nLab semi-topological D=4 Chern-Simons theory

Idea

In mathematical/theoretical physics, semi-topological or semi-holomorphic D=4 Chern-Simons theory is a variant of ordinary D=3 Chern-Simons theory where, roughly, one of the three coordinate functions is promoted from an ordinary real-valued coordinate to a holomorphic coordinate.

Hence semi-topological/holomorphic D=4 Chern-Simons theory is a field theory defined on spacetimes of the form $\Sigma^2 \times C$ where $\Sigma^2$ is a real smooth manifold, while $C$ is a complex curve, and whose Lagrangian density is schematically the ordinary Chern-Simons form, but with one of the wedge factors promoted from $d x^i$ to $d z$.

Accordingly, this field theory is a topological field theory (only) with respect to $\Sigma^2$, while it behaves like a conformal field theory with respect to $C$, whence “semi-topological/holomorphic” Chern-Simons theory: If one instead promotes all three coordinates of ordinary D=3 Chern-Simons theory to holomorphic coordinates, one obtains full 6d holomorphic Chern-Simons theory.

But beware that in the literature this is often referred to just as D=4 Chern-Simons theory.

Properties

Wilson lines and relation to integrable lattice models

Due to this particular property of the theory, Wilson line observables that form a network on $\Sigma^2$ but with each line located at a distinct point in the complex curve $C$ play a special role: Arranging them along the lines of a rectangular grid makes these observables behave much like those of a lattice model with their intersection points as vertices, but the topological invariance of the theory with respect to $\Sigma^2$ implies certain symmetry operations on these observables, coming from invariance of the theory under moving Wilson lines across each other. These symmetries turn out to behave like R-matrices making the corresponding lattice model an integrable system.

Yangian symmetry

… Yangian …

Consider the example when the spacetime is of the form $\mathbb{C}_w \times \mathbb{C}_z$ with subscripts to denote the respective coordinates. Pushing forward the factorization algebra for gives a locally constant factorization algebra on $\mathbb{C}_w$.

Applying Koszul duality gives a Hopf algebra because it is augmented as an $E_1$ algebra. This is a result of Tamarkin. In this case the Hopf algebra is the linear dual of the completed Yangian. This is seen by evaluating the classical limit and the first order bialgebra structure and then invoking a uniqueness result of Drinfeld.

References

The idea of semi-topological 4d CS theory, and its relation to Yangians and integrable lattice models, is due to

• Kevin Costello, Supersymmetric gauge theory and the Yangian (arXiv:1303.2632)

• Edward Witten, Integrable Lattice Models From Gauge Theory, Adv.Theor.Math.Phys. 21 (2017) 1819-1843 (arXiv:1611.00592)

• Kevin Costello, Edward Witten, Masahito Yamazaki, Gauge Theory and Integrability, I, ICCM Not. 6, 46-119 (2018) (arXiv:1709.09993)

• Kevin Costello, Edward Witten, Masahito Yamazaki, Gauge Theory and Integrability, II, ICCM Not. 6, 120-146 (2018) (arXiv:1802.01579)

• Kevin Costello, Masahito Yamazaki, Gauge Theory And Integrability, III (arxiv:1908.02289)

• Meer Ashwinkumar, Integrable Lattice Models and Holography, JHEP 02 (2021) 227 (arXiv:2003.08931)

• Meer Ashwinkumar, Jun-ichi Sakamoto, Masahito Yamazaki, Dualities and Discretizations of Integrable Quantum Field Theories from 4d Chern-Simons Theory [arXiv:2309.14412]

Rigorous discussion using homotopy theory (see at homotopical AQFT):

• Marco Benini, Alexander Schenkel, Benoit Vicedo, Homotopical analysis of 4d Chern-Simons theory and integrable field theories (arXiv:2008.01829)

Lecture notes:

• Sylvain Lacroix, 4-dimensional Chern-Simons theory and integrable field theories (arXiv:2109.14278)

Discussion of realizations of semi-holomorphic 4d Chern-Simons theory as the worldvolume theory of intersecting D-brane/NS5-brane models:

• Meer Ashwinkumar, Meng-Chwan Tan, Qin Zhao, Branes and Categorifying Integrable Lattice Models (arXiv:1806.02821)

and further relating to the quantum geometric Langlands correspondence:

• Meer Ashwinkumar, Meng-Chwan Tan, Unifying Lattice Models, Links and Quantum Geometric Langlands via Branes in String Theory (arXiv:1910.01134)

Relation to the Berkovits superstring:

• Nathan Berkovits, Rodrigo S. Pitombo, 4D Chern-Simons and the pure spinor $AdS_5 \times S^5$ superstring [arXiv:2401.03976]

A higher gauge D=5 Chern-Simons theory analogous to semi-topological D=4 Chern-Simons theory:

• Alexander Schenkel, Benoit Vicedo: 5d 2-Chern-Simons theory and 3d integrable field theories [arXiv:2405.08083]

Relation of 4d semi-topological CS to special solutions of 4d gravity:

• Lewis T. Cole, Peter Weck: Integrability in Gravity from Chern-Simons Theory [arXiv:2407.08782]

Relation to 4d WZW theory:

• Masashi Hamanaka, Shan-Chi Huang: Solitons in 4d Wess-Zumino-Witten models – Towards unification of integrable systems [arXiv:2408.16554]

See also

• Wikipedia, Four-dimensional Chern–Simons theory