\text{AdS}_2\times \text{S}^2\times \text{CY}_2 solutions in Type IIB with 8 supersymmetries
AAdS × S × CY solutions in Type IIB with 8supersymmetries Yolanda Lozano a, , Carlos Nunez b, and Anayeli Ramirez a, a : Department of Physics, University of Oviedo, Avda. Federico Garcia Lorca s/n, 33007 Oviedo,Spain b : Department of Physics, Swansea University, Swansea SA2 8PP, United Kingdom Abstract
We present a new infinite family of Type IIB supergravity solutions preserving eight super-charges. The structure of the space is AdS × S × CY × S fibered over an interval. Thesesolutions can be related through double analytical continuations with those recently constructedin [1]. Both types of solutions are however dual to very different superconformal quantum me-chanics. We show that our solutions fit locally in the class of AdS × S × CY solutions fiberedover a 2d Riemann surface Σ constructed by Chiodaroli, Gutperle and Krym, in the absenceof D3 and D7 brane sources. We compare our solutions to the global solutions constructedby Chiodaroli, D’Hoker and Gutperle for Σ an annulus. We also construct a cohomogeneity-two family of solutions using non-Abelian T-duality. Finally, we relate the holographic centralcharge of our one dimensional system to a combination of electric and magnetic fluxes. Wepropose an extremisation principle for the central charge from a functional constructed out ofthe RR fluxes. [email protected] [email protected] [email protected] a r X i v : . [ h e p - t h ] J a n ontents × S × CY backgrounds 2 × S × CY × Σ backgrounds of Chiodaroli-Gutperle-Krym 10 geometries and the “plus-solution” . . . . . . . . . . . . . . . . . . . . 133.3 The LNRS geometries and the “minus-solution” . . . . . . . . . . . . . . . . . . 143.4 The annulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.5 Zoom-in to our solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 × S × CY × Σ solutions with Σ an infinite strip 20 × S × CY solutions . . . . . . . . . . . . . . . . . . . . . . . 204.2 The NATD solution as a CGK geometry . . . . . . . . . . . . . . . . . . . . . . 23 The Maldacena conjecture [2] and its extensions motivate the search for AdS backgroundspreserving different amounts of Supersymmetry (SUSY), in different dimensions.The half-maximal SUSY case is especially fructiferous. The correspondence between linearquiver conformal field theories preserving half-maximal SUSY and half-maximal BPS solutionswith an AdS factor, leads to a precise map between infinite families of string backgrounds andtheir dual super-conformal field theories (SCFTs). Indeed, various works have developed thedictionary between d -dimensional SCFTs, the associated Hanany-Witten brane set-ups [3] andthe dual AdS d +1 string theory backgrounds.For the case d = 6, for which the strongly coupled conformal point is reached at highenergies, the papers [4, 5, 6, 7] have outlined the holographic dictionary and many other works1ave developed it. For d = 5, the works [8, 9, 10, 11, 12, 13, 14] presented backgrounds with anAdS factor and their UV-dual SCFTs. The dictionary for the case of four dimensional N = 2linear quiver SCFTs and their AdS dual backgrounds was studied in [15, 16, 17, 18] amongother works. The case of d = 3 SCFTs (arising at low energies after a RG flow) and the dualAdS backgrounds is studied in [19, 20, 21] among other works. The correspondence for the caseof two-dimensional (half-maximal BPS) low-energy SCFTs is particularly rich and has receiveda lot of attention recently. With the lens described above (linear quivers, Hanany-Witten set-ups and dual backgrounds), we encounter the works [22, 23, 24, 25, 26, 27, 28, 29, 30] amongvarious other papers.The study of AdS backgrounds in string/M-theory has a long and illustrious history.With the point of view described above, partial aspects of the correspondence between super-conformal quantum mechanics theories (SCQMs) of the quiver type and half-maximal BPSbackgrounds containing an AdS factor were initially studied in [31, 32, 33, 34, 35, 36, 37]. Therecent works [38, 1, 39] made precise and concrete the viewpoint advertised above for differentinfinite families of string backgrounds containing an AdS factor.This work presents a new infinite family of backgrounds with an AdS factor. We focusour presentation mostly on geometrical aspects of the new type IIB solutions. The contents ofthis paper are distributed as follows. In Section 2, we present the new backgrounds preservingeight supersymmetries (four Poincar´e and four conformal SUSYs). We study the conservedbrane charges and deduce the associated brane set-up, consisting on D1 and D5 ’colour’ branes(dissolved into fluxes) with D3 and D7 ’source’ branes (present in the background and violatingBianchi identities). NS-five branes and fundamental strings complete this configuration. Wedefine the holographic central charge following the procedure and physical meaning advanced in[1]. The section is closed with a brief discussion of the dual SCQM. In Section 3 we connect ourbackgrounds with those presented in [34, 35]. We point out that the presence of sources in oursolutions extend (for the AdS fixed point) the results of [34, 35]. We also link the solutions in[1] with those of [34, 35] (under the above mentioned restrictions). These links require a zoom-inprocedure that we discuss in detail. In Section 4 we uncover a new and explicit infinite family ofsolutions of cohomogeneity-two, by applying non-Abelian T-duality on the AdS backgroundsof [23, 24, 25, 26]. The study of these backgrounds and their ’completion’ following the ideasof [40, 41, 42, 43, 14] is reserved for a future study. We extend to the families of backgroundsdiscussed in this work a relation uncovered in [1],[39] between the Ramond-Ramond sector ofthe backgrounds and the holographic central charge. Such relation is discussed In Section 5.A functional whose extremisation yields the central charge is also presented. Finally, Section 6gives a short summary of the work, together with some ideas to work on the future. × S × CY backgrounds In this section we present a new family of AdS solutions with N = 4 Poincar´e supersymmetryin Type IIB supergravity. These geometries are foliations of AdS × S × CY × S over an interval.Alternatively, they can be considered as foliations of AdS × S × CY over a 2d Riemann surface2 with the topology of an annulus. The NS-NS sector of our solutions reads,d s st = u (cid:113)(cid:98) h h (cid:98) h h − ( u (cid:48) ) d s + u (cid:113)(cid:98) h h d s + (cid:115)(cid:98) h h d s + (cid:113)(cid:98) h h u (d ψ + d ρ ) ,e − φ = h (cid:98) h (cid:0) (cid:98) h h − ( u (cid:48) ) (cid:1) ,H = −
12 d (cid:18) ρ + uu (cid:48) (cid:98) h h − ( u (cid:48) ) (cid:19) ∧ vol AdS + 1 h d ρ ∧ H + 12 vol S ∧ d ψ . (2.1)Here φ is the dilaton, H the NS-NS three-form and the metric is given in string frame. Aprime denotes a derivative with respect to ρ . The two-form H is defined on the CY . Thecoordinate ψ ranges in [0 , π ], while the ρ coordinate describes an interval that we will take tobe bounded between 0 and 2 π ( P + 1) (see below). Note that u ≥ (cid:98) h h − ( u (cid:48) ) ≥ F = h (cid:48) d ψ ,F = − h H ∧ d ψ − (cid:16) h + h (cid:48) u (cid:48) u h (cid:98) h − ( u (cid:48) ) (cid:17) vol AdS ∧ d ψ + 14 (cid:18) − d (cid:18) u (cid:48) u (cid:98) h (cid:19) + 2 h d ρ (cid:19) ∧ vol S ,F = − (1 + (cid:63) ) (cid:32) ∂ ρ (cid:98) h vol CY + h u (cid:98) (cid:63) d h ∧ dρ − u (cid:48) u h (4 (cid:98) h h − ( u (cid:48) ) ) H ∧ vol AdS (cid:33) ∧ d ψ.F = (cid:32) (cid:32)(cid:98) h + uu (cid:48) (cid:98) h (cid:48) h (cid:98) h − ( u (cid:48) ) (cid:33) vol AdS ∧ d ψ − (cid:18) (cid:98) h d ρ − d (cid:18) uu (cid:48) h (cid:19)(cid:19) ∧ vol S (cid:33) ∧ vol CY − (cid:63) (cid:18) h H ∧ d ψ (cid:19) ,F = (cid:98) h h (cid:48) u (cid:98) h (4 (cid:98) h h − ( u (cid:48) ) ) vol AdS ∧ vol CY ∧ vol S ∧ d ρ . (2.2)Supersymmetry holds whenever, u (cid:48)(cid:48) = 0 , H + (cid:98) (cid:63) H = 0 , (2.3)where (cid:98) (cid:63) is the Hodge dual on CY . In turn, the Bianchi identities of the fluxes impose–awayfrom localised sources–that, h (cid:48)(cid:48) = 0 , d H = 0 , h u ∇ (cid:98) h + ∂ ρ (cid:98) h − h H ∧ H = 0 . (2.4)In what follows we will concentrate on backgrounds for which H = 0 and (cid:98) h = (cid:98) h ( ρ ). Thesebackgrounds are supersymmetric solutions of the Type IIB equations of motion if the warpingfunctions satisfy (away from localised sources), (cid:98) h (cid:48)(cid:48) = 0 , h (cid:48)(cid:48) = 0 , u (cid:48)(cid:48) = 0 , (2.5)3hich makes them linear functions of ρ .We focus on the solutions defined by the piecewise linear functions (cid:98) h , h considered in [24,25]. These are continuous functions with discontinuous derivatives. These imply discontinuitiesin the RR-sector that are interpreted as generated by sources in the background. The solutionsin [24, 25] have well-defined 2d dual CFTs. This requires a global definition of the ρ -interval.We achieve this imposing that (cid:98) h and h vanish at both ends of the ρ -interval, that we takeat ρ = 0 , π ( P + 1). Ending the space in this fashion, introduces extra source branes in theconfiguration. For the backgrounds to be trustable (in view of holographic applications), weneed to impose that the sources are ’sparse’, namely