathbf{J} in def. is

θ:G/TΩ 1(,𝔤)//T \mathbf{\theta} : G/T \stackrel{}{\to} \Omega^1(-,\mathfrak{g})//T

given over a test manifold UU \in CartSp by the map

θ U:C (U,G/T)Ω 1(U,𝔤) \mathbf{\theta}_U : C^\infty(U,G/T) \to \Omega^1(U,\mathfrak{g})

which sends gg *θg \mapsto g^* \theta, where θ\theta is the Maurer-Cartan form on GG.

Proof

We compute the homotopy pullback of J\mathbf{J} along the point inclusion by the factorization lemma as discussed at homotopy pullback – Constructions.

This says that with J\mathbf{J} presented canonically as a map of presheaves of groupoids via the above definitions, its homotopy fiber is presented by the presheaf of groupids hofib(J)hofib(\mathbf{J}) which is the limit cone in

hofib(J) Ω 1(,𝔤) (BG conn) I BG conn * BG conn. \array{ hofib(\mathbf{J}) &\to& &\to& \Omega^1(-, \mathfrak{g}) \\ \downarrow && \downarrow && \downarrow \\ && (\mathbf{B}G_{conn})^I &\to& \mathbf{B}G_{conn} \\ \downarrow && \downarrow \\ * &\stackrel{}{\to}& \mathbf{B}G_{conn} } \,.

Unwinding the definitions shows that hofib(J)hofib(\mathbf{J}) has

  1. objects over a UU \in CartSp are equivalently morphisms 0gg *θ0 \stackrel{g}{\to} g^* \theta in Ω 1(U,𝔤)//C (U,G)\Omega^1(U,\mathfrak{g})//C^\infty(U,G), hence equivalently elements gC (U,G)g \in C^\infty(U,G);

  2. morphisms are over UU commuting triangles

    g 1 *θ t g 2 *θ g 1 g 2 0 \array{ g_1^* \theta &&\stackrel{t}{\to}&& g_2^* \theta \\ & {}_{\mathllap{g_1}}\nwarrow && \nearrow_{\mathrlap{g_2}} \\ && 0 }

    in Ω 1(U,𝔤)//C (U,G)\Omega^1(U,\mathfrak{g})//C^\infty(U,G) with tC (U,T)t \in C^\infty(U,T), hence equivalently morphisms

    g 1tg 2 g_1 \stackrel{t}{\to} g_2

    in C (U,G)//C (U,T)C^\infty(U,G)//C^\infty(U,T).

  3. The canonical map hofib(J)Ω 1(,𝔤)//Thofib(\mathbf{J}) \to \Omega^1(-,\mathfrak{g})//T picks the top horizontal part of these commuting triangles hence equivalently sends gg to g *θg^* \theta.

Proposition

If λ,Γ wt𝔤 *\langle \lambda ,- \rangle \in \Gamma_{wt} \hookrightarrow \mathfrak{g}^* is in the weight lattice, then there is a morphism of moduli stacks

λ,:Ω 1(,𝔤)//TBU(1) conn \langle \lambda, - \rangle \;\colon\; \Omega^1(-,\mathfrak{g})//T \to \mathbf{B}U(1)_{conn}

in H\mathbf{H} given over a test manifold UU \in CartSp by the functor

λ, U:Ω 1(U,𝔤)//C (U,G)Ω 1(U)//C (U,U(1)) \langle \lambda, - \rangle_U \;:\; \Omega^1(U,\mathfrak{g})//C^\infty(U,G) \to \Omega^1(U)//C^\infty(U,U(1))

which is given on objects by

Aλ,A A \mapsto \langle \lambda, A\rangle

and which maps morphisms labeled by exp(ξ)T\exp(\xi) \in T, ξC (,𝔱)\xi \in C^\infty(-,\mathfrak{t}) as

exp(ξ)exp(iλ,ξ). \exp(\xi) \mapsto \exp( i \langle \lambda, \xi \rangle ) \,.
Proof

That this construction defines a map *//T*//U(1)*//T \to *//U(1) is the statement of prop. . It remains to check that the differential 1-forms gauge-transform accordingly.

