Today I summarize part of the relation between integrable systems and gauge theories that was presented in 2009 by Nekrasov and Shatashvili in a series of papers (see references below).

The Bethe-Gauge correspondence

The first correspondence that I review applies to two-dimensional gauge theories. Consider for instance a two-dimensional theory with $\mathcal{N} = (2,2)$ supersymmetry. The supersymmetric vacua of this theory correspond to the stationary eigenstates of a quantum integrable system, and we have a quite complete dictionary between the two sides of the correspondence:
\begin{tikzpicture}
[+preamble]
\usepackage{tikz}
\usetikzlibrary{arrows}
\tikzstyle{arrow} = [thin,->,>=angle 60]
[/preamble]
\node[draw,text width=6cm,align=center](1) at (0cm,0cm) {$2d$ gauge theory with $\mathcal{N} = (2,2)$ supersymmetry};
\node[draw,text width=6cm,align=center](2) at (9cm,0cm) {Intrinsically quantum integrable system (no continuous Planck constant)};
\node[text width=6cm,align=center](3) at (0cm,-2cm) {Finite-dimensional space of susy vacua};
\node[text width=6cm,align=center](4) at (9cm,-2cm) {Finite-dimensional Hilbert space};
\node[text width=6cm,align=center](5) at (0cm,-4cm) {Generators of the twisted chiral ring $\mathcal{O}_k$ and their expectation values};
\node[text width=6cm,align=center](6) at (9cm,-4cm) {Hamiltonians of the integrable system and their eigenvalues};
\node[text width=6cm,align=center](7) at (0cm,-6cm) {Coulomb branch moduli $\sigma_i$};
\node[text width=6cm,align=center](8) at (9cm,-6cm) {Spectral parameters $\lambda_i$};
\node[text width=6cm,align=center](9) at (0cm,-8cm) {Effective twisted superpotential $\tilde{W}^{\textrm{eff}} (\sigma)$};
\node[text width=6cm,align=center](10) at (9cm,-8cm) {Yang-Yang function $Y (\lambda)$};
\draw[] (1) -- (2);
\draw[] (3) -- (4);
\draw[] (5) -- (6);
\draw[arrow] (5) -- node[anchor=east]{acts on}(3);
\draw[arrow] (6) -- node[anchor=west]{acts on}(4);
\draw[] (7) -- (8);
\draw[] (9) -- (10);
\end{tikzpicture}

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Let us explain briefly how this is obtained. In a theory with $\mathcal{N} = (2,2)$, we can call $Q_{\pm , \pm}$ the four real supercharges, and define $$Q_A = Q_{+,+} + Q_{-,-} \qquad Q_B = Q_{+,-} + Q_{-,-} . $$ Those two combinations correspond to two inequivalent twists, called the $A$-twist and the $B$-twist. Here we focus on the former. One crucial property is $$Q_{A}^2 = 0 \qquad \{Q_A , Q_A^{\dagger} \} = H $$, so we can consider the cohomology $ \mathcal{H}^{\textrm{quantum}} = \textrm{Ker} Q_A / \textrm{Im} Q_A$. The name we have given to this space reflects the fact that it is the space of states of some quantum integrable system.

To understand this, one has to introduce the twisted chiral ring, which is generated by the operators that (anti-)commute with $Q_A$. Let us call $\mathcal{O}_i$ a basis of such operators. The fact that they generate a ring comes from the operator product expansion $$ \mathcal{O}_i \mathcal{O}_j = c_{ij}^k \mathcal{O}_k + \{ Q_A , \cdots \} . $$ So indeed when we neglect all $Q_A$-exact quantities, we have an honest ring. But now if $| 0 \rangle$ is a vacuum state of the Hamiltonian, $H | 0 \rangle = 0$, then because of the commutation relations $|i\rangle = \mathcal{O}_i | 0 \rangle$ is as well, and moreover $$ \mathcal{O}_i |j\rangle = c_{ij}^k |k \rangle \, . $$ The supersymmetric vacua form a representation of the ring. This gives a hint of why the $\mathcal{O}_i$ will be identified with the Hamiltonians of the quantum integrable system.
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We still have to explain why the name "Bethe" is used here! This comes from the fact that in most interesting cases, at low energy on the Coulomb branch the theory becomes abelian. It is then possible to compute exactly an effective twisted superpotential $\tilde{W}^{\textrm{eff}} (\sigma)$, which allows to determine the supersymmetric vacua by solving an extremization problem. But one has to be careful because of integer-valued electric fluxes, and this is taken care of by solving $$ \boxed{\exp \left( \frac{\partial \tilde{W}^{\textrm{eff}} (\sigma)}{\partial \sigma^i}\right) = 1 } $$ This equation coincides with the Bethe equation determining the exact spectrum of the quantum integrable system. In this context $\tilde{W}^{\textrm{eff}} (\sigma)$ is known as the Yang-Yang function $Y(\lambda)$.

The four-dimensional story

There is a nice analog of the two-dimensional discussion above to four-dimensional theories. Here I simply quote the main ideas one has to remember.

Note first that a $4d$ theory can be seen as a $2d$ theory with an infinite number of fields! This is in essence why the space of vacua and the Hilbert space of the corresponding integrable system will be infinite-dimensional. An other very important point here is that we have a continuous parameter $\epsilon$ that plays the role of the Planck constant $\hbar$. On the gauge theory side, it appears in the partial $\Omega$-deformation. When this Planck constant is set to zero, we recover a classical integrable system associated to a field theory on $\mathbb{R}^4$: this is simply the Donagi-Witten integrable system.

We can sketch an analog diagram, without entering further into details here:
\begin{tikzpicture}
[+preamble]
\usepackage{tikz}
[/preamble]
\node[draw,text width=6cm,align=center](1) at (0cm,0cm) {$4d$ gauge theory with $\mathcal{N} = 2$ supersymmetry on $\Omega$-background with $\epsilon_1 = \epsilon$ and $\epsilon_2 = 0$};
\node[draw,text width=6cm,align=center](2) at (9cm,0cm) {Quantum integrable system with Planck constant $\hbar = \epsilon$};
\node[text width=6cm,align=center](3) at (0cm,-2cm) {Infinite dimensional space of supersymmetric vacua};
\node[text width=6cm,align=center](4) at (9cm,-2cm) {Infinite dimensional space of stationary eigenstates};
\draw[] (1) -- (2);
\draw[] (3) -- (4);
\end{tikzpicture}
As nicely summarized in section 5.2 of this review by Teschner, this correspondence was the starting point of a series of developments on both sides. Note in particular this paper by Nekrasov and Witten where the link between the $\Omega$-deformation and the AGT correspondence is elucidated.

References

  • Supersymmetric vacua and Bethe ansatz, Nikita A. Nekrasov and Samson L. Shatashvili. Jan 2009. arXiv:0901.4744.
  • Quantization of Integrable Systems and Four Dimensional Gauge Theories, Nikita A. Nekrasov and Samson L. Shatashvili. Aug 2009. arXiv:0908.4052.