In mathematical physics, four-dimensional Chern–Simons theory, also known as semi-holomorphic or semi-topological Chern–Simons theory, is a quantum field theory initially defined by Nikita Nekrasov,[1] rediscovered and studied by Kevin Costello,[2] and later by Edward Witten and Masahito Yamazaki.
[3][4][5] It is named after mathematicians Shiing-Shen Chern and James Simons who discovered the Chern–Simons 3-form appearing in the theory.
The gauge theory has been demonstrated to be related to many integrable systems, including exactly solvable lattice models such as the six-vertex model of Lieb and the Heisenberg spin chain[3][4] and integrable field theories such as principal chiral models, symmetric space coset sigma models and Toda field theory, although the integrable field theories require the introduction of two-dimensional surface defects.
[5] The theory is also related to the Yang–Baxter equation and quantum groups such as the Yangian.
The theory is similar to three-dimensional Chern–Simons theory which is a topological quantum field theory, and the relation of 4d Chern–Simons theory to the Yang–Baxter equation bears similarities to the relation of 3d Chern–Simons theory to knot invariants such as the Jones polynomial discovered by Witten.
is a complex curve (hence has real dimension 2) endowed with a meromorphic one-form
A heuristic puts strong restrictions on the
This theory is studied perturbatively, in the limit that the Planck constant
In the path integral formulation, the action will contain a ratio
, at which point perturbation theory breaks down.
A corollary of the Riemann–Roch theorem relates the degree of the canonical divisor defined by
(equal to the difference between the number of zeros and poles of
the complex projective line.
is either a complex plane, cylinder or torus.
must be a parallelizable 2d manifold, which is also a strong restriction: for example, if
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.
, is a 2-dimensional 'object' which is considered to be localized at a point
on the complex curve but covers
for engineering integrable field theories.
is then the space on which a 2-dimensional field theory lives, and this theory couples to the bulk gauge field
Supposing the bulk gauge field
, the field theory on the defect can interact with the bulk gauge field if it has global symmetry group
In general, one can have multiple defects
, and the action for the coupled theory is then
There are two distinct classes of defects: Order defects are easier to define, but disorder defects are required to engineer many of the known 2-dimensional integrable field theories.
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',[8] and the two theories have been shown to be closely related.
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.