D4-branes wrapped on a spindle

Abstract

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.

A tale of (M)2 twists

“…As observed in [5] and confirmed in [8] for roots of the same sign, σ = 1 supersymmetry should be preserved by a topological topological twist, whilst for roots of differing sign σ = −1 supersymmetry is not preserved by such a topological twist but instead by the so-called anti-twist in the language of [4,8]. 6 Solutions with σ = −1 were studied in [2] for the Einstein-Maxwell truncation and in [5,6] for the X 0 X 1 truncation (pairwise equal gauge fields and constrained scalars).…”

“…For the M5-brane and D4-brane geometries, supersymmetry is preserved by yet another mechanism, [4,11] dubbed a topological topological twist, or twist from now on. As in the usual topological twist the charge of the field strength of a background R-symmetry vector through the spindle is the Euler character of the spindle, however the spinor is not globally a constant as in the usual topological twist.…”

“…(2.1) with the four scalars X (I) subject to the constraint X (1) X (2) X (3) X (4) = 1. One may obtain the above Lagrangian from the general form of 4d N = 2 gauged supergravity coupled to three vector multiplets with prepotential (4) .…”

“…The AdS 2 solution of 4d U(1) 4 gauged STU supergravity that we are interested in was originally found in 11d supergravity on a squashed S 7 in [19] using a double Wick rotation of the 4d black hole solutions with spherical horizon studied in [20]. We will be interested in the 4d solutions obtained by truncating the theory on the S 7 as studied in [5], see also [6].…”

“…There has been a recent surge in studying supersymmetric AdS d−2 × Σ solutions of d = 4, 5, 6, 7 gauged supergravity, where Σ is a Riemann surface admitting a non-constant curvature metric. Solutions where Σ is the weighted projective space WCP 1 n + ,n − , also known as a spindle, were first found in [1], and later extended in [2][3][4][5][6][7][8][9][10]. These solutions are naturally interpreted as arising from compactifying M2-, D3-, D4-, and M5-branes on the spindle and can be uplifted to 10-or 11-dimensional supergravity.…”