that they occur separated enough in the ρ -interval. This imposes that P (the length of the ρ -interval) is large.The functions (cid:98) h and h are then defined as, (cid:98) h ( ρ ) = Υ h ( ρ ) =Υ β π ρ ≤ ρ ≤ π,α k + β k π ( ρ − πk ) 2 πk ≤ ρ ≤ π ( k + 1) , k = 1 , ..., P − α P − α P π ( ρ − πP ) 2 πP ≤ ρ ≤ π ( P + 1) , (2.6) h ( ρ ) = ν π ρ ≤ ρ ≤ π,µ k + ν k π ( ρ − πk ) 2 πk ≤ ρ ≤ π ( k + 1) , k = 1 , ..., P − µ P − µ P π ( ρ − πP ) 2 πP ≤ ρ ≤ π ( P + 1) . (2.7)The choice of constants is imposed by continuity of the metric and dilaton. This implies that α k = k − (cid:88) j =0 β j , µ k = k − (cid:88) j =0 ν j . (2.8)In turn, β k and ν k must be integer numbers to give well defined quantised charges (see the nextsubsection). In (2.6) the number Υ is chosen such that,ΥVol CY = 16 π . (2.9)In most of our analysis in this paper we will concentrate on solutions for which u = u =constant. In that case the behaviour of the metric and dilaton at both ends of the ρ -interval isd s ∼ x (d s + d s ) + d s + x (d x + d ψ ) , e − φ ∼ x , (2.10)where x = ρ close to ρ = 0 and x = 2 π ( P + 1) − ρ close to ρ = 2 π ( P + 1). This correspondsto a superposition of D3-branes, extended on AdS × S and smeared on ψ and the CY , andD7-branes, extended on AdS × S × CY and smeared on ψ .The backgrounds in eqs.(2.1)-(2.2) can be obtained applying the usual T duality rules overthe Hopf fibre of the three sphere of the AdS × S backgrounds in [39]. Additionally, thesesolutions have the same structure as the geometries in [1], namely AdS × S × CY × S foliatedover an interval. The relation with the backgrounds in [1] is through an analytic continuation,d s → − d s , d s → − d s , e φ → ie φ , F i → − iF i ,u → − iu, (cid:98) h → i (cid:98) h , h → ih , ρ → iρ, ψ → − iψ, g i → ig i . (2.11)4 dS × S × CY × I ρ AdS × S × CY × I ρ AdS × S × CY × I ρ × S ψ AdS × S × CY × I ρ × S ψ T-duality T-dualityAnalyticContinuationAnalyticContinuation
AdS ↔ S S ↔ AdS AdS ↔ S S ↔ AdS Figure 1: Relations between the infinite family of AdS backgrounds to massive IIA constructedin [23] (top left), the IIB AdS backgrounds studied in [1] (bottom left), the IIA AdS back-grounds constructed in [39] (top right), and the new AdS solutions in Type IIB given byeqs.(2.1)-(2.2) (bottom right).These relations are summarised in Figure 1.Next we study the charges associated with the backgrounds in eqs. (2.1)-(2.2) and theassociated brane set-up. We compute the charges associated to our backgrounds using that the magnetic charge for aDp brane is given by, Q m Dp = 1(2 π ) − p (cid:90) M − p (cid:98) F − p , (2.12)where M − p is any (8 − p )-dimensional compact manifold transverse to the branes. In turn,the electric charge of Dp branes is defined by, Q e Dp = 1(2 π ) p +1 (cid:90) AdS × Σ p (cid:98) F p +2 , (2.13)where Σ p is the p -dimensional manifold on which the brane extends. In both expressions wehave set α (cid:48) = g s = 1. For the electric charges we need to regularise the volume of the AdS space. We take it to be the analytical continuation of the volume of the two-sphere,Vol AdS = 4 π . (2.14)5n the previous expressions (cid:98) F are the Page fluxes, defined as (cid:98) F = F ∧ e − B . They read, for ourbackgrounds (cid:98) F = h (cid:48) d ψ , (cid:98) F = 12 ( h (cid:48) ( ρ − πk ) − h ) vol AdS ∧ d ψ + 14 (cid:18) h + u (cid:48) ( u (cid:98) h (cid:48) − (cid:98) h u (cid:48) )2 (cid:98) h (cid:19) vol S ∧ d ρ , (cid:98) F = 14 (cid:32) h ( ρ − πk ) − ( u − ( ρ − πk ) u (cid:48) ) ( u (cid:98) h (cid:48) − (cid:98) h u (cid:48) )4 (cid:98) h (cid:33) vol AdS ∧ vol S ∧ d ρ − (cid:98) h (cid:48) vol CY ∧ d ψ , (cid:98) F = 12 ( (cid:98) h − ( ρ − πk ) (cid:98) h (cid:48) ) vol AdS ∧ vol CY ∧ d ψ − (cid:18) (cid:98) h + u (cid:48) ( uh (cid:48) − h u (cid:48) )2 h (cid:19) vol S ∧ vol CY ∧ d ρ , (cid:98) F = − (cid:18) ( ρ − πk ) (cid:98) h − ( u − ( ρ − πk ) u (cid:48) ) ( uh (cid:48) − h u (cid:48) )4 h (cid:19) vol AdS ∧ vol S ∧ vol CY ∧ dρ. (2.15)In these expressions we have allowed for large gauge transformations of the B -field, B → B + kπ vol AdS , as in [39] (see this reference for more details).Before calculating the quantised charges associated to these fluxes it is useful to computethe following quantities,d (cid:98) F = h (cid:48)(cid:48) d ρ ∧ d ψ , d (cid:98) F = 12 h (cid:48)(cid:48) ( ρ − πk )vol AdS ∧ d ρ ∧ d ψ , d (cid:98) F = − h (cid:48)(cid:48) vol CY ∧ d ρ ∧ d ψ , d (cid:98) F = − h (cid:48)(cid:48) ( ρ − πk )vol AdS ∧ vol CY ∧ d ρ ∧ d ψ , d (cid:98) F = 0 . (2.16)In these expressions (cid:98) h (cid:48)(cid:48) and h (cid:48)(cid:48) are the ones that follow from eqs.