semi-holomorphic 4d Chern-Simons theory



Algebraic Quantum Field Theory

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)



field theory:

Lagrangian field theory


quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization



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Local QFT

Perturbative QFT



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 Σ 2×C\Sigma^2 \times C where Σ 2\Sigma^2 is a real smooth manifold, while CC is a complex curve, and whose Lagrangian density is schematically the ordinary Chern-Simons form, but with one of the wedge factors promoted from dx id x^i to dzd z.

Accordingly, this field theory is a topological field theory (only) with respect to Σ 2\Sigma^2, while it behaves like a conformal field theory with respect to CC, 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.


Wilson lines and relation to integrable lattice models

Due to this particular property of the theory, Wilson line observables that form a network on Σ 2\Sigma^2 but with each line located at a distinct point in the complex curve CC 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 Σ 2\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



The theory and its relation to Yangians and integrable lattice models is due to

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.