For this the key point is that since TG λT \simeq G_\lambda stabilizes λ,\langle \lambda , - \rangle under the coadjoint action, the gauge transformation law for points A:UBG connA : U \to \mathbf{B}G_{conn}, which for gC (U,G)g \in C^\infty(U,G) is

AAd gA+g *θ, A \mapsto Ad_g A + g^* \theta \,,

maps for g=exp(ξ)C (U,T)C (U,G)g = exp( \xi ) \in C^\infty(U,T) \hookrightarrow C^\infty(U,G) to the gauge transformation law in BU(1) conn\mathbf{B}U(1)_{conn}:

λ,A λ,Ad gA+λ,g *θ =λ,A+dλ,ξ \begin{aligned} \langle \lambda, A \rangle & \mapsto \langle \lambda, Ad_g A\rangle + \langle \lambda, g^* \theta\rangle \\ & = \langle \lambda, A \rangle + d \langle\lambda, \xi \rangle \end{aligned}
Remark

The composite of the canonical maps of prop. and prop. modulates a canonical circle bundle with connection on the coset space/coadjoint orbit:

λ,θ:G/TθΩ 1(,𝔤)//Tλ,BU(1) conn. \langle \lambda, \mathbf{\theta}\rangle : G/T \stackrel{\mathbf{\theta}}{\to} \Omega^1(-,\mathfrak{g})//T \stackrel{\langle \lambda, - \rangle}{\to} \mathbf{B}U(1)_{conn} \,.
Proposition

The curvature 2-form of the circle bundle λ,θ\langle \lambda, \mathbf{\theta}\rangle from remark is the symplectic form of prop. . Therefore λ,θ\langle \lambda, \mathbf{\theta}\rangle is a prequantization of the coadjoint orbit (𝒪 λG/T,ν λ)(\mathcal{O}_\lambda \simeq G/T, \nu_\lambda).

Proof

The curvature 2-form is modulated by the composite

ω:G/TθΩ 1(,𝔤)//Tλ,BU(1) connF ()Ω cl 2. \omega : G/T \stackrel{\mathbf{\theta}}{\to} \Omega^1(-,\mathfrak{g})//T \stackrel{\langle \lambda, - \rangle}{\to} \mathbf{B}U(1)_{conn} \stackrel{F_{(-)}}{\to} \Omega^2_{cl} \,.

Unwinding the above definitions and propositions, one finds that this is given over a test manifold UU \in CartSp by the map

ω U:C (G/T)Ω cl 2(U) \omega_U : C^\infty(G/T) \to \Omega^2_{cl}(U)

which sends

[g]dλ,g *θ. [g] \mapsto d \langle \lambda, g^* \theta \rangle \,.
Nonabelian charged particle trajectories – Wilson loops

Let Σ\Sigma be an oriented closed smooth manifold of dimension 3 and let

C:S 1Σ C \;\colon\; S^1 \hookrightarrow \Sigma

be a submanifold inclusion of the circle: a knot in Σ\Sigma.

Let RR be an irreducible unitary representation of GG and let λ,\langle \lambda,-\rangle be a weight corresponding to it by the Borel-Weil-Bott theorem.

Regarding the inclusion CC as an object in the arrow (∞,1)-topos H Δ 1\mathbf{H}^{\Delta^1}, say that a gauge field configuration for GG-Chern-Simons theory on Σ\Sigma with Wilson loop CC and labeled by the representation RR is a map

ϕ:CJ \phi \;\colon\; C \to \mathbf{J}

in the arrow (∞,1)-topos H (Δ 1)\mathbf{H}^{(\Delta^1)} of the ambient cohesive (∞,1)-topos. Such a map is equivalently by a square

S 1 (A| S 1) g Ω 1(,𝔤)//T C g J Σ A BG conn \array{ S^1 &\stackrel{(A|_{S^1})^g}{\to}& \Omega^1(-,\mathfrak{g})//T \\ \downarrow^{\mathrlap{C}} &\swArrow_{g}& \downarrow^{\mathrlap{\mathbf{J}}} \\ \Sigma &\stackrel{A}{\to}& \mathbf{B}G_{conn} }

in H\mathbf{H}. In components this is

which fixes the field on the circle defect to be (A| S 1) g(A|_{S^1})^g, as indicated.