(2.6)-(2.7), (cid:98) h (cid:48)(cid:48) = 12 π P (cid:88) k =1 ( β k − − β k ) δ ( ρ − πk ) , h (cid:48)(cid:48) = 12 π P (cid:88) k =1 ( ν k − − ν k ) δ ( ρ − πk ) , (2.17) (cid:98) h (cid:48)(cid:48) × ( ρ − πk ) = h (cid:48)(cid:48) × ( ρ − πk ) = xδ ( x ) = 0 . We then obtain d (cid:98) F = d (cid:98) F = 0 , (2.18)and d (cid:98) F = 12 π P (cid:88) k =1 ( ν k − − ν k ) δ ( ρ − πk ) d ρ ∧ d ψ (2.19)d (cid:98) F = − π P (cid:88) k =1 ( β k − − β k ) δ ( ρ − πk ) vol CY ∧ d ρ ∧ d ψ. (2.20)These results can be put in correspondence with the brane set-up summarised in Table 1. Thefact that d (cid:98) F = 0 and d (cid:98) F = 0 indicates that the D5 and D1 branes play the role of colourbranes (dissolved in fluxes) in the brane set-up. On the other hand, d (cid:98) F and d (cid:98) F being nonzero6 = t x x x x x = ρ x = r x = θ x = θ x = ψ D1 x xD3 x x x xD5 x x x x x xD7 x x x x x x x xNS5 x x x x x xF1 x xTable 1: Brane set-up associated to our solutions. x corresponds to the time direction of theten dimensional spacetime, x , . . . , x are the coordinates spanned by the CY and x , x arethe coordinates parametrising the S .indicate that the D7 and D3 branes are flavour branes, that is, explicit sources with dynamicsdescribed by the Born-Infeld-Wess-Zumino action.Substituting (cid:98) h and h as defined by eqs.(2.6) and (2.7), together with eqs.(2.9) and (2.14),we find, in each ρ -interval [2 πk, π ( k + 1)] Q e D1 = 1(2 π ) (cid:90) AdS × S ψ (cid:98) F = (cid:18) Vol
AdS π (cid:19) (cid:18) Vol ψ π (cid:19) h − h (cid:48) ( ρ − πk )2 = µ k ,Q m D3 = 116 π (cid:90) CY × S ψ (cid:98) F = 116 π (cid:90) CY × S ψ × I ρ d (cid:98) F = (cid:18) ΥVol CY π (cid:19) × Vol ψ (cid:90) h (cid:48)(cid:48) d ρ = ( β k − − β k ) ,Q e D5 = 1(2 π ) (cid:90) AdS × CY × S ψ (cid:98) F = (cid:18) Vol
AdS π (cid:19) (cid:18) ΥVol CY π (cid:19) (cid:18) Vol ψ π (cid:19) h − h (cid:48) ( ρ − πk )2 = α k ,Q m D7 = (cid:90) S ψ (cid:98) F = (cid:90) S ψ × I ρ d (cid:98) F = Vol ψ (cid:90) h (cid:48)(cid:48) d ρ = ( ν k − − ν k ) . (2.21)Further, in the brane set-up the F1-strings are electrically charged with respect to the NS-NS3-form H while the NS5 branes are magnetically charged, Q e F1 = 1(2 π ) (cid:90) AdS × I ρ H = (cid:18) Vol
AdS π (cid:19) (cid:18) π (cid:19) (cid:90) π ( k +1)2 πk d ρ = 1 ,Q m NS5 = 1(2 π ) (cid:90) S × S ψ H = (cid:18) Vol S π (cid:19) (cid:18) Vol ψ π (cid:19) = 1 . (2.22) To close this part of our study we compute the holographic central charge associated to oursolutions. Being the field theory zero-dimensional, the previous quantity should be interpretedas the number of vacuum states in the dual superconformal quantum mechanics (see [1, 39] fora further discussion of the physical meaning of this quantity). We follow the prescription in744, 45]. We get for the internal volume, V int = (cid:90) d x (cid:113) e − φ det g ,ind = Vol CY Vol S Vol ψ (cid:90) π ( P +1)0 (4 (cid:98) h h − ( u (cid:48) ) ) d ρ , (2.23)and, finally, for the central charge c hol,1d = 3 V int πG N = 3 π (cid:90) π ( P +1)0 (cid:18) h h − ( u (cid:48) ) (cid:19) d ρ . (2.24)We have used that G N = 8 π and set units so that α (cid:48) = g s = 1.We would like to stress that in the usual calculations, such as the previous one, givingrise to the holographic central charge, only the NS-NS sector of the backgrounds needs to betaken into account. We will point out an interesting relation between the holographic centralcharge and the RR sector of our AdS solutions in Section 5. Such relation has been previouslyencountered in the AdS solutions constructed in [1, 39]. Whilst the main focus of this work in not the Quantum Mechanical analysis of the duals tothe backgrounds in eqs.(2.1)-(2.2), we add below some thoughts along this direction.In the papers [24, 25, 1, 39] concrete quivers were proposed as UV-descriptions of weaklycoupled 2d QFTs or 1d Quantum Mechanics. It was conjectured that these quivers becomestrongly coupled at low energies and a conformal fixed point arises. Checks for these proposalswere presented in each of the works [24, 25, 1, 39], for the different systems under study.These checks deal with RG-invariant quantities that can be well-identified in the UV and IRdescriptions.As we indicated around eq.