Yang monopole




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\infty-Chern-Weil theory

Differential cohomology



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.


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

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

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

which is the integration of the curvature differential 2-form F F_\nabla of any principal connection \nabla on this bundle over any sphere wrapping the removed origin of 3\mathbb{R}^3. We may think of the Dirac monopole as being an effective magnetic monopoleparticle” with worldline {0}× time\{0\} \times \mathbb{R}_{time}. But in the above description we can just as well remove a little 3-ball D ϵ 3D^3_\epsilon from space, instead of just a point, and when viewed as such the Dirac monopole is a 2-brane with worldvolume (D ϵ)× times(\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( space 5{0})× time X_6 \coloneqq (\mathbb{R}^5_{space} - \{0\}) \times \mathbb{R}_{time}

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

Q= S 4F F 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 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 5\mathbb{R}^5.

As before, we may equivalently think of removing instead of just a point a small 5-ball D ϵ 5D_{\epsilon}^5 and then the Yang monopole appears as a 4-brane with worldvolume (D ϵ 5)× time(\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 F F \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)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:

c 2:BSU(N) connB 3U(1) conn \mathbf{c}_2 \colon \mathbf{B}SU(N)_{conn} \to \mathbf{B}^3 U(1)_{conn}

(this is discussed at differential string structure). Hence c 2()\mathbf{c}_2(\nabla) is a 3-connection with curvature

F c 2()=F F . 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)SU(N)-principal connection with corresponding instanton number:

Q= S 4F c 2(). Q = \int_{S^4} F_{\mathbf{c}_2(\nabla)} \in \mathbb{Z} \hookrightarrow \mathbb{R} \,.


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).



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)SU(2) Gauge Fields, J. Math. Phys. 19, 320 (1978).

More general discussion is in

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

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)

Last revised on June 20, 2013 at 14:15:13. See the history of this page for a list of all contributions to it.