four-dimensional Chern–Simons theory

The theory is defined on a 4-dimensional manifold which is a product of two 2-dimensional manifolds: $M\; =\; \backslash Sigma\; \backslash times\; C$, where $\backslash Sigma$ is a smooth orientable 2-dimensional manifold, and $C$ is a complex curve (hence has real dimension 2) endowed with a meromorphic one-form $\backslash omega$.

The field content is a gauge field $A$. The action is given by wedging the Chern–Simons 3-form $CS(A)$ with $\backslash omega$:

$$S\_\{4d\}\; =\; \backslash frac\{1\}\{2\backslash pi\}\; \backslash int\_M\; \backslash omega\; \backslash wedge\; CS(A).$$

A heuristic puts strong restrictions on the $C$ to be considered. This theory is studied perturbatively, in the limit that the Planck constant . In the path integral formulation, the action will contain a ratio $\backslash omega/\backslash hbar$. Therefore, zeroes of $\backslash omega$ naïvely correspond to points at which $\backslash hbar\; \backslash rightarrow\; \backslash infty$, at which point perturbation theory breaks down. So $\backslash omega$ may have poles, but not zeroes. A corollary of the Riemann–Roch theorem relates the degree of the canonical divisor defined by $\backslash omega$ (equal to the difference between the number of zeros and poles of $\backslash omega$, with multiplicity) to the genus $g$ of the curve $C$, giving{{Cite book |last=Donaldson |first=Simon |author-link=Simon Donaldson |url=https://www.ma.imperial.ac.uk/~skdona/RSPREF.PDF |title=Riemann Surfaces |publisher=Oxford University Press |year=2011 |isbn=978-0-19-852639-1 |pages=88, Proposition 16 |language=en}}

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

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

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

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 $D$, is a 2-dimensional 'object' which is considered to be localized at a point $z$ on the complex curve but covers $\backslash Sigma,$ which is fixed to be $\backslash mathbb\{R\}^2$ for engineering integrable field theories. This defect $D$ is then the space on which a 2-dimensional field theory lives, and this theory couples to the bulk gauge field $A$.

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

In general, one can have multiple defects $D\_\backslash alpha$ with $\backslash alpha\; =\; 1,\; \backslash cdots,\; n$, and the action for the coupled theory is then

$$S\_\{4d-2d\}\; =\; \backslash frac\{1\}\{2\backslash hbar\backslash pi\}\; \backslash int\_\{\backslash mathbb\; R^2\; \backslash times\; C\}\; \backslash omega\; \backslash wedge\; CS(A)\; +\; \backslash sum\_\{\backslash alpha\; =\; 1\}^\{n\}\; \backslash frac\{1\}\{\backslash hbar\}\; \backslash int\_\{\backslash mathbb\{R\}^2\; \backslash times\; z\_\backslash alpha\}\; \backslash mathcal\{L\}\_\backslash alpha\; (\backslash phi\_\backslash alpha;$$

A_w|_{z_\alpha}, A_{\overline w}|_{z_\alpha}),

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

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.

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

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

4d Chern–Simons theory is a 'master theory' for integrable systems, providing a framework that incorporates many integrable systems. Another theory which shares this feature, but with a Hamiltonian rather than Lagrangian description, is classical affine Gaudin models with a 'dihedral twist',{{cite journal

|last1=Vicedo |first1=Benoît

|title=On Integrable Field Theories as Dihedral Affine Gaudin Models

|journal=International Mathematics Research Notices

|date=4 August 2020

|volume=2020 |issue=15 |pages=4513–4601

|doi=10.1093/imrn/rny128

|url=https://academic.oup.com/imrn/article-abstract/2020/15/4513/5045617?redirectedFrom=fulltext|arxiv=1701.04856

}} and the two theories have been shown to be closely related.{{cite journal

|last1=Vicedo |first1=Benoît

|title=4D Chern–Simons theory and affine Gaudin models

|journal=Letters in Mathematical Physics |date=24 February 2021

|volume=111 |issue=1 |pages=24

|doi=10.1007/s11005-021-01354-9

|bibcode=2021LMaPh.111...24V

|s2cid=254800771

|language=en |issn=1573-0530|doi-access=free

}}

Another 'master theory' for integrable systems is the anti-self-dual Yang–Mills (ASDYM) system. Ward's conjecture is the conjecture that in fact all integrable ODEs or PDEs come from ASDYM. A connection between 4d Chern–Simons theory and ASDYM has been found so that they in fact come from a six-dimensional holomorphic Chern–Simons theory defined on twistor space. The derivation of integrable systems from this 6d Chern–Simons theory through the alternate routes of 4d Chern–Simons theory and ASDYM in fact fit into a commuting square.

{{cite journal

|last1=Bittleston |first1=Roland

|last2=Skinner |first2=David

|title=Twistors, the ASD Yang-Mills equations and 4d Chern-Simons theory

|journal=Journal of High Energy Physics

|date=22 February 2023

|volume=2023 |issue=2 |page=227

|doi=10.1007/JHEP02(2023)227|s2cid=226281535

|arxiv=2011.04638

|bibcode=2023JHEP...02..227B

}}

• Chern–Simons theory
• Infinite-dimensional Chern–Simons theory
• Integrable system
• Classical Gaudin model
• Anti-self-dual Yang–Mills equations

• [https://ncatlab.org/nlab/show/semi-topological+4d+Chern-Simons+theory nLab page]

{{reflist}}

{{Integrable systems}}

Category:Quantum field theory

Category:Integrable systems