(2.11) and summarised in Figure 1, the backgrounds of Section2 arise after an Abelian T-duality on the backgrounds of [39]. This suggests that the quantummechanical system proposed in [39] should also apply here. We are in fact T-dualising acrossa non-R-symmetry-direction, hence we expect the amount of SUSY to be the same. The R-symmetry of the quivers in [39] is SU (2) R , and there is also a global SU (2) g symmetry. We arechoosing a U (1) g inside SU (2) g for our dualisation. Therefore, our dual quantum mechanicalsystem should have SU (2) R × U (1) g symmetry. This is in fact geometrically realised by thepresence of the round S and the circle S ψ in the backgrounds of Section 2.Since the string sigma model in a background and in its T-dual is the same, we expect thesame dual quantum mechanical systems for our backgrounds as those for the backgrounds [39](only that perhaps it will be written in a different language).Using this reasoning, we may think about the SCQM as that arising in the very low energylimit of a system of D3-D7 branes—dual to a four dimensional N = 2 QFT. This system is’polluted’ by one-dimensional defects. These are Wilson loops (arising from F1-D5) and ’t Hooft8oops (arising from NS5-D1) added to the background, see for example [46]. Note that both theD1’s and the D5’s extend on the ψ -isometric direction. From the discussion above, it followsthat the dual SCQM to our backgrounds is the description of these one-dimensional defectsinside a four dimensional N = 2 QFT. In fact, in the IR the gauge symmetry on both D7 andD3 branes should become global. This implies that these branes must be sources/flavours, as itoccurs in the backgrounds of Section 2. By the same token we have two lines of one dimensionalgauge groups: Π Pi =1 U ( α i ) and Π Pi =1 U ( µ i ) realised on D5 and D1 branes in each ρ -interval. Thisis reflected by the counting of branes of eq.(2.21). The nodes in the [2 πk, π ( k + 1)] intervalwill have SU ( β k − β k +1 ) and SU ( ν k − ν k +1 ) flavour groups, realised on the D3 and D7 branes,as also reflected by eq.(2.21). The brane set-up is the one described in Table 1.As was found in [39], our 1d system should also have Wilson lines (in an antisymmetricrepresentation) inserted in the different gauge nodes of the quiver. These Wilson lines arisefrom the massive fermionic strings that stretch between D1s in the k -th interval and D7s in allother intervals. The Wilson lines would be in the ( ν , . . . , ν k − ) antisymmetric representationof the U ( µ k ) gauge group. The same applies to the massive D3-D5 fermionic strings and theantisymmetric Wilson lines on the U ( α k ) groups. As in [39], this information can be encodedin Young diagrams.We would also have a dynamical CS-term of each gauge group. This comes from the masslessfermionic strings stretched between D1-D7 and D5-D3 branes. The coefficient can be extractedstudying the WZ action for a D1 along [ t, ψ ] and a D5 along [ t, CY , ψ ]. As expected, thesecoefficients are quantised.The field content of the UV-quantum mechanical quiver follows directly from the analysisof Appendix B in [39]. In fact, each node contains a (4 ,
4) vector multiplet and a (4 ,
4) adjointhyper, (0 ,
4) bifundamental hypers join the two types of colour, D5 and D1, branes, (4,4)bifundamental hypers join the D7 sources with D5-branes, and the D3 sources with D1 branes,respectively. Finally, (0 ,
2) Fermi multiplets join source D7 with colour D1s and source D3 withcolour D5 branes. The quiver diagram is depicted in Figure 2. β − β β − β ν − ν ν − ν µ α α µ β k − − β k ν k − − ν k µ k α k Figure 2: The proposed quantum mechanical quiver. This follows from the analysis of openstrings in the Hanany-Witten set-up. 9
Connection with the AdS × S × CY × Σ backgroundsof Chiodaroli-Gutperle-Krym In this section we relate our backgrounds to the general class of AdS × S × CY × Σ solutionsto Type IIB supergravity with 8 supercharges found by Chiodaroli, Gutperle and Krym (CGK)in [34]. We show that our solutions fit locally in their classification in the absence of D3 and D7brane sources (in this sense our backgrounds extend those in [34] at the AdS point). A similaranalysis shows that the family of AdS solutions to Type IIB supergravity recently found in [1]also fits in their general class. The CGK backgrounds are dual to one dimensional conformal interfaces inside the two dimen-sional CFT associated to the D1-D5 system. These solutions (unlike ours) interpolate betweenAdS in the IR (at the interface) and the AdS × S × CY solution of Type IIB supergravity inthe UV. We shall focus on the AdS fixed points and compare them with both the backgroundsdiscussed in section 2 and the solutions found in [1].In [34], the authors used techniques developed in [19, 47] to find half BPS solutions thatpreserve eight of the sixteen supersymmetries of the AdS × S × CY vacuum, and are locallyasymptotic to this vacuum solution. They provided an ansatz for the bosonic fields in Type IIBsupergravity for a foliation of AdS × S × CY over a two-dimensional Riemann surface Σ witha boundary, and