# Genus one topological string amplitude and Ray-Singer torsion

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In BCOV's paper http://arxiv.org/abs/hep-th/9309140 the genus one string amplitude of a Calabi-Yau 3-fold was explained in the B-model as the Ray-Singer torsion (there is a similar discussion in the Mirror Symmetry Book chapter 35). Here are my questions.

1. The first step in deriving this equivalence is to see that

$F_1 = \frac{1}{2} \int \frac{d \tau}{\tau_2} {\rm Tr} (-1)^F F_L F_R q^{H_L} \overline{q}^{H_R}$

This is an exercise in the Mirror Symmetry Book (exercise 31.1.2). However I am a mathematician and don't quite see why this is true. Can anyone explain in more details how to derive the above identity? (I sort of know what path integral and correlation function mean and know the meanings of the operators in the above formula and on which Hilbert space they act.)

2. My real question is that in Landau-Ginzburg B-model, does the above formula still mean something physically interesting? This is why I would like to understand the above formula in the Calabi-Yau case, and to see if the same argument works for LG model. asked Apr 26, 2015

What are $F_L$ and $F_R$?

@Ryan Thorngren, $F_L = F_V + F_A, F_R = F_V- F_A$, where $F_V$ and $F_A$ are the operators corresponding to the vector and axial R-symmetries.

Cool. My expectation is that those are there because of the topological twist. The rest of the trace is the standard expression for the supersymmetric index on a torus. The integral is integration over the hyperbolic upper half plane, which covers the moduli of genus 1 Riemann surfaces.

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This answer assumes some knowledge of bosonic string theory.

In section 7.3 of volume 1 of Polchinski's textbook on string theory, it is proved that the genus 1 amplitude for the bosonic string without any vertex operator insertion is given by

$F_1 \sim \frac{1}{2} \int_F \frac{d\tau d\overline{\tau}}{Im(\tau)} <b(0) \overline{b}(0)\bar{c}(0)c(0)>$

where $b$, $c$ (resp $\overline{b}$, $\overline{c}$) are the left (resp. right) antighost and ghost fields and where the integration is over the fundamental domain $F$ for the action of $SL_2(\mathbb{Z})$ on the upper half-plane, i.e. the moduli space of elliptic curves. Essentially, there is one insertion of the antighosts $b$, $\overline{b}$ because a genus 1 complex curve has one complex moduli and there is one insertion of the ghosts $c$, $\overline{c}$ because a genus 1 complex curve has a one complex dimensional group of automorphisms.

Now the key point is that the algebraic structure of the topological twist of a $N=2$ two dimensional conformal field theory is similar to the  algebraic structure appearing in the covariant BRST quantization of the bosonic string. In the topological twisted theory, we are looking at the cohomology of a nilpotent operator $Q=\oint G^+$, the stress tensor is $Q$-exact: $T(z) = \{Q,G^-(z)\}$ and there is a $U(1)$ R-symmetry  current $J(z)$ with respect to which $G^+$ and $G^-$ are respectively of charge $1$ and $-1$. In the BRST formulation of the bosonic string, we are looking at the cohomology of the nilpotent BRST operator $Q_B=\oint j_B$ where $j_B$ is the ghost current, the stress tensor is $Q_B$-exact: $T(z) =\{Q_B, b(z)\}$ and there is a $U(1)$ ghost number current $J_B(z) \sim b(z)c(z)$ with respect to which $j_B$ and $b$ have respectively of charge $1$ and $-1$. This suggests the identification of $Q$ with $Q_B$, $G^+$ with $j_B$, $G^-$ with $b$ and $J$ with $J_B$. As we have for the bosonic string

$F_1 \sim \frac{1}{2} \int_F \frac{d\tau d\overline{\tau}}{Im \tau}<J_B(0)\overline{J_B}(0)>$

we have for the topological string that

$F_1 \sim \frac{1}{2} \int_F \frac{d\tau d\overline{\tau}}{Im \tau}<J(0)\overline{J}(0)>$

As $F_L$ and $F_R$ are the charges associated to the currents $J$ and $\overline{J}$, the correlation function $<J(0)\overline{J}(0)>$ is given in operator formalism by $Tr (-1)^F F_L F_R q^{H_L}\overline{q}^{H_R}$.

The preceding derivation works for any topological twist of a $N=(2,2)$ two dimensional conformal field theory. It applies as well to a Calabi-Yau sigma model as to a LG model.

answered May 4, 2015 by (5,120 points)
edited May 4, 2015 by 40227
@40227, I guess in "operator formalism", F_L, F_R should act on certain Hilbert space of some loop space (b.c. in (1+d)-d QFT). And its relation to Ray-Singer torsion is because in B-model path integral localizes to the space of constant maps (inside the loop space), right? (localization also respects the correspondence between path integral and operator formalism, right?) In LG B-model the path integral localizes to constant maps to the critical points of the superpotential; but if we think it is first localized to all constant maps, and then localizes to critical points, the first reduction would still give something similar to Ray-Singer torsion (which is something in (0+1)-d QFT), right?

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