Moreover, a gauge transformation between two such fields κ:ϕϕ\kappa : \phi \Rightarrow \phi' is a GG-gauge transformation of AA and a TT-gauge transformation of A| S 1A|_{S^1} such that these intertwine the component maps gg and gg'. If we keep the bulk gauge field AA fixed, then his means that two fields ϕ\phi and ϕ\phi' as above are gauge equivalent precisely if there is a function t:S 1Tt \;\colon\; S^1 \to T such that g=gtg = g' t, hence gauge equivalence classes of fields for fixed bulk gauge field AA are parameterized by their components [g]=[g][S 1,G/T][g] = [g'] \in [S^1, G/T] with values in the coset space, hence in the coadjoint orbit.

For every such field configuration we can evaluate two action functionals:

  1. that of 3d Chern-Simons theory, whose extended Lagrangian is c:BG connB 3U(1) conn\mathbf{c} : \mathbf{B}G_{conn} \to \mathbf{B}^3 U(1)_{conn};

  2. that of the 1-dimensional Chern-Simons theory discussed above whose extended Lagrangian is λ,:Ω 1(,𝔤)//TBU(1) conn\langle \lambda, -\rangle : \Omega^1(-,\mathfrak{g})//T \to \mathbf{B}U(1)_{conn}, by prop. .

These are obtained by postcomposing the above square on the right by these extended Lagrangians

S 1 (A| S 1) g Ω 1(,𝔤)//T λ, BU(1) conn C g J Σ A BG conn c BU(1) conn \array{ S^1 &\stackrel{(A|_{S^1})^g}{\to}& \Omega^1(-,\mathfrak{g})//T &\stackrel{\langle \lambda, -\rangle}{\to}& \mathbf{B}U(1)_{conn} \\ \downarrow^{\mathrlap{C}} &\swArrow_{g}& \downarrow^{\mathrlap{\mathbf{J}}} \\ \Sigma &\stackrel{A}{\to}& \mathbf{B}G_{conn} &\stackrel{\mathbf{c}}{\to}& \mathbf{B}U(1)_{conn} }

and then preforming the fiber integration in ordinary differential cohomology over S 1S^1 and over Σ\Sigma, respectively.

For the bottom map this gives the ordinary action functional of Chern-Simons theory. For the top map inspection of the proof of prop. shows that this gives the 1d Chern-Simons action whose partition function is the Wilson loop observable by prop. above.

2d CS-theory, WZW-term and Chan-Paton gauge fields

In the context of string theory, the background gauge field for the open string sigma-model over a D-brane in bosonic string theory or type II string theory is a unitary principal bundle with connection, or rather, by the Kapustin-part of the Freed-Witten-Kapustin anomaly cancellation mechanism, a twisted unitary bundle, whose twist is the restriction of the ambient B-field to the D-brane.

We considered these fields already above. Here we discuss the corresponding action functional for the open string coupled to these fields

The first hint for the existence of such background gauge fields for the open string 2d-sigma-model comes from the fact that the open string’s endpoint can naturally be taken to carry labels i{1,n}i \in \{1, \cdots n\}. Further analysis then shows that the lowest excitations of these (i,j)(i,j)-strings behave as the quanta of a U(n)U(n)-gauge field, the (i,j)(i,j)-excitation being the given matrix element of a U(n)U(n)-valued connection 1-form AA.

This original argument goes back work by Chan and Paton. Accordingly one speaks of Chan-Paton factors and Chan-Paton bundles .

We discuss the Chan-Paton gauge field and its quantum anomaly cancellation in extended prequantum field theory.

Throughout we write H=\mathbf{H} = Smooth∞Grpd for the cohesive (∞,1)-topos of smooth ∞-groupoids.

{}
  1. The B-field as a prequantum 2-bundle

  2. The Chan-Paton gauge field

  3. The open string sigma-model

  4. The anomaly-free open string coupling to the Chan-Paton gauge field

The BB-field as a prequantum 2-bundle

For XX a type II supergravity spacetime, the B-field is a map

B:XB 2U(1). \nabla_B \;\colon\; X \to \mathbf{B}^2 U(1) \,.

If X=GX = G is a Lie group, this is the prequantum 2-bundle of GG-Chern-Simons theory. Viewed as such we are to find a canonical ∞-action of the circle 2-group BU(1)\mathbf{B}U(1) on some VHV \in \mathbf{H}, form the corresponding associated ∞-bundle and regard the sections of that as the prequantum 2-states? of the theory.

The Chan-Paton gauge field is such a prequantum 2-state.

