# nLab self-dual Yang-Mills theory

Contents

## Surveys, textbooks and lecture notes

#### Differential cohomology

differential cohomology

# Contents

## Idea

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.

## Definition

### Via an action functional

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)

$\Omega^2(X) \simeq \Omega^2(X)_+ \oplus \Omega^2(X)_-$

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

$\Omega^2(X, \mathfrak{g}) \simeq \Omega^2(X, \mathfrak{g})_+ \oplus \Omega^2(X, \mathfrak{g})_- \,.$

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

$(\nabla, \mathcal{G}) \mapsto \int_X \langle (F_\nabla)_- \wedge \mathcal{G} \rangle dvol_g \,.$

### Via BV-complexes

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

$\array{ & \Omega^0(X, \mathfrak{g}_P) &\stackrel{d_{\nabla_0}}{\to}& \Omega^1(X, \mathfrak{g}_P) &\stackrel{P_- \circ d_{\nabla_0}}{\to}& \Omega^2_-(X, \mathfrak{g}_P) \\ \\ deg = & 1 & & 0 && -1 } \,,$

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

$A \mapsto A + d_{\nabla_0} \lambda \,.$

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

$\array{ & \Omega^0(X, \mathfrak{g}_P) &\stackrel{d_{\nabla_0}}{\to}& \Omega^1(X, \mathfrak{g}_P) &\stackrel{P_- \circ d_{\nabla_0}}{\to}& \Omega^2_-(X, \mathfrak{g}_P) \\ & && \oplus && \oplus \\ & && \Omega^2_-(X, \mathfrak{g}_P) &\to& \Omega^3(X, \mathfrak{g}_P) &\to& \Omega^4(X, \mathfrak{g}_P) \\ \\ deg = & 1 & & 0 && -1 && -2 } \,,$

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”.

## Properties

### Of the action functional

If one changes the action functional of self-dual Yang-Mills theory by adding a term

$\cdots + \epsilon \int_X \langle \mathcal{G} \wedge \mathcal{G}\rangle$

for some non-vanishing $\epsilon \in \mathbb{R}$, then it becomes equivalent to that of ordinary Yang-Mills theory in the form

$\nabla \mapsto \frac{1}{\epsilon} \int_X \left( \langle F_\nabla \wedge \star F_\nabla \rangle - \langle F_\nabla \wedge F_\nabla \rangle \right) dvol_g \,.$

### 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.

## References

### 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 $N=4$ supersymmetric self-dual Yang-Mills theory (arXiv:hep-th/9509099)