Supersymmetry Conditions Research Articles

D-terms in generalised complex geometry

Generalised Complex Geometry provides a natural interpretation of the N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 supersymmetry conditions for warped solutions of type II supergravity as differential equations on polyforms on the internal manifold. Written in this language the supersymmetry conditions correspond to calibration conditions for probe D-branes: D-string, domain-wall or space-filling branes, depending on the directions they span in the non-compact four-dimensional space. The BPS condition corresponding to the calibration of space-filling D-branes has been reformulated by Tomasiello, eliminating the explicit dependence on the metric. We generalise this derivation to the case of non-supersymmetric backgrounds violating the domain-wall and D-string calibration conditions. We use this reformulation to derive constraints that the ten-dimensional solutions with BPS space-filling sources must respect in order to dimensionally reduce to solutions of four-dimensional N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 supergravity with non-vanishing F-terms and potentially non-vanishing D-terms. We give the equations of motion for the class of type II vacua satisfying these constraints in the language of pure spinors. We investigate how restrictive these constraints are for the class of type IIB SU(3) backgrounds with BPS space-filling O5-planes.

A story of non-conformal branes: spindles, disks, circles and black holes

We consider the (p + 2)-dimensional gauged supergravities arising as a consistent truncation of type II on S8−p, which are associated with the near-horizon limit of Dp-branes, for p = 2, 4, 5, 6 (and NS5-branes for p = 5). In a truncation of these theories with only abelian gauge fields and scalars, we find several classes of new solutions, with and without supersymmetry. Our ansatz for such backgrounds is inspired by the recent progress in the study of branes wrapped on orbifolds, but unlike those examples we consider “non-conformal branes”, i.e. no Anti de Sitter factors in the metric. Focusing on cases with non-trivial gauge fields, we can divide the solutions that we present in three categories: 1) branes wrapping Riemann surfaces, spindles and disks, 2) branes wrapped on a circle with a holonomy for the gauge field along the circle and 3) electrically charged black holes in gauged supergravity, which uplift to rotating branes in ten dimensions. We carefully analyze the conditions for supersymmetry in all these cases.

New non-supersymmetric flux vacua from generalised calibrations

We construct a new class of non-supersymmetric ten-dimensional type II flux vacua, by studying first order differential equations which are deformations of the \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{N}$$\\end{document} = 1 supersymmetry conditions. We do so within the context of Generalised Complex Geometry, where there is a natural interpretation of the \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{N}$$\\end{document} = 1 supersymmetry conditions in terms of calibration conditions for probe D-branes, called D-string, domain-wall or space-filling branes, depending on them wrapping two, three or four non-compact dimensions. We focus on the class of non-supersymmetric vacua violating the D-string calibration condition, and write down their general equations of motion in the language of pure spinors. We solve them for a subclass of vacua, where the deformation of the calibration condition is dictated by the foliated geometry of the internal space. We also construct backgrounds violating both the domain-wall and D-string calibration conditions, generalising the one-parameter DWSB class of backgrounds introduced in Lüst et al. We present several explicit solutions with SU(2) and SU(3) structures, and we investigate briefly their associated low energy effective theories.

Localization on AdS3× S2. Part I. The 4d/5d connection in off-shell Euclidean supergravity

We begin to develop the formalism of localization for the functional integral of supergravity on AdS3× S2. We show how the condition of supersymmetry in the Euclidean ℍ3/ℤ × S2 geometry naturally leads to a twist of the S2 around the time direction of AdS3. The twist gives us a five-dimensional Euclidean supergravity background dual to the elliptic genus of (0, 4) SCFT2 at the semiclassical level. On this background we set up the off-shell BPS equations for one of the Killing spinors, such that the functional integral of five-dimensional Euclidean supergravity on ℍ3/ℤ × S2 localizes to its space of solutions. We obtain a class of solutions to these equations by lifting known off-shell BPS solutions of 4-dimensional supergravity on AdS2× S2. In order to do this consistently, we construct and use a Euclidean version of the off-shell 4d/5d lift of arXiv:1112.5371, which could be of independent interest.

