Restrictions on underlying manifolds

A heuristic puts strong restrictions on the ${\displaystyle C}$ to be considered. This theory is studied perturbatively, in the limit that the Planck constant ${\displaystyle \hbar <<1}$. In the path integral formulation, the action will contain a ratio ${\displaystyle \omega /\hbar }$. Therefore, zeroes of ${\displaystyle \omega }$ naïvely correspond to points at which ${\displaystyle \hbar \rightarrow \infty }$, at which point perturbation theory breaks down. So ${\displaystyle \omega }$ may have poles, but not zeroes. A corollary of the Riemann–Roch theorem relates the degree of the canonical divisor defined by ${\displaystyle \omega }$ (equal to the difference between the number of zeros and poles of ${\displaystyle \omega }$, with multiplicity) to the genus ${\displaystyle g}$ of the curve ${\displaystyle C}$, giving

$${\displaystyle {\text{number of zeros of }}\omega -{\text{number of poles of }}\omega =2g-2}$$

Then imposing that ${\displaystyle \omega }$ has no zeroes, ${\displaystyle g}$ must be ${\displaystyle 0}$ or ${\displaystyle 1}$. In the latter case, ${\displaystyle \omega }$ has no poles and ${\displaystyle C=\mathbb {C} /\Lambda }$ a complex torus (with ${\displaystyle \Lambda }$ a 2d lattice). If ${\displaystyle g=0}$, then ${\displaystyle C}$ is ${\displaystyle \mathbb {CP} ^{1}}$ the complex projective line. The form ${\displaystyle \omega }$ has two poles; either a single pole with multiplicity 2, in which case it can be realized as ${\displaystyle \omega =dz}$ on ${\displaystyle \mathbb {C} }$, or two poles of multiplicity one, which can be realized as ${\displaystyle \omega ={\frac {dz}{z}}}$ on ${\displaystyle \mathbb {C} ^{\times }\cong \mathbb {C} /\mathbb {Z} }$. Therefore ${\displaystyle C}$ is either a complex plane, cylinder or torus.

There is also a topological restriction on ${\displaystyle \Sigma }$, due to a possible framing anomaly. This imposes that ${\displaystyle \Sigma }$ must be a parallelizable 2d manifold, which is also a strong restriction: for example, if ${\displaystyle \Sigma }$ is compact, then it is a torus.

Surface defects and field theories

The above is sufficient to obtain spin chains from the theory, but to obtain 2-dimensional integrable field theories, one must introduce so-called surface defects. A surface defect, often labelled ${\displaystyle D}$, is a 2-dimensional 'object' which is considered to be localized at a point ${\displaystyle z}$ on the complex curve but covers ${\displaystyle \Sigma ,}$ which is fixed to be ${\displaystyle \mathbb {R} ^{2}}$ for engineering integrable field theories. This defect ${\displaystyle D}$ is then the space on which a 2-dimensional field theory lives, and this theory couples to the bulk gauge field ${\displaystyle A}$.

Supposing the bulk gauge field ${\displaystyle A}$ has gauge group ${\displaystyle G}$, the field theory on the defect can interact with the bulk gauge field if it has global symmetry group ${\displaystyle G}$, so that it has a current ${\displaystyle J}$ which can couple via a term which is schematically ${\displaystyle \int JA}$.

In general, one can have multiple defects ${\displaystyle D_{\alpha }}$ with ${\displaystyle \alpha =1,\cdots ,n}$, and the action for the coupled theory is then

$${\displaystyle S_{4d-2d}={\frac {1}{2\hbar \pi }}\int _{\mathbb {R} ^{2}\times C}\omega \wedge CS(A)+\sum _{\alpha =1}^{n}{\frac {1}{\hbar }}\int _{\mathbb {R} ^{2}\times z_{\alpha }}{\mathcal {L}}_{\alpha }(\phi _{\alpha };A_{w}|_{z_{\alpha }},A_{\overline {w}}|_{z_{\alpha }}),}$$

with ${\displaystyle \phi _{\alpha }}$ the collection of fields for the field theory on ${\displaystyle D_{\alpha }}$, and coordinates ${\displaystyle w,{\overline {w}}}$ for ${\displaystyle \mathbb {R} ^{2}}$.

There are two distinct classes of defects:

1. Order defects, which introduce new degrees of freedom on the defect which couple to the bulk gauge field.
2. Disorder defects, where the bulk gauge field has some singularities.

Order defects are easier to define, but disorder defects are required to engineer many of the known 2-dimensional integrable field theories.

Spin chains

• Six-vertex model
• Eight-vertex model
• XXZ Heisenberg spin-chain

Integrable field theories

• Gross–Neveu model
• Thirring model
• Wess–Zumino–Witten model
• Principal chiral model and deformations
• Symmetric space coset sigma models