The Chan-Paton gauge field

We discuss the Chan-Paton gauge fields over D-branes in bosonic string theory and over Spin cSpin^c-D-branes in type II string theory.

We fix throughout a natural number nn \in \mathbb{N}, the rank of the Chan-Paton gauge field.

Proposition

The extension of Lie groups

U(1)U(n)PU(n) U(1) \to U(n) \to PU(n)

exhibiting the unitary group as a circle group-extension of the projective unitary group sits in a long homotopy fiber sequence of smooth ∞-groupoids of the form

U(1)U(n)PU(n)BU(1)BU(n)BPU(n)dd nB 2U(1), U(1) \to U(n) \to PU(n) \to \mathbf{B}U(1) \to \mathbf{B}U(n) \to \mathbf{B}PU(n) \stackrel{\mathbf{dd}_n}{\to} \mathbf{B}^2 U(1) \,,

where for GG a Lie group BG\mathbf{B}G is its delooping Lie groupoid, hence the moduli stack of GG-principal bundles, and where similarly B 2U(1)\mathbf{B}^2 U(1) is the moduli 2-stack of circle 2-group principal 2-bundles (bundle gerbes).

Proposition

Here

dd n:BPU(n)B 2U(1) \mathbf{dd}_n \;\colon\; \mathbf{B} PU(n) \to \mathbf{B}^2 U(1)

is a smooth refinement of the universal Dixmier-Douady class

dd n:BPU(n)K(,3) dd_n \;\colon\; B PU(n) \to K(\mathbb{Z}, 3)

in that under geometric realization of cohesive ∞-groupoids ||:{\vert- \vert} \colon Smooth∞Grpd \to ∞Grpd we have

|dd n|dd n. {\vert \mathbf{dd}_n \vert} \simeq dd_n \,.
Remark

By the discussion at ∞-action the homotopy fiber sequence in prop.

BU(n) BPU(n) B 2U(1) \array{ \mathbf{B} U(n) &\to& \mathbf{B} PU(n) \\ && \downarrow \\ && \mathbf{B}^2 U(1) }

in H\mathbf{H} exhibits a smooth∞-action of the circle 2-group on the moduli stack BU(n)\mathbf{B}U(n) and it exhibits an equivalence

BPU(n)(BU(n))//(BU(1)) \mathbf{B} PU(n) \simeq (\mathbf{B}U(n))//(\mathbf{B} U(1))

of the moduli stack of projective unitary bundles with the ∞-quotient of this ∞-action.

Proposition

For XHX \in \mathbf{H} a smooth manifold and c:XB 2U(1)\mathbf{c} \;\colon\; X \to \mathbf{B}^2 U(1) modulating a circle 2-group-principal 2-bundle, maps

cdd n \mathbf{c} \to \mathbf{dd}_n

in the slice (∞,1)-topos H /B 2U(1)\mathbf{H}_{/\mathbf{B}^2 U(1)}, hence diagrams of the form

X BPU(n) c dd n B 2U(1) \array{ X &&\stackrel{}{\to}&& \mathbf{B} PU(n) \\ & {}_{\mathllap{\mathbf{c}}}\searrow &\swArrow& \swarrow_{\mathrlap{\mathbf{dd}_n}} \\ && \mathbf{B}^2 U(1) }

in H\mathbf{H} are equivalently rank-nn unitary twisted bundles on XX, with the twist being the class [c]H 3(X,)[\mathbf{c}] \in H^3(X, \mathbb{Z}).

Proposition

There is a further differential refinement

(BU(n))//(BU(1)) conn dd^ n B 2U(1) conn (BU(n))//(BU(1)) dd^ n B 2U(1), \array{ (\mathbf{B}U(n))//(\mathbf{B}U(1))_{conn} &\stackrel{\widehat \mathbf{dd}_n}{\to}& \mathbf{B}^2 U(1)_{conn} \\ \downarrow && \downarrow \\ (\mathbf{B}U(n))//(\mathbf{B}U(1)) &\stackrel{\widehat \mathbf{dd}_n}{\to}& \mathbf{B}^2 U(1) } \,,

where B 2U(1) conn\mathbf{B}^2 U(1)_{conn} is the universal moduli 2-stack of circle 2-bundles with connection (bundle gerbes with connection).

