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''N'' = 4 supersymmetric Yang–Mills (SYM) theory is a relativistic conformally invariant Lagrangian
gauge theory 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 ...
describing the interactions of
fermion In particle physics, a fermion is a subatomic particle that follows Fermi–Dirac statistics. Fermions have a half-integer spin (spin 1/2, spin , Spin (physics)#Higher spins, spin , etc.) and obey the Pauli exclusion principle. These particles i ...
s via gauge field exchanges. In ''D''=4
spacetime In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...
dimensions, ''N''=4 is the maximal number of supersymmetries or supersymmetry charges. SYM theory is a toy theory based on
Yang–Mills theory Yang–Mills theory is a quantum field theory for nuclear binding devised by Chen Ning Yang and Robert Mills in 1953, as well as a generic term for the class of similar theories. The Yang–Mills theory is a gauge theory based on a special un ...
; it does not model the real world, but it is useful because it can act as a proving ground for approaches for attacking problems in more complex theories. It describes a universe containing boson fields and fermion fields which are related by four supersymmetries (this means that transforming bosonic and fermionic fields in a certain way leaves the theory invariant). 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 Superspace is the coordinate space of a theory exhibiting supersymmetry. In such a formulation, along with ordinary space dimensions ''x'', ''y'', ''z'', ..., there are also "anticommuting" dimensions whose coordinates are labeled in Grassmann num ...
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. This is also known as the super-ambitwistor correspondence. A similar super-ambitwistor characterization holds for ''D''=10, ''N''=1 dimensional super Yang–Mills theory, and the lower dimensional cases ''D''=6, ''N''=2 and ''D''=4, ''N''=4 may be derived from this via dimensional reduction.


Meaning of ''N'' and numbers of fields

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."''N'' = 4: Maximal Particles for Maximal Fun", from ''4 gravitons'' blog (2013)
/ref> In an analogy with symmetries under rotations, ''N'' would be the number of independent rotations, ''N'' = 1 in a plane, ''N'' = 2 in 3D space, etc... That is, in a ''N'' = 4 SYM theory, the gauge boson can be "rotated" into ''N'' = 4 different supersymmetric fermion partners. 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. Because in 3D space one may use different rotations to reach a same point (or here the same spin-0 boson), each spin-0 boson is superpartners of two different spin-1/2 fermions, not just one. So in total, one has only 6 spin-0 bosons, not 16. Therefore, ''N'' = 4 SYM has 1 + 4 + 6 = 11 fields, namely: one vector field (the spin-1 gauge boson), four spinor fields (the spin-1/2 fermions) and six scalar fields (the spin-0 bosons). ''N'' = 4 is the maximum number of independent supersymmetries: starting from a spin-1 field and using more supersymmetries, e.g., ''N'' = 5, only rotates between the 11 fields. To have ''N'' > 4 independent supersymmetries, one needs to start from a gauge field of spin higher than 1, e.g., a spin-2
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...
field such as that of the
graviton In theories of quantum gravity, the graviton is the hypothetical elementary particle that mediates the force of gravitational interaction. There is no complete quantum field theory of gravitons due to an outstanding mathematical problem with re ...
. This is the ''N'' = 8 supergravity theory.


Lagrangian

The Lagrangian for the theory is : L = \operatorname \left\, where g and \theta_I are coupling constants (specifically g is the gauge coupling and \theta_I is the instanton angle), the
field strength In physics, field strength refers to a value in a vector-valued field (e.g., in volts per meter, V/m, for an electric field ''E''). For example, an electromagnetic field has both electric field strength and magnetic field strength. Field str ...
is F^k_ = \partial_\mu A^k_\nu-\partial_\nu A^k_\mu+f^A^l_\mu A^m_\nu with A^k_\nu the gauge field and indices ''i'',''j'' = 1, ..., 6 as well as ''a'', ''b'' = 1, ..., 4, and f represents the structure constants of the particular gauge group. The \lambda^a are left Weyl fermions, \sigma^\mu are the
Pauli matrices In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices that are traceless, Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () ...
, D_\mu is the gauge covariant derivative, X^i are real scalars, and C_i^ represents the structure constants of the R-symmetry group SU(4), which rotates the four supersymmetries. As a consequence of the nonrenormalization theorems, this supersymmetric field theory is in fact a superconformal field theory.


