nLab geometry of physics -- prequantum gauge theory and gravity

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

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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

   • String field theory

     • Gauge fields and gravity – Einstein-Maxwell-Yang-Mills theory

       • Kaluza-Klein compactification

       • Standard model of particle physics

       • standard model of cosmology

1d Chern-Simons theory

• 1d Chern-Simons theory

2d 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 $G$ be a semisimple compact Lie group. For some considerations below we furthermore assume it to be simply connected.

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

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

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

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

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

The coadjoint orbit and the coset space/ flag manifold

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

Write

1. $T \hookrightarrow G$ inclusion of the maximal torus of $G$.

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

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

Definition

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

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

for its coadjoint orbit

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

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

Proposition

There is an equivalence

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

given by

$$g G_\lambda \mapsto Ad_g^* \langle\lambda,-\rangle \,.$$

Definition

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

Example

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

The symplectic form

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

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

Definition

Write

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

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

Write

$$\nu_\lambda := d_{dR} \Theta_\lambda$$

for its de Rham differential.

Proposition

The 2-form $\nu_\lambda$ from def.

1. satisfies

   $$\nu_\lambda = \frac{1}{2}\langle \lambda, [\theta\wedge \theta]\rangle \,.$$

2. it descends to a closed $G$-invariant 2-form on the coset space, to be denoted by the same symbol

   $$\nu_\lambda \in \Omega^2_{cl}(G/G_\lambda)^G \,.$$

3. this is non-degenerate and hence defines a symplectic form on $G/G_\lambda$.

The prequantum bundle

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

Assume now that $G$ is simply connected.

Proposition

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

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

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

$$\rho_\alpha : \exp(\xi) \mapsto \exp(i \langle \alpha, \xi\rangle) \,.$$

Proposition

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

The Hamiltonian $G$-action / coadjoint moment map

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

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

Above (…) we discussed how an irreducible unitary representation of $G$ is encoded by the prequantization of a coadjoint orbit $(\mathcal{O}_\lambda, \nu_\lambda)$. Here we discuss how to express Wilson loops/holonomy of $G$-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 $A|_{S^1} \in \Omega^1(S^1, \mathfrak{g})$ be a Lie algebra valued 1-form on the circle, equivalently a $G$-principal connection on the circle.

For

$$\rho : G \to Aut(V)$$

a representation of $G$, write

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

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

Definition

Let the action functional

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

be given by sending $g T : S^1 \to G/T$ represented by $g : S^1 \to G$ to

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

where

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

is the gauge transformation of $A$ under $g$.

Proposition

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

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

2. action functional the functional of def. :

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

See for instance (Beasley, (4.55)).

Remark

Notice that since $\mathcal{O}_\lambda$ 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

• 3d Chern-Simons theory

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

• higher dimensional Chern-Simons theory

7d Chern-Simons theory

• 7d Chern-Simons theory

$\infty$-Chern-Simons theory

• infinity-Chern-Simons theory

String field theory

• string field theory

Gauge fields and gravity – Einstein-Maxwell-Yang-Mills theory

• gravity

• Yang-Mills theory

  • Yang-Mills instanton number = second Chern class

• Einstein-Yang-Mills theory

Kaluza-Klein compactification

• Kaluza-Klein compactification

Standard model of particle physics

• standard model of particle physics

Standard model of cosmology

• standard model of cosmology

Semantic Layer

exposition and survey is in (FSS 13).

Examples

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 $n \in \mathbb{N}$ let

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

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

$$\mathbf{B}det \;\colon\; \mathbf{B}U(n) \to \mathbf{B}U(1)$$

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

$${\vert \mathbf{B}det\vert} \simeq c_1 \;\colon\; B U(n) \to B U(1) \simeq K(\mathbb{Z},2) \,.$$

Moreiver this has the evident differential refinement

$$\widehat {\mathbf{B} det} \;\colon\; \mathbf{B} U(n)_{conn} \to \mathbf{B} U(1)_{conn}$$

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

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

So we get a 1d Chern-Simons theory with $\widehat{\mathbf{B}det}$ 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.

Examples

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 $\lambda \in \mathfrak{g}$ a regular element, there is a canonical diagram of smooth moduli stacks of the form

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

where

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

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

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

4. the top map denoted $\langle \lambda,- \rangle$ 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 $\Omega_\lambda \simeq G/T$.

Definitions and constructions

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

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

As usual, write

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

for the moduli stack of $G$-principal connections.

Definition

Write

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

for the canonical map, as indicated.

Remark

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

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

By the discussion at ∞-action this exhibits the canonical action $\rho$ of $G$ 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 $\mathbf{J}$ in def. is

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

given over a test manifold $U \in$ CartSp by the map

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

which sends $g \mapsto g^* \theta$, where $\theta$ is the Maurer-Cartan form on $G$.

