[Paper Review] Exact results in a N=2 superconformal gauge theory at strong coupling

Matteo Beccaria, M. Billó+3 more|arXiv (Cornell University)|May 31, 2021

Black Holes and Theoretical Physics58 references8 citations

TL;DR

This paper derives exact analytical expressions for a class of observables in an N=2 superconformal SU(N) gauge theory with symmetric and anti-symmetric matter via a full Lie algebra approach to the matrix model obtained through localization. It shows that these observables are encoded in an infinite matrix depending on the 't Hooft coupling λ, enabling high-order perturbative expansions and exact strong-coupling behavior, validated by Monte Carlo simulations and Padé resummation.

ABSTRACT

We consider the $\mathcal{N}=2$ SYM theory with gauge group SU($N$) and a matter content consisting of one multiplet in the symmetric and one in the anti-symmetric representation. This conformal theory admits a large-$N$ 't Hooft expansion and is dual to a particular orientifold of $AdS_{5} imes S^{5}$. We analyze this gauge theory relying on the matrix model provided by localization a la Pestun. Even though this matrix model has very non-trivial interactions, by exploiting the full Lie algebra approach to the matrix integration, we show that a large class of observables can be expressed in a closed form in terms of an infinite matrix depending on the 't Hooft coupling $λ$. These exact expressions can be used to generate the perturbative expansions at high orders in a very efficient way, and also to study analytically the leading behavior at strong coupling. We successfully compare these predictions to a direct Monte Carlo numerical evaluation of the matrix integral and to the Padé resummations derived from very long perturbative series, that turn out to be extremely stable beyond the convergence disk $|λ|

Motivation & Objective

• To derive exact analytical expressions for observables in an N=2 superconformal gauge theory with SU(N) gauge group and symmetric/anti-symmetric matter content.
• To develop a method based on the full Lie algebra approach to handle the non-trivial interactions in the localized matrix model, avoiding the Cartan subalgebra approximation.
• To compute high-order perturbative expansions efficiently and extract exact strong-coupling behavior from the matrix model.
• To validate analytical predictions against direct Monte Carlo simulations and Padé resummations of long perturbative series.

Proposed method

• Utilization of the matrix model derived via Pestun's localization for N=2 superconformal theories on S^4, which reduces the path integral to a finite-dimensional matrix integral.
• Application of the full Lie algebra approach to perform matrix integration without reducing to eigenvalue integrals, preserving non-Abelian structure.
• Derivation of an infinite matrix representation of observables in terms of the 't Hooft coupling λ, enabling exact closed-form expressions.
• Use of recursion relations and determinant identities for infinite matrices to compute sub-matrix determinants and inverse elements.
• Explicit computation of matrix elements of the inverse matrix Y⁻¹ and S⁻¹ using cofactor expansions and block-triangular decomposition.
• Validation of analytical results through comparison with Monte Carlo numerical evaluations and Padé resummation of high-order perturbative series.

Experimental results

Research questions

• RQ1 Can exact strong-coupling behavior be extracted from a strongly coupled N=2 superconformal gauge theory with non-trivial matter content?
• RQ2 Can the full Lie algebra approach yield closed-form expressions for observables in a matrix model with complex interactions?
• RQ3 How do analytical predictions from the infinite matrix model compare with numerical evaluations and Padé resummations of perturbative series?
• RQ4 What is the leading strong-coupling scaling behavior of BPS observables in this N=2 theory?
• RQ5 Can high-order perturbative expansions be generated efficiently from exact matrix model expressions?

Key findings

• The paper derives exact expressions for a large class of observables in terms of an infinite matrix depending on the 't Hooft coupling λ, enabling analytical study of strong-coupling behavior.
• The matrix model's structure allows for efficient generation of high-order perturbative expansions beyond the convergence radius |λ| < π².
• The strong-coupling behavior of twisted and untwisted observables is analytically computed and shown to scale with powers of λ, consistent with holographic expectations.
• The analytical results for the inverse matrix elements are derived explicitly as S⁻¹_{k,ℓ} = 2√[(2k+1)(2ℓ+1)] × min(k(k+1), ℓ(ℓ+1)) for k ≤ ℓ and k ≥ ℓ respectively.
• The derived expressions are validated by direct Monte Carlo simulations of the matrix integral, showing excellent agreement.
• Padé resummations of very long perturbative series confirm the stability and accuracy of the analytical predictions beyond the convergence disk.