S-duality
[1] In quantum field theory, S-duality generalizes a well established fact from classical electrodynamics, namely the invariance of Maxwell's equations under the interchange of electric and magnetic fields.
Recent work of Anton Kapustin and Edward Witten suggests that Montonen–Olive duality is closely related to a research program in mathematics called the geometric Langlands program.
For example, the strength of gravity is described by a number called Newton's constant, which appears in Newton's law of gravity and also in the equations of Albert Einstein's general theory of relativity.
Similarly, the strength of the electromagnetic force is described by a coupling constant, which is related to the charge carried by a single proton.
In perturbation theory, quantities called probability amplitudes, which determine the probability for various physical processes to occur, are expressed as sums of infinitely many terms, where each term is proportional to a power of the coupling constant
: In order for such an expression to make sense, the coupling constant must be less than 1 so that the higher powers of
S-duality is a particular example of a general notion of duality in physics.
The term duality refers to a situation where two seemingly different physical systems turn out to be equivalent in a nontrivial way.
Working in the language of vector calculus and assuming that no electric charges or currents are present, these equations can be written[3] Here
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
This situation is the most basic manifestation of S-duality in a field 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.
[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.
denotes the Langlands dual group which is in general different from
[8] An important quantity in quantum field theory is complexified coupling constant.
is the theta angle, a quantity appearing in the Lagrangian that defines the theory,[9] and
[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] 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.
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.
The two N=1 theories appearing in Seiberg duality are not identical, but they give rise to the same physics at large distances.
In the mid 1990s, physicists noticed that these five string theories are actually related by highly nontrivial dualities.
The existence of S-duality in string theory was first proposed by Ashoke Sen in 1994.
[14][failed verification] It was shown that type IIB string theory with the coupling constant
is equivalent via S-duality to the same string theory with the coupling constant
[15] Witten's proposal was based on the observation that type IIA and E8×E8 heterotic string theories are closely related to a gravitational theory called eleven-dimensional supergravity.
His announcement led to a flurry of work now known as the second superstring revolution.
