We consider supersymmetric solutions of $\textit{D}$ = 4, $$ \mathcal{N} $$ = 2 Euclidean gauged supergravity coupled to an arbitrary number of vector multiplets. Such solutions admit an R-symmetry Killing vector, $\textit{ξ}$, constructed as a bilinear in the Killing spinor. The Killing spinor bilinears can also be used to construct polyforms that are equivariantly closed under the action of the equivariant exterior derivative d = d – $\textit{ξ}$⌟ . This allows one to compute various flux integrals and the on-shell action using localization, without solving any supergravity equations, just assuming the supersymmetric solutions exist. The flux integrals allow one to obtain important UV-IR relations, relating fixed point data in the bulk to data on the asymptotic AdS boundary, allowing one to express the gravitational free energy in terms of boundary SCFT data. We illustrate the formalism with a number of examples, including classes of solutions which are unlikely to be ever constructed in closed form.

Switzerland

Pietro Genolini

United Kingdom

Jerome Gauntlett

Yusheng Jiao

Alice Lüscher

James Sparks

Equivariant localization for $\textit{D}$ = 4 gauged supergravity

We study $\textit{D}$ = 11 supergravity solutions which are dual to one-dimensional superconformal defects in $\textit{d}$ = 3 SCFTs. We consider defects in ABJM theory with monodromy for U(1) ⊂ SO(8) global symmetry, as well as in 𝒩 = 2 mABJM SCFT, which arises from the RG flow of a mass deformation of ABJM theory, with monodromy for U(1) ⊂ SU(3) × U(1) global symmetry. We show that the defects of the two SCFTs are connected by a line of bulk marginal mass deformations and argue that they are also related by bulk RG flow. In all cases we allow for the possibility of conical singularities at the location of the defect. Various physical observables of the defects are computed including the defects conformal weight and the partition function, as well as associated supersymmetric Renyi entropies.

Belgium

Igal Arav

South Korea

Matthew Roberts

Greece

Christopher Rosen

Superconformal monodromy defects in ABJM and mABJM theory

We study type IIB supergravity solutions that are dual to two-dimensional superconformal defects in d = 4 SCFTs which preserve   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = (0, 2) supersymmetry. We consider solutions dual to defects in   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 SYM theory that have non-trivial monodromy for U(1)3 ⊂ SO(6) global symmetry and we also allow for the possibility of conical singularities. In addition, we consider the addition of fermionic and bosonic mass terms that have non trivial dependence on the spatial directions transverse to the defect, while preserving the superconformal symmetry of the defect. We compute various physical quantities including the central charges of the defect expressed as a function of the monodromy, the on-shell action as well as associated supersymmetric Rényi entropies. Analogous computations are carried out for superconformal defects in the   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1, d = 4 Leigh-Strassler SCFT. We also show that the defects of the two SCFTs are connected by a line of bulk marginal mass deformations and argue that they are also related by bulk RG flow.

Superconformal monodromy defects in   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$ =4 SYM and LS theory

We use equivariant localization to construct off-shell entropy functions for supersymmetric black holes in  $$\mathcal{N}$$  = 2, D = 4 gauged supergravity coupled to matter. This allows one to compute the black hole entropy without solving the supergravity equations of motion and provides a novel generalization of the attractor mechanism. We consider magnetically charged black holes in AdS 4 which have an AdS 2 × M 2 near horizon geometry, where M 2 is a sphere or a spindle, and we also obtain entropy functions for ungauged supergravity as a simple corollary. We derive analogous results for black strings and rings in D = 5 supergravity which have an AdS 3 × M 2 near horizon geometry, and in this setting we derive an off-shell expression for the central charge of the dual  $$\mathcal{N}$$  = (0, 2), d = 2 SCFT.

