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A variant of Yang-Mills theory in which the field strength/curvature 2-form of the Yang-Mills field is constrained to be Hodge self-dual.
Let $(X, g)$ be a (pseudo) Riemannian manifold of dimension 4. Write $\star \colon \Omega^2(X) \to \Omega^2(X)$ for the corresponding Hodge star operator. Its square is $\star^2 = +1$ for Euclidean signature and $\star^2 = -1$ for Lorentzian signature. Decompose (possibly after complexification)
into the direct sum of eigenspaces of $\star$, the self-dual and the anti-self-dual forms.
Let $G$ be a Lie group. Write $\mathfrak{g}$ for the corresponding Lie algebra. Let $\langle -,-\rangle$ be a binary invariant polynomial on the Lie algebra.
Accordingly we have
The configuration space of self-dual Yang-Mills theory on $(X,g)$ is that of pairs $(\nabla, \mathcal{G})$ with
$\nabla \in \mathbf{H}^1_{conn}(X,G)$ is a $G$-principal connection over $X$;
$\mathcal{G} \in \Omega^2(X, \mathfrak{g})_-$ is an anti-self-dual 2-form.
The action functional of the theory is
Let $X$ be an oriented smooth manifold of dimension 4 equipped with a conformal structure with Hodge star operator $\star_g$. Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$.
Let $P \to X$ be a $G$-principal bundle and write $\mathfrak{g}_P \coloneqq P \times_G \mathfrak{g}$ for the associated bundle via the adjoint action of the group on its Lie algebra. Fix a $G$-principal connection $\nabla_0$ on $P$ with self dual curvature $F_{\nabla_0} = 0 \in \Omega_-^2(X, \mathfrak{g}_P)$.
Consider then the chain complex
where
$d_\nabla$ is the de Rham differential coupled to the connection, hence the covariant derivative of $\nabla$;
$P_-$ is the projection onto anti-self dual 2-forms.
This is a derived L-infinity algebroid model for perturbations of self-dual connections about $\nabla_0$:
a field configuration is an element in degree 0, hence a differential 1-form $A \in \Omega^1(X, \mathfrak{g}_P)$, which is in the kernel of the differential, hence of self-dual curvature. A gauge transformation of this is an element $\lambda \Omega^1(X, \mathfrak{g}_P)$ 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 $X$) 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 $\Omega^1(X,\mathfrak{g}_P)$ is in degree 0, whereas what is displayed above is the “delooped deived $L_\infty$-algebra”.
If one changes the action functional of self-dual Yang-Mills theory by adding a term
for some non-vanishing $\epsilon \in \mathbb{R}$, then it becomes equivalent to that of ordinary Yang-Mills theory in the form
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.
The action functional above is due to
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
See also
The description via BV-complexes is amplified in the context of factorization algebras of observables in section 4.12.3 of
Last revised on October 14, 2019 at 14:26:30. See the history of this page for a list of all contributions to it.