# Spacetime interpretation of gravitational descendants

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Perturbative string theory is obtained by coupling an appropriate two dimensional conformal field theory with two dimensional gravity i.e. S-matrix elements in perturbative string theory are obtained by summation over the genus and integration over the moduli space of Riemann surfaces of correlation functions in the worldsheet conformal field theory.

Topological string theory follows a similar approach: it is defined by coupling an appropriate two dimensional topological quantum field theory with two dimensional topological gravity. Let $X$ be a Calabi-Yau 3-fold. By topological twist of the supersymmetric sigma model of target space $X$, it is possible to construct two topological quantum field theories respectively called $A$ and $B$-model. Coupling with two dimensional gravity defines the $A$ and $B$-model topological string of target $X$ (see for example http://arxiv.org/abs/hep-th/0410178 for a review).

Let me concentrate on the $A$-model. The simplest quantity of interest is the genus $g$ free energy:

$F_g(q) = \sum_{d \in H_2(X,\mathbb{Z})} N_{g,d}q^d$

where $q$  is the exponentional of the (complexified) Kaehler parameters of $X$ and where $N_{g,d}$ is the number of holomorphic genus $g$ degree $d$ curves on $X$. Formally, one can write

$N_{g,d} = \int_{\bar{M}_g(X,d)} 1$

where $\bar{M}_g(X,d)$ is the moduli space of holomorphic genus $g$ degree $d$ curves on $X$. Mathematically the numbers $N_{g,d}$ are called the Gromov-Witten invariants of $X$ (here I skipped details having to do with "virtual" counting...)

It is remarkable that the quantities $F_g(q)$ defined in topological string theory have in fact an interpretation in physical string theory. If we consider type $IIA$ superstring theory compactified on $X$ then we obtain a $N=2$ four dimensional supergravity at low energy. What is remarkable is that the functions $F_g(q)$ appear in the higher derivatives expansion of the low energy action via terms of the form

$F_g(q) R_+^2 F_+^{2g-2}$

where $R_+$ is the (selfdual part of ) the Riemann curvature tensor and $F_+$ is the (selfdual part of) the graviphoton fied strength (the $N=2$ gravitational multiplet contains a $U(1)$ gauge field called the graviphoton). The Kaehler parameters $q$ of $X$ corresponds to vev of complex scalars in the four dimensional theory.

This is very nice but one could ask: is there a similar interpretation in physical strings of more complicated objects in topological strings ? Examples of such objects are insertions of so called gravitational descendants, which are the natural observables in two dimensional gravity. There are defined by

$<\tau_{k_1}...\tau_{k_n}>_g = \int_{\bar{M}_{g,n}(X,d)} \psi_1^{k_1}...\psi_n^{k_n}$

where $\bar{M}_{g,n}(X,d)$ is the moduli space of holomorphic genus $g$ degree $d$ curves on $X$ with $n$ punctures and where the $\psi_i$'s are natural operators inserted at the punctures (mathematically there are first Chern classes of natural complex line bundles on $\bar{M}_{g,n}(X,d)$ )

Introducing formal variables $t_1$, $t_2$,..., one can form generating functions

$F_g(q, t_1,t_2,...)=\sum_{d \in H_2(X,\mathbb{Z})}\sum_{k_1,k_2,...}<\tau_{k_1}...\tau_{k_n}>_g q^d t_1^{k_1}t_2^{k_2}...$

specializing to $F_g(q)$ when the $t_i$'s are put to zero.

My question is: is there an interpretation of $F_g(q, t_1,t_2,...)$ in the four dimensional $N=2$ supergravity obtained by compactifying type $IIA$ superstring theory on $X$? In particular, what is the spacetime interpretation of the hierarchy of the infinite number of new variables $t_1, t_2,...$?

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