quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...

, Seiberg duality, conjectured by

Nathan Seiberg Nathan "Nati" Seiberg (; ; born September 22, 1956) is an Israeli American theoretical physicist who works on quantum field theory and string theory. He is currently a professor at the Institute for Advanced Study in Princeton, New Jersey, Unit ...

in 1994, is an

S-duality In theoretical physics, S-duality (short for strong–weak duality, or Sen duality) is an equivalence of two physical theories, which may be either quantum field theories or string theories. S-duality is useful for doing calculations in theore ...

relating two different supersymmetric QCDs. The two theories are not identical, but they agree at low energies. More precisely under a

renormalization Renormalization is a collection of techniques in quantum field theory, statistical field theory, and the theory of self-similar geometric structures, that is used to treat infinities arising in calculated quantities by altering values of the ...

group flow they flow to the same

IR fixed point In physics, an infrared fixed point is a set of coupling constants, or other parameters, that evolve from arbitrary initial values at very high energies (short distance) to fixed, stable values, usually predictable, at low energies (large distance ...

, and so are in the same

universality class In statistical mechanics, a universality class is a collection of mathematical models which share a single scale-invariant limit under the process of renormalization group flow. While the models within a class may differ dramatically at finite sc ...

. It is an extension to nonabelian

gauge theories In physics, a gauge theory is a type of field theory in which the Lagrangian, and hence the dynamics of the system itself, does not change under local transformations according to certain smooth families of operations (Lie groups). Formally, t ...

with N=1 supersymmetry of

Montonen–Olive duality Montonen–Olive duality or electric–magnetic duality is the oldest known example of strong–weak duality or S-duality according to current terminology. It generalizes the electro-magnetic symmetry of Maxwell's equations by stating that magn ...

in N=4 theories and electromagnetic duality in

abelian Abelian may refer to: Mathematics Group theory * Abelian group, a group in which the binary operation is commutative ** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms * Metabelian group, a group ...

theories.

The statement of Seiberg duality

Seiberg duality is an equivalence of the

s in an ''N''=1 theory with SU(N_c) as the

gauge group A gauge group is a group of gauge symmetries of the Yang–Mills gauge theory of principal connections on a principal bundle. Given a principal bundle P\to X with a structure Lie group G, a gauge group is defined to be a group of its vertical ...

and N_f

flavors Flavour or flavor is either the sensory perception of taste or smell, or a flavoring in food that produces such perception. Flavour or flavor may also refer to: Science * Flavors (programming language), an early object-oriented extension to L ...

of fundamental chiral multiplets and N_f flavors of

antifundamental In mathematics differential geometry, an antifundamental representation of a Lie group is the complex conjugate of the fundamental representation,. although the distinction between the fundamental and the antifundamental representation is a matter o ...

chiral multiplets in the chiral limit (no

bare mass In quantum field theory, specifically the theory of renormalization, the bare mass of an elementary particle is the limit of its mass as the scale of distance approaches zero or, equivalently, as the energy of a particle collision approaches infini ...

es) and an N=1 chiral QCD with N_f-N_c colors and N_f flavors, where N_c and N_f are positive integers satisfying ::

N_f>N_c+1

. A stronger version of the duality relates not only the chiral limit but also the full deformation space of the theory. In the special case in which :

N_f < N_c < N_f

the IR fixed point is a nontrivial interacting superconformal field theory. For a superconformal field theory, the

anomalous scaling dimension In theoretical physics, the scaling dimension, or simply dimension, of a local operator in a quantum field theory characterizes the rescaling properties of the operator under spacetime Dilation (affine geometry), dilations x\to \lambda x. If the q ...

of a chiral superfield

D=\frac R

where R is the R-charge. This is an exact result. The dual theory contains a fundamental "meson" chiral superfield M which is color neutral but transforms as a bifundamental under the flavor symmetries. The dual theory contains the

superpotential In theoretical physics, the superpotential is a function in supersymmetric quantum mechanics. Given a superpotential, two "partner potentials" are derived that can each serve as a potential in the Schrödinger equation. The partner potentials hav ...