found that the local solutions of the BPS equations can be written in termsof two harmonic and two holomorphic functions defined on Σ. The solutions corresponds toa D1-D5 configuration with extra NS5 branes and fundamental strings, but vanishing D3 andD7 brane charges. We will see that our solutions fit locally within this class of solutions in theabsence of D3 and D7 brane sources. Our D3 and D7 sources are localised in ρ and smeared inthe ψ -coordinate. The mapping explained below is valid at points in ρ where the sources arenot present.We start summarising the local solutions constructed in [34]. The metric for the ten-dimensional spacetime is given, in Einstein frame, byd s = f d s + f d s + f d s + (cid:101) ρ d z d z , (3.1)where the warping factors f i ( i = 1 , . . . ,
3) and (cid:101) ρ are functions of z and z , the local holomorphiccoordinates of Σ. The orthonormal frames can be written as, f d s = η i i e i ⊗ e i , i , = 0 , ,f d s = δ j j e j ⊗ e j , j , = 2 , ,f d s = δ k k e k ⊗ e k , k , = 4 , , , , (cid:101) ρ d z d z = δ ab e a ⊗ e b , a, b = 8 , . (3.2)10he NS-NS and RR three-forms are written as a complex three-form, defined as G = e Φ H + ie − Φ ( F − χ H ). This form is given by, G = g (1) a e a + g (2) a e a . (3.3)In turn, the self-dual five-form flux is, F = h a e a + (cid:101) h a e a , a = z, z , (3.4)where the self-duality condition implies h a = − (cid:15) a b (cid:101) h b .The local solutions of the BPS equations and Bianchi identities admit a description interms of four functions, A , B , H and K . The analysis in [34] shows that the functions A and B must be holomorphic on the Riemann surface Σ, whilst H and K must be harmonic. Thesupergravity fields can be written in terms of these functions as, f = e − | H | f K (cid:0) ( A + A ) K − ( B − B ) (cid:1) , f = e − | H | f K (cid:0) ( A + A ) K − ( B + B ) (cid:1) ,f = 4 e KA + A , e = 14 K (cid:0) ( A + A ) K − ( B + B ) (cid:1) (cid:0) ( A + A ) K − ( B − B ) (cid:1) ,χ = 12 iK (cid:16) B − B − ( A − A ) K (cid:17) , (cid:101) ρ = e K ( A + A ) H | ∂ z H | | B | . (3.5)Here Φ = − φ/
2, where φ is the dilaton. For the five-form field strength, we define a four-formpotential, along CY , C CY = − i B − B A + A − (cid:101) K, ∂ z C CY = f (cid:101) ρ (cid:101) h z , (3.6)where (cid:101) K is the harmonic function conjugate to K .The potentials for the field strengths in equation (3.3) are written in terms of the holomor-phic and harmonic functions as b (1) = − H ( B + B )( A + A ) K − ( B + B ) − h , h = 12 (cid:90) ∂ z HB + c.c. ,b (2) = − i H ( B − B )( A + A ) K − ( B − B ) + (cid:101) h , (cid:101) h = − i (cid:90) ∂ z HB + c.c. ,c (1) = − i H ( AB − AB )( A + A ) K − ( B + B ) + (cid:101) h , (cid:101) h = − i (cid:90) A ∂ z HB + c.c. ,c (2) = − H ( AB + AB )( A + A ) K − ( B − B ) + h , h = 12 (cid:90) A ∂ z HB + c.c. , (3.7)where (cid:101) h i and h i are harmonic functions conjugate to each other. In the previous expression b (1) and b (2) are the potentials of the NS-NS three-form H and c (1) and c (2) are the potentials The harmonic conjugate of g is denoted as (cid:101) g and satisfies i∂ z (cid:101) g = ∂ z g . F . These read, H =d b (1) ∧ vol AdS + d b (2) ∧ vol S F =d C − χH = (d c (1) − χ d b (1) ) ∧ vol AdS + (d c (2) − χ d b (2) ) ∧ vol S . (3.8)The existence of sensible regular solutions imposes the following conditions on the functions A , B , H and K , • The harmonic functions A + A, B + B and K must have common singularities. • No singular points should appear in the bulk of the Riemann surface Σ. • The functions A + A, K and H cannot have any zero in the bulk of the Riemann surface. • The holomorphic functions B and ∂ z H must have common zeros.The previous conditions guarantee a non-vanishing and finite everywhere f (except at isolatedsingular points at the boundary), a finite f in the interior of the Riemann surface and vanishingat the boundary, and, finally, finite and non-vanishing f and e functions everywhere on theRiemann surface, including the boundary.The equations in (3.5) can be inverted to find A , B , H and K in terms of f i ( i = 1 , . . . , χ and Φ. One finds two possibilities, that we will refer as the “plus and minus solutions” ,Sol + : H = f f f , K + = f f f , A + = f f e − iχ, B + = e Φ f f (cid:113) f − f , (3.9)Sol − : H = f f f , K − = f f f , A − = f f e − iχ, B − = i e Φ f f (cid:113) f − f . (3.10)Inserting the “plus-solution”, equation (3.9), or the “minus-solution”, equation (3.10), in thefirst expression of (3.6) one obtains, in both cases, the function associated to the 4-form po-tential, C CY = − (cid:101) K, (3.11)where (cid:101) K is the harmonic function conjugate to K , according the footnote 1.In the next subsection we obtain the harmonic and holomorphic functions that give rise toour backgrounds in eqs. (2.1)-(2.2), as well as to the geometries in [1]. The “plus solution” corresponds to our AdS backgrounds and the “ minus solutions” to the AdS geometriesof [1]. Both these solutions are related through an analytical continuation, as explained around eq.