R-symmetry Charges Research Articles

Spectrum of a supersymmetric color superconductor

We have recently shown that the ground state of mathcal{N} = 4, SU(Nc) super Yang- Mills coupled to Nf ≪ Nc flavors, in the presence of non-zero isospin and R-symmetry charges, is a supersymmetric, superfluid, color superconductor. The holographic description consists of Nf D7-brane probes in AdS5×S5 with electric and instantonic fields on their worldvolume. These correspond to fundamental strings and D3-branes dissolved on the D7-branes, respectively. Here we use this description to determine the spectrum of mesonic excitations. As expected for a charged superfluid we find non-relativistic, massless Goldstone modes. We also find extra ungapped modes that are not associated to the breaking of any global symmetries but to the supersymmetric nature of the ground state. If the quark mass is much smaller than the scale of spontaneous symmetry breaking a pseudo-Goldstone boson is also present. We highlight some new features that appear only for Nf> 2. We show that, in the generic case of unequal R-symmetry charges, the dissolved strings and D3-branes blow up into a D5-brane supertube stretched between the D7-branes.

A supersymmetric color superconductor from holography

We use holography to study d = 4, mathcal{N}=4 , SU(Nc) super Yang-Mills coupled to Nf ≪ Nc quark flavors. We place the theory at finite isospin density nI by turning on an isospin chemical potential μI = Mq, with Mq the quark mass. We also turn on two R-symmetry charge densities n1 = n2. We show that the ground state is a supersymmetric, superfluid, color superconductor, namely a finite-density state that preserves a fraction of supersymmetry in which part of the global symmetries and part of the gauge symmetries are spontaneously broken. The holographic description consists of Nf D7-brane probes in AdS5 × S5. The symmetry breaking is due to the dissolution of some D3-branes inside the D7-branes triggered by the electric field associated to the isospin charge. The massless spectrum contains Goldstone bosons and their fermionic superpartners. The massive spectrum contains long-lived, mesonic quasi-particles if nI ≪ μI3, and no quasi-particles otherwise. We discuss the possibility that, despite the presence of mass scales and charge densities in the theory, conformal and relativistic invariance arise as emergent symmetries in the infrared.

Massive spin 2 excitations in AdS6 \xd7 S2 warped spacetimes

We study (massive) spin-2 fluctuations around warped AdS6 solutions with 16 supersymmetries in type IIB supergravity. We identify two classes of fluctuations which are universally present for all solutions of this form. The holographically dual operators have scaling dimensions Δ = 5 + 3ℓ and Δ = 6 + 3ℓ, where the integer ℓ encodes the R-symmetry charge. They are identified as descendant states in respective BPS multiplets (the current multiplet for Δ = 5). We also compute the normalization of the energy-momentum tensor two-point function and show that it is related to the S5 partition function of the dual 5d SCFTs.

Multiparticle one-loop amplitudes and S-duality in closed superstring theory

Explicit expressions for one-loop five supergraviton scattering amplitudes in both type II superstring theories are determined by making use of the pure spinor formalism. The type IIB amplitude can be expressed in terms of a doubling of ten-dimensional super Yang-Mills tree amplitude, while the type IIA amplitude has additional pieces that cannot be expressed in that manner. We evaluate the coefficients of terms in the analytic part of the low energy expansion of the amplitude, which correspond to a series of terms in an effective action of the schematic form D 2k R 5 for 0 ≤ k ≤ 5 (where R is the Riemann curvature). Comparison with earlier analyses of the tree amplitudes and of the four-particle one-loop amplitude leads to an interesting extension of the action of SL(2, $ \mathbb{Z} $ ) S-duality on the moduli-dependent coefficients in the type IIB theory. We also investigate closed-string five-particle amplitudes that violate conservation of the U (1) R-symmetry charge — processes that are forbidden in supergravity. The coefficients of their low energy expansion are shown to agree with S-duality systematics. A less detailed analysis is also given of the six-point function, resulting in the vanishing of the analytic parts of the R 6 and D 4 R 6 interactions in the ten-dimensional effective action, but not in lower dimensions.