Fermionic CFTs from classical codes over finite fields

We construct a class of chiral fermionic CFTs from classical codes over finite fields whose order is a prime number. We exploit the relationship between classical codes and Euclidean lattices to provide the Neveu–Schwarz sector of fermionic CFTs. On the other hand, we construct the Ramond sector using the shadow theory of classical codes and Euclidean lattices. We give various examples of chiral fermionic CFTs through our construction. We also explore supersymmetric CFTs in terms of classical codes by requiring the resulting fermionic CFTs to satisfy some necessary conditions for supersymmetry.

Superspaces for heterotic pure spinor string compactifications

We study supersymmetry conditions for the heterotic pure superstring preserving N=1 supersymmetry in four dimensions directly from the curved superspace defined by the Berkovits–Howe constraints.

Scale‐Separated AdS3 Vacua from G2‐Orientifolds Using Bispinors

AbstractIn this paper, minimal supersymmetric AdS3 solutions on G2‐orientifolds of type IIA and IIB supergravity are studied. These are obtained by techniques of the bispinor formalism. A straightforward argument is derived against AdS3 solutions in massless type IIA string theory on such G2‐backgrounds. Furthermore, it is shown that some of the AdS3 solutions in massive type IIA do admit a separation of scales, whereas the same cannot be said for the type IIB vacua. Finally, a formula for the 3d superpotential in the bispinor language is given that is valid for both type IIA and type IIB configurations. It is shown that the supersymmetry conditions of this superpotential lead back to the bispinor equations.

The complete search for the supersymmetric Pati-Salam models from intersecting D6-branes

We construct a systematic method to build all the possible three-family mathcal{N} = 1 supersymmetric Pati-Salam models from Type IIA orientifolds on \U0001d54b6/(ℤ2× ℤ2) with intersecting D6-branes, in which the SU(4)C× SU(2)L× SU(2)R gauge symmetry can be broken down to the SU(3)C× SU(2)L× U(1)Y Standard Model gauge symmetry by the D-brane splitting and supersymmetry preserving Higgs mechanism. This is essentially achieved by solving all the common solutions for three generation conditions, RR tadpole cancellation conditions, and mathcal{N} = 1 supersymmetry conditions with deterministic algorithm. We find that there are 202752 possible supersymmetric Pati-Salam models in total, and show that there are only 33 independent models with different gauge coupling relations at GUTs scale after modding out equivalent relations, such as T-dualities, etc. In particular, there is one and only one independent model which has gauge coupling unification. Furthermore, one can construct other types of intersecting D-brane models utilizing such deterministic algorithm, and therefore we suggest a brand new method for D-brane model building.

Exactness of semiclassical quantization rule for broken supersymmetry

Semiclassical methods provide important tools for approximating solutions in quantum mechanics. In several cases these supposedly approximate solutions are intriguingly exact, as has been shown by direct calculations on particular systems. In this paper we prove that the long-conjectured exactness of the supersymmetry-based semiclassical quantization condition for broken supersymmetry is a consequence of the additive shape invariance for the corresponding potentials.

The near-horizon geometry of supersymmetric rotating AdS4 black holes in M-theory

We classify the necessary and sufficient conditions to obtain the near-horizon geometry of extremal supersymmetric rotating black holes embedded in 11d supergravity which are associated to rotating M2-branes. Such rotating black holes admit an AdS2 near-horizon geometry which is fibered by the transverse spacetime directions. In this paper we allow for the most general fibration over AdS2 with a flux configuration permitting rotating M2-branes. Using G-structure techniques we rewrite the conditions for supersymmetry in terms of differential equations on an eight-dimensional balanced space. The 9d compact internal space is a U(1)-fibration over this 8d base. The geometry is constrained by a master equation reminiscent of the one found in the non-rotating case. We give a Lagrangian from which the equations of motion may be derived, and show how the asymptotically AdS4 electrically charged Kerr-Newman black hole in 4d mathcal{N} = 2 supergravity is embedded in the classification. In addition, we present the conditions for the near-horizon geometry of rotating black strings in Type IIB by using dualities with the 11d setup.