Localization and attraction

We explain how equivariant localization may be applied to AdS/CFT to compute various BPS observables in gravity, such as central charges and conformal dimensions of chiral primary operators, without solving the supergravity equations. The key ingredient is that supersymmetric AdS solutions with an R-symmetry are equipped with a set of equivariantly closed forms. These may in turn be used to impose flux quantization and compute observables for supergravity solutions, using only topological information and the Berline-Vergne-Atiyah-Bott fixed point formula. We illustrate the formalism by considering AdS 5 × M 6 and AdS 3 × M 8 solutions of D = 11 supergravity. As well as recovering results for many classes of well-known supergravity solutions, without using any knowledge of their explicit form, we also compute central charges for which explicit supergravity solutions have not been constructed.

Equivariant localization for AdS/CFT

We construct a new class of supersymmetric AdS3 × Y 7 solutions of type IIB supergravity, where Y 7 is an S 5 fibration over a spindle, which are dual to d = 2,   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = (0, 2) SCFTs. The solutions are constructed in a sub-truncation of D = 5, SO(6) maximal gauged supergravity and they all lie within the anti-twist class. We show that the central charge computed from the gravity solutions agrees with an anomaly polynomial calculation associated with compactifying the   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1, d = 4 Leigh-Strassler SCFT on a spindle.

Netherlands

Leigh-Strassler compactified on a spindle

We construct supersymmetric AdS 3 solutions of D = 11 supergravity, dual to d = 2,   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = (0, 2) SCFTs, that are associated with M5-branes wrapping two different four-dimensional orbifolds. In one case the orbifold is a spindle fibred over another spindle, while in the other it is a spindle fibred over a Riemann surface with genus g &gt; 1. We show that the central charges of the d = 2 SCFTs calculated from the gravity solutions agree with field theory computations using anomaly polynomials. The new D = 11 solutions are obtained after constructing a new consistent Kaluza-Klein truncation of maximal D = 7 gauged supergravity reduced on a spindle down to D = 5 minimal gauged supergravity.

K. Cheung

Jacob Fry

M5-branes wrapped on four-dimensional orbifolds

In the context of holography, we analyse aspects of supersymmetric geometries based on two-dimensional orbifolds known as spindles. By analysing spin c  spinors on a spindle with an azimuthal rotation symmetry we show that under rather general conditions there are just two possibilities, called the ‘twist’ and the ‘anti-twist’, which are determined by the quantized magnetic flux through the spindle. A special case of the twist is the standard topological twist which is associated with constant and chiral spinors. We construct solutions of D = 5 and D = 4 STU gauged supergravity theories that are dual to D3-branes and M2-branes wrapping spindles, respectively, which realize both the anti-twist, as seen before, but also the twist. For the wrapped D3-brane solutions we reproduce the central charge of the gravity solution from the dual field theory by analysing the anomaly polynomial of   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 SYM theory. We also discuss M5-branes wrapped on spindles both from a gravity and a field theory point of view.

Pietro Ferrero

Supersymmetric spindles

We construct supersymmetric AdS 5 × Σ solutions of D = 7 gauged supergravity, where Σ is a two-dimensional orbifold known as a spindle. These uplift on S 4 to solutions of D = 11 supergravity which have orbifold singularites. We argue that the solutions are dual to d = 4,   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1 SCFTs that arise from N M5-branes wrapped on a spindle, embedded as a holomorphic curve inside a Calabi-Yau three-fold. In contrast to the usual topological twist solutions, the superconformal R-symmetry mixes with the isometry of the spindle in the IR, and we verify this via a field theory calculation, as well as reproducing the gravity formula for the central charge.

Italy

Dario Martelli

M5-branes wrapped on a spindle

We construct a continuous one parameter family of AdS4 × S 1 × S 5 S-fold solutions of type IIB string theory which have nontrivial SL(2, ℤ) monodromy in the S 1 direction. The solutions span a subset of a conformal manifold that contains the known   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 S-fold SCFT in d = 3, and generically preserve   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 supersymmetry. We also construct RG flows across dimensions, from AdS5 × S 5, dual to   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4, d = 4 SYM compactified with a twisted spatial circle, to various AdS4 ×S 1 ×S 5 S-fold solutions, dual to d = 3 SCFTs. We construct additional flows between the AdS5 dual of the Leigh-Strassler SCFT and an   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 S-fold as well as RG flows between various S-folds.