W=\alpha M \tilde\tilde

Relations between the original and dual theories

Being an S-duality, Seiberg duality relates the strong coupling regime with the weak coupling regime, and interchanges chromoelectric fields (

gluon A gluon ( ) is a type of Massless particle, massless elementary particle that mediates the strong interaction between quarks, acting as the exchange particle for the interaction. Gluons are massless vector bosons, thereby having a Spin (physi ...

s) with chromomagnetic fields (gluons of the dual gauge group), and chromoelectric charges (

quark A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nucleus, atomic nuclei ...

s) with nonabelian

't Hooft–Polyakov monopole __NOTOC__ In theoretical physics, the t Hooft–Polyakov monopole is a topological soliton similar to the Dirac monopole but without the Dirac string. It arises in the case of a Yang–Mills theory with a gauge group G, coupled to a Higgs field ...

s. In particular, the

Higgs phase In theoretical physics, it is often important to consider gauge theory that admits many physical phenomena and "phases", connected by phase transitions, in which the vacuum may be found. Global symmetries in a gauge theory may be broken by the Hi ...

is dual to the

confinement Confinement may refer to: * With respect to humans: ** An old-fashioned or archaic synonym for childbirth ** Postpartum confinement (or postnatal confinement), a system of recovery after childbirth, involving rest and special foods ** Civil confi ...

phase as in the

dual superconducting model In the theory of quantum chromodynamics, dual superconductor models attempt to explain confinement of quarks in terms of an electromagnetic dual theory of superconductivity. Overview In an electromagnetic dual theory the roles of electric and ma ...

. The

meson In particle physics, a meson () is a type of hadronic subatomic particle composed of an equal number of quarks and antiquarks, usually one of each, bound together by the strong interaction. Because mesons are composed of quark subparticles, the ...

s and

baryon In particle physics, a baryon is a type of composite particle, composite subatomic particle that contains an odd number of valence quarks, conventionally three. proton, Protons and neutron, neutrons are examples of baryons; because baryons are ...

s are preserved by the duality. However, in the electric theory the meson is a quark bilinear (

M \equiv Q^c Q

), while in the magnetic theory it is a fundamental field. In both theories the baryons are constructed from quarks, but the number of quarks in one baryon is the rank of the gauge group, which differs in the two dual theories. The gauge symmetries of the theories do not agree, which is not problematic as the gauge symmetry is a feature of the formulation and not of the fundamental physics. The global symmetries relate distinct physical configurations, and so they need to agree in any dual description.

Evidence for Seiberg duality

The

moduli space In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme (mathematics), scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of suc ...

s of the dual theories are identical. The global symmetries agree, as do the charges of the mesons and baryons. In certain cases it reduces to ordinary electromagnetic duality. It may be embedded in

string theory In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera ...

via Hanany–Witten brane cartoons consisting of intersecting

D-brane In string theory, D-branes, short for Dirichlet membrane, are a class of extended objects upon which open strings can end with Dirichlet boundary conditions, after which they are named. D-branes are typically classified by their spatial dimensi ...

s. There it is realized as the motion of an

NS5-brane In string theory In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate throug ...

which is conjectured to preserve the universality class. Six nontrivial anomalies may be computed on both sides of the duality, and they agree as they must in accordance with

Gerard 't Hooft Gerardus "Gerard" 't Hooft (; born July 5, 1946) is a Dutch theoretical physicist and professor at Utrecht University, the Netherlands. He shared the 1999 Nobel Prize in Physics with his thesis advisor Martinus J. G. Veltman "for elucidating t ...

anomaly matching condition In quantum field theory, the anomaly matching condition by Gerard 't Hooft states that the calculation of any chiral anomaly for the flavor symmetry must not depend on what scale is chosen for the calculation if it is done by using the degrees of ...

s. The role of the additional fundamental meson superfield M in the dual theory is very crucial in matching the anomalies. The global gravitational anomalies also match up as the parity of the number of chiral fields is the same in both theories. The R-charge of the Weyl fermion in a chiral superfield is one less than the R-charge of the superfield. The R-charge of a gaugino is +1. Another evidence for Seiberg duality comes from identifying the superconformal index, which is a generalization of the

Witten index In quantum field theory and statistical mechanics, the Witten index at the inverse temperature β is defined as a modification of the standard partition function: :\textrm -1)^F e^/math> Note the (-1)F operator, where F is the fermion number o ...

, for the electric and the magnetic phase. The identification gives rise to complicated integral identities which have been studied in the mathematical literature.

Generalizations

Seiberg duality has been generalized in many directions. One generalization applies to quiver gauge theories, in which the flavor symmetries are also gauged. The simplest of these is a super QCD with the flavor group gauged and an additional term in the

. It leads to a series of Seiberg dualities known as a duality cascade, introduced by

Igor Klebanov Igor R. Klebanov (born 1962) is an American theoretical physicist. Since 1989, he has been a faculty member at Princeton University, where he is currently a Eugene Higgins Professor of Physics and the director of the Princeton Center for Theoret ...

and Matthew Strassler. Whether Seiberg duality exists in 3-dimensional nonabelian gauge theories with only 4 supercharges is not known, although it is conjectured in some special cases with Chern–Simons terms.

References

{{Reflist

The statement of Seiberg duality

Relations between the original and dual theories

Evidence for Seiberg duality

Generalizations

References

Further reading