N = 4 supersymmetric Yang–Mills (SYM) theory is a relativistic conformally invariant Lagrangian gauge theory describing the interactions of fermions via gauge field exchanges.
It is one of the simplest (in the sense that it has no free parameters except for the gauge group) and one of the few ultraviolet finite quantum field theories in 4 dimensions.
It can be thought of as the most symmetric field theory that does not involve gravity.
Like all supersymmetric field theories, SYM theory may equivalently be formulated as a superfield theory on an extended superspace in which the spacetime variables are augmented by a number of Grassmann variables which, for the case N=4, consist of 4 Dirac spinors, making a total of 16 independent anticommuting generators for the extended ring of superfunctions.
The field equations are equivalent to the geometric condition that the supercurvature 2-form vanish identically on all super null lines.
In N supersymmetric Yang–Mills theory, N denotes the number of independent supersymmetric operations that transform the spin-1 gauge field into spin-1/2 fermionic fields.
In turns, each fermion can be rotated into four different bosons: one corresponds to the rotation back to the spin-1 gauge field, and the three others are spin-0 boson fields.
represents the structure constants of the particular gauge group.
represents the structure constants of the R-symmetry group SU(4), which rotates the four supersymmetries.
of the gauge field for i = 4 to 9 become scalars upon eliminating the extra dimensions.
This also gives an interpretation of the SO(6) R-symmetry as rotations in the extra compact dimensions.
By compactification on a T6, all the supercharges are preserved, giving N = 4 in the 4-dimensional theory.
naturally pair together into a single coupling constant The theory has symmetries that shift
This theory is also important[1] in the context of the holographic principle.
There is a duality between Type IIB string theory on AdS5 × S5 space (a product of 5-dimensional AdS space with a 5-dimensional sphere) and N = 4 super Yang–Mills on the 4-dimensional boundary of AdS5.
Despite this, the AdS/CFT correspondence is the most successful realization of the holographic principle, a speculative idea about quantum gravity originally proposed by Gerard 't Hooft, who was expanding on work on black hole thermodynamics, and was improved and promoted in the context of string theory by Leonard Susskind.
[9] As the number of colors (also denoted N) goes to infinity, the amplitudes scale like
, so that only the genus 0 (planar graph) contribution survives.
Planar Feynman diagrams are graphs in which no propagator cross over another one, in contrast to non-planar Feynman graphs where one or more propagator goes over another one.
compared to planar ones which therefore dominate in the large N limit.
Likewise, a planar limit is a limit in which scattering amplitudes are dominated by Feynman diagrams which can be given the structure of planar graphs.
vanishes and a perturbative formalism is therefore well-suited for large N calculations.
Therefore, planar graphs are associated to the domain where perturbative calculations converge well.
Beisert et al. [12] give a review article demonstrating how in this situation local operators can be expressed via certain states in spin chains (in particular the Heisenberg spin chain), but based on a larger Lie superalgebra rather than
These spin chains are integrable in the sense that they can be solved by the Bethe ansatz method.
They also construct an action of the associated Yangian on scattering amplitudes.
Nima Arkani-Hamed et al. have also researched this subject.
Using twistor theory, they find a description (the amplituhedron formalism) in terms of the positive Grassmannian.
[13] N = 4 super Yang–Mills can be derived from a simpler 10-dimensional theory, and yet supergravity and M-theory exist in 11 dimensions.
The connection is that if the gauge group U(N) of SYM becomes infinite as