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

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

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 November 20, 2019 at 02:23:11. See the history of this page for a list of all contributions to it.