A symmetry of Maxwell's equations

In classical physics, the behavior of the electric and magnetic field is described by a system of equations known as Maxwell's equations. Working in the language of vector calculus and assuming that no electric charges or currents are present, these equations can be written^[3]

${\displaystyle {\begin{aligned}\nabla \cdot \mathbf {E} &=0,\\\nabla \cdot \mathbf {B} &=0,\\\nabla \times \mathbf {E} &=-{\frac {\partial \mathbf {B} }{\partial t}},\\\nabla \times \mathbf {B} &={\frac {1}{c^{2}}}{\frac {\partial \mathbf {E} }{\partial t}}.\end{aligned}}}$

Here ${\displaystyle \mathbf {E} }$ is a vector (or more precisely a vector field whose magnitude and direction may vary from point to point in space) representing the electric field, ${\displaystyle \mathbf {B} }$ is a vector representing the magnetic field, ${\displaystyle t}$ is time, and ${\displaystyle c}$ is the speed of light. The other symbols in these equations refer to the divergence and curl, which are concepts from vector calculus.

An important property of these equations^[4] is their invariance under the transformation that simultaneously replaces the electric field ${\displaystyle \mathbf {E} }$ by the magnetic field ${\displaystyle \mathbf {B} }$ and replaces ${\displaystyle \mathbf {B} }$ by ${\displaystyle -1/c^{2}\mathbf {E} }$:

${\displaystyle {\begin{aligned}\mathbf {E} &\rightarrow \mathbf {B} \\\mathbf {B} &\rightarrow -{\frac {1}{c^{2}}}\mathbf {E} .\end{aligned}}}$

In other words, given a pair of electric and magnetic fields that solve Maxwell's equations, it is possible to describe a new physical setup in which these electric and magnetic fields are essentially interchanged, and the new fields will again give a solution of Maxwell's equations. This situation is the most basic manifestation of S-duality in a field theory.

Montonen–Olive duality

In quantum field theory, the electric and magnetic fields are unified into a single entity called the electromagnetic field, and this field is described by a special type of quantum field theory called a gauge theory or Yang–Mills theory. In a gauge theory, the physical fields have a high degree of symmetry which can be understood mathematically using the notion of a Lie group. This Lie group is known as the gauge group. The electromagnetic field is described by a very simple gauge theory corresponding to the abelian gauge group U(1), but there are other gauge theories with more complicated non-abelian gauge groups.^[5]

It is natural to ask whether there is an analog in gauge theory of the symmetry interchanging the electric and magnetic fields in Maxwell's equations. The answer was given in the late 1970s by Claus Montonen and David Olive,^[6] building on earlier work of Peter Goddard, Jean Nuyts, and Olive.^[7] Their work provides an example of S-duality now known as Montonen–Olive duality. Montonen–Olive duality applies to a very special type of gauge theory called N = 4 supersymmetric Yang–Mills theory, and it says that two such theories may be equivalent in a certain precise sense.^[1] If one of the theories has a gauge group ${\displaystyle G}$, then the dual theory has gauge group ${\displaystyle {^{L}}G}$ where ${\displaystyle {^{L}}G}$ denotes the Langlands dual group which is in general different from ${\displaystyle G}$.^[8]

An important quantity in quantum field theory is complexified coupling constant. This is a complex number defined by the formula^[9]

${\displaystyle \tau ={\frac {\theta }{2\pi }}+{\frac {4\pi i}{g^{2}}}}$

where ${\displaystyle \theta }$ is the theta angle, a quantity appearing in the Lagrangian that defines the theory,^[9] and ${\displaystyle g}$ is the coupling constant. For example, in the Yang–Mills theory that describes the electromagnetic field, this number ${\displaystyle g}$ is simply the elementary charge ${\displaystyle e}$ carried by a single proton.^[1] In addition to exchanging the gauge groups of the two theories, Montonen–Olive duality transforms a theory with complexified coupling constant ${\displaystyle \tau }$ to a theory with complexified constant ${\displaystyle -1/\tau }$.^[9]

Relation to the Langlands program

The geometric Langlands correspondence is a relationship between abstract geometric objects associated to an algebraic curve such as the elliptic curves illustrated above.

In mathematics, the classical Langlands correspondence is a collection of results and conjectures relating number theory to the branch of mathematics known as representation theory.^[10] Formulated by Robert Langlands in the late 1960s, the Langlands correspondence is related to important conjectures in number theory such as the Taniyama–Shimura conjecture, which includes Fermat's Last Theorem as a special case.^[10]

In spite of its importance in number theory, establishing the Langlands correspondence in the number theoretic context has proved extremely difficult.^[10] As a result, some mathematicians have worked on a related conjecture known as the geometric Langlands correspondence. This is a geometric reformulation of the classical Langlands correspondence which is obtained by replacing the number fields appearing in the original version by function fields and applying techniques from algebraic geometry.^[10]

In a paper from 2007, Anton Kapustin and Edward Witten suggested that the geometric Langlands correspondence can be viewed as a mathematical statement of Montonen–Olive duality.^[11] Starting with two Yang–Mills theories related by S-duality, Kapustin and Witten showed that one can construct a pair of quantum field theories in two-dimensional spacetime. By analyzing what this dimensional reduction does to certain physical objects called D-branes, they showed that one can recover the mathematical ingredients of the geometric Langlands correspondence.^[12] Their work shows that the Langlands correspondence is closely related to S-duality in quantum field theory, with possible applications in both subjects.^[10]

Seiberg duality

Another realization of S-duality in quantum field theory is Seiberg duality, first introduced by Nathan Seiberg around 1995.^[13] Unlike Montonen–Olive duality, which relates two versions of the maximally supersymmetric gauge theory in four-dimensional spacetime, Seiberg duality relates less symmetric theories called N=1 supersymmetric gauge theories. The two N=1 theories appearing in Seiberg duality are not identical, but they give rise to the same physics at large distances. Like Montonen–Olive duality, Seiberg duality generalizes the symmetry of Maxwell's equations that interchanges electric and magnetic fields.