# A Problem with fields and operators in holographic duality

1. Aug 22, 2016

### ShayanJ

I'm reading McGreevy's lecture notes on holographic duality but I have two problems now: (See here!)

1) The author considers a matrix field theory for large N expansion. At first I thought its just a theory considered as a simple example and has nothing to do with the $\mathcal N=4$ SYM SU(N) theory which is going to be analyzed later. But if you see the linked pdf, the author says the following just above equation (1):
Does this mean there is a connection between this matrix field theory and the SYM? What is this connection?
Also, only SU(N) gauge fields are NxN matrices and not the matter fields. So what is he talking about?

2) In section 3, he considers single-trace operators, defined by equation (4). But I have no idea what kind of an operator this is. The field/operator correspondence in AdS/CFT is supposed to be between quantities like e.g. bulk metric and boundary theory SEM tensor or bulk gauge fields and boundary theory global currents. But what is this single trace operator? Can anyone clarify and give an example?

Thanks

2. Aug 23, 2016

### ShayanJ

I found the answer to my questions. So I post it here for any future student wandering around in hope of finding the answer to the same questions:
The key point here is that the fields of $\mathcal N=4$ SYM SU(N) are a SU(N) gauge field and some matter fields which are superpartners to this SU(N) gauge field and so they are themselves matrices in the internal space but scalars and spinors under the Lorentz group. The author of the above document realizes that any term appearing in the theory's Lagrangian is a matrix product and so he can define a larger matrix containing all the above fields whose derivative and products give all the terms in the Lagrangian. He just doesn't do it explicitly and writes a schematic Lagrangian because that's all he needs.
The answer to the question 2 is now clear too. Aside from operators like SEM tensor and global currents, (matter) field products can be considered too but because (matter) fields here are also matrices, these product terms should actually be the trace of the field products.

Last edited: Aug 23, 2016