Black holes in higher spin supergravity

We study black hole solutions in Chern-Simons higher spin supergravity based on the superalgebra sl(3|2). These black hole solutions have a U(1) gauge field and a spin 2 hair in addition to the spin 3 hair. These additional fields correspond to the R-symmetry charges of the supergroup sl(3|2). Using the relation between the bulk field equations and the Ward identities of a CFT with $ \mathcal{N} $ = 2 super- $ {{\mathcal{W}}_3} $ symmetry, we identify the bulk charges and chemical potentials with those of the boundary CFT. From these identifications we see that a suitable set of variables to study this black hole is in terms of the charges present in three decoupled bosonic sub-algebras of the $ \mathcal{N} $ = 2 super- $ {{\mathcal{W}}_3} $ algebra. The entropy and the partition function of these R-charged black holes are then evaluated in terms of the charges of the bulk theory as well as in terms of its chemical potentials. We then compute the partition function in the dual CFT and find exact agreement with the bulk partition function.

Supersymmetric quantum criticality supported by baryonic charges

In the context of the $AdS_4\times Q^{111}$ solution of D=11 supergravity we construct supersymmetric zero temperature black brane solutions that interpolate between $AdS_4$ in the UV and $AdS_2\times \mathbb{R}^2$ in the IR. The dual N=2 SCFT has a $U(1)^2$ baryonic symmetry and the solutions carry electric charge with respect to one of the U(1) factors and magnetic charge with respect to the other. The solutions describe stable zero temperature ground states of the deformed SCFT which have finite entropy density. We also construct analogous supersymmetric solutions that flow to $AdS_2\times S^2$ and to $AdS_2\times H^2/\Gamma$ in the IR which, in addition, carry magnetic R-symmetry charge similar to other known wrapped brane solutions.

Critical phenomena in [formula omitted] SYM plasma

Strongly coupled N = 4 supersymmetric Yang–Mills plasma at finite temperature and chemical potential for an R-symmetry charge undergoes a second order phase transition. We demonstrate that this phase transition is of the mean field theory type. We explicitly show that the model is in the dynamical universality class of ‘model B’ according to the classification of Hohenberg and Halperin, with dynamical critical exponent z = 4 . We study bulk viscosity in the mass deformed version of this theory in the vicinity of the phase transition. We point out that all available models of bulk viscosity at continuous phase transition are in conflict with our explicit holographic computations.

Thermodynamics of large N gauge theories with chemical potentials in a 1/D expansion

In order to understand thermodynamical properties of N D-branes with chemical potentials associated with R-symmetry charges, we study a one dimensional large N gauge theory (bosonic BFSS type model) as a first step. This model is obtained through a dimensional reduction of a 1+D dimensional SU(N) Yang-Mills theory and we use a 1/D expansion to investigate the phase structure. We find three phases in the \mu-T plane. We also show that all the adjoint scalars condense at large D and obtain a mass dynamically. This dynamical mass protects our model from the usual perturbative instability of massless scalars in a non-zero chemical potential. We find that the system is at least meta-stable for arbitrary large values of the chemical potentials in D \to \infty limit. We also explore the existence of similar condensation in higher dimensional gauge theories in a high temperature limit. In 2 and 3 dimensions, the condensation always happens as in one dimensional case. On the other hand, if the dimension is higher than 4, there is a critical chemical potential and the condensation happens only if the chemical potentials are below it.

Counting BPS operators in [formula omitted] SYM

The free field partition function for a generic U ( N ) gauge theory, where the fundamental fields transform in the adjoint representation, is analysed in terms of symmetric polynomial techniques. It is shown by these means how this is related to the cycle polynomial for the symmetric group and how the large N result may be easily recovered. Higher order corrections for finite N are also discussed in terms of symmetric group characters. For finite N, the partition function involving a single bosonic fundamental field is recovered and explicit counting of multi-trace quarter BPS operators in free N = 4 super-Yang–Mills discussed, including a general result for large N. The partition function for quarter BPS operators in the chiral ring of N = 4 super-Yang–Mills is analysed in terms of plane partitions. Asymptotic counting of BPS primary operators with differing R-symmetry charges is discussed in both free N = 4 super-Yang–Mills and in the chiral ring. Also, general and explicit expressions are derived for SU ( 2 ) gauge theory partition functions, when the fundamental fields transform in the adjoint, for free field theory.