(2.11). .2 Our AdS geometries and the “plus-solution” In order to compare the generic backgrounds given by eqs.(2.1)-(2.2) with the solutions in[34, 35] we express our solutions in Einstein frame, to agree with their conventions. We obtain, f = u √ (cid:32) (cid:98) h h (4 (cid:98) h h − ( u (cid:48) ) ) (cid:33) / , f = u √ (cid:32) (cid:98) h h − ( u (cid:48) ) (cid:98) h h (cid:33) / ,f = (cid:32)(cid:98) h (4 (cid:98) h h − ( u (cid:48) ) )2 h (cid:33) / , e = e − φ = 12 (cid:115) h (cid:98) h (cid:113) (cid:98) h h − ( u (cid:48) ) ,χ = h (cid:48) ψ, (cid:101) ρ = 1 √ u (cid:16)(cid:98) h h (4 (cid:98) h h − ( u (cid:48) ) ) (cid:17) / , C CY = − h (cid:48) ψ. (3.12)We emphasise that these expressions are valid at the points where h (cid:48)(cid:48) = (cid:98) h (cid:48)(cid:48) = 0.We take the ρ and ψ coordinates to define the real and imaginary parts of the z variable.With this parametrisation, Σ is an annulus, defined in the complex plane (see Figure 3), z = ψ + iρ with ψ ∈ [0 , π ] and ρ ∈ [0 , π ( P + 1)] . (3.13)Locally, our solutions are defined by the three functions u, (cid:98) h , h , which must be linear in ρ . ρ = 0 ψ = 2 πρ = 2 π ( P + 1) z ∂ Σ ∂ Σ ψρ Figure 3: Riemann surface associated to our AdS geometries. Given the periodicity of ψ itdefines an annulus.We take u = u + u ρ, h = µ + νρ, (cid:98) h = α + βρ. (3.14)Substituting (3.12) in (3.9) and taking into account (3.14), we find for the functions A, B, H, K , A = h − iψh (cid:48) = µ − iνz, B = u (cid:48) u ,H = u u − i u z − z ) , K = (cid:98) h α − i β z − z ) . (3.15)It is easy to check that H , K , A + A and B + B are harmonic functions and A and B areholomorphic. The harmonic function (cid:101) K reads, in turn, (cid:101) K = − (cid:98) h (cid:48) z + z ) = − β z + z ) , (3.16)13hich is the harmonic function conjugate to the expression for K in (3.15).From the equations (3.7), and using (3.15), we can then obtain the harmonic functions andpotentials associated with the NS-NS three-form, h = − i z − z ) , (cid:101) h = −
14 ( z + z ) ,b (1) = u (2 u − iu ( z − z )) u + (2 iα + ( z − z ) β )(2 iµ + ( z − z ) ν ) − h , b (2) = −
14 ( z + z ) , (3.17)as well as those associated with the RR three-form, h = − ν z + z ) − i µ ( z − z ) , (cid:101) h = i ν z − z ) − µ z + z ) ,c (1) = u (2 u − iu ( z − z ))( z + z ) ν u + (2 iα + ( z − z ) β )(2 iµ + ( z − z ) ν )) + (cid:101) h , c (2) = − u (2 iu + u ( z − z ))8( β ( z − z ) + 2 iα ) + h . (3.18)From these expressions we can recover H and F as given in eqs. (2.1)-(2.2). Note that h i and (cid:101) h i are harmonic functions conjugate to each other.We have thus shown that our solutions can be obtained, locally, from the class of solutionsconstructed in [34]. Note that in our analysis we have implicitly assumed that h (cid:48)(cid:48) = 0 and (cid:98) h (cid:48)(cid:48) = 0 also hold globally. This is necessary in order to match the axion and the 4-form RRpotential given in (3.12). This assumption –translated to our geometries– indicates that we arenot allowing for D7 and D3 brane sources, according to equations (2.16)-(2.17). This agreeswith the analysis in [34], which does not include either these types of branes.We will show in subsection 3.4 that D3-brane sources can be included in the two boundariesof the annulus following the formalism for the annulus derived in [35]. This allows to recover thesolutions in our class where D3-branes terminate the space at ρ = 2 π ( P +1). Quite surprisingly,we will also see that, even if not included in the analysis in [35], D7-brane sources can also beallowed at the end of the space. We will show that they also manifest as (smeared) singularitiesof the basic harmonic function defined in the annulus in [35].Before that, we show in the next subsection that the AdS geometries found in [1], that wewill refer as LNRS geometries, fit as well in the CGK class. As we already mentioned in section 2, our class of geometries can be obtained through adouble analytic continuation from the AdS solutions studied in [1]. In this section we showthat the latter fit within the class of solutions referred as “minus solutions” in [34].The warping factors, dilaton, axion and RR 4-form potential associated to the AdS geome-14ries