Definition

Write

((BU(n)//BU(1)) connFieldsB 2U(1) conn)H /B 2U(1) conn \left( \left(\mathbf{B}U\left(n\right)//\mathbf{B}U\left(1\right)\right)_{conn} \stackrel{\mathbf{Fields}}{\to} \mathbf{B}^2 U\left(1\right)_{conn} \right) \;\; \in \mathbf{H}_{/\mathbf{B}^2 U(1)_{conn}}

for the differential smooth universal Dixmier-Douady class of prop. , regarded as an object in the slice (∞,1)-topos over B 2U(1) conn\mathbf{B}^2 U(1)_{conn}.

Definition

Let

ι X:QX \iota_X \;\colon\; Q \hookrightarrow X

be an inclusion of smooth manifolds or of orbifolds, to be thought of as a D-brane worldvolume QQ inside an ambient spacetime XX.

Then a field configuration of a B-field on XX together with a compatible rank-nn Chan-Paton gauge field on the D-brane is a map

ϕ:ι XFields \phi \;\colon\; \iota_X \to \mathbf{Fields}

in the arrow (∞,1)-topos H (Δ 1)\mathbf{H}^{(\Delta^1)}, hence a diagram in H\mathbf{H} of the form

Q gauge (BU(n)//BU(1)) ι X dd^ n X B B 2U(1) conn \array{ Q &\stackrel{\nabla_{gauge}}{\to}& (\mathbf{B}U(n)//\mathbf{B}U(1)) \\ {}^{\iota_X}\downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\hat \mathbf{dd}_n}} \\ X &\stackrel{\nabla_B}{\to}& \mathbf{B}^2 U(1)_{conn} }

This identifies a twisted bundle with connection on the D-brane whose twist is the class in H 3(X,)H^3(X, \mathbb{Z}) of the bulk B-field.

This relation is the Kapustin-part of the Freed-Witten-Kapustin anomaly cancellation for the bosonic string or else for the type II string on Spin cSpin^c D-branes. (FSS)

Remark

If we regard the B-field as a background field for the Chan-Paton gauge field, then remark determines along which maps of the B-field the Chan-Paton gauge field may be transformed.

Y X (BU(n)//BU(1)) conn B 2U(1) conn. \array{ Y &\stackrel{}{\to}& X &\stackrel{}{\to}& (\mathbf{B}U(n)//\mathbf{B}U(1))_{conn} \\ & \searrow & \downarrow & \swarrow \\ &&\mathbf{B}^2 U(1)_{conn} } \,.

On the local connection forms this acts as

AA+α. A \mapsto A + \alpha \,.
BB+dα B \mapsto B + d \alpha

This is the famous gauge transformation law known from the string theory literature.

The open string sigma-model
Remark

The D-brane inclusion Qι XXQ \stackrel{\iota_X}{\to} X is the target space for an open string with worldsheet Σι ΣΣ\partial \Sigma \stackrel{\iota_\Sigma}{\hookrightarrow} \Sigma: a field configuration of the open string sigma-model is a map

ϕ:ι Σι X \phi \;\colon\; \iota_\Sigma \to \iota_X

in H Δ 1\mathbf{H}^{\Delta^1}, hence a diagram of the form

Σ ϕ bdr Q ι Σ ι X Σ ϕ bulk X. \array{ \partial \Sigma &\stackrel{\phi_{bdr}}{\to}& Q \\ \downarrow^{\mathrlap{\iota_\Sigma}} &\swArrow& \downarrow^{\mathrlap{\iota_X}} \\ \Sigma &\stackrel{\phi_{bulk}}{\to}& X } \,.

For XX and QQ ordinary manifolds just says that a field configuration is a map ϕ bulk:ΣX\phi_{bulk} \;\colon\; \Sigma \to X subject to the constraint that it takes the boundary of Σ\Sigma to QQ. This means that this is a trajectory of an open string in XX whose endpoints are constrained to sit on the D-brane QXQ \hookrightarrow X.

If however XX is more generally an orbifold, then the homotopy filling the above diagram imposes this constraint only up to orbifold transformations, hence exhibits what in the physics literature are called “orbifold twisted sectors” of open string configurations.

