U-duality Invariants Research Articles

Jordan duality and Freudenthal duality

We define a Jordan duality A → A* on the quantised charges A of five dimensional black holes and strings and an analogous Freudenthal duality x → x̃ on the quantised charges x of four dimensional stringy black holes, both of which are distinct from U-duality but leave the lowest order entropy invariant. We discuss the action of these maps on discrete U-duality invariants. Based on joint work with L. Borsten, M. J. Duff, and W. Rubens.

Branes, weights and central charges

We study the properties of half-supersymmetric branes in string theory with 32 supercharges from a purely group-theoretical point of view using the U-duality symmetry of maximal supergravity and the R-symmetry of the corresponding supersymmetry algebra. In particular, we show that half-supersymmetric branes are always associated to the longest weights of the U-duality representation of the potentials coupling to them. We compare the features of branes with three or more transverse directions (that we call "standard" branes) to those with two or less transverse directions (that we denominate "non-standard" branes). We show why the BPS condition of the non-standard branes is in general degenerate and for each case we calculate this degeneracy. We furthermore show how the orbits of multi-charge configurations of non-standard branes can be calculated and give the U-duality invariants describing these orbits. We show that different orbits of non-standard branes can have the same BPS condition.

Observations on integral and continuous U-duality orbits in supergravity

One would often like to know when two a priori distinct extremal black p-brane solutions are in fact related by U-duality. In the classical supergravity limit the answer for a large class of theories has been known for some time now. However, in the full quantum theory the U-duality group is broken to a discrete subgroup, a consequence of the Dirac–Zwanziger–Schwinger charge quantization conditions. The question of U-duality orbits in this case is a nuanced matter. In the present work we address this issue in the context of supergravity in four, five and six dimensions. The purpose of this paper is to present and clarify what is currently known about these orbits while at the same time filling in some of the details not yet appearing in the literature. For the continuous case we present the cascade of relationships existing between the orbits, generated as one descends from six to four dimensions, together with the corresponding implications for the associated moduli spaces. In addressing the discrete case we exploit the mathematical framework of integral Jordan algebras, the integral Freudenthal triple system and, in particular, the work of Krutelevich. The charge vector of the dyonic black string in D = 6 is related to a two-charge reduced canonical form uniquely specified by a set of two arithmetic U-duality invariants. Similarly, the black hole (string) charge vectors in D = 5 are equivalent to a three-charge canonical form, again uniquely fixed by a set of three arithmetic U-duality invariants. However, the situation in four dimensions is, perhaps predictably, less clear. While black holes preserving more than 1/8 of the supersymmetries may be fully classified by the known arithmetic invariants, 1/8-BPS and non-BPS black holes yield increasingly subtle orbit structures, which remain to be properly understood. However, for the very special subclass of projective black holes a complete classification is known. All projective black holes are related to a four- or five-charge canonical form determined uniquely by the set of known arithmetic U-duality invariants. Moreover, acts transitively on the charge vectors of projective black holes with a given leading-order entropy.

R4 couplings and automorphic unipotent representations

Four-graviton, eight-derivative couplings in the low energy effective action of toroidal type II string compactifications are tightly constrained by U-duality invariance and by supersymmetry. In this note, we revisit earlier proposals for the automorphic form governing these couplings in dimension D=3,4,5,6, and propose that the correct automorphic form is the minimal theta series for the corresponding U-duality group. Evidence for this proposal comes from i) the matching of infinitesimal characters, ii) the fact that the Fourier coefficients have support on 1/2-BPS charges and iii) decompactification limits. In particular, we show that non-perturbative effects can be interpreted as 1/2-BPS instantons, or 1/2-BPS particles in one dimension higher (together with Taub-NUT instantons in the D=3 case). Based on similar considerations, we also conjecture the form of 1/4-BPS saturated couplings such as $\nabla^4 R^4$ couplings in the same dimensions.

Universality of the superpotential for [formula omitted] extremal black holes

We provide a general strategy to obtain the superpotential W for both BPS and non-BPS extremal black holes in N = 2 four-dimensional supergravities based on symmetric spaces. This extends the construction of W in terms of U-duality invariants that was presented in previous work on the t 3 model. As an application, we explicitly provide W and the solutions to the related gradient flows for the st 2 and stu models. The procedure is shown to hold also for the full N = 8 theory. The role of flat directions in moduli space is clarified.

Black holes admitting a Freudenthal dual

The quantized charges x of four-dimensional stringy black holes may be assigned to elements of an integral Freudenthal triple system whose automorphism group is the corresponding U duality and whose U-invariant quartic norm {delta}(x) determines the lowest-order entropy. Here, we introduce a Freudenthal duality x{yields}x-tilde, for which x-tilde-tilde=-x. Although distinct from U duality, it nevertheless leaves {delta}(x) invariant. However, the requirement that x-tilde be an integer restricts us to the subset of black holes for which {delta}(x) is necessarily a perfect square. The issue of higher-order corrections remains open as some, but not all, of the discrete U-duality invariants are Freudenthal invariant. Similarly, the quantized charges A of five-dimensional black holes and strings may be assigned to elements of an integral Jordan algebra, whose cubic norm N(A) determines the lowest-order entropy. We introduce an analogous Jordan dual A*, with N(A) necessarily a perfect cube, for which A**=A and which leaves N(A) invariant. The two dualities are related by a 4D/5D lift.

