Extremal Correlators and Random Matrix Theory

We study the correlation functions of Coulomb branch operators of four-dimensional $\mathcal{N} = 2$ Superconformal Field Theories (SCFTs). We focus on rank-one theories, such as the SU(2) gauge theory with four fundamental hypermultiplets."Extremal"correlation functions, involving exactly one anti-chiral operator, are perhaps the simplest nontrivial correlation functions in four-dimensional Quantum Field Theory. We show that the large charge limit of extremal correlators is captured by a"dual"description which is a chiral random matrix model of the Wishart-Laguerre type. This gives an analytic handle on the physics in some particular excited states. In the limit of large random matrices we find the physics of a non-relativistic axion-dilaton effective theory. The random matrix model also admits a 't Hooft expansion in which the matrix is taken to be large and simultaneously the coupling is taken to zero. This explains why the extremal correlators of SU(2) gauge theory obey a nontrivial double scaling limit in states of large charge. We give an exact solution for the first two orders in the 't Hooft expansion of the random matrix model and compare with expectations from effective field theory, previous weak coupling results, and we analyze the non-perturbative terms in the strong 't Hooft coupling limit. Finally, we apply the random matrix theory techniques to study extremal correlators in rank-1 Argyres-Douglas theories. We compare our results with effective field theory and with some available numerical bootstrap bounds.


Introduction
In this talk we analyze the large global-R charge expansion for a special set of correlation functions in 4D N = 2 SCFTs. Setup SU(2) SW theory with N f = 4 flavors: • Vector Multiplet: G = SU(2), ( 1 , 2 ) Adjoint fermions, complex-Adjoint scalar.
This field content implies that = 0 exactly. In particular, the theory has two marginal parameters (g 2 YM , ✓) which can be packaged into: In this model there is a global U(1) R symmetry acting on the following operators The dimension of O n is protected by supersymmetry with n labeling the U(1) R charge. These operators are special chiral operators also known as Coulomb branch operators satisfyinḡ In particular, they are both SU(2) R and Lorentz singlets.
The O n should be thought of as order parameters for the spontaneously broken conformal symmetry on the Coulomb branch moduli space of vacua.

SCFT
In order to compute their correlation function we choose a normalization in which: In particular, any correlation function consisting of purely chiral operators is automatically vanishing.
Conversely, if we insert a single anti-chiral operatorŌ n we get a non-vanishing result when m = P n k=1 i k .
Performing all the OPEs we are left with the following observable The G 2n are the simplest non-trivial correlation functions known in 4D interacting theories. They are commonly referred to as "extremal correlators".
For example, let's write down the explicit expression for G 2 An e cient algorithm for computing G 2n has been proposed in [GGIKKP]. In fact, if we define a matrix

Extremal Correlation Function
where Z S 4 [⌧,⌧ ] denotes the N = 2 four-sphere partition function computed by [Pestun] then The large-charge limit corresponds to large values of n. Computing G 2n becomes technically very challenging.
A surprising conjecture by [Bourget-R.Gomez-Russo; Beccaria] suggests that log G 2n could be organized as a familiar 't Hooft expansion log G 2n = n C 0 (g 2 n) + C 1 (g 2 n) + 1 n C 2 (g 2 n) + . . . , where C 0 = 2 log(g 2 n) and C 1 (g 2 n) is known up to some order in perturbative expansion.
Our goal is to establish a novel double scaling limit, which justify such genus expansion, where g 2 ! 0, n ! 1 with ⌘ g 2 n fixed.
The emergent dual description is given in terms of a matrix ensemble which is well known in the study of random matrix theory (RMT).
The basic idea is that we need to reformulate the problem in terms of an integral over matrices where W = A † A and A is a general complex matrix. The matrices W are called "Wishart Matrices".
This model should not be confused with the standard Hermitian ensemble.
In order to proceed, we need to specify the potential V (W ). ◆ .

Comments on the Potential
• The interaction is suppressed at leading order in the planar limit.
• It is also single trace, large-n expansion is thus well defined.
We can write down Y (x) quite explicitly as: where H(x) ⌘ G (1 + x)G (1 x).

Evaluatingthetlatrixlntegrol
In terms of eigenvalues a 2n = ÷ ! :* FI iii. gie a irxis Using the explicit expression we con show thot : Gym = detcmtnxcntts-Zmtzdetn.sn M Zn Using a Wishart matrix ensemble with potential V (W ) we are able to show that the extremal correlators admit a 't Hooft expansion: this is a strict large-charge limit where no double scaling limit is performed. Since is no longer fixed, we have Z [DW ] e 1 g 2 Tr(W )+ 1 2 Tr log W +Tr (log Y (W )) .
The eigenvalue distribution is given by the so called "Marchenko-Pastur distribution" . At large-n it goes like In the large-n limit, a typical eigenvalue of W is of order n. The integral over Wishart matrices can be further simplified to Z [DW ] e 1 g 2 Tr(W )+Tr log W .
The large charge EFT in this setup consists of a U(1) superfluid plus an axion-dilaton e↵ective action for the spontaneously broken conformal symmetry on the Coulomb branch.
The O(1/n) term can also be matched exactly by studying eigenvalues close to the lower edge of the distribution. When = g 2 n is fixed there is no suppression for the eigenvalues and we cannot ignore Tr log(Y (W )).
To compute C 1 ( ) we can use the following useful identity which is due to the correlation functions of single-trace operators being disconnected in the large-n limit.
The leading e↵ect of 1 n Tr log(Y (W )) can be captured by Hmmmm .

Back to the Double Scaling Limit
The complete expression for C 1 ( ) is given by: where the last term in this expression comes from the quadratic interaction term.
The integral C 1 ( ) can be solved exactly. At small-we have showing perfect agreement with the perturbative computation of [Bourget-R.Gomez-Russo; Beccaria].

Non-Perturbative E↵ects
The function F inst ( ) contains exponentially suppressed terms in .
These non-perturbative e↵ects should be thought of as wordline instanton contributions from hypermultiplets and W-bosons whose mass remains finite in the double-scaling limit.
• We have shown how extremal correlators at large charge are described by a dual matrix model.
• Such duality holds for other interesting examples of rank-1 N = 2 SCFTs.
• A new 't Hooft coupling emerges naturally in this description.
• The RMT approach allows us to confirm predictions of EFT and also to uncover new degrees of freedom.
• Our analysis suggests that perturbation theory for highly excited states should be organized as a genus expansion.
• The Wilson-Fisher O(2) model in d = 4 ✏ has anomalous dimension for the n operator given by