Proof

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

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

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

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

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

2. morphisms are over $U$ commuting triangles

   $$\array{ g_1^* \theta &&\stackrel{t}{\to}&& g_2^* \theta \\ & {}_{\mathllap{g_1}}\nwarrow && \nearrow_{\mathrlap{g_2}} \\ && 0 }$$

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

   $$g_1 \stackrel{t}{\to} g_2$$

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

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

Proposition

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

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

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

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

which is given on objects by

$$A \mapsto \langle \lambda, A\rangle$$

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

$$\exp(\xi) \mapsto \exp( i \langle \lambda, \xi \rangle ) \,.$$

Proof

That this construction defines a map $*//T \to *//U(1)$ 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 $T \simeq G_\lambda$ stabilizes $\langle \lambda , - \rangle$ under the coadjoint action, the gauge transformation law for points $A : U \to \mathbf{B}G_{conn}$, which for $g \in C^\infty(U,G)$ is

$$A \mapsto Ad_g A + g^* \theta \,,$$

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

$$\begin{aligned} \langle \lambda, A \rangle & \mapsto \langle \lambda, Ad_g A\rangle + \langle \lambda, g^* \theta\rangle \\ & = \langle \lambda, A \rangle + d \langle\lambda, \xi \rangle \end{aligned}$$

Remark

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

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

Proposition

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

Proof

The curvature 2-form is modulated by the composite

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

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

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

which sends

$$[g] \mapsto d \langle \lambda, g^* \theta \rangle \,.$$

Nonabelian charged particle trajectories – Wilson loops

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

$$C \;\colon\; S^1 \hookrightarrow \Sigma$$

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

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

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

$$\phi \;\colon\; C \to \mathbf{J}$$

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

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

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

• a $G$-principal connection $A$ on $\Sigma$;

• a $G$-valued function $g$ on $S^1$

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

Moreover, a gauge transformation between two such fields $\kappa : \phi \Rightarrow \phi'$ is a $G$-gauge transformation of $A$ and a $T$-gauge transformation of $A|_{S^1}$ such that these intertwine the component maps $g$ and $g'$. If we keep the bulk gauge field $A$ fixed, then his means that two fields $\phi$ and $\phi'$ as above are gauge equivalent precisely if there is a function $t \;\colon\; S^1 \to T$ such that $g = g' t$, hence gauge equivalence classes of fields for fixed bulk gauge field $A$ are parameterized by their components $[g] = [g'] \in [S^1, G/T]$ 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 $\mathbf{c} : \mathbf{B}G_{conn} \to \mathbf{B}^3 U(1)_{conn}$;

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

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

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

and then preforming the fiber integration in ordinary differential cohomology over $S^1$ and over $\Sigma$, 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 $i \in \{1, \cdots n\}$. Further analysis then shows that the lowest excitations of these $(i,j)$-strings behave as the quanta of a $U(n)$-gauge field, the $(i,j)$-excitation being the given matrix element of a $U(n)$-valued connection 1-form $A$.

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

Examples

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 $B$-field as a prequantum 2-bundle

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

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

If $X = G$ is a Lie group, this is the prequantum 2-bundle of $G$-Chern-Simons theory. Viewed as such we are to find a canonical ∞-action of the circle 2-group $\mathbf{B}U(1)$ on some $V \in \mathbf{H}$, 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 $Spin^c$-D-branes in type II string theory.

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

Proposition

The extension of Lie groups

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

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

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

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

Proposition

Here

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

is a smooth refinement of the universal Dixmier-Douady class

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

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

$${\vert \mathbf{dd}_n \vert} \simeq dd_n \,.$$

Remark

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

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

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

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

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

Proposition

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

$$\mathbf{c} \to \mathbf{dd}_n$$

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

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

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

Proposition

There is a further differential refinement

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

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

Definition

Write

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

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

Definition

Let

$$\iota_X \;\colon\; Q \hookrightarrow X$$

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

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

$$\phi \;\colon\; \iota_X \to \mathbf{Fields}$$

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

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

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

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

On the local connection forms this acts as

$$A \mapsto A + \alpha \,.$$

$$B \mapsto B + d \alpha$$

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

The open string sigma-model

Remark

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

$$\phi \;\colon\; \iota_\Sigma \to \iota_X$$

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

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

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

If however $X$ 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 $[\iota_\Sigma, \iota_X]$ of such field configurations is the homotopy pullback

$$\array{ [\iota_{\Sigma}, \iota_X] &\to& [\Sigma, X] \\ \downarrow &\swArrow& \downarrow \\ [S^1, Q] &\to& [S^1, X] } \,.$$

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

Proposition

For $\Sigma$ a smooth manifold with boundary $\partial \Sigma$ of dimension $n$ and for $\nabla \;\colon \; X \to \mathbf{B}^n U(1)_{conn}$ a circle n-bundle with connection on some $X \in \mathbf{H}$, then the transgression of $\nabla$ to the mapping space $[\Sigma, X]$ 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

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

Remark

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

Proposition

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

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

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 $\exp(2 \pi i \int_{S^1} [S^1, \widehat\mathbf{dd}_n])$ 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 $\mathbb{C}//U(1)$-valued action functionals coming from the bulk field and the boundary field

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

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:

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

Therefore the product action functional is a well-defined function

$$[\iota_\Sigma, \iota_X] \stackrel{ \exp(2 \pi i \int_{\Sigma} [\Sigma, \nabla_b] ) \cdot hol_{S^1}( [S^1, \widehat {\mathbf{dd}}_n] )^{-1} }{\to} U(1) \,.$$

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 $G$-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 $\phi : C \to \mathbf{J}$ in $\mathbf{H}^{(\Delta^1)}$ as above is the homotopy pullback

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

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

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

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

and

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

to yield

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

and then forming the product yields the action functional

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

This is the action functional of 3d $G$-Chern-Simons theory on $\Sigma$ with Wilson loop $C$ in the representation determined by $\lambda$.

Similarly, in codimension 1 let $\Sigma_2$ now be a 2-dimensional closed manifold, thought of as a slice of $\Sigma$ above, and let $\coprod_i {*} \to \Sigma_2$ 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

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

and

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

and hence a total prequantum bundle

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

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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