self-dual Yang-Mills theory




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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 self-dual.


Via an action functional

Let (X,g)(X, g) be a (pseudo) Riemannian manifold of dimension 4. Write :Ω 2(X)Ω 2(X)\star \colon \Omega^2(X) \to \Omega^2(X) for the corresponding Hodge star operator. Its square is 2=+1\star^2 = +1 for Euclidean signature and 2=1\star^2 = -1 for Lorentzian signature. Decompose (possibly after complexification)

Ω 2(X)Ω 2(X) +Ω 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 GG be a Lie group. Write 𝔤\mathfrak{g} for the corresponding Lie algebra. Let ,\langle -,-\rangle be a binary invariant polynomial on the Lie algebra.

Accordingly we have

Ω 2(X,𝔤)Ω 2(X,𝔤) +Ω 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)(X,g) is that of pairs (,𝒢)(\nabla, \mathcal{G}) with

  • H conn 1(X,G)\nabla \in \mathbf{H}^1_{conn}(X,G) is a GG-principal connection over XX;

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

The action functional of the theory is

(,𝒢) X(F ) 𝒢dvol g. (\nabla, \mathcal{G}) \mapsto \int_X \langle (F_\nabla)_- \wedge \mathcal{G} \rangle dvol_g \,.

Via BV-complexes

Let XX be an oriented smooth manifold of dimension 4 equipped with a conformal structure with Hodge star operator g\star_g. Let GG be a Lie group with Lie algebra 𝔤\mathfrak{g}.

Let PXP \to X be a GG-principal bundle and write 𝔤 PP× G𝔤\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 GG-principal connection 0\nabla_0 on PP with self dual curvature F 0=0Ω 2(X,𝔤 P)F_{\nabla_0} = 0 \in \Omega_-^2(X, \mathfrak{g}_P).

Consider then the chain complex

Ω 0(X,𝔤 P) d 0 Ω 1(X,𝔤 P) P d 0 Ω 2(X,𝔤 P) deg= 1 0 1, \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 } \,,


This is a derived L-infinity algebroid model for perturbations of self-dual connections about 0\nabla_0:

a field configuration is an element in degree 0, hence a differential 1-form AΩ 1(X,𝔤 P)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 λΩ 1(X,𝔤 P)\lambda \Omega^1(X, \mathfrak{g}_P) transforming

AA+d 0λ. 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 XX) is

Ω 0(X,𝔤 P) d 0 Ω 1(X,𝔤 P) P d 0 Ω 2(X,𝔤 P) Ω 2(X,𝔤 P) Ω 3(X,𝔤 P) Ω 4(X,𝔤 P) deg= 1 0 1 2, \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 Ω 1(X,𝔤 P)\Omega^1(X,\mathfrak{g}_P) is in degree 0, whereas what is displayed above is the “delooped deived L L_\infty-algebra”.


Of the action functional

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

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

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

1ϵ X(F F F F )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 \,.

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 N=4N=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

Last revised on December 6, 2018 at 10:08:22. See the history of this page for a list of all contributions to it.