S-duality
Encyclopedia
In theoretical physics
, S-duality (also a strong–weak duality) is an equivalence of two quantum field theories
or string theories
. An S-duality transformation maps states and vacua with coupling constant
in one theory to states and vacua with coupling constant 
in the dual theory. This has permitted the use of perturbation theory
, normally useful only for "weakly coupled" theories with
less than 1, to also describe the "strongly coupled" (
greater than 1) regimes of string theory, by mapping them onto dual, weakly coupled regimes.
In the case of four-dimensional quantum field theories, S-duality was understood by Ashoke Sen
, Nathan Seiberg
, and others. In this context, it usually exchanges the electric
and magnetic field
s (and the electrically charged particles with magnetic monopole
s).
See Montonen-Olive duality
, Seiberg duality
.
Many more examples come from string theory: S-duality relates type IIB string theory with the coupling constant
to the same type IIB string theory with the coupling constant 
. Similarly, type I string theory
with the coupling
is equivalent to the SO(32) heterotic string
theory with the coupling constant
. Perhaps most amazing are the S-dualities of type IIA string theory and E8 heterotic string theory with coupling constant 
to the higher dimensional M-theory with a compact dimension of size 
.
S-duality has been rigorously shown to hold in some lattice models
. It depends on the Pontryagin dual group.
In particular, in 2 dimensions, if the vertices can take on values in a locally compact abelian group
G and the action
/energy
only depends on the edges (e.g. the Ising model
for Z2, the Potts model
for Zn, the XY model
for U(1) ), then its dual via the Kramers-Wannier duality
to a model where the vertices take on values in the dual group G'.
In 3 dimensions, such a model would be dual to a lattice gauge model over the dual group G'.
In 4 dimensions, a lattice gauge model with G as the gauge group would be dual to a lattice gauge model with G' as the gauge group (with the electric and magnetic fields interchanged).
S-duality typically exchanges local charges with topological charges.
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, S-duality (also a strong–weak duality) is an equivalence of two quantum field theories
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
or string theories
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
. An S-duality transformation maps states and vacua with coupling constant
Coupling constant
In physics, a coupling constant, usually denoted g, is a number that determines the strength of an interaction. Usually the Lagrangian or the Hamiltonian of a system can be separated into a kinetic part and an interaction part...
in one theory to states and vacua with coupling constant in the dual theory. This has permitted the use of perturbation theoryPerturbation theory (quantum mechanics)
In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. The idea is to start with a simple system for which a mathematical solution is known, and add an...
, normally useful only for "weakly coupled" theories with
less than 1, to also describe the "strongly coupled" ( greater than 1) regimes of string theory, by mapping them onto dual, weakly coupled regimes.In the case of four-dimensional quantum field theories, S-duality was understood by Ashoke Sen
Ashoke Sen
Ashoke Sen , FRS, is an Indian theoretical physicist. He has made a number of major original contributions to the subject of string theory, including his landmark paper on strong-weak coupling duality or S-duality, which was influential in changing the course of research in the field...
, Nathan Seiberg
Nathan Seiberg
Nathan "Nati" Seiberg is an Israeli American theoretical physicist who works on string theory. He was recipient of a 1996 MacArthur Fellowship and the Dannie Heineman Prize for Mathematical Physics in 1998. He is currently a professor at the Institute for Advanced Study in Princeton, New Jersey, USA...
, and others. In this context, it usually exchanges the electric
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
and magnetic field
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
s (and the electrically charged particles with magnetic monopole
Magnetic monopole
A magnetic monopole is a hypothetical particle in particle physics that is a magnet with only one magnetic pole . In more technical terms, a magnetic monopole would have a net "magnetic charge". Modern interest in the concept stems from particle theories, notably the grand unified and superstring...
s).
See Montonen-Olive duality
Montonen-Olive duality
In theoretical physics, Montonen–Olive duality is the oldest known example of S-duality or a strong-weak duality. It generalizes the electro-magnetic symmetry of Maxwell's equations...
, Seiberg duality
Seiberg duality
In quantum field theory, Seiberg duality, conjectured by Nathan Seiberg, is an S-duality relating two different supersymmetric QCDs. The two theories are not identical, but they agree at low energies. More precisely under a renormalization group flow they flow to the same IR fixed point, and so...
.
Many more examples come from string theory: S-duality relates type IIB string theory with the coupling constant
to the same type IIB string theory with the coupling constant . Similarly, type I string theoryType I string theory
In theoretical physics, type I string theory is one of five consistent supersymmetric string theories in ten dimensions. It is the only one whose strings are unoriented and which contains not only closed strings, but also open strings.The classic 1976 work of Ferdinando Gliozzi, Joel Scherk and...
with the coupling
is equivalent to the SO(32) heterotic stringHeterotic string
In physics, a heterotic string is a peculiar mixture of the bosonic string and the superstring...
theory with the coupling constant
. Perhaps most amazing are the S-dualities of type IIA string theory and E8 heterotic string theory with coupling constant to the higher dimensional M-theory with a compact dimension of size .S-duality has been rigorously shown to hold in some lattice models
Lattice model (physics)
In physics, a lattice model is a physical model that is defined on a lattice, as opposed to the continuum of space or spacetime. Lattice models originally occurred in the context of condensed matter physics, where the atoms of a crystal automatically form a lattice. Currently, lattice models are...
. It depends on the Pontryagin dual group.
In particular, in 2 dimensions, if the vertices can take on values in a locally compact abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
G and the action
Action (physics)
In physics, action is an attribute of the dynamics of a physical system. It is a mathematical functional which takes the trajectory, also called path or history, of the system as its argument and has a real number as its result. Action has the dimension of energy × time, and its unit is...
/energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...
only depends on the edges (e.g. the Ising model
Ising model
The Ising model is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables called spins that can be in one of two states . The spins are arranged in a graph , and each spin interacts with its nearest neighbors...
for Z2, the Potts model
Potts model
In statistical mechanics, the Potts model, a generalization of the Ising model, is a model of interacting spins on a crystalline lattice. By studying the Potts model, one may gain insight into the behaviour of ferromagnets and certain other phenomena of solid state physics...
for Zn, the XY model
XY model
The classical XY model is a model of statistical mechanics. It is the special case of the n-vector model for n=2.-Definition:...
for U(1) ), then its dual via the Kramers-Wannier duality
Kramers-Wannier duality
The Kramers–Wannier duality is a symmetry in statistical physics. It relates the free energy of a two-dimensional square-lattice Ising model at a low temperature to that of another Ising model at a high temperature. It was discovered by Hendrik Kramers and Gregory Wannier in 1941...
to a model where the vertices take on values in the dual group G'.
In 3 dimensions, such a model would be dual to a lattice gauge model over the dual group G'.
In 4 dimensions, a lattice gauge model with G as the gauge group would be dual to a lattice gauge model with G' as the gauge group (with the electric and magnetic fields interchanged).
S-duality typically exchanges local charges with topological charges.
See also
- T-dualityT-dualityT-duality is a symmetry of quantum field theories with differing classical descriptions, of which the relationship between small and large distances in various string theories is a special case. Discussion of the subject originated in a paper by T. S. Buscher and was further developed by Martin...
- U-dualityU-dualityIn physics, U-duality is a symmetry of string theory or M-theory combining S-duality and T-duality transformations. The term is most often met in the context of the "U-duality group" of M-theory as defined on a particular background space . This is the union of all the S- and T-dualities...
- Jordan–Wigner transformation