algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
quantum mechanical system, quantum probability
interacting field quantization
In mathematical/theoretical physics, semi-holomorphic 4d Chern-Simons theory is a variant of ordinary 3d 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-holomorphic 4d 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-holomorphic” Chern-Simons theory: If one instead promotes all three coordinates of ordinary 3d 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 4d Chern-Simons theory.
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 the the 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 …
The 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)
Discussion of realizations of semi-holomorphic 4d Chern-Simons theory as the worldvolume theory of intersecting D-brane/NS5-brane models
Created on May 7, 2019 at 11:51:20. See the history of this page for a list of all contributions to it.