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We describe the asymptotic behavior of minimal area submanifolds in product spacetimes of an asymptotically hyperbolic space times a compact internal manifold. In particular, we find that unlike the case of a minimal area submanifold just in an asymptotically hyperbolic space, the internal part of the boundary submanifold is constrained to be itself a minimal area submanifold. For applications to holography, this tells us what are the allowed flavor branes that can be added to a holographic field theory. We also give a compact geometric expression for the spectrum of operator dimensions associated with the slipping modes of the submanifold in the internal space. We illustrate our results with several examples, including some that havent appeared in the literature before.
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Minimal area surfaces in AdS$_3$ ending on a given curve at the boundary are dual to planar Wilson loops in N=4 SYM. In previous work it was shown that the problem of finding such surfaces can be recast as the one of finding an appropriate parameterization of the boundary contour that corresponds to conformal gauge. A. Dekel was able to find such reparameterization in a perturbative expansion around a circular contour. In this work we show that for more general contours such reparameterization can be found using a numerical procedure that does not rely on a perturbative expansion. This provides further checks and applications of the integrability method. An interesting property of the method is that it uses as data the Schwarzian derivative of the contour and therefore it has manifest global conformal invariance. Finally, we apply Shanks transformation to extend the near circular expansion to larger deformations, the results are in agreement with the new method.
We apply an arbitrary number of dressing transformations to a static minimal surface in AdS(4). Interestingly, a single dressing transformation, with the simplest dressing factor, interrelates the latter to solutions of the Euclidean non linear sigma model in dS(3). We present an expression for the area element of the dressed minimal surface in terms of that of the initial one and comment on the boundary region of the dressed surface. Finally, we apply the above formalism to the elliptic minimal surfaces and obtain new ones.
We define a holographic dual to the Donaldson-Witten topological twist of $mathcal{N}=2$ gauge theories on a Riemannian four-manifold. This is described by a class of asymptotically locally hyperbolic solutions to $mathcal{N}=4$ gauged supergravity in five dimensions, with the four-manifold as conformal boundary. Under AdS/CFT, minus the logarithm of the partition function of the gauge theory is identified with the holographically renormalized supergravity action. We show that the latter is independent of the metric on the boundary four-manifold, as required for a topological theory. Supersymmetric solutions in the bulk satisfy first order differential equations for a twisted $Sp(1)$ structure, which extends the quaternionic Kahler structure that exists on any Riemannian four-manifold boundary. We comment on applications and extensions, including generalizations to other topological twists.
By work of Uhlenbeck, the largest principal curvature of any least area fiber of a hyperbolic $3$-manifold fibering over the circle is bounded below by one. We give a short argument to show that, along certain families of fibered hyperbolic $3$-manifolds, there is a uniform lower bound for the maximum principal curvatures of a least area minimal surface which is greater than one.
The S-matrix of a theory often exhibits symmetries which are not manifest from the viewpoint of its Lagrangian. For instance, powerful constraints on scattering amplitudes are imposed by the dual conformal symmetry of planar 4d $mathcal{N}=4$ super Yang-Mills theory and the ABJM theory. Motivated by this, we investigate the consequences of dual conformal symmetry in six dimensions, which may provide useful insight into the worldvolume theory of M5-branes (if it enjoys such a symmetry). We find that 6d dual conformal symmetry uniquely fixes the integrand of the one-loop 4-point amplitude, and its structure suggests a Lagrangian with more than two derivatives. On integrating out the loop momentum in $6-2 epsilon$ dimensions, the result is very similar to the corresponding amplitude of $mathcal{N}=4$ super Yang-Mills theory. We confirm this result holographically by generalizing the Alday-Maldacena solution for a minimal area string in Anti-de Sitter space to a minimal volume M2-brane ending on pillow-shaped Wilson surface in the boundary whose seams correspond to a null-polygonal Wilson loop. This involves careful treatment of a prefactor which diverges as $1/epsilon$, and we comment on its possible interpretation. We also study 2-loop 4-point integrands with 6d dual conformal symmetry and speculate on the existence of an all-loop formula for the 4-point amplitude.
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