# Do Gauge Theories (CFTs) Have Phase Transitions as the 't Hooft Coupling is Varied?

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With an eye toward AdS/CFT, I'm wondering if large $N$ CFTs have a (quantum) phase transition as the 't Hooft coupling is varied. To be more specific -- if I look at correlation functions of low-dimension, single trace operators as a function of $\lambda$, are they analytic, continuous, discontinuous in $\lambda$? How much do we know / any references? I'm already aware of 0811.3001.

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retagged Apr 19, 2014
I can't edit, but making stuff like 0811.3001 hyperlinks would be nice.

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I'm not sure the question is very well-posed, since most large-N theories aren't going to exist at multiple values of the 't Hooft coupling. (QCD is asymptotically free, for example, so it's always a "small-$\lambda$" theory in this sense.) ${\cal N}=4$ SYM is an example where the gauge coupling is an exactly marginal direction, so it really exists at all $\lambda$. That's the only setting where I think the question is really well-defined, and as far as I'm aware there is no quantum phase transition, just a smooth change as $\lambda$ varies.

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@MattReece Can you give some references which analyze the phase-transitions of $\cal{N}=4$ SYM? To start-off I have seen - http://arxiv.org/PS_cache/hep-th/pdf/0612/0612073v2.pdf, http://arxiv.org/PS_cache/hep-th/pdf/0303/0303207v1.pdf I would be glad to see something more expository.

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The example i know does not use 't Hooft's coupling, but i think it may address your question in a more straightforward way (i'll be "loose" with constants and numerical pre-factors, but will keep all the relevant info and details).

Think of a 0-dimensional scalar field (bosonic $D0$-brane) with a quartic potential à la $V(\phi) = \mu\,\phi^2 + \lambda\,\phi^4$. And, if you allow me some poetic license with polynomials, let me rewrite this potential in the following form: $V_g(\phi) = \phi^2 + g\,\phi^4$, where $g=\lambda/\mu$, and it should be clear that "large $g$" means "strong coupling", while "small $g$" means "weak coupling".

The Partition Function (Feynman Path Integral) for this function is given by,

$$\mathcal{Z}_g[J] = \int e^{i\, S_g(\phi)}\,e^{i\,J\,\phi}\,\mathrm{d}\phi < \infty\; ;$$ where $S_g[\phi] = V_g(\phi)$ is the Action of the system (where i'm keeping the coupling constant explicit), and the requirement is that the integral converges (so the Partition Function is "well defined" in some sense).

But, there's a differential version of the above, called the Schwinger-Dyson equation, given by,

$$\frac{\partial S_g(\phi)}{\partial\phi} = 0 \;\Longleftrightarrow\; \mu\,\phi + \lambda\,\phi^3 = 0 \;\Longleftrightarrow\; \phi\,(1 + g\,\phi^2) = 0\;;$$ remembering that $\phi\mapsto\partial_J=\partial/\partial J$, giving us the following:

$$(\partial_J + g\,\partial_J^3)\,\mathcal{Z}(J) = J\;.$$

From this differential equation you clearly know one thing: there are 3 solutions to the above problem. This implies that each solution admits an integral representation, which is the Partition Function associated with that particular solution.

In fact, each $\mathcal{Z}_g(J)$ is defined within a certain Stokes' wedge, meaning that when you cross a Stokes' line you pick up some non-trivial constant contribution (very much à la wall-crossing phenomena).

Moreover, you can write the solutions to the above in terms of [confluent] hypergeometric functions (or, if you will, in terms of Meijer's G-function, or Fox's H-function) and distinguish its PolyLogarithm contribution, which is related to its singularity (pole) structure, and can be a big deal when talking about perturbation theory.

Anyway, this is just a "crash-core-dump" version of what i believe attacks your question; the point being that $g$ labels your "quantum phases". Note that it's possible to extend the field from scalar-valued to vector-, matrix-, tensor-, and Lie Algebra-valued: all the results mentioned go through with minor modifications (such as tracing appropriate variables, etc).

Hope this helps.

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answered Sep 21, 2011 by (685 points)
Just to be clear: When you say "each $Z_g(J)$ is defined within a certain Stokes' wedge", do you mean in the source $J$ or in the (complexified) coupling $g$? Are the multiple solutions connected via $g$ alone, or is it also necessary to change the boundary conditions?

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