Ten-dimensional Lagrangian

The above Lagrangian can be found by beginning with the simpler ten-dimensional Lagrangian : L = \operatorname \left\, where I and J are now run from 0 through 9 and \Gamma^I are the 32 by 32 gamma matrices ( 32=2^ ), followed by adding the term with \theta_I which is a topological term. The components A_i 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 ''T''6, all the supercharges are preserved, giving ''N'' = 4 in the 4-dimensional theory. A Type IIB string theory interpretation of the theory is the worldvolume theory of a stack of D3-branes.


S-duality

The coupling constants \theta_I and g naturally pair together into a single coupling constant : \tau := \frac+\frac. The theory has symmetries that shift \tau by integers. The S-duality conjecture says there is also a symmetry which sends \tau \mapsto \frac as well as switching the group G to its Langlands dual group.


AdS/CFT correspondence

This theory is also important 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 A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
) and ''N'' = 4 super Yang–Mills on the 4-dimensional boundary of AdS5. However, this particular realization of the AdS/CFT correspondence is not a realistic model of gravity, since gravity in our universe is 4-dimensional. Despite this, the AdS/CFT correspondence is the most successful realization of the holographic principle, a speculative idea about
quantum gravity Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics. It deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the v ...
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.


Integrability

There is evidence that '' N'' = 4 supersymmetric Yang–Mills theory has an integrable structure in the planar large ''N'' limit (see below for what "planar" means in the present context). As the number of colors (also denoted ''N'') goes to infinity, the amplitudes scale like N^, so that only the genus 0 (planar graph) contribution survives. Planar
Feynman diagrams In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduced ...
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. A non-planar graph has a smaller number of possible gauge loops compared to a similar planar graph. Non-planar graphs are thus suppressed by factors 1/N^ compared to planar ones which therefore dominate in the large ''N'' limit. Consequently, a planar Yang–Mills theory denotes a theory in the large ''N'' limit, with ''N'' usually the number of
color Color (or colour in English in the Commonwealth of Nations, Commonwealth English; American and British English spelling differences#-our, -or, see spelling differences) is the visual perception based on the electromagnetic spectrum. Though co ...
s. Likewise, a planar limit is a limit in which scattering amplitudes are dominated by
Feynman diagrams In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduced ...
which can be given the structure of planar graphs.planar limit in nLab
/ref> In the large ''N'' limit, the coupling g 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. 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 \mathfrak(2) for ordinary spin. 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 In mathematics, the Grassmannian \mathbf_k(V) (named in honour of Hermann Grassmann) is a differentiable manifold that parameterizes the set of all k-dimension (vector space), dimensional linear subspaces of an n-dimensional vector space V over a ...
.


Relation to 11-dimensional M-theory

''N'' = 4 super Yang–Mills can be derived from a simpler 10-dimensional theory, and yet supergravity and
M-theory In physics, M-theory is a theory that unifies all Consistency, consistent versions of superstring theory. Edward Witten first conjectured the existence of such a theory at a string theory conference at the University of Southern California in 1 ...
exist in 11 dimensions. The connection is that if the gauge group U(''N'') of SYM becomes infinite as N\rightarrow \infty it becomes equivalent to an 11-dimensional theory known as matrix theory.


See also

* 4D N = 1 global supersymmetry * 6D (2,0) superconformal field theory * Extended supersymmetry * N = 1 supersymmetric Yang–Mills theory * N = 8 supergravity * Seiberg–Witten theory


References


Citations


Sources

* {{DEFAULTSORT:N = 4 supersymmetric Yang-Mills theory Supersymmetric quantum field theory Conformal field theory