N=4 SYM: Extended Superspace, Coloumb Phase

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In summary, the conversation discusses the attempt to extend the usual N=1 superspace to a superspace that corresponds to extended (N=2, 4) supersymmetry. This extension leads to the appearance of additional bosonic coordinates, causing complications in the N=2 and N=4 cases. The use of "analytic superspace" allows for an "on-shell" formulation of N=4 SYM, where observables can be described but a path-integral cannot be written. There is also a discussion about the non-renormalization properties of the quantum potential for the scalar fields in N=4 SYM and the theory's behavior in the Coloumb phase.
  • #1
Squark
Hello everyone,

Two question on N=4 Super-Yang-Mills:

1) As far as I understand, the attempt to extend the usual N=1
superspace to a superspace that corresponds to extended (N=2, 4)
supersymmetry meets the difficulty that additional bosonic coordinates
appear, corresponding to central charges. This means we have an
infinite set of auxilary fields in the theory. This complication may in
certain cases be overcomed in the N=2 case but not in the N=4 case. In
fact, we do have the so-called "analytic superspace" which allows
bypassing the problem for N=4 but it only works on-shell.

A) What does it mean to have an "on-shell" formulation of the theory?
That we can describe observables, but we cannot write down a
path-integral? What is the "on-shell" formulation of N=4 SYM in
analytic superspace?

B) Nevertheless, can we write a path-integral for N=4 SYM in extended
superspace (with the infinite set of auxilary fields included)?

2) The quantum potential for the scalar fields in N=4 SYM has
non-renormalization properties due to supersymmetry. In particular,
there is a non-trivial moduli space of vacua (with dimension 6 times
the rank of the gauge group) corresponding to the vanishing of the
quantum potential. In a generic sector the gauge group is broken down
to a maximal Abelian subgroup (the so-called Coloumb phase). All the
fields of the theory are in the adjoint representation. In
hep-th/9908171, p. 47, the conclusion is drawn that in the Coloumb
phase theory is in fact _free_ (since the adjoint representation is
trivial for an Abelian group). However, what about the interactions
associated with massive gauge bosons?

Thx in advance for any help!Squark
 
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  • #2
I wrote:

> 2) The quantum potential for the scalar fields in N=4 SYM has
> non-renormalization properties due to supersymmetry. In particular,
> there is a non-trivial moduli space of vacua (with dimension 6 times
> the rank of the gauge group) corresponding to the vanishing of the
> quantum potential. In a generic sector the gauge group is broken down
> to a maximal Abelian subgroup (the so-called Coloumb phase). All the
> fields of the theory are in the adjoint representation. In
> hep-th/9908171, p. 47, the conclusion is drawn that in the Coloumb
> phase theory is in fact _free_ (since the adjoint representation is
> trivial for an Abelian group). However, what about the interactions
> associated with massive gauge bosons?


Maybe the author meant that the theory flows to a free one in the IR.
That would make sense.Squark
 
  • #3
I wrote:

> 2) The quantum potential for the scalar fields in N=4 SYM has
> non-renormalization properties due to supersymmetry. In particular,
> there is a non-trivial moduli space of vacua (with dimension 6 times
> the rank of the gauge group) corresponding to the vanishing of the
> quantum potential. In a generic sector the gauge group is broken down
> to a maximal Abelian subgroup (the so-called Coloumb phase). All the
> fields of the theory are in the adjoint representation. In
> hep-th/9908171, p. 47, the conclusion is drawn that in the Coloumb
> phase theory is in fact _free_ (since the adjoint representation is
> trivial for an Abelian group). However, what about the interactions
> associated with massive gauge bosons?


Maybe the author meant that the theory flows to a free one in the IR.
That would make sense.Squark
 

What is N=4 SYM?

N=4 SYM (Super Yang-Mills theory with N=4 supersymmetry) is a type of quantum field theory that describes the behavior of particles and their interactions. It is a supersymmetric extension of Yang-Mills theory, which is a classical gauge theory. N=4 SYM is a highly symmetric and mathematically elegant theory that has many applications in theoretical physics and has been extensively studied by scientists.

What is extended superspace in N=4 SYM?

In N=4 SYM, extended superspace refers to a mathematical framework that is used to describe the theory. It is an extension of ordinary spacetime that includes additional dimensions, known as supersymmetric coordinates, which are needed to account for the presence of supersymmetry. In this extended superspace, the equations of motion for the theory can be written in a more compact and elegant form.

What is the Coulomb phase in N=4 SYM?

The Coulomb phase in N=4 SYM refers to a particular phase of the theory where the gauge fields are described by the Coulomb branch of the moduli space. In this phase, the gauge fields have a purely electric field configuration and do not have any magnetic fields. This phase is characterized by a high degree of symmetry and is important for understanding the behavior of the theory at high energies.

What are the implications of N=4 SYM for string theory?

N=4 SYM has played a crucial role in the development of string theory, which is a theoretical framework that aims to unify all the forces of nature. The theory has a close relationship with string theory, and many important insights about string theory have been obtained by studying N=4 SYM. In particular, the AdS/CFT correspondence, which relates N=4 SYM to a specific type of string theory, has been extensively studied and has led to many important results.

What are some current research topics related to N=4 SYM?

There are many active research topics related to N=4 SYM, including the study of its mathematical properties, applications to string theory and quantum gravity, and connections to other areas of physics such as condensed matter physics. Some specific topics of interest include the study of integrability and dualities in the theory, higher loop calculations, and the behavior of the theory at high energies. Researchers are also exploring new techniques and approaches for solving problems in N=4 SYM, which may have implications for other areas of theoretical physics.

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