F-theory and AdS3/CFT2 (2, 0)

Abstract

We continue to develop the program initiated in [1] of studying supersymmetric AdS 3 backgrounds of F-theory and their holographic dual 2d superconformal field theories, which are dimensional reductions of theories with varying coupling. Imposing 2d N = (0, 2) supersymmetry, we derive the general conditions on the geometry for Type IIB AdS 3 solutions with varying axio-dilaton and five-form flux. Locally the compact part of spacetime takes the form of a circle fibration over an eight-fold Y τ 8 , which is elliptically fibered over a base M 6 . We construct two classes of solutions given in terms of a product ansatz M 6 = Σ × M 4 , where Σ is a complex curve and M 4 is locally a Kähler surface. In the first class M 4 is globally a Kähler surface and we take the elliptic fibration to vary non-trivially over either of these two factors, where in both cases the metrics on the total space of the elliptic fibrations are not Ricci-flat. In the second class the metric on the total space of the elliptic fibration over either curve or surface are Ricci-flat. This results in solutions of the type AdS 3 × K3 × M τ 5 , dual to 2d (0, 2) SCFTs, and AdS 3 × S 3 /Γ × CY 3 , dual to 2d (0, 4) SCFTs, respectively. In all cases we compute the charges for the dual field theories with varying coupling and find agreement with the holographic results. We also show that solutions with enhanced 2d N = (2, 2) supersymmetry must have constant axio-dilaton. Allowing the internal geometry to be non-compact leads to the most general class of Type IIB AdS 5 solutions with varying axio-dilaton, i.e. F-theoretic solutions, that are dual to 4d N = 1 SCFTs.In [1] this approach was initiated by studying the F-theory solutions dual to 2d (0, 4) SCFTs, which were shown to be of the typeis an elliptically fibered Calabi-Yau three-fold, and the complex structure of the elliptic fiber E τ is identified with the axio-dilaton τ IIB . This provides a generalization of the known solutions with (4, 4) supersymmetry where Y 6 = Y 4 × E τ with Y 4 = K3 or T 4 and the axio-dilaton is constant. The discrete subgoup Γ of SU(2) can be modded out, whilst retaining (0, 4) supersymmetry. In fact, these solutions were shown to be the most general ones dual to 2d (0, 4) SCFTs, supported by five-form flux.The dual field theories are closely related to the MSW string [9], and have a dual description in terms of D3-branes wrapped on a curve inside the base Kähler B 2 of the elliptically fibered Calabi-Yau three-fold: C ⊂ B 4 ⊂ Y τ 6 . The varying axio-dilaton induces a varying coupling of the 4d N = 4 Super-Yang Mills theory on the D3-brane, along the curve C. The resulting theory is supersymmetric when a particular topological twist, the so-called topological duality twist [3,4], is applied along the curve C [5, 6, 10]. The dual M-theory setup is the MSW string wrapped on an elliptic surface, and a dual M-theory solution confirms the F-theoretic results in [1], including the holographic comparison of the central charges to leading and subleading orders. In this context t...

Supersymmetric indices of 3d S-fold SCFTs

“…As usual in localisation computations, the wall. According to their finding, this boundary condition has to be accompanied by a flip of the coordinates (x 3,4,5 , x 7,8,9 ) → (x 7,8,9 , −x 3,4,5 ). As a result, such a configuration is referred to as an S-flip.…”

“…The latter can only come from two copies of the N = 3 extra SUSY-current multiplet A 2 [0] The term x 1/2 indicates that this theory contains a free part due to the fact that the R-charge of the basic monopole operators hits the unitary bound. The above index can be rewritten as where the index of a free hypermultiplet is given by 8 We remark that one has to use the sufficient condition stated in [52, sec. 4.3] with great care.…”

3d printing of 2d $$ \mathcal{N}=\left(0,2\right) $$ gauge theories

“…The corresponding Green-Schwarz mechanism for anomaly cancellation was studied in [21]. AdS 3 /CFT 2 pairs were constructed in [22,23]. Finally, 2d theories with exotic SUSY were constructed in [24].…”

“…The (246) coordinates are periodically identified to form a T 3 . It is natural to pair (23), (45) and (67) to form three complex variables x, y and z, such that the periodic directions (246) are their arguments. The NS5-brane extends along (01) and wraps a holomorphic surface Σ embedded in (234567), which is defined as the zero locus of the Newton polynomial associated to the toric diagram of the CY 4 .…”

AdS3 solutions with from S3 × S3 fibrations

“…Recently, more attention has been given to populating the space of supergravity solutions with various superconformal algebras. Most efforts have focused on type IIB with partial results for solutions with small  = (4, 0) in [18,19], large  = (4, 0) in [35,36] and  = (2, 0) in [20][21][22][23] in [24]. Exceptions to this trend include a classification of purely NS  = (2, 2) solutions in [25], large  = (4, 0) in Mtheory [26] and massive IIA, [36] and small  = (4, 0) in massive IIA.…”