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self-dual Yang-Mills theory

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 $\star :{\Omega }^2(X)\to {\Omega }^2(X)$ 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)

$${\Omega }^2(X)\simeq {\Omega }^2(X{)}_{+}\oplus {\Omega }^2(X{)}_{-}$$\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 $𝔤$ for the corresponding Lie algebra. Let $⟨-,-⟩$ be a binary invariant polynomial on the Lie algebra.

Accordingly we have

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

• $\nabla \in H_{\mathrm{conn}}^1(X,G)$ is a $G$-principal connection over $X$;

• $𝒢\in {\Omega }^2(X,𝔤{)}_{-}$ is an anti-self-dual 2-form.

The action functional of the theory is

$$(\nabla ,𝒢)\mapsto {\int }_X⟨(F_{\nabla }{)}_{-}\wedge 𝒢⟩{\mathrm{dvol}}_g.$$(\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 $𝔤$.

Let $P\to X$ be a $G$-principal bundle and write ${𝔤}_P≔P{\times }_G𝔤$ 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 $F_{{\nabla }_0}=0\in {\Omega }_{-}^2(X,{𝔤}_P)$.

Consider then the chain complex

$$\begin{array}{cccccc}& {\Omega }^0(X,{𝔤}_P)& \stackrel{d_{{\nabla }_0}}{\to }& {\Omega }^1(X,{𝔤}_P)& \stackrel{P_{-}\circ d_{{\nabla }_0}}{\to }& {\Omega }_{-}^2(X,{𝔤}_P)\\ \\ \mathrm{deg}=& 1& & 0& & -1\end{array},$$\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 $A\in {\Omega }^1(X,{𝔤}_P)$, which is in the kernel of the differential, hence of self-dual curvature. A gauge transformation of this is an element $\lambda {\Omega }^1(X,{𝔤}_P)$ transforming

$$A\mapsto A+d_{{\nabla }_0}\lambda .$$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

$$\begin{array}{cccccc}& {\Omega }^0(X,{𝔤}_P)& \stackrel{d_{{\nabla }_0}}{\to }& {\Omega }^1(X,{𝔤}_P)& \stackrel{P_{-}\circ d_{{\nabla }_0}}{\to }& {\Omega }_{-}^2(X,{𝔤}_P)\\ & & & \oplus & & \oplus \\ & & & {\Omega }_{-}^2(X,{𝔤}_P)& \to & {\Omega }^3(X,{𝔤}_P)& \to & {\Omega }^4(X,{𝔤}_P)\\ \\ \mathrm{deg}=& 1& & 0& & -1& & -2\end{array},$$\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 ${\Omega }^1(X,{𝔤}_P)$ is in degree 0, whereas what is displayed above is the “delooped deived $L_{\mathrm{\infty }}$-algebra”.

Properties

Of the action functional

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

$$\cdots +ϵ{\int }_X⟨𝒢\wedge 𝒢⟩$$\cdots + \epsilon \int_X \langle \mathcal{G} \wedge \mathcal{G}\rangle

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

$$\nabla \mapsto \frac{1}{ϵ}{\int }_X(⟨F_{\nabla }\wedge \star F_{\nabla }⟩-⟨F_{\nabla }\wedge F_{\nabla }⟩){\mathrm{dvol}}_g.$$\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 \,.

Of the BV-complex

(…)

Related concepts

• self-dual higher gauge theory

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)

See also

• 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

• Kevin Costello, Owen Gwilliam, Factorization algebras in perturbative quantum field theory (wiki, early/partial draft pdf)