Non-Kähler deformed conifold, ultraviolet completion and supersymmetric constraints in the baryonic branch

Gravity duals for a class of UV complete minimally supersymmetric nonconformal gauge theories require deformed conifolds with fluxes. However these manifolds do not allow for the standard Kähler or conformally Kähler metrics on them, instead the metrics are fully non-Kähler. We take a generic such configuration of a non-Kähler deformed conifold with fluxes and ask what constraints do supersymmetry impose in the baryonic branch. We study the supersymmetry conditions and show that for the correct choices of the vielbeins and the complex structure all the equations may be consistently solved. The constraints now lead not only to the known cases in the literature but also to some new backgrounds. We also show how geometric features of these backgrounds, including the overall warp-factor and the resolution parameters, can be seen on the field theory side from perturbative “probe-brane” type calculations by Higgsing the theory and studying one-loop 4-point functions of vector and chiral multiplets. Finally we discuss how UV completions of these gauge theories may be seen from our setup, both from type IIB as well as from the T-dual type IIA brane constructions.

On AdS3 solutions of Type IIB

We study mathcal{N} = 1 supersymmetric AdS3× M7 backgrounds of Type IIB supergravity, with non-vanishing axio-dilaton, three-form and five-form fluxes, and a “strict” SU(3)-structure on M7. We derive the necessary and sufficient conditions for supersymmetry as a set of constraints on the torsion classes of the SU(3)-structure. Given an Ansatz for the three-form fluxes, the problem of also solving the equations of motion involves a “master equation”, which generalizes ones that have previously appeared in the literature.

Boundary conditions and localization on AdS. Part II. General analysis

We develop the method of Green’s function to evaluate the one loop determinants that arise in localization of supersymmetric field theories on AdS spaces. The theories we study have at least mathcal{N} = 2 supersymmetry and normalisable boundary conditions are consistent with supersymmetry. We then show that under general assumptions the variation of the one loop determinant with respect to the localizing background reduces to a total derivative. Therefore it receives contributions only from the origin of AdS and from asymptotic infinity. From expanding both the Greens function and the quadratic operators at the origin of AdS and asymptotic infinity, we show that the variation of the one loop determinant is proportional to an integer. Furthermore, we show that this integer is an index of a first order differential operator. We demonstrate that these assumptions are valid for Chern-Simons theories coupled to chiral multiplets on AdS2 × S1. Finally we use our results to show that U(Nc) Chern-Simons theory at level k coupled to Nf chiral multiplets and Nf anti-chiral multiplets in the fundamental obeys level-rank duality on AdS2 × S1.

Mathcal{N}=2 supersymmetric AdS4 solutions of type IIB supergravity

We analyze general mathcal{N}=2 supersymmetric AdS4 solutions of type IIB supergravity. Utilizing a set of pure spinor equations directly adapted to mathcal{N}=2 , the necessary and sufficient conditions for supersymmetry are reduced to a concise system of partial differential equations for two functions which determine the solutions. We show that using this system analytic solutions can be generated, thus potentially expanding the rather limited set of known AdS4 solutions in type IIB supergravity.

Geometric constraints on the space of mathcal{N} = 2 SCFTs. Part I: physical constraints on relevant deformations

We initiate a systematic study of four dimensional mathcal{N} = 2 superconformal field theories (SCFTs) based on the analysis of their Coulomb branch geometries. Because these SCFTs are not uniquely characterized by their scale-invariant Coulomb branch geometries we also need information on their deformations. We construct all inequivalent such deformations preserving mathcal{N} = 2 supersymmetry and additional physical consistency conditions in the rank 1 case. These not only include all the ones previously predicted by S-duality, but also 16 additional deformations satisfying all the known mathcal{N} = 2 low energy consistency conditions. All but two of these additonal deformations have recently been identified with new rank 1 SCFTs; these identifications are briefly reviewed.Some novel ingredients which are important for this study include: a discussion of RG-flows in the presence of a moduli space of vacua; a classification of local mathcal{N} = 2 supersymmetry-preserving deformations of unitary mathcal{N} = 2 SCFTs; and an analysis of charge normalizations and the Dirac quantization condition on Coulomb branches.This paper is the first in a series of three. The second paper [1] gives the details of the explicit construction of the Coulomb branch geometries discussed here, while the third [2] discusses the computation of central charges of the associated SCFTs.

