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May 13, 2006

Actions for Self-dual Gauge Fields

In 4k+24k+2 dimensions, with Minkowski signature, The Hodge **-operator, * 2=1*^2=1, when acting on Ω 2k+1\Omega^{2k+1}. One therefore has theories with 2k2k-form gauge fields, whose 2k+12k+1-form field strength is (anti)self-dual. The classic examples are the chiral scalar in 2 dimensions and the 4-form gauge field of type IIB supergravity in 10 dimensions.

There isn’t a wholly satisfactory action principle for such fields, which make constructing the quantum theory somewhat less than straightforward. Recently, Moore and Belov came out with a beautiful paper on constructing an action principle for such fields and its connection to spin-Chern-Simons Theory in 4k+34k+3 dimensions.

In constructing the partition function, one typically works in Euclidean signature, where * E 2=1*_E^2=-1 on Ω 2k+1\Omega^{2k+1}. This endows Ω 2k+1\Omega^{2k+1} with a complex structure, J=* EJ=-*_E, and we can decompose into the ±i\pm i eigenspaces Ω 2k+1=V +V \Omega^{2k+1} \otimes \mathbb{C} = V^+\oplus V^- Any vector ϕ +V +\phi^+\in V^+ can be written uniquely as ϕ +=12(ϕ+i* Eϕ) \phi^+= \frac{1}{2} (\phi +i*_E\phi) for some ϕΩ 2k+1\phi\in \Omega^{2k+1}. There’s a natural metric on Ω 2k+1\Omega^{2k+1}, g(ϕ 1,ϕ 2)=ϕ 1* Eϕ 2 g(\phi_1,\phi_2) = \int \phi_1 \wedge *_E \phi_2 which, using the complex structure, gives us a symplectic form as well ω(ϕ 1,ϕ 2)=g(Jϕ 1,ϕ 2)=ϕ 1ϕ 2 \omega(\phi_1,\phi_2) = g(J\phi_1,\phi_2)= \int \phi_1 \wedge \phi_2 Together, these induce a Hermitian form on V +V^+ H(ϕ +,ψ +)=2iω(ϕ +,ψ +¯)=g(ϕ,ψ)+iω(ϕ,ψ) H(\phi^+,\psi^+) = 2 i \omega (\phi^+, \overline{\psi^+}) = g(\phi,\psi) +i\omega(\phi,\psi)

To define a partition function, one actually needs a bilinear form on V +V^+. This, they obtain by virtue of a choice of a Lagrangian subspace, V LΩ 2k+1V_L \subset \Omega^{2k+1}. Letting V L =J(V L)V_L^\perp=J(V_L), Ω 2k+1=V LV L \Omega^{2k+1}=V_L\oplus V_L^\perp is a Lagrangian decomposition. Every ϕΩ 2k+1\phi\in \Omega^{2k+1} can be uniquely decomposed as ϕ=ϕ L+ϕ L \phi= \phi_L+ \phi_L^\perp. We can define an involution, II which acts as +1+1 on V LV_L and as 1-1 on V L V_L^\perp. Then I(ϕ)=ϕ Lϕ L I(\phi)= \phi_L-\phi_L^\perp and the desired bilinear form is B(ϕ +,ψ +)=g(ϕ,I(ψ))+iω(ϕ,I(ψ)) B(\phi^+,\psi^+) = g(\phi, I(\psi))+ i\omega(\phi, I(\psi)) The partition function involves a Θ\Theta-function constructed using the symmetric quadratic form, HBH-B.

Paradoxically, having gone through the construction of the quantum partition function, one can then go back and construct a classical action by extending this quadratic form from the cohomology to the space of closed forms.

Given a choice of Lagrangian subspace, V LV_L, we get another Lagrangian subspace, V 1Ω 2k+1V_1\subset \Omega^{2k+1}, as follows. Let Γ LH DR 2k+1\Gamma_L\subset H_{\text{DR}}^{2k+1} be the Lagrangian subspace of the de Rham cohomology corresponding to V LΩ 2k+1V_L\subset \Omega^{2k+1}. Choose a Lagrangian decomposition H DR 2k+1=Γ LΓ 1H_{\text{DR}}^{2k+1}= \Gamma_L\oplus\Gamma_1 and let V 1Ω 2k+1V_1\subset \Omega^{2k+1} consists of all closed 2k+12k+1-forms whose DeRham cohomology class lies in Γ 1\Gamma_1.

The Euclidean action is now defined for RV 1R\in V_1. One can decompose R=R L+R L R= R_L+R_L^\perp. S E(R +)=π(R L * ER L iR LR L ) S_E(R^+) = \pi \int (R_L^\perp \wedge *_E R_L^\perp -i R_L\wedge R_L^\perp) and the Minkowskian action1, related to this by Wick rotation is S M(R)=π(R L *R L +R LR L ) S_M(R) = \pi \int (R_L^\perp \wedge * R_L^\perp + R_L\wedge R_L^\perp) The equations of motion that follow from this action are d(*R L R L)=0 d(* R_L^\perp -R_L)=0 and the Bianchi identity, dR=d(R L+R L )=0dR = d(R_L+R_L^\perp)=0. It follows that F +(R)=R L +*R L F^+(R)= R_L^\perp + * R_L^\perp is self-dual and closed.

This action has, in addition to the usual gauge invariance under shifting the 2k2k-form gauge potential by a closed 2k2k-form with integral periods, another gauge invariance, under which RR+vR\mapsto R +v, for v(V LV 1) cptv\in (V_L\cap V_1)^{\text{cpt}}, a compactly-supported, exact 2k+12k+1-form in V 2V_2. This extra gauge invariance doesn’t play any role classically, but is important in the quantization of the theory.

At least, at the classical level, Moore and Belov’s approach is closely related to the old work of Henneaux and Teitelboim


1 There’s a caveat here. Whereas, in Euclidean signature, a choice of Lagrangian subspace, V LV_L, automatically defined a Lagrangian decomposition, Ω 2k+1=V L* EV L\Omega^{2k+1}= V_L\oplus *_E V_L, in Minkowski signature, one has to impose, by hand, the condition V L* MV L={0} V_L\cap *_M V_L = \{0\} There’s always a nowhere-vanishing timelike vector field, ξ\xi on a Lorentzian manifold, MM. An example of a Lagrangian subspace satisfying this constraint is V L(ξ)={ϕΩ 2k+1|i ξϕ=0} V_L(\xi) = \{ \phi\in \Omega^{2k+1}| i_\xi \phi =0\}

Posted by distler at May 13, 2006 3:12 AM

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