Proposition

The moduli stack [ι Σ,ι X][\iota_\Sigma, \iota_X] of such field configurations is the homotopy pullback

[ι Σ,ι X] [Σ,X] [S 1,Q] [S 1,X]. \array{ [\iota_{\Sigma}, \iota_X] &\to& [\Sigma, X] \\ \downarrow &\swArrow& \downarrow \\ [S^1, Q] &\to& [S^1, X] } \,.
The anomaly-free open string coupling to the Chan-Paton gauge field
Proposition

For Σ\Sigma a smooth manifold with boundary Σ\partial \Sigma of dimension nn and for :XB nU(1) conn\nabla \;\colon \; X \to \mathbf{B}^n U(1)_{conn} a circle n-bundle with connection on some XHX \in \mathbf{H}, then the transgression of \nabla to the mapping space [Σ,X][\Sigma, X] yields a section of the complex line bundle associated to the pullback of the ordinary transgression over the mapping space out of the boundary: we have a diagram

[Σ,X] exp(2πi Σ) //U(1) conn [Σ,X] ρ¯ conn [Σ,X] exp(2πi Σ) BU(1) conn. \array{ [\Sigma, X] &\stackrel{\exp(2 \pi i \int_{\Sigma})}{\to}& \mathbb{C}//U(1)_{conn} \\ \downarrow^{\mathrlap{[\partial \Sigma, X]}} && \downarrow^{\mathrlap{\overline{\rho}}_{conn}} \\ [\partial \Sigma, X] &\stackrel{\exp(2 \pi i \int_{\partial \Sigma})}{\to}& \mathbf{B} U(1)_{conn} } \,.
Remark

This is the higher parallel transport of the nn-connection \nabla over maps ΣX\Sigma \to X.

Proposition

The operation of forming the holonomy of a twisted unitary connection around a curve fits into a diagram in H\mathbf{H} of the form

[S 1,(BU(n))//(BU(1)) conn] hol S 1 //U(1) conn [S 1,dd^ n] ρ¯ conn [S 1,B 2U(1) conn] exp(2πi S 1) BU(1) conn. \array{ [S^1, (\mathbf{B}U(n))//(\mathbf{B}U(1))_{conn}] &\stackrel{hol_{S^1}}{\to}& \mathbb{C}//U(1)_{conn} \\ \downarrow^{\mathrlap{[S^1, \widehat\mathbf{dd}_n]}} &\swArrow_{\simeq}& \downarrow^{\mathrlap{\overline{\rho}_{conn}}} \\ [S^1, \mathbf{B}^2 U(1)_{conn}] &\stackrel{\exp(2 \pi i \int_{S^1})}{\to}& \mathbf{B}U(1)_{conn} } \,.
Remark

By the discussion at ∞-action the diagram in prop. says in particular that forming traced holonomy of twisted unitary bundles constitutes a section of the complex line bundle on the moduli stack of twisted unitary connection on the circle which is the associated bundle to the transgression exp(2πi S 1[S 1,dd^ n])\exp(2 \pi i \int_{S^1} [S^1, \widehat\mathbf{dd}_n]) of the universal differential Dixmier-Douady class.

It follows that on the moduli space of the open string sigma-model of prop. above there are two //U(1)\mathbb{C}//U(1)-valued action functionals coming from the bulk field and the boundary field

[ι Σ,ι X] [Σ,X] exp(2πi Σ[Σ, B]) //U(1) conn [S 1,Q] [S 1,X] hol S 1([S 1, gauge]) //U(1) conn. \array{ [\iota_{\Sigma}, \iota_X] &\to& [\Sigma, X] &\stackrel{exp(2 \pi i \int_{\Sigma}[\Sigma, \nabla_B] ) }{\to}& \mathbb{C}//U(1)_{conn} \\ \downarrow &\swArrow& \downarrow \\ [S^1, Q] &\to& [S^1, X] \\ \downarrow^{\mathrlap{hol_{S^1}([S^1, \nabla_{gauge}])}} \\ \mathbb{C}//U(1)_{conn} } \,.

