The Hitchin functionals and the topological B-model at one loop

TL;DR

This work quantizes Hitchin functionals at quadratic order to probe their connection to the topological B-model. Using the BV formalism, it finds that the one-loop factor for the minimal Hitchin theory, Z_{H,1-loop}, equals I_1/I_0 and does not reproduce the B-model, while the extended Hitchin functional Z_{HE,1-loop} matches Z_{B,1-loop}, resolving the classical discrepancy. The analysis reveals a gravitational anomaly under metric variations for both the B-model and extended Hitchin theory, which reduces to a volume factor in the Ricci-flat Kahler sector and motivates a higher-dimensional embedding reinterpretation. Overall, the extended Hitchin functional emerges as the correct quantum analog to the B-model at one loop, with implications for generalized Calabi-Yau and flux-background contexts.

Abstract

The quantization in quadratic order of the Hitchin functional, which defines by critical points a Calabi-Yau structure on a six-dimensional manifold, is performed. The conjectured relation between the topological B-model and the Hitchin functional is studied at one loop. It is found that the genus one free energy of the topological B-model disagrees with the one-loop free energy of the minimal Hitchin functional. However, the topological B-model does agree at one-loop order with the extended Hitchin functional, which also defines by critical points a generalized Calabi-Yau structure. The dependence of the one-loop result on a background metric is studied, and a gravitational anomaly is found for both the B-model and the extended Hitchin model. The anomaly reduces to a volume-dependent factor if one computes for only Ricci-flat Kahler metrics.

Figures (1)

• Figure 1: To depict the Hodge decomposition of a real form of type $(p,p)$, we introduce a vertical line and draw only one half of a square, symbolizing that the form is self-conjugate. The Hodge decomposition of a real form of type $(1,1)$ is sketched on the left; since ${\overset{ \bullet}{\Omega}}{}^{11}$ and ${\underset{ \bullet}{\Omega}}^{11}$ are real, they are depicted by little triangles rather than little squares, and as ${\Omega \bullet}{}^{11}$ is the complex conjugate of ${{\bullet} \Omega}^{11}$, only one of them is shown. More generally, we will usually only depict one of each complex conjugate pair of fields. Another subtlety arises in the Hodge decomposition when $\Omega^{pq}$ is complex but one of the spaces arising in its Hodge decomposition is real. On the right, we show this for $\Omega^{10}$. The little square depicting ${\Omega \bullet}^{10}$ is divided into upper and lower triangles representing $\partial$ of a real or imaginary $(0,0)$-form, respectively. Alternatively, they represent images of real forms of type $(0,0)$ and $(1,1)$, as shown.