nLab electric-magnetic duality

Idea

Electric-magnetic duality is a lift of Hodge duality from de Rham cohomology to ordinary differential cohomology.

Description

Consider a circle n-bundle with connection $\nabla$ on a space $X$. Its higher parallel transport is the action functional for the sigma-model of $(n-1)$-dimensional objects ($(n-1)$-branes) propagating in $X$.

For $n = 1$ this is the coupling of the electromagnetic field to particles. For $n = 2$ this is the coupling of the Kalb-Ramond field to strings.

The curvature $F_\nabla \in \Omega^{n+1}(X)$ is a closed $(n+1)$-form. The condition that its image $\star F_\nabla$ under the Hodge star operator is itself closed

$$d_{dR} \star F_\nabla = 0$$

is the Euler-Lagrange equation for the standard (abelian Yang-Mills theory-action functional on the space of circle n-bundle with connection.

If this is the case, it makes sense to ask if $\star F_\nabla$ itself is the curvature $(d-(n+1))$-form of a circle $(d-(n+1)-1)$-bundle with connection $\tilde \nabla$, where $d = dim X$ is the dimension of $X$.

If such $\tilde \nabla$ exists, its higher parallel transport is the gauge interaction action functional for $(d-n-3)$-dimensional objects propagating on $X$.

In the special case of ordinary electromagnetism with $n=1$ and $d = 4$ we have that electrically charged 0-dimensional particles couple to $\nabla$ and magnetically charged $(4-(1+1)-2) = 0$-dimensional particles couple to $\tilde \nabla$.

In analogy to this case one calls generally the $d-n-3$-dimensional objects coupling to $\tilde \nabla$ the magnetic duals of the $(n-1)$-dimensional objects coupling to $\nabla$.

Generalizations

For $d= 4$ EM-duality is the special abelian case of S-duality for Yang-Mills theory. Witten and Kapustin argued that this is governed by the geometric Langlands correspondence.

Examples

• In N=2 D=4 super Yang-Mills theory electric-magnetic duality is studied as Seiberg-Witten theory.

• In heterotic string theory one considers 1-dimensional objects in $d=10$-dimensional spaces electrically charged (under the Kalb-Ramond field). Their magnetic duals are 5-dimensional objects (fivebranes), studied in dual heterotic string theory.

electric-magnetic duality of D-branes/RR-fields in type II string theory:

electric charge	magnetic charge
D0-brane	D6-brane
D1-brane	D5-brane
D2-brane	D4-brane
D3-brane	D3-brane

• M2-brane and M5-brane: the 4-form $G_4$ which the M2-brane couples to is the Hodge-dual of the 7-form $G_7$ which the M5-brane couples to

$$G_7 = \star G_4$$

as imposed by the equations of motion for 11d supergravity (see here). The associated C3-field and C6-field are electric-magnetic duals.

Related concepts

duality in physics, duality in string theory

• parent action functional

• S-duality

  • electro-magnetic duality

    • Montonen-Olive duality

    • dual graviton

    • dual photon

  • Seiberg duality

  • geometric Langlands correspondence

  • quantum geometric Langlands correspondence

References

Detailed review is in

• José Figueroa-O'Farrill, Electromagnetic Duality for Children (1998) (pdf)

It was originally noticed in

• Peter Goddard, Jean Nuyts, David Olive, Gauge Theories And Magnetic Charge, Nucl. Phys. B 125 (1977) 1-28 [doi:10.1016/0550-3213(77)90221-8, spire:111435]

that where electric charge in Yang-Mills theory takes values in the weight lattice of the gauge group, then magnetic charge takes values in the lattice of what is now called the Langlands dual group.

This led to the electric/magnetic duality conjecture formulation in

• Claus Montonen, David Olive, Magnetic Monopoles As Gauge Particles? Phys. Lett. B72 (1977) 117-120 (spire:121372, doi:10.1016/0370-2693(77)90076-4)

According to (Kapustin-Witten 06, pages 3-4) the observation that the Montonen-Olive dual charge group coincides with the Langlands dual group is due to

• Michael Atiyah, private communication to Edward Witten, 1977

See also the references at S-duality.

The insight that the Montonen-Olive duality works more naturally in super Yang-Mills theory is due to

• David Olive, Edward Witten, Supersymmetry Algebras That Include Topological Charges, Phys. Lett. B78 (1978) 97-101.

and that it works particularly for N=4 D=4 super Yang-Mills theory is due to

• H. Osborn, Topological Charges For $N = 4$ Supersymmetric Gauge Theories And Monopoles Of Spin 1, Phys. Lett. B83 (1979) 321-326.

The observation that the $\mathbb{Z}_2$ electric/magnetic duality extends to an $SL(2,\mathbb{Z})$-action in this case is due to

• John Cardy, E. Rabinovici, Phase Structure Of Zp Models In The Presence Of A Theta Parameter, Nucl. Phys. B205 (1982) 1-16;

• John Cardy, Duality And The Theta Parameter In Abelian Lattice Models, Nucl. Phys. B205 (1982) 17-26.

• A. Shapere and Frank Wilczek, Selfdual Models With Theta Terms, Nucl. Phys. B320 (1989) 669-695.

and specifically the embedding of this into string theory S-duality originates in

• Ashoke Sen, Dyon - Monopole Bound States, Self-Dual Harmonic Forms on the Multi-Monopole Moduli Space, and $SL(2,\mathbb{Z})$ Invariance in String Theory (arXiv:hep-th/9402032)

The understanding of this $SL(2,\mathbb{Z})$-symmetry as a remnant conformal transformation on a 6-dimensional principal 2-bundle-theory – the 6d (2,0)-superconformal QFT – compactified on a torus is described in

• Edward Witten, pages 4-5 of Some Comments On String Dynamics, Proceedings of String95 (arXiv:hepth/9507121)

• Edward Witten, On S-Duality in Abelian Gauge Theory (arXiv:hep-th/9505186)

• Cumrun Vafa, Geometric Origin of Montonen-Olive Duality, Adv.Theor.Math.Phys.1:158-166, 1998 (arXiv:hep-th/9707131)

• Edward Witten, Conformal field theory in four and six dimensions, in Ulrike Tillmann, Topology, Geometry and Quantum Field Theory: Proceedings of the 2002 Oxford Symposium in Honour of the 60th Birthday of Graeme Segal, London Mathematical Society Lecture Note Series (2004) (arXiv:0712.0157)

The relation of S-duality to geometric Langlands duality was understood in

• Anton Kapustin, Edward Witten, Electric-Magnetic Duality And The Geometric Langlands Program , Communications in number theory and physics, Volume 1, Number 1, 1–236 (2007) (arXiv:hep-th/0604151)

Exposition of this is in

• Edward Frenkel, What Do Fermat’s Last Theorem and Electro-magnetic Duality Have in Common? KITP talk 2011 (web)

A perspective unifying EM-duality of higher gauge theories with non-abelian T-duality:

• Ján Pulmann, Pavol Ševera, Fridrich Valach, A non-abelian duality for (higher) gauge theories (arXiv:1909.06151)

An arithmetic version of EM-duality:

• David Ben-Zvi, Yiannis Sakellaridis, Akshay Venkatesh, Relative Langlands Duality (pdf)