Intersecting attractors

We apply the entropy formalism to the study of the near-horizon geometry of extremal black p-brane intersections in D>5 dimensional supergravities. The scalar flow towards the horizon is described in terms an effective potential given by the superposition of the kinetic energies of all the forms under which the brane is charged. At the horizon active scalars get fixed to the minima of the effective potential and the entropy function is given in terms of U-duality invariants built entirely out of the black p-brane charges. The resulting entropy function reproduces the central charges of the dual boundary CFT and gives rise to a Bekenstein-Hawking like area law. The results are illustrated in the case of black holes and black string intersections in D=6, 7, 8 supergravities where the effective potentials, attractor equations, moduli spaces and entropy/central charges are worked out in full detail.

First order description of black holes in moduli space

We show that the second order field equations characterizing extremal solutions for spherically symmetric, stationary black holes are in fact implied by a system of first order equations given in terms of a prepotential W. This confirms and generalizes the results in [14]. Moreover we prove that the squared prepotential function shares the same properties of a c-function and that it interpolates between M^2_{ADM} and M^2_{BR}, the parameter of the near-horizon Bertotti-Robinson geometry. When the black holes are solutions of extended supergravities we are able to find an explicit expression for the prepotentials, valid at any radial distance from the horizon, which reproduces all the attractors of the four dimensional N>2 theories. Far from the horizon, however, for N-even our ansatz poses a constraint on one of the U-duality invariants for the non-BPS solutions with Z \neq 0. We discuss a possible extension of our considerations to the non extremal case.

Magic supergravities,N=8and black hole composites

We present explicit U-duality invariants for the $\mathbb{R}$, $\mathbb{C}$, $\mathbb{Q}$, $\mathbb{O}$ (real, complex, quaternionic, and octonionic) magic supergravities in four and five dimensions using complex forms with a reality condition. From these invariants we derive an explicit entropy function and corresponding stabilization equations which we use to exhibit stationary multicenter $1/2$ Bogomol'nyi-Prasad-Sommerfield (BPS) solutions of these $N=2$ $d=4$ theories, starting with the octonionic one with ${E}_{7(\ensuremath{-}25)}$ duality symmetry. We generalize to stationary $1/8$ BPS multicenter solutions of $N=8$, $d=4$ supergravity, using the consistent truncation to the quaternionic magic $N=2$ supergravity. We present a general solution of non-BPS attractor equations of the STU truncation of magic models. We finish with a discussion of the BPS-non-BPS relations and attractors in $N=2$ versus $N=5$, 6, 8.

T-duality, and the K-theoretic partition function of type IIA superstring theory

We study the partition function of type IIA string theory on 10-manifolds of the form T 2× X, where X is 8-dimensional, compact, and spin. We pay particular attention to the effects of the topological phases in the supergravity action implied by the K-theoretic formulation of RR fields, and we use these to check the T-duality invariance of the partition function. We find that the partition function is only T-duality invariant when we take into account the T-duality anomalies in the RR sector, the fermionic path integral (including 4-fermi interaction terms), and 1-loop corrections including worldsheet instantons. We comment on applications of our computation to speculations about the role of the Romans mass in M-theory. We also discuss some issues which arise when one attempts to extend these considerations to checking the full U-duality invariance of the theory.

Microscopic Entropy of the Most General BPS Black Hole for Type II/M-Theory on Torii

In the present dissertation we shall review the statistical computation of the entropy for the most general static BPS black hole solution in the framework of toroidally compactified type II/M-theory. This achievement is inscribed within a research project aimed to the study of the microscopic properties of this kind of solutions in relation to U-duality invariants (e.g. the entropy) computed on the corresponding macroscopic (supergravity) description.

MONOPOLES AND BLACK HOLE ENTROPY

We consider the entropy of a black hole which has zero area horizon. The microstates appear as monopole solutions of the effective theory on the corresponding brane configurations. The resulting entropy formula coincides with the one expected from stringy calculation and agrees with U-duality invariance of entropy.

U-duality invariance of the four-dimensional Born-Infeld theory

We calculate the Hamiltonian of a compactified D4-brane, with general fluxes and moduli, and find the BPS-mass. The results are invariant under the complete U-duality SO(5,5,Z).

A note on non-perturbative R4 couplings

Exact non-perturbative results have been conjectured for R 4 couplings in type II maximally supersymmetric string theory. Strong evidence has already been obtained, but contributions of cusp forms, invisible in perturbation theory, have remained an open possibility. In this note, we use the D=8 N=2 superfield formalism of Berkovits to prove that supersymmetry requires the exact R 4 threshold to be an eigenmode of the Laplacian on the scalar manifold with a definite eigenvalue. Supersymmetry and U-duality invariance then identify the exact result with the order-3/2 Eisenstein series, and rule out cusp form contributions.

Branes, central charges and U-duality invariant BPS conditions

In extended supergravity theories there are p-brane solutions preserving different numbers of supersymmetries, depending on the charges, the spacetime dimension and the number of original supersymmetries (8, 16 or 32). We find U-duality invariant conditions on the quantized charges which specify the number of supersymmetries preserved with a particular charge configuration. These conditions relate U-duality invariants to the picture of intersecting branes. The analysis is carried out for all extended supergravities with 16 or 32 supersymmetries in various dimensions.