Quantization of Integrable Systems and Four Dimensional Gauge Theories (0908.4052v1)

Abstract: We study four dimensional N=2 supersymmetric gauge theory in the Omega-background with the two dimensional N=2 super-Poincare invariance. We explain how this gauge theory provides the quantization of the classical integrable system underlying the moduli space of vacua of the ordinary four dimensional N=2 theory. The epsilon-parameter of the Omega-background is identified with the Planck constant, the twisted chiral ring maps to quantum Hamiltonians, the supersymmetric vacua are identified with Bethe states of quantum integrable systems. This four dimensional gauge theory in its low energy description has two dimensional twisted superpotential which becomes the Yang-Yang function of the integrable system. We present the thermodynamic-Bethe-ansatz like formulae for these functions and for the spectra of commuting Hamiltonians following the direct computation in gauge theory. The general construction is illustrated at the examples of the many-body systems, such as the periodic Toda chain, the elliptic Calogero-Moser system, and their relativistic versions, for which we present a complete characterization of the L^2-spectrum. We very briefly discuss the quantization of Hitchin system.

• The paper establishes that 4D gauge theories in an Ω-background yield exact quantization of integrable systems by connecting twisted superpotentials to Bethe ansatz conditions.
• It employs localization to derive TBA-like integral equations that capture non-perturbative contributions in the gauge theory partition function.
• The framework is demonstrated on models like the periodic Toda chain and elliptic Calogero-Moser system, highlighting its broad applicability to integrable models.

Overview of "Quantization of Integrable Systems and Four Dimensional Gauge Theories"

The paper by Nikita A. Nekrasov and Samson L. Shatashvili explores the interplay between four-dimensional supersymmetric quantum field theories and integrable systems. It sheds light on how the quantization of classical integrable systems emerges from four-dimensional gauge theories subjected to a specific background deformation, known as the $\Omega$-background. This research provides a framework connecting these gauge theories to quantum integrable systems, with particular emphasis on the role of supersymmetric vacua corresponding to Bethe states.

Key Contributions

• $\Omega$-Background and Supersymmetric Gauge Theories: The authors begin by considering four-dimensional $\mathcal{N}=2$ supersymmetric gauge theories in an $\Omega$-background, characterized by two parameters $\epsilon_1$ and $\epsilon_2$. This background breaks some of the Poincaré symmetries in spacetime, leading to nontrivial modifications of the quantum field theory. In the particular case where $\epsilon_2 = 0$, the system retains two-dimensional $\mathcal{N}=2$ supersymmetry, and the gauge theory can be effectively studied as a two-dimensional theory with a twisted superpotential $W(\sigma)$.
• Connection to Quantum Integrable Systems: The central claim is that the quantization parameter in the resulting integrable system can be identified with the deformation parameter $\epsilon$. This leads to eigenstates of the gauge theory's low-energy effective twisted superpotential being interpreted as eigenstates of quantum integrable systems. Specifically, the Bethe ansatz equations, which characterize the spectra of these integrable systems, emerge naturally from the condition of supersymmetric vacua in the deformed gauge theory.
• Thermodynamic Bethe Ansatz and Twisted Superpotential: A highlight of the paper is the formulation of thermodynamic Bethe ansatz (TBA)-like integral equations, which help compute the twisted superpotential in exact terms. These results are derived from the localization of the gauge theory partition function in the $\Omega$-background, effectively capturing the non-perturbative contributions.
• Examples of Integrable Systems: The authors apply their framework to several well-known integrable systems. They thoroughly investigate the periodic Toda chain and the elliptic Calogero-Moser system, both of which relate to specific $\mathcal{N}=2$ gauge theories. The paper also includes the relativistic analogues of these systems, demonstrating the breadth of applicability of their formalism.

Implications and Future Directions

This work has several important theoretical and practical implications:

1. Quantum Spectra from Gauge Theory: The methodology developed in the paper offers a way to derive quantum mechanical spectra of integrable systems directly from the properties of supersymmetric gauge theories. This provides a novel perspective on the quantization of integrable models and enriches the understanding of Bethe ansatz solutions.
2. Gauge Theory as an Insight: By leveraging the insights from gauge theory, the results potentially guide the quantization of more complex integrable systems, including the Hitchin systems. This opens up new avenues for exploring dualities between seemingly unrelated physical models.
3. Mathematical Structures: The connection between the gauge theory's geometry and the algebraic structures of integrable systems facilitates a deeper exploration of symplectic geometry and the thermodynamic properties of these systems. It also aligns with the growing area of research into the Langlands program and related geometric representation theories.
4. Speculative Extensions: Future research directions could explore extensions to other background geometries and dimensions, or investigate connections with string theory and holography. The formalism might also be adapted to address quantum field theories with lower supersymmetries or those that incorporate non-trivial defects.

In summary, this research provides crucial insights into the linkage between quantum field theories and integrable systems, encapsulating elegant mathematical structures that highlight the intricate web of relationships at the heart of theoretical physics.