Connections on bundles
Higher abelian differential cohomology
Higher nonabelian differential cohomology
Application to gauge theory
A variant of Yang-Mills theory in which the field strength/curvature 2-form of the Yang-Mills field is constrained to be self-dual.
Via an action functional
Let be a (pseudo) Riemannian manifold of dimension 4. Write for the corresponding Hodge star operator. Its square is for Euclidean signature and for Lorentzian signature. Decompose (possibly after complexification)
into the direct sum of eigenspaces of , the self-dual and the anti-self-dual forms.
Let be a Lie group. Write for the corresponding Lie algebra. Let be a binary invariant polynomial on the Lie algebra.
Accordingly we have
The configuration space of self-dual Yang-Mills theory on is that of pairs with
is a -principal connection over ;
is an anti-self-dual 2-form.
The action functional of the theory is
Let be an oriented smooth manifold of dimension 4 equipped with a conformal structure with Hodge star operator . Let be a Lie group with Lie algebra .
Let be a -principal bundle and write for the associated bundle via the adjoint action of the group on its Lie algebra. Fix a -principal connection on with self dual curvature .
Consider then the chain complex
This is a derived L-infinity algebroid model for perturbations of self-dual connections about :
a field configuration is an element in degree 0, hence a differential 1-form , which is in the kernel of the differential, hence of self-dual curvature. A gauge transformation of this is an element transforming
Consider then the action functional on this complex of fields which is simply zero. Then the corresponding local BV-complex (with local antibracket taking values in the densities on ) is
This formulaton of self-dual Yang-Mills theory is considered in (Costello-Gwilliam, section 4.12.3). There the grading is such that the Lie algebra of gauge transformations is in degree 0, whereas what is displayed above is the “delooped deived -algebra”.
Of the action functional
If one changes the action functional of self-dual Yang-Mills theory by adding a term
for some non-vanishing , then it becomes equivalent to that of ordinary Yang-Mills theory in the form
Via the Penrose-Ward twistor transform
Solutions to the equations of motion of self-dual Yang-Mills theory are naturally produced by seding cohomology classes on twistor space through the Penrose-Ward twistor transform. See there for more details.
Via action functional
The action functional above is due to
- Gordon Chalmers, Warren Siegel, T-Dual Formulation of Yang-Mills Theory, (1997) (arXiv:hep-th/9712191)
briefly reviewed at the beginning of
M.V. Movshev, A note on self-dual Yang-Mills theory, arXiv:0812.0224
N.J. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. (3) 55 (1987), no. 1, 59–126, MR887284 doi
For self-dual super Yang-Mills theory a discusion is in
- E. Sokatchev, An action for supersymmetric self-dual Yang-Mills theory (arXiv:hep-th/9509099)
- H. J. de Vega, Nonlinear multiplane wave solutions of self-dual Yang-Mills theory (EUCLID)
Via BV complex
The description via BV-complexes is amplified in the context of factorization algebras of observables in section 4.12.3 of