nLab Chan-Paton bundle

under construction

Contents

Idea

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.

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 .

Definition

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.

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. 1

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

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. 4, 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 \ref{PullbackAlongGeneralizedLocalDiffeomorphisms} 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. 7 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. 5 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.

References

In the traditional physicist’s string theory introductions one finds Chan-Paton bundles discussed for instance

section 2.4 of

or around p. 66 of

These lectures tend to ignore most of the subtleties though. For traditional discussion of the Freed-Witten-Kapustin anomaly, see there. The above account in terms of higher geometry and extended prequantum field theory is due to section 5.4 of

Lecture notes along these lines are at

Revised on August 22, 2013 07:22:42 by Urs Schreiber (150.212.95.105)