### Actions for Self-dual Gauge Fields

In $4k+2$ dimensions, with Minkowski signature, The Hodge $*$-operator, $*^2=1$, when acting on $\Omega^{2k+1}$. One therefore has theories with $2k$-form gauge fields, whose $2k+1$-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 $4k+3$ dimensions.

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

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

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

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

This action has, in addition to the usual gauge invariance under shifting the $2k$-form gauge potential by a closed $2k$-form with integral periods, another gauge invariance, under which $R\mapsto R +v$, for $v\in (V_L\cap V_1)^{\text{cpt}}$, a compactly-supported, exact $2k+1$-form in $V_2$. 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, $V_L$, automatically defined a Lagrangian decomposition, $\Omega^{2k+1}= V_L\oplus *_E V_L$, in Minkowski signature, one has to impose, by hand, the condition
$V_L\cap *_M V_L = \{0\}$
There’s always a nowhere-vanishing timelike vector field, $\xi$ on a Lorentzian manifold, $M$. An example of a Lagrangian subspace satisfying this constraint is
$V_L(\xi) = \{ \phi\in \Omega^{2k+1}| i_\xi \phi =0\}$