Partition function on torus can be defined using a generalized Witten like index as given below:

$$F_1=\int_\mathbb{T}\frac{d^2\tau}{\tau_2} Tr(-1)^F F_LF_R \;q^{L_0} \bar{q}^{\bar{L_0}},$$ where $\mathbb{T}$ is the fundamental region for $\tau$ defined in upper half plane.

My question is pretty simple and that is to evaluate the large volume limit of this generalized index which is the partition function on torus for topological string.

By large volume limit one means $t, \bar t\rightarrow \infty$ where $t$ and $\bar{t}$ are complexified Kaehler moduli. In a more simple terms, I need only zero modes contributions for evaluation of this Witten like index which gives leading contribution to evaluation of $F_1$ for constant maps like B model. This is done in BCOV1.

Now at disposal we have equation number (5) from BCOV1 for calculation:

$$Tr[(-1)^FF_LF_R \;q^{L_0} \bar{q}^{\bar{L_0}}]_{\infty}=\frac{1}{(2\pi\tau_2)^n}\int d\mu(\prod_{r}d\psi^{\bar r}d\psi^{r}d\chi^{\bar{r}}d\chi^{r})g_{i\bar j}\psi^i\psi^{\bar{j}}g_{k\bar{l}}\chi^k\chi^{\bar{l}}\exp[-\tau_2 R_{i\bar j k\bar l}\psi^i\psi^{\bar{j}}\chi^k\chi^{\bar{l}}]$$

$n$ is complex dimension of target space, $d\mu$ is volume form defined on it and usual expression for left and right fermion numbers $F_L$ and $F_R$ are substituted. This looks very much like the analogue of Witten index calculation but has it's differences also. My problem is basically the factor $1/(2\pi\tau_2)^n$ in front of the whole expression for which I can't seem to find a proper justification. First I thought it is just some normalization factor I do not need to worry about but actually it has huge consequences like if one writes down the Gromov-Witten A model correlation function then to obtain the number of elliptic curves counting in, for example, a quintic CY this factor is responsible for a factor of 50/12 which needs to be used for relation to A model of GW to Kodaira-Spencer of B model in classical intersection theory just like for genus 0 case, one needs to know $\int k\wedge k\wedge k=5$ on quintic CY in CP^4 on A side to count rational curves.

So my question is seemingly very simple: How does one get the factor of $1/(2\pi\tau_2)^n$ in front of the whole expression above.


1 Answer 1


Sorry, I almost forgot that I posted this here. I actually figured out how to get that factor albeit with help.

The thing is that I was assuming that no bosonic modes contribute in the limit I was considering but in fact they do cause of the bosonic part of Hamiltonian which is just P^2/2 coming from constant modes of complex chiral bosons on torus which are constant for the case and not zero.

To account for supersymmetry in conjunction with \psi and \chi, I need 2n of these real bosons of which each contributes a factor $\int\frac{dp}{2\pi}exp(-\tau_2\frac{p^2}{2})$ giving $1/\sqrt(2\pi\tau_2)$ and for 2n of these 1 dimensional contribution, one has the desired factor.

Thanks to Cumrun Vafa for the hint.


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