Neither is a well-defined \mathbb{C}-valued function by itself. But by pasting the above diagrams, we see that both these constitute sections of the same complex line bundle on the moduli stack of fields:

[ι Σ,ι X] [Σ,X] [Σ, B] [S 1,B 2U(1) conn] exp(2πi Σ) //U(1) conn [S 1,Q] [S 1,X] [S 1, gauge] [S 1, B] [S 1,(BU(n))//(BU(1)) conn] [S 1,dd^ n] [S 1,B 2U(1) conn] hol S 1 exp(2πi S 1()) //U(1) conn BU(1) conn. \array{ [\iota_{\Sigma}, \iota_X] &\to& [\Sigma, X] &\stackrel{[\Sigma, \nabla_B]}{\to}& [S^1, \mathbf{B}^2 U(1)_{conn}] &\stackrel{\exp(2 \pi i \int_{\Sigma})}{\to}& \mathbb{C}//U(1)_{conn} \\ \downarrow &\swArrow& \downarrow && && \downarrow \\ [S^1, Q] &\to& [S^1, X] \\ \downarrow^{\mathrlap{[S^1, \nabla_{gauge}]}} && & \searrow^{\mathrlap{[S^1, \nabla_B]}} & && \downarrow \\ [S^1, (\mathbf{B}U(n))//(\mathbf{B}U(1))_{conn}] & &\stackrel{[S^1, \widehat \mathbf{dd}_n]}{\to}& & [S^1, \mathbf{B}^2 U(1)_{conn}] \\ \downarrow^{\mathrlap{hol_{S^1}}} && && & \searrow^{\mathrlap{\exp(2 \pi i \int_{S^1}(-))}} \\ \mathbb{C}//U(1)_{conn} &\to& &\to& &\to& \mathbf{B}U(1)_{conn} } \,.

Therefore the product action functional is a well-defined function

[ι Σ,ι X]exp(2πi Σ[Σ, b])hol S 1([S 1,dd^ n]) 1U(1). [\iota_\Sigma, \iota_X] \stackrel{ \exp(2 \pi i \int_{\Sigma} [\Sigma, \nabla_b] ) \cdot hol_{S^1}( [S^1, \widehat {\mathbf{dd}}_n] )^{-1} }{\to} U(1) \,.

This is the Kapustin anomaly-free action functional of the open string.

3d Chern-Simons theory with Wilson loops

We discuss how an extended Lagrangian for GG-Chern-Simons theory with Wilson loop defects is naturally obtained from the above higher geometric formulation of the orbit method. In particular we discuss how the relation between Wilson loops and 1-dimensional Chern-Simons theory sigma-models with target space the coadjoint orbit, as discussed above is naturally obtained this way.

More formally, we have an extended Chern-Simons theory as follows.

The moduli stack of fields ϕ:CJ\phi : C \to \mathbf{J} in H (Δ 1)\mathbf{H}^{(\Delta^1)} as above is the homotopy pullback

Fields(S 1Σ) [S 1,Ω 1(,𝔤)//T] [Σ,BG conn] [S 1,BG conn] \array{ \mathbf{Fields}(S^1 \hookrightarrow \Sigma) &\stackrel{}{\to}& [S^1, \Omega^1(-,\mathfrak{g})//T] \\ \downarrow &\swArrow_\simeq& \downarrow \\ [\Sigma, \mathbf{B}G_{conn}] &\to& [S^1, \mathbf{B}G_{conn}] }

in H\mathbf{H}, where square brackets indicate the internal hom in H\mathbf{H}.

Postcomposing the two projections with the two transgressions of the extended Lagrangians

exp(2πi Σ[Σ,c]):[Σ,BG conn][Σ,c][Σ,B 3U(1) conn]exp(2πi Σ())U(1) \exp(2 \pi i \int_\Sigma[\Sigma, \mathbf{c}]) \;\colon\; [\Sigma, \mathbf{B}G_{conn}] \stackrel{[\Sigma, \mathbf{c}]}{\to} [\Sigma, \mathbf{B}^3 U(1)_{conn}] \stackrel{\exp(2 \pi i \int_\Sigma (-))}{\to} U(1)

and

exp(2πi Σ[S 1,λ,]):[S 1,Ω 1(,𝔤)//T][Σ,λ,][S 1,BU(1) conn]exp(2πi S 1())U(1) \exp(2 \pi i \int_\Sigma[S^1, \langle \lambda, -\rangle]) \;\colon\; [S^1, \Omega^1(-,\mathfrak{g})//T] \stackrel{[\Sigma, \langle \lambda , -\rangle]}{\to} [S^1, \mathbf{B} U(1)_{conn}] \stackrel{\exp(2 \pi i \int_{S^1} (-))}{\to} U(1)

