ADS/CFT
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Strings ADS/CFT |
I have been working on this project in collaboration with Ansar Fayyazuddin of the University of Stockholm. We are interested in configurations of M5-branes wrapped on a Riemann surface. In general these configurations are related to four-dimensional N=2 supersymmetric field theories, with the Riemann surface being the Seiberg-Witten curve. These Hanany-Witten brane constructions give a geometric interpretation of many features of the gauge theories. However, another relation between string/M-theory and gauge theories is via the AdS/CFT correspondence. To pursue this approach we need to know the supergravity description of such brane configurations.
In [1] we analysed the BPS equations which must be satisfied for such a supersymmetric solution of 11-dimensional supergravity. We solved the problem for a general Riemann surface, giving the metric and four-form field strength explicitly in terms of a Kähler potential. However, we were unable to solve explicitly for the Kähler potential itself. The equation for the Kähler potential is related to the complex Monge-Ampère equation. We considered the simplification of smearing some of the branes in the special configuration where the (degenerate) Riemann surface describes orthogonally intersecting branes. This corresponds to the case where the gauge theory is conformal and we could relate our solution to the previously known T-dual type IIB solution (after compactifying on a circle to type IIA.) Even with the smearing of some branes, we could only solve in the near-horizon limit. This is all that is required for the AdS/CFT correspondence but it is nevertheless an interesting unsolved problem to find the full supergravity solution.
To find supergravity duals of more general N=2 supersymmetric gauge theories, we need to find the near-horizon solutions for a general Riemann surface. To do this we first considered [2] the problem of finding a fully localised near-horizon solution for intersecting branes. We knew from the AdS/CFT correspondence that this solution, dual to a conformal field theory, must be of the form of a warped AdS produce space. By requiring that our general form of the solution could be expressed as a warped AdS product, we derived some extra constraints. These constraints were consistent with the equations determining the Kähler potential and enabled us to find the solution. This is the first fully localised solution for intersecting branes, although it is still only known in the near-horizon limit.
The Riemann surfaces we are considering are determined by the zeroes of a holomorphic function of two complex variables (they are embedded in a complex two-dimensional space.) The intersecting branes correspond to degenerate surfaces where this holomorphic function factorises into a product of holomorphic functions of each variable. However, it was fairly simple to see exactly where this function appeared in the solution for intersecting branes. So in [3] we showed that we could fairly easily generalise the solution to arbitrary Riemann surfaces by allowing an arbitrary holomorphic in essentially the same form of the solution. This means that we now have the near-horizon supergravity dual description of a large class of N=2 four-dimensional gauge theories.
We are currently investigating how we can use these supergravity solutions to directly analyse the dual gauge theories. For example, dimensions of field theory operators can be calculated by calculating the mass of perturbations around the supergravity background. Since the general field theories are non-conformal, there are also interesting issues about how the renormalisation group flow can be understood in terms of supergravity.
We are also considering possible generalisations of this construction. There are many known brane configurations corresponding to various types of gauge theories in different dimensions, with certain amounts of supersymmetry. It would be interesting to find near-horizon supergravity solutions of these brane configurations too. We are currently looking (see [4]) at the most similar construction which would describe N=1 four-dimensional gauge theories. This simply corresponds to an M5-brane wrapped on a Riemann surface which is now embedded in a complex three-dimensional space. We hope that our solutions for the N=2 case (which are a special case of this problem) will provide some clues to enable us to solve this more complicated problem.
[1] Ansar Fayyazuddin and Douglas J. Smith, Localized intersections of M5-branes and four-dimensional superconformal field theories, JHEP 04(1999)030, hep-th/9902210.
[2] Ansar Fayyazuddin and Douglas J. Smith, Warped AdS near-horizon geometry of completely localized intersections of M5-branes, JHEP 10(2000)023, hep-th/0006060.
[3] Björn Brinne, Ansar Fayyazuddin, Subir Mukhopadhyay and Douglas J. Smith, Supergravity M5-branes wrapped on Riemann surfaces and their QFT duals, JHEP 12(2000)013, hep-th/0009047.
[4] Björn Brinne, Ansar Fayyazuddin, Tasneem Zehra Husain and Douglas J. Smith, N=1 M5-brane geometries, JHEP 0103 (2001) 052, hep-th/0012194.