Wilson lines for AdS5 black strings

Received: December Wilson lines for AdS5 black strings 0 Piazza della Scienza 3 , I-20126 Milano , Italy 1 Dipartimento di Fisica, Universita` di Milano-Bicocca 2 Open Access , c The Authors We describe a simple method of extending AdS5 black string solutions of 5d gauged supergravity in a supersymmetric way by addition of Wilson lines along a circular direction in space. When this direction is chosen along the string, and due to the specific form of 5d supergravity that features Chern-Simons terms, the existence of magnetic charges automatically generates conserved electric charges in a 5d analogue of the Witten effect. Therefore we find a rather generic, model-independent way of adding electric charges to already existing solutions with no backreaction from the geometry or breaking of any symmetry. We use this method to explicitly write down more general versions of the Benini-Bobev black strings [1, 2] and comment on the implications for the dual field theory and the similarities with generalizations of the Cacciatori-Klemm black holes [3] in AdS4. ArXiv ePrint: 1411.2432 Black Holes in String Theory; AdS-CFT Correspondence; Supergravity - Extending Benini-Bobev solutions Near-horizon BTZ with Wilson lines Relation to axions in 4d Dual field theory interpretation Wilson lines along the string/axions Wilson lines along the Riemann surface A Explicit form of the attractor solution in 4D In this short note we present a method of adding Wilson lines to already existing 5d black string solutions in a supersymmetry preserving way. It is a well-known feature of black string and black ring solutions in ungauged supergravity [46] that Wilson lines are zero-modes of the equations of motion and supersymmetry variations, which lead to nontrivial electric charges in the full asymptotically flat solution. This fact seems to have been overlooked in the literature dealing with asymptotically AdS solutions in 5d gauged supergravity [1, 2, 714]. Here we aim to close this gap and show that, apart from a few subtleties, one can extend the known black string solutions in the same way. This amounts to introducing a non-vanishing background gauge fields number of gauge fields. We will focus on black string solutions supported by magnetic charges, whose horizons can be Riemann surfaces of any genus. One obvious choice for asymptotic AdS5 appropriately). An alternative choice1 arises for black strings with a throughout spacetime, since there exist nontrivial one-cycles in these cases (assuming again an appropriate foliation of AdS5). In both cases the Wilson lines (1.1) cannot be gauged away, but only periodically identified, as wI The addition of these Wilson lines, trivial as it might seem, actually introduces inter1We thank Nikolay Bobev for pointing out this possibility. string, the Wilson lines lead to nonvanishing conserved electric charges in the spacetime due to the Chern-Simons terms present in 5d supergravity and the fact that black strings are already supported by magnetic charge. The resulting electric charges are therefore proportional to both the Wilson lines and the magnetic charges, similarly to the Witten effect [15] in 4d that requires a nonvanishing theta-angle. In fact, as can be seen explicitly by dimensional reduction [16, 17], the Wilson lines can be thought of as a 5d analogue of surface, one obtains a deformation of the original magnetically charged solution (cf. [18] for analogous results in 7d) with no electric charges but an interesting dual field theory interpretation discussed in section 4. conventions [7] is given by e1 e1 L = with a gauge coupling constant g and a scalar potential depending on the constant FI fields when the condition V = 6 CIJK XI XJ XK = 1 , is satisfied. The Lagrangian is completely specified by the constant symmetric tensor of coefficients CIJK . All the physical quantities in (1.2) can be expressed in terms of the homogeneous cubic polynomial V, i.e. one can uniquely determine the scalar and gauge Let us now consider a black string that is already satisfying the equations of motion. In the known examples in gauged supergravity [1, 2, 7, 914] (and similar to the ones in ungauged supergravity [46]), the metric and field strengths are of the form ds2 = f (r)2 and asymptotically locally AdS5 foliated in R Now we want to argue that the addition of the Wilson lines (1.1) to the gauge field solution above immediately solves the equations of motion without changing any other correspond to z). First note that the field strengths do not change upon the addition 2One can naturally extend the following arguments to more general theories with hypermultiplets with virtually no difference. of Wilson lines, and bare gauge fields only enter in the Lagrangian (1.2) via the Chern This term does not couple with either the metric or scalars, therefore one just needs to make sure that the Maxwell equations are satisfied to conclude that we have found a new solution. But even this t (...truncated)