to yield

Fields(S 1Σ) [S 1,Ω 1(,𝔤)//T] exp(2πi S 1[S 1,λ,]) U(1) [Σ,BG conn] [S 1,BG conn] exp(2πi Σ 2[Σ 3,c]) U(1) \array{ \mathbf{Fields}(S^1 \hookrightarrow \Sigma) &\stackrel{}{\to}& [S^1, \Omega^1(-,\mathfrak{g})//T] &\stackrel{\exp(2 \pi i \int_{S^1} [S^1, \langle \lambda, -\rangle] ) }{\to}& U(1) \\ \downarrow &\swArrow_\simeq& \downarrow \\ [\Sigma, \mathbf{B}G_{conn}] &\to& [S^1, \mathbf{B}G_{conn}] \\ \downarrow^{\mathrlap{\exp(2\pi i \int_{\Sigma_2} [\Sigma_3, \mathbf{c}])}} \\ U(1) }

and then forming the product yields the action functional

exp(2πi S 1[S 1,])exp(2πi Σ[Σ,c]):Fields(S 1Σ)U(1). \exp(2 \pi i \int_{S^1}[S^1, \langle -\rangle]) \cdot \exp(2 \pi i \int_{\Sigma}[\Sigma, \mathbf{c}]) \;:\; \mathbf{Fields}(S^1 \hookrightarrow \Sigma) \to U(1) \,.

This is the action functional of 3d GG-Chern-Simons theory on Σ\Sigma with Wilson loop CC in the representation determined by λ\lambda.

Similarly, in codimension 1 let Σ 2\Sigma_2 now be a 2-dimensional closed manifold, thought of as a slice of Σ\Sigma above, and let i*Σ 2\coprod_i {*} \to \Sigma_2 be the inclusion of points, thought of as the punctures of the Wilson line above through this slice. Then we have prequantum bundles given by transgression of the extended Lagrangians to codimension 1

exp(2πi Σ 2[Σ,c]):[Σ 2,BG conn][Σ 2,c][Σ 2,B 3U(1) conn]exp(2πi Σ 2())BU(1) conn \exp\left(2 \pi i \int_{\Sigma_2}\left[\Sigma, \mathbf{c}\right]\right) \;\colon\; \left[\Sigma_2, \mathbf{B}G_{conn}\right] \stackrel{\left[\Sigma_2, \mathbf{c}\right]}{\to} \left[\Sigma_2, \mathbf{B}^3 U(1)_{conn}\right] \stackrel{\exp\left(2 \pi i \int_{\Sigma_2} \left(-\right)\right)}{\to} \mathbf{B}U\left(1\right)_{conn}

and

exp(2πi i*[ i*,λ,]):[ i*,Ω 1(,𝔤)//T][ i*,λ,][ i*,BU(1) conn]exp(2πi i*())BU(1) conn \exp\left(2 \pi i \int_{\coprod_i {*}}\left[\coprod_i {*}, \left\langle \lambda, -\right\rangle\right]\right) \;\colon\; \left[\coprod_i {*}, \Omega^1\left(-,\mathfrak{g}\right)//T\right] \stackrel{[\coprod_i {*}, \langle \lambda , -\rangle]}{\to} \left[\coprod_i {*}, \mathbf{B} U(1)_{conn}\right] \stackrel{\exp\left(2 \pi i \int_{\coprod_i {*}} \left(-\right)\right)}{\to} \mathbf{B}U(1)_{conn}

and hence a total prequantum bundle

exp(2πi i*[ i*,β,])exp(2πi Σ 2[Σ 2,c]):Fields( i*Σ)BU(1) conn. \exp\left(2 \pi i \int_{\coprod_i {*}}\left[\coprod_i {*}, \langle \beta, -\rangle\right]\right) \otimes \exp\left(2 \pi i \int_{\Sigma_2}\left[\Sigma_2, \mathbf{c}\right]\right) \;:\; \mathbf{Fields}\left(\coprod_i {*} \hookrightarrow \Sigma\right) \to \mathbf{B}U\left(1\right)_{conn} \,.

One checks that this is indeed the correct prequantization as considered in (Witten 98, p. 22).

Chan-Paton gauge fields on D-branes

Syntactic Layer

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Created on May 13, 2015 at 13:33:59. See the history of this page for a list of all contributions to it.