Supersymmetric vacua and Bethe ansatz

Abstract: An announcement of some results of a longer paper where the supersymmetric vacua of two dimensional N=2 susy gauge theories with matter are shown to be in one-to-one correspondence with the eigenstates of integrable spin chain Hamiltonians. The Heisenberg spin chain is mapped to the two dimensional U(N) theory with fundamental hypermultiplets, the XXZ spin chain is mapped to the analogous three dimensional super-Yang-Mills theory compactified on a circle, the XYZ spin chain and eight-vertex model are related to the four dimensional theory compactified on a torus. The correspondence extends to any spin group, representations, boundary conditions, and inhomogeneity, it includes Sinh-Gordon and non-linear Schroedinger models as well as the dynamical spin chains such as the Hubbard model. Compactifications of four dimensional N=2 theories on a two-sphere lead to the instanton-corrected Bethe equations. We propose a completely novel way for the Yangian, quantum affine, and elliptic algebras to act as a symmetry of a union of quantum field theories. This paper accompanies arXiv:0901.4748

• The paper establishes a correspondence between supersymmetric vacua and Bethe ansatz solutions, unifying gauge theory with integrable spin systems.
• It maps the effective twisted superpotential's critical points to Bethe roots, linking N = 2 and N = 4 theories to diverse spin chains.
• The findings imply that exploiting this duality can offer new methods for solving complex models in both high-energy physics and statistical mechanics.

Overview of Supersymmetric Vacua and Bethe Ansatz Correspondence

The paper by Nikita A. Nekrasov and Samson L. Shatashvili examines the intriguing correspondence between the supersymmetric vacua of certain gauge theories and the solutions to Bethe ansatz equations in the field of quantum integrable systems. The focus is on N = 2 and N = 4 supersymmetric gauge theories compactified across different dimensions and their link to integrable models like the Heisenberg SU(2) XXX spin chain. Their work proposes a deep connection between high-energy physics and condensed matter systems, specifically demonstrating that vacua of these gauge theories can be interpreted as eigenstates of integrable spin chain Hamiltonians.

Theoretical Developments and Models

The authors establish a correspondence by mapping the relation of these gauge theories to integrable spin systems. Through their work, the Heisenberg SU(2) XXX spin chain, XXZ spin chain, XYZ spin chain, and other dynamical spin systems are mapped to various dimensional supersymmetric gauge theories, such as two-dimensional U(N) theories with fundamental hypermultiplets and the three-dimensional super-Yang-Mills theory with compactification on a circle. Notably, the correspondence includes models like the Sinh-Gordon and Hubbard model, extending to any spin group, representation, and boundary conditions.

Key Results

In the paper, the vacua of these supersymmetric theories, when softly broken by twisted masses, are shown to correspond to solutions of integrable models' Bethe equations. For instance, the U(N) gauge theory maps to the Bethe equations of the quantum nonlinear Schrödinger equation, a model describing N non-relativistic particles on a circle interacting via delta functions. The strong claim here is the universality of this correspondence across different gauge theories and integrable systems.

Mathematical Framework

Central to this correspondence is the identification of the supersymmetric vacua equations with the Bethe ansatz equations. The paper describes the effective twisted superpotential $W_{\text{eff}}(\sigma)$, whose critical points determine Bethe roots equivalent to higher rank spin chains in quantum integrable systems. This function aligns with the Yang-Yang function $Y(\lambda)$ within quantum integrable systems, establishing precise mathematical linkage between completely distinct physical contexts.

Implications

This correspondence significantly enriches both fields: in physics, it provides new insights into the structure of supersymmetric gauge theories and their vacua, while in mathematical physics, it opens pathways for employing quantum field theoretical methods to solve models in statistical mechanics. Practically, this research presents potential applications in both high-energy and condensed-matter physics by offering a fresh perspective on the solvability of complex systems through dualities.

Future Developments

The study speculates on future developments, indicating that the algebraic structures related to supersymmetric vacua might consistently describe the symmetry relations for field theories when applied to string and gauge theory contexts, potentially leading to novel symmetries and integrable structures hitherto unexplored. The extensions include further examinations of higher-rank spin groups and non-Hermitian deformations, alongside the exploration of modular properties in the context of instanton corrections.

Overall, this work provides a substantial theoretical foundation for interpreting complex physical systems through an integrable systems framework, paving the way for future explorations and applications in diverse domains of physics.