Marginal deformations of heterotic G2 sigma models

Recently, the infinitesimal moduli space of heterotic G2 compactifications was described in supergravity and related to the cohomology of a target space differential. In this paper we identify the marginal deformations of the corresponding heterotic nonlinear sigma model with cohomology classes of a worldsheet BRST operator. This BRST operator is nilpotent if and only if the target space geometry satisfies the heterotic supersymmetry conditions. We relate this to the supergravity approach by showing that the corresponding cohomologies are indeed isomorphic. We work at tree-level in α′ perturbation theory and study general geometries, in particular with non-vanishing torsion.

CCNV Space-Times as Potential Supergravity Solutions

It is of interest to study supergravity solutions preserving a nonminimal fraction of supersymmetries. A necessary condition for supersymmetry to be preserved is that the space-time admits a Killing spinor and hence a null or time-like Killing vector field. Any space-time admitting a covariantly constant null vector (CCNV) field belongs to the Kundt class of metrics and more importantly admits a null Killing vector field. We investigate the existence of additional non-space-like isometries in the class of higher-dimensional CCNV Kundt metrics in order to produce potential solutions that preserve some supersymmetries.

Supersymmetric gauge theory with space-time-dependent couplings

We study deformations of ${\cal N}=4$ supersymmetric Yang-Mills theory with couplings and masses depending on space-time. The conditions to preserve part of the supersymmetry are derived and a lot of solutions of these conditions are found. The main example is the case with $ISO(1,1)\times SO(3)\times SO(3)$ symmetry, in which couplings, as well as masses and the theta parameter, can depend on two spatial coordinates. In the case $ISO(1,1)$ is enhanced to $ISO(1,2)$, it reproduces the supersymmetric Janus configuration found by Gaiotto and Witten. When $SO(3)\times SO(3)$ is enhanced to $SO(6)$, it agrees with the world-volume theory of D3-branes embedded in F-theory (a background with 7-branes in type IIB string theory). We also find the general solution of the supersymmetry conditions for the cases with $ISO(1,1)\times SO(2)\times SO(4)$ symmetry. The cases with time dependent couplings and/or masses are also considered.

(2,2) and (0,4) supersymmetric boundary conditions in 3d N=4 theories and type IIB branes

The half-BPS boundary conditions preserving $\mathcal{N}=(2,2)$ and $\mathcal{N}=(0,4)$ supersymmetry in 3d $\mathcal{N}=4$ supersymmetric gauge theories are examined. The BPS equations admit decomposition of the bulk supermultiplets into specific boundary supermultiplets of preserved supersymmetry. Nahm-like equations arise in the vector multiplet BPS boundary condition preserving $\mathcal{N}=(0,4)$ supersymmetry and Robin-type boundary conditions appear for the hypermultiplet coupled to vector multiplet when $\mathcal{N}=(2,2)$ supersymmetry is preserved. The half-BPS boundary conditions are realized in the brane configurations of Type IIB string theory.

Supersymmetric branes and instantons on curved spaces

We discuss non-linear instantons in supersymmetric field theories on curved spaces arising from D-branes. Focussing on D3-branes and four-dimensional field theories, we derive the supersymmetry conditions and show the intimate relation between the instanton solutions and the non-linearly realized supersymmetries of the field theory. We demonstrate that field theories with non-linearly realized supersymmetries are coupled to supergravity backgrounds in a similar fashion as those with linearly realized supersymmetries, and provide details on how to derive such couplings from a type II perspective.