nLab Yang monopole

Idea

The Yang monopole A monopole in Yang-Mills theory. The generalization of the Dirac monopole from 3+1 dimensional spacetime to 5+1 dimensional spacetime.

Definition

Recalling the Dirac monopole

For comparison, first note that the Dirac monopole is a circle group principal bundle with non-trivial first Chern class on a spacetime of the form

$$X_4 \;\coloneqq\; (\mathbb{R}^3_{space} - \{0\}) \times \mathbb{R}_{time}$$

witnessed (via Chern-Weil theory) by the magnetic charge

$$Q = \int_{S^2} F_\nabla \in \mathbb{Z} \hookrightarrow \mathbb{R}$$

which is the integration of the curvature differential 2-form $F_\nabla$ of any principal connection $\nabla$ on this bundle over any sphere wrapping the removed origin of $\mathbb{R}^3$. We may think of the Dirac monopole as being an effective magnetic monopole “particle” with worldline $\{0\} \times \mathbb{R}_{time}$. But in the above description we can just as well remove a little 3-ball $D^3_\epsilon$ from space, instead of just a point, and when viewed as such the Dirac monopole is a 2-brane with worldvolume $(\partial D_\epsilon) \times \mathbb{R}_{times}$.

The Yang monopole

Now analogously, a Yang monopole is a nontrivial special unitary group-principal bundle (for which there is no non-trivial first Chern class) with non-trivial second Chern class on

$$X_6 \coloneqq (\mathbb{R}^5_{space} - \{0\}) \times \mathbb{R}_{time}$$

witnessed (via Chern-Weil theory) by the instanton number

$$Q = \int_{S^4} \langle F_\nabla \wedge F_\nabla\rangle \in \mathbb{Z} \hookrightarrow \mathbb{R}$$

which is the integration of the curvature differential 2-form $F_\nabla$ wedge-squared and evaluated in the canonical Killing form invariant polynomial to a differential 4-form of any principal connection $\nabla$ on this bundle over any 4-sphere wrapping the removed origin of $\mathbb{R}^5$.

As before, we may equivalently think of removing instead of just a point a small 5-ball $D_{\epsilon}^5$ and then the Yang monopole appears as a 4-brane with worldvolume $(\partial D_\epsilon^5) \times \mathbb{R}_{time}$. This is how the Yang monopole appears in string theory/M-theory (Bergsgoeff-Gibbons-Townsend 06).

In terms of higher gauge theory the Yang monopole is seen to be more directly analogous to the Dirac monopole: the 4-form $\langle F_\nabla \wedge F_\nabla\rangle$ is actually the curvature 4-form of a circle 3-bundle with connection on spacetime which is induced from the given $SU(N)$-principal connection, namely the Chern-Simons circle 3-bundle which is modulated by the universal characteristic class that is the differential refinement of the second Chern class:

$$\mathbf{c}_2 \colon \mathbf{B}SU(N)_{conn} \to \mathbf{B}^3 U(1)_{conn}$$

(this is discussed at differential string structure). Hence $\mathbf{c}_2(\nabla)$ is a 3-connection with curvature

$$F_{\mathbf{c}_2(\nabla)} = \langle F_\nabla \wedge F_\nabla \rangle \,.$$

Now the analog of the first Chern class as one passes to such circle n-bundles is called the higher Dixmier-Douady class, and the Yang monopole charge is just this 2-Dixmier-Douady class of the Chern-Simons circle 3-bundle induced by a $SU(N)$-principal connection with corresponding instanton number:

$$Q = \int_{S^4} F_{\mathbf{c}_2(\nabla)} \in \mathbb{Z} \hookrightarrow \mathbb{R} \,.$$

Examples

In M-brane theory

The end-surface of an M5-brane ending on an M9-brane is a Yang-monopole in the M5 worldvolume (Bergsgoeff-Gibbons-Townsend 06).

References

General

The Yang monopole was introduced as a generalization to 5+1 dimensional spacetime of the Dirac monopole in 3+1 dimensional spacetime in

• Chen Ning Yang, Generalization of Dirac’s Monopole to $SU(2)$ Gauge Fields, J. Math. Phys. 19, 320 (1978).

More general discussion is in

• Tigran Tchrakian, Dirac-Yang monopoles and their regular counterparts (arXiv:hep-th/0612249)

See also

• Frederik Nørfjand, Nikolaj Thomas Zinner, Non-existence theorems and solutions of the Wu-Yang monopole equation (arxiv:1911.08140)

In string theory

Appearance of Yang monopoles in string theory goes back to

• Adil Belhaj, Pablo Diaz, Antonio Segui, On the Superstring Realization of the Yang Monopole (arXiv:hep-th/0703255)

The appearance of the Yang monopole as the boundary of an open M5-brane ending on an M9-brane is discussed in

• Eric Bergshoeff, Gary Gibbons, Paul Townsend, Open M5-branes, Phys.Rev.Lett.97:231601 2006 (arXiv:hep-th/0607193)

Based on this a Yang monopole is realized in type IIA string theory in section 2 of

• Adil Belhaj, Pablo Diaz, Antonio Segui, The Yang Monopole in IIA Superstring: Multi-charge Disease and Enhancon Cure (arXiv:1102.1538)