Generating charge from diffeomorphisms

We unravel some subtleties involving the definition of sphere angular momentum charges in AdSq × Sp spacetimes, or equivalently, R-symmetry charges in the dual boundary CFT. In the AdS3 context, it is known that charges can be generated by coordinate transformations, even though the underlying theory is diffeomorphism invariant. This is the bulk version of spectral flow in the boundary CFT. We trace this behavior back to special properties of the p-form field strength supporting the solution, and derive the explicit formulas for angular momentum charges. This analysis also reveals the higher dimensional origin of three dimensional Chern-Simons terms and of chiral anomalies in the boundary theory.

Semi-classical string solutions for [formula omitted] SYM

We study semi-classically the dynamics of string solitons in the Maldacena–Nuñez background, dual in the infra-red to N=1 , d=4 SYM. For closed string configurations rotating in the S 2× R space wrapped by the stack of N D-branes we find a behaviour that indicates the decoupling of the stringy Kaluza–Klein modes with sufficiently large R-symmetry charge. We show that the spectrum of a pulsating string configuration in S 2 coincides with that of a N=2 super-sine-Gordon model. Closed string configurations spinning in the transversal S 3 give a relation of the energy and the conserved angular momentum identical to that obtained for configurations spinning in the S 5 of the AdS 5× S 5, dual to N=4 SYM. In order to obtain non-trivial relations between the energy and the spin, we also consider conical-like configurations stretching along a radial variable in the unwrapped directions of the system of D-branes and simultaneously along the transversal direction. We find that in this precise case, these configurations are unstable—contrary to other backgrounds, where we show that they are stable. We point out that in the Poincaré-like coordinates used for the Maldacena–Nuñez background it seems that it is not possible to reproduce the well-known field theory relation between the energy and the angular momentum. We reach a similar conclusion for the Klebanov–Strassler background, by showing that the conical-like configurations are also unstable.

Perturbative and non-perturbative partial supersymmetry breaking: N = 4 → N = 2 → N = 1

We show the existence of a supersymmetry breaking mechanism in string theory, where N = 4 supersymmetry is broken spontaneously to N = 2 and N = 1 with moduli-dependent gravitino masses. The spectrum of the spontaneously broken theory with lower supersymmetry is in one-to-one correspondence with the spectrum of the heterotic N = 4 string. The mass splitting of the N = 4 spectrum depends on the compactification moduli as well as the three R-symmetry charges. We also show that, in string theory, chiral theories can be obtained after spontaneous breaking of extended supersymmetry. This was impossible at the level of field theory. In the large-moduli limit a restoration of the N = 4 supersymmetry is obtained. As expected the graviphotons and some of the gauge bosons become massive in N = 1 vacua. At some special points of the moduli space some of the N = 4 states with non-zero winding numbers and with spin 0 and 1 2 become massless chiral superfields of the unbroken N = 1 supersymmetry. Such vacua have a dual type II description, in which there are magnetically charged states with spin 0 and 1 2 that become massless. The heterotic-type II duality suggests some novel non-perturbative transitions on the type II side. Such transitions do not seem to have a geometric interpretation, since they relate type II vacua with symmetric world-sheet structure to asymmetric ones. The heterotic interpretation of such a transition is an ordinary Higgsing of an SU(2) factor. In the case of N = 4 → N = 2, the perturbative N = 2 prepotential is determined by the perturbative N = 4 BPS states. This observation permits us to suggest a method to determine the exact non-perturbative prepotential of the effective N = 2 supergravity using the shifted spectrum of the non-perturbative BPS states of the underlying N = 4 theory.

BPS states in superstrings with spontaneously broken SUSY ( N = 4 → N = 2 → N = 1)

We show the existence of a supersymmetry-breaking mechanism in string theory, where N = 4 supersymmetry is broken spontaneously to N = 2 and N = 1 with moduli-dependent gravitino masses. The BPS spectrum of the theory with lower supersymmetry is in one-to-one correspondence with the spectrum of the heterotic N = 4 string. The mass splitting of the N = 4 spectrum depends on the moduli as well as the three R-symmetry charges. In the case of N = 4 → N = 2, the perturbative N = 2 prepotential is determined by the perturbative N = 4 BPS states. This observation led us to suggest a method that determines the exact non-perturbative prepotential of the effective N = 2 supergravity using the shifted spectrum of the non-perturbative BPS states of the underlying N = 4 theory.