# Argyres-Douglas CFT

Adding of mass in supersymmetric gauge theories will affect structure of moduli space by creating new singular point (picture and some statements from Matteo Bertolini: Lectures on Supersymmetry): Novel phenomenon which can occur when the gauge group rank n > 1 and/or when matter is added: the existence of special points on the moduli space, known as Argyres-Douglas points, where the theory enjoys an interacting (as opposed to free) conformal phase (this CFT haven't Lagrangian description!).

Physically, this corresponds to mutually non-local objects (see this), as e.g. a dyon and a monopole, or a dyon and an electrically charged object, becoming simultaneously massless.

I have some questions, related Argyres-Douglas theory, that describe such interacting CFT.

1) Why Argyres-Douglas points correspondence to CFT? Is this related to mutually non-locality of of emergent massless objects? How we lose all dimension parameters in such points?

2) Which RG flow have this points as fixed?

3) Also I here that this theory is in strong coupling, and so haven't Lagrangian description. How to understand, that this theories in strong coupling? What can we now say about this theories??

I will be very appreciate for answers!

• Your questions are very broad, and the answers to them will also be very broad. You say that you need "details" in response to my answer, but what exactly do you need details about? Please make your questions sharp enough so they can have good answers. – Bruce Lee Apr 1 '20 at 14:04
• @BruceLee, For example, "Why Argyres-Douglas points correspondence to CFT?". It is very concrete question. About bootstrap: "What is operator spectrum?" – Nikita Apr 1 '20 at 14:38

The $$SU(N)$$ gauge theory with $$\mathcal{N} = 2$$ SUSY and $$F$$ hypermultiplets in fundamental representation has $$\beta$$ function

$$\beta(g) = \frac{g^3}{16\pi^2} (F-2N)$$

One trivial set of fixed points which can be seen is $$F=2N$$, which doesn't depend on the coupling.

Now consider that there are two species of massless particles, one is electrically charged and one is magnetically charged at the same point in the moduli space of the $$\mathcal{N} =2$$ gauge theory. The renormalization of the coupling from the electric charge drives the IR coupling to zero, while renormalization from the magnetic charge drives the IR coupling to $$\infty$$. For an appropriate set of charges, it was first pointed out by Argyres and Douglas for $$SU(3)$$ gauge theories (and later others for SU(2) etc.) that the IR coupling flows to an IR fixed point. Since this is a fixed point with $$\beta =0$$, its a CFT. For the specifics of the RG flow, have a look at the linked paper.

Since this is a CFT, one can understand features of this theory even without a Lagrangian description, using the CFT data (scaling dimensions and the OPE coefficients) (See Conformal Bootstrap). You start with the CFT data, and can "solve" the CFT as you can calculate all possible correlation functions.

• Thank you for your answer! But I wanna more concrete answer. How describe two species of massless particles? Why magnetic charge in IR goes to infinity? What information can one extract from conformal bootstrap for AD theories? – Nikita Mar 31 '20 at 14:06
• @Nikita I edited my answer a bit. For a specific example of how this works, see Section 3 of the linked paper, in particular equations 6.3-6.7. Also I think it is a good exercise to work out why the renormalization of the coupling due to the magnetic charge drives it in IR to $\infty$. Regarding the bootstrap, it is one of the powerful ways to work with CFTs, and doesn't require a Lagrangian description, so I thought it is worth mentioning. – Bruce Lee Apr 1 '20 at 8:40
• I know this information about conformal bootstrap and it's general philosophy:))) , but I don't know final result and how powerful is this method in AD. My question about AD, not about bootstrap. Please, if you know, clarify this more concrete details. – Nikita Apr 1 '20 at 11:32
• I am sorry, but your answer is fully not informative, and don't contain anything, that answer my questions. It is only general philosophy, but I wanna concrete statements and details. – Nikita Apr 1 '20 at 11:36
• @Nikita I understand that, but your questions are themselves very vaguely posed. You say that you need "details", but what exactly do you need details about? – Bruce Lee Apr 1 '20 at 14:00