Type II AdS Wormholes

Construction of numerical wormhole solutions of type II SUGRA for both massive type IIA on $S^3\times S^3$ and type IIB on $T^{1,1}$ using a shooting method. These solutions provide explicit uplifts of regular 4D/5D AdS wormholes to 10D.

For more information see our preprint:

Gregory J. Loges, Gary Shiu and Thomas Van Riet, A 10d construction of Euclidean axion wormholes in flat and AdS space, arXiv:2302.03688.

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Brief background

Consistent truncations are identified for each model in the reduction to 4D/5D. To find wormhole solutions these necessarily include massive scalars. The actions for the relevant modes are $\ldots$

• Massive type IIA on $S^3\times S^3$:

$$ \begin{align*} S_4 &= \frac{1}{2\kappa_4^2}\int\Big({\star R} - \frac{1}{2}\mathrm{d}\phi\wedge{\star\mathrm{d}\phi} - 6\mathrm{d}u\wedge{\star\mathrm{d}u} + \frac{1}{2}e^{-3u+\phi/2}\mathrm{d}\chi\wedge{\star\mathrm{d}\chi} - {\star\mathcal{V}}\Big)\\ \mathcal{V} &= -12e^{-4u} + 5e^{-9u-\phi/2} + e^{-3u+5\phi/2} \end{align*} $$

• Type IIB on $T^{1,1}$:

$$ \begin{align*} S_5 &= \frac{1}{2\kappa_5^2}\int\Big({\star R} - \frac{1}{2}\mathrm{d}\phi\wedge{\star\mathrm{d}\phi} + \frac{1}{2}e^{2\phi}\mathrm{d}\chi\wedge{\star\mathrm{d}\chi}\\ &\qquad\qquad\qquad -\frac{1}{2}e^{-4u-\phi}\mathrm{d}b\wedge{\star\mathrm{d}b} + \frac{1}{2}e^{-4u+\phi}(\mathrm{d}c - \chi\mathrm{d}b)\wedge{\star(\mathrm{d}c-\chi\mathrm{d}b)}\\ &\qquad\qquad\qquad - \frac{28}{3}\mathrm{d}u\wedge{\star\mathrm{d}u} - \frac{8}{3}\mathrm{d}u\wedge{\star\mathrm{d}v} - \frac{4}{3}\mathrm{d}v\wedge{\star\mathrm{d}v} - {\star\mathcal{V}} \Big)\\ \mathcal{V} &= 2e^{-\frac{8}{3}(4u+v)}\left( 2e^{4u+4v} - 12e^{6u+2v} + 4 \right) \end{align*} $$

In both cases the metric ansatz is taken to be

$$ \mathrm{d}s_d^2 = f^2\mathrm{d}{r^2} + q^2\mathrm{d}\Omega_{d-1}^2 \qquad q(r) = \sqrt{q_0^2 + r^2} $$

so that the coordinate $r$ covers the full wormhole. Numerical AdS wormhole solutions to the equations of motion are found using a shooting method; initial values for $f$ and the scalars at $r=0$ are chosen so that solutions are smooth across $r=0$ (care is needed because $q'(0)=0$) and are parity even/odd. These initial values are then adjusted so that integrating out to large $r$ the solutions may be matched onto AdS boundary conditions, $rf\to 1$ and $u,v,\phi\to0$ for $r\to\infty$.

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Contents overview

• typeIIA_S3xS3.nb and typeIIB_T11.nb: Mathematica notebooks with derivations of the 4D/5D equations of motion directly from 10D, including cross-checks with the 4D/5D actions above, as well as the construction of perturbative solutions used to set the initial conditions at $r=0$ in the shooting method.

• wormholes.py: Implementation of the equations of motion and shooting method, as well as some functionality for extracting the dual one-point functions.

• wh_examples.ipynb: Demonstrates functionality of wormholes.py with examples of numerically constructed wormholes, both with and without the shooting method.

• wh_plots.ipynb: High-res images of wormhole profiles and two-dimensional parameter space, for use in paper.

• wh_animation.ipynb: Plots of $T^{1,1}$ wormholes along a rectangular path in parameter space, for use in an animation.

The functions solve_S3S3() and solve_T11() solve the equations of motion for given wormhole size and initial conditions. Without fine-tuning the initial conditions the geometry quickly becomes singular or the scalars diverge. The functions wormhole_S3S3() and wormhole_T11() return wormhole solutions of size $q_0$ after using a shooting method to optimize the initial conditions.