# nLab enhanced gauge symmetry

## Surveys, textbooks and lecture notes

#### Chern-Weil theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory

# Contents

## Idea

Generally, an enhanced gauge symmetry referes to the appearance of gauge theory with a larger gauge group than may be suprficially apparent.

### On coincident D-branes

A famous case is the enhance gauge symmetry in the Chan-Paton gauge fields of coincident D-branes. A priori each D-brane carries a complex line bundle/circle principal bundle, but as $n$ of them coincide the gauge group is expected to enhance from $(U(1))^n$ to the unitary group $U(n)$ (or the special unitary group $SU(n)$).

Mathematically this gauge enhancement on D-branes is modeled by the Baum-Douglas geometric cycle model for K-homology (Reis-Szabo 05, def. 1.5 and p. 16, Szabo 08, p. 4): One such cycle is given by a submanifold $W \overset{\phi}{\hookrightarrow} X$ of spacetime $X$ equipped with spin^c structure and with a complex vector bundle $E \to X$, and given two such cycles $(W,\phi,E_1)$ and $(W,\phi,E_2)$ with the same underlying manifold (“coincident D-brane worldvolume”) then their formal direct sum is identified with the single cycle carrying the direct sum of vector bundles

$(W,\phi, E_1) + (W, \phi, E_2) \;\sim\; (W, \phi, E_1 \oplus E_2) \,.$

If here $E_i$ has rank $n_i$ and hence structure group $U(n_i)$, then the direct sum of vector bundles $E_1 \oplus E_2$ has structure group the product group $U(n_1) \times U(n_2)$.

In particular the sum of $n$ coincidnt cycles each carrying a line bundle yields a rank-$n$ vector bundle with structure group $(U(1))^n$. In this sense then a cycle carrying a general rank $n$-vector bundle, hence with structure group $U(n)$, may be thought of as a “gauge enhancement” of making $n$ D-branes coincide.

### From massless BPS states

In the context of the realization of super Yang-Mills theories as effective field theories of superstring theory, BPS branes in the string background becomes BPS states of the SYM theory. These become massless as the cycles on the which BPS charge is supported shrink away (at boundary points of the given moduli space). In this case these additional massless states may appear as additional gauge bosons in the compactified gauge theory which thereby develops a bigger gauge group. Typically a previously abelian gauge group becomes non-abelian this way.

Examples include KK-compactification of M-theory on K3 fibers and on G2-manifolds with ADE singularities, corresponding to F-theory on elliptic fibrations making K3-fibrations with singular fibers, corresponding to heterotic string theory on singular elliptic fibrations. See at

## References

Discussion of gauge enhancement on coincident D-branes in terms of Baum-Douglas geometric cycles for K-homology is in

reviewed in

Review of the standard lore of gauge enhancement in M-theory includes

Original articles includ

Revised on December 28, 2016 12:51:59 by Urs Schreiber (185.117.214.13)