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Universality of the topological string at large radius and NS-brane resurgence

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Abstract

We show that there is a natural universal limit of the topological string free energies at the large radius point. The new free energies keep a nonholomorphic dependence on the complex structure moduli space and their functional form is the same for all Calabi–Yau geometries, compact and noncompact alike. The asymptotic nature of the free energy expansion changes in this limit due to a milder factorial growth of its coefficients, and this implies a transseries extension with instanton effects in \(\exp (- 1/g_s^2)\), of NS-brane type, rather than \(\exp (-1/g_s)\), of D-brane type. We show a relation between the instanton action of NS-brane type and the volume of the Calabi–Yau manifold which points to a possible interpretation in terms of NS5-branes. A similar rescaling limit has been considered recently leading to an Airy equation for the partition function which is here used to explain the resurgent properties of the rescaled transseries.

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Notes

  1. We include a superindex \({}^{(0)}\) to specify that a quantity is related to perturbation theory.

  2. The constant map contribution to the free energies is present for every geometry and it implies a constant instanton action, \(4\pi ^2{i}\). We subtract the constant map contribution to get rid of this action without losing generality.

  3. We can still rescale u and \(H^{(0),u}_g\) by geometry-independent numbers. Here, we make a particular choice such that the leading coefficients of \(H^{(0),u}_g\) are the rational numbers \(a^{(0)}_g\) relevant to [7]. Also, the integration constants arising from (36) must be zero to agree with \(H^{(0),u}(u=0) = 0\). Recall that we subtracted the constant map contribution and \(u=0\) represents the holomorphic limit in the large radius frame.

  4. In the language of the nonholomorphic generators introduced in [6], \(T_2\) takes the role of \(S^{zz}\). A similar definition for u applies in that case with \(T_2 =: T_{2,{\text {hol}}} + \tilde{b}{\kappa }^{-1}u\). Note that no rescaling by \(z^2\) is involved.

  5. If \(f(\xi ) = \sum f_n \, \xi ^n\), then \([\xi ^n](f(\xi )) := f_n\).

  6. In (50), the transseries parameter \(\sigma \) is one of the two integration constants of the equation. The other has been implicitly fixed to reproduce the familiar perturbative series.

  7. Note how the Stokes constant, \(S_1\), and the one-instanton coefficient \(H^{(1),u}_0\) appear multiplying each other, so that a change of normalization in \(H^{(1),u}_0\) will change \(S_1\) accordingly.

  8. The equation for \(\tilde{H}^{(0),u}\) in that case is

    $$\begin{aligned} \theta _{\tau _s}^2 \tilde{H}^{(0),u} + \left( \theta _{\tau _s} \tilde{H}^{(0),u}\right) ^2 + \left( 1- \frac{\zeta - 1/3}{\tau _s} \right) \theta _{\tau _s} \tilde{H}^{(0),u} + \frac{5}{36} = \frac{1}{\lambda _s^4} \frac{(\zeta -1)^2(4\zeta -1)}{36}, \end{aligned}$$
    (70)

    where the combination \(\zeta = 1 - 2 \tau _s/u\) is kept fixed.

  9. The actual definition for the alien derivative takes into account the choice of analytic continuation of the Borel transform around the singularities. In this case, there is only one singularity and the definition collapses to the one we are using.

  10. Note that the Borel transform leaves out \(c_0\) which reappears as the proper residue times \(-{i}\).

  11. The instanton action \(\tilde{A} = 4/3\) becomes \(A = 2/3\) once we take the factor of 2 in (70) between x and \(\tau _s\). Removing the constraint that \(\zeta \) is fixed to go from \(\tilde{H}\) to H takes some extra work. Alternatively, one can take a resurgence approach to (49); see also the comments below.

  12. Then again, Stokes phenomenon has the generic property of turning on the value of the transseries parameter \(\sigma _\text {NS}\) so there may be values of z and \(S^{zz}\) for which these sectors are visible through resummation.

  13. For example, in local \(\mathbb {P}^2\) the propagator in essentially proportional to \(\hat{E}_2(\tau ,\overline{\tau }) = E_2(\tau ) - \frac{6{i}}{\pi (\tau -\overline{\tau })}\) and \(\tau = \partial _T^2 F_0\) is proportional to T to leading order.

  14. Both W and its classical truncation are only approximate solutions the equation for the instanton action.

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Acknowledgements

I would like to thank David Sauzin, Emanuel Scheidegger, Ricardo Schiappa, and Marcel Vonk for helpful discussions and comments. I also appreciate useful comments and observations by Marcos Mariño, Boris Pioline, Ricardo Schiappa, and Marcel Vonk on a draft of this paper. This research is supported by the FCT-Portugal Grant EXCL/MAT-GEO/0222/2012.

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  1. CAMGSD, Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1, 1049-001, Lisbon, Portugal

    Ricardo Couso-Santamaría

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Correspondence to Ricardo Couso-Santamaría.

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Couso-Santamaría, R. Universality of the topological string at large radius and NS-brane resurgence. Lett Math Phys 107, 343–366 (2017). https://doi.org/10.1007/s11005-016-0906-y

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