[Paper Review] OPE of Wilson-'t Hooft operators in N=4 and N=2 SYM with gauge group G=PSU(3)

Ruxandra Moraru, Natalia Saulina|arXiv (Cornell University)|Jun 28, 2012

Black Holes and Theoretical Physics20 references7 citations

TL;DR

This paper computes the first non-trivial Operator Product Expansion (OPE) of Wilson-'t Hooft loop operators in ${\mathcal{N}}=4$ and ${\mathcal{N}}=2$ Super-Yang-Mills theories with gauge group $G=PSU(3)$, using holomorphic-topological twist and BPS state moduli spaces. The OPE structure is determined by computing holomorphic Euler characteristics of vector bundles over compact and non-compact moduli spaces, yielding explicit decomposition into electric and magnetic representations with signs derived from cohomological data.

ABSTRACT

We compute the simplest non-trivial Operator Product Expansion of Wilson-'t Hooft loop operators in N=4 and N=2 Super-Yang-Mills theory with gauge group G=PSU(3). This amounts to finding the Euler characters of certain vector bundles, describing electric degrees of freedom of loop operators entering the OPE, over moduli spaces of BPS states in the presence of loop operators.

Motivation & Objective

• To compute the first non-trivial OPE of Wilson-'t Hooft operators in ${\mathcal{N}}=4$ and ${\mathcal{N}}=2$ SYM with $G=PSU(3)$, extending prior results for $G=SU(2)$.
• To determine the spectrum of electric and magnetic states contributing to the OPE by computing holomorphic Euler characteristics of vector bundles over moduli spaces of BPS states.
• To verify the consistency of the OPE structure with S-duality and representation theory, particularly the role of the Langlands dual group ${{}^{L}G}=SU(3)$.
• To provide a geometric and cohomological framework for computing OPEs in supersymmetric gauge theories with non-Abelian gauge groups.

Proposed method

• Utilizes the holomorphic-topological twist of ${\mathcal{N}}=2$ gauge theory to reduce the problem to a supersymmetric quantum mechanics on a Riemann surface.
• Computes the holomorphic Euler characteristics $\mathbf{I}_{{\mathcal{N}}=4}\bigl{(}{\mathcal{M}},{\mathcal{V}}_{a,b}\bigr{)}$ and $\mathbf{I}_{{\mathcal{N}}=2}\bigl{(}{\mathcal{M}},{\mathcal{V}}_{a,b}\bigr{)}$ on compact moduli space ${\mathcal{M}}$, which encodes ground states of the quantum mechanics.
• Analyzes the non-compact 'bulk' part $X$ of the moduli space via $\mathbf{I}_{{\mathcal{N}}=4}\bigl{(}X,{\mathcal{V}}^{bulk}_{a,b}\bigr{)}$ and $\mathbf{I}_{{\mathcal{N}}=2}\bigl{(}X,{\mathcal{V}}^{bulk}_{a,b}\bigr{)}$, using $L^2$ Dolbeault cohomology on singular spaces.
• Employs the connection between BPS configurations and solutions to 3d Bogomolny equations with magnetic sources to define the moduli spaces ${\mathcal{M}}$ and $X$.
• Applies $L^2$ Dolbeault cohomology techniques to compute cohomology groups $H^{j}_{\overline{D},L^2}(X, {\mathcal{O}}_X(n))$ and checks for square-integrable harmonic forms to determine physical states.
• Uses explicit ansatz for differential forms and checks norm convergence to rule out spurious states, especially in $H^2_{\overline{D},L^2}(X, {\mathcal{O}}_X(-3))$ and $H^2_{\overline{D},L^2}(X, {\mathcal{O}}_X(3))$.

Experimental results

Research questions

• RQ1 What is the structure of the OPE of Wilson-'t Hooft operators in ${\mathcal{N}}=4$ SYM with $G=PSU(3)$, and how does it reflect S-duality?
• RQ2 How do the electric and magnetic quantum numbers of the intermediate states in the OPE arise from the geometry of BPS moduli spaces?
• RQ3 What is the role of the compact moduli space ${\mathcal{M}}$ and its non-compact 'bulk' part $X$ in computing the OPE coefficients?
• RQ4 Why do certain irreducible representations fail to appear in $L^2$ cohomology groups, and how does this affect the physical spectrum?
• RQ5 How can $L^2$ Dolbeault cohomology on singular spaces be computed consistently to extract physical states?

Key findings

• The OPE of two Wilson-'t Hooft operators with magnetic charge $w_1$ and electric charges $\nu = aw_1 + bw_2$ ($a+2b \equiv 0 \mod 3$) decomposes as $WT_{w_1,\nu} \times WT_{w_1,0} = WT_{2w_1,\nu} + \sum_j (-)^{s_j} WT_{w_2,\nu_j}$, with explicit determination of $\nu_j$ and signs $(-)^{s_j}$ for specific $a,b$.
• The coefficient $\mathbf{I}_{{\mathcal{N}}=4}\bigl{(}{\mathcal{M}},{\mathcal{V}}_{a,b}\bigr{)}$ is computed via $R^p\pi_*\Omega^q_{{\mathcal{M}}}$ and $H^j({\mathcal{M}}, \Omega^p \otimes {\mathcal{V}})$, with results matching expected representation-theoretic structures.
• The $L^2$ Dolbeault cohomology $H^2_{\overline{D},L^2}(X, {\mathcal{O}}_X(-3))$ contains no $\mathbb{V}_{(0,0)}$ representation, as confirmed by divergent norm behavior in both $s \to 0$ and $s \to \infty$ limits.
• Similarly, $H^2_{\overline{D},L^2}(X, {\mathcal{O}}_X(3))$ contains no $\mathbb{V}_{(3,0)}$, $\mathbb{V}_{(1,1)}$, or $\mathbb{V}_{(0,0)}$ representations due to lack of square-integrable solutions.
• The computation of $\mathbf{I}_{{\mathcal{N}}=4}\bigl{(}X, {\mathcal{V}}^{bulk}_{a,b}\bigr{)}$ relies on $L^2$ cohomology on $T\mathbb{P}^2$, with explicit formulae for $L^2$ cohomology of $\Omega^j_X(3m)$ for $m \geq 1$.
• Norm integrals are evaluated using polar coordinates and standard trigonometric integrals, with key formulae for $\int d\Phi / (a + b\cos\Phi)^n$ and radial integrals over $T_v$, ensuring convergence checks for physical states.