Musings

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 action¹, 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