We construct supersymmetric "Image missing"solutions of D = 6 gauged supergravity, where "Image missing" is a two-dimensional orbifold known as a spindle. These uplift to solutions of massive type IIA supergravity using a general prescription, that we describe. We argue that these solutions correspond to the near-horizon limit of a system of Nf D8-branes, together with N D4-branes wrapped on a spindle, embedded as a holomorphic curve inside a Calabi-Yau three-fold. The dual field theories are d = 3, $$ \mathcal{N} $$ N = 2 SCFTs that arise from a twisted compactification of the d = 5, $$ \mathcal{N} $$ N = 1 USp(2N) gauge theory. We show that the holographic free energy associated to these solutions is reproduced by extremizing an off- shell free energy, that we conjecture to arise in the large N limit of the localized partition function of the d = 5 theories on "Image missing". We formulate a universal proposal for a class of off-shell free energies, whose extremization reproduces all previous results for branes wrapped on spindles, as well as on genus g Riemann surfaces Σg. We further illustrate this proposal discussing D4-branes wrapped on "Image missing", for which we present a supersymmetric "Image missing" solution of D = 6 gauged supergravity along with the associated entropy function.

We study AdS3$$ \times {S}^3/{\mathrm{\mathbb{Z}}}_k\times {\tilde{S}}^3/{\mathrm{\mathbb{Z}}}_{k^{\prime }} $$ × S 3 / ℤ k × S ˜ 3 / ℤ k ′ solutions to M-theory preserving $$ \mathcal{N} $$ N = (0, 4) supersymmetries, arising as near-horizon limits of M2-M5 brane intersections ending on M5’-branes, with both types of five-branes placed on A-type singularities. Solutions in this class asymptote locally to AdS7$$ /{\mathrm{\mathbb{Z}}}_k\times {\tilde{S}}^3/{\mathrm{\mathbb{Z}}}_{k^{\prime }} $$ / ℤ k × S ˜ 3 / ℤ k ′ , and can thus be interpreted as holographic duals to surface defect CFTs within the $$ \mathcal{N} $$ N = (1, 0) 6d CFT dual to this solution. Upon reduction to Type IIA, we obtain a new class of solutions of the form AdS3× S3/ℤk× S2×Σ2 preserving (0,4) supersymmetries. We construct explicit 2d quiver CFTs dual to these solutions, describing D2-D4 surface defects embedded within the 6d (1,0) quiver CFT dual to the AdS7/ℤk solution to massless IIA. Finally, in the massive case, we show that the recently constructed AdS3× S2× CY2 solutions with $$ \mathcal{N} $$ N = (0, 4) supersymmetries gain a defect interpretation as surface CFTs originating from D2-NS5-D6 defects embedded within the 5d CFT dual to the Brandhuber-Oz AdS6 background.

We derive exact relations to all orders in the α ′ expansion for the charges of a bound system of heterotic strings, solitonic 5-branes and, optionally, a Kaluza-Klein monopole. The expressions, which differ from those of the zeroth-order supergravity approximation, coincide with the values obtained when only the corrections of quadratic order in curvature are included. Our computation relies on the consistency of string theory as a quantum theory of gravity; the relations follow from the matching of the Wald entropy with the microscopic degeneracy. In the heterotic frame, the higher-curvature terms behave as delocalized sources that introduce a shift between near-horizon and asymptotic charges. On the other hand, when described in terms of lower-dimensional effective fields, the solution carries constant charges over space which coincide with those of the asymptotic heterotic fields. In addition, we describe why the Gauss-Bonnet term, which only captures a subset of the relevant corrections of quadratic order in curvature, in some cases succeeds to reproduce the correct value for the Wald entropy, while fails in others.