constructed in [1] (in Einstein frame) are given by, f = u √ (cid:32) (cid:98) h h + ( u (cid:48) ) (cid:98) h h (cid:33) / , f = u √ (cid:32) (cid:98) h h (4 (cid:98) h h + ( u (cid:48) ) ) (cid:33) / ,f = 1 √ (cid:32)(cid:98) h (4 (cid:98) h h + ( u (cid:48) ) ) h (cid:33) / , e = e − φ = 12 (cid:115) h (cid:98) h (cid:113) (cid:98) h h + ( u (cid:48) ) ,χ = h (cid:48) ψ , (cid:101) ρ = 1 √ u (cid:16)(cid:98) h h (4 (cid:98) h h + ( u (cid:48) ) ) (cid:17) / , C CY = − h (cid:48) ψ . (3.19)The Riemann surface is the same one defined in equation (3.13) and Figure 3, and, as in theprevious subsection, we are also taking h (cid:48)(cid:48) = 0 and (cid:98) h (cid:48)(cid:48) = 0 globally, i.e. solutions withoutD7 and D3 brane sources. This is needed to obtain the axion and RR 4-form potential of theprevious equations.In this case the matching with the solutions in [34] is with the “minus-solutions” defined byequation (3.10). Taking into account (3.14), the harmonic and holomorphic functions read, A = h − iψh (cid:48) = µ − i νz , B = i u (cid:48) i u ,H = u u − i u z − z ) , K = (cid:98) h α − i β z − z ) . (3.20)As in the previous subsection, the functions H , K , A + A B + B are harmonic and A and B holomorphic. The harmonic function (cid:101) K reads exactly as in (3.16).In turn, the harmonic functions that give rise to the NS-NS and RR three-forms read, h = −
14 ( z + z ) , (cid:101) h = i z − z ) ,h = − µ z + z ) + i ν z − z ) , (cid:101) h = i µ z − z ) + ν z + z ) ,b (1) = 14 ( z + z ) , b (2) = u (2 u − iu ( z − z ))4( u − (2 iµ + ν ( z − z ))(2 iα + β ( z − z ))) + (cid:101) h ,c (1) = − u ( u ( z − z ) + 2 iu )8(2 iα + β ( z − z )) + (cid:101) h , c (2) = νu (2 u − iu ( z − z ))( z + z )8( u − (2 iµ + ν ( z − z ))(2 iα + β ( z − z ))) + h . (3.21)From these expressions we recover the NS-NS and RR field strengths, H and F , of the solutionsin [1]. As we have already mentioned, the class of solutions constructed in [34] have vanishing D3 andD7-brane charges. Those solutions have a Riemann surface with a single boundary component.In the follow-up paper [35], the authors constructed solutions in which the Riemann surface Σ15as an arbitrary number of boundaries and non-vanishing D3 brane charges. The D3-branesoccur as poles of a basic harmonic function at the boundaries. In this section we consider thesimplest case of a Riemann surface with two disconnected boundary components, namely theannulus. We will then see in subsection 3.5 that we can recover the solutions with D3-branesources at ρ = 2 π ( P + 1), the end of the space. Quite surprisingly, we will see that D7-branesseem also allowed at the end of the space.The annulus is defined as,Σ ≡ (cid:26) w ∈ C, ≤ Re( w ) ≤ , ≤ Im( w ) ≤ t (cid:27) , (3.22)with t ∈ R + . The points w + 1 and w are identified, thus giving the topology of an annulus.Its two boundaries, ∂ Σ , , are located at Im( w ) = 0 and Im( w ) = t . The annulus can beconstructed from a double surface (cid:98) Σ, which is defined as a rectangular torus with periods 1 and τ , where τ is a purely imaginary parameter, τ = it . The original surface Σ is obtained as thequotient Σ = (cid:98) Σ / J where J ( z ) = z .The construction of the solutions for the annulus in [35] proceeds in three steps. First,a basic harmonic function with singularities and suitable boundary conditions is constructed.Second, the harmonic functions, A + A , H and K , are expressed as linear superpositions ofthe basic harmonic function, evaluated at the various poles in the two boundaries. Finally,the meromorphic function B is constructed such that it satisfies certain regularity conditions.Some of these conditions come from imposing that the solutions asymptote locally to the AdS × S × CY background. These regularity conditions will not be satisfied by our solutions, firstbecause they do not asymptote to this geometry and, second, because the D3-branes (also theD7-branes) are smeared in the ψ direction. This introduces significant changes in the regularityanalysis. For this reason we will not give a detailed account of the regularity conditions imposedin [35]. We will see however in the next subsection that our solutions can still be recoveredfrom the general formalism in [35] in an appropriate limit.The construction in [35] of the basic harmonic function is carried out in terms of ellipticfunctions and their related Jacobi theta function of the first kind, θ ( w | τ ) = 2 ∞ (cid:88) n =0 ( − n e iπτ ( n + ) sin[(2 n + 1) w ] , (3.23)as follows, h ( w, w ) = i (cid:18) ∂ w θ ( πw | τ ) θ ( πw | τ ) + 2 πiwτ (cid:19) + c.c. (3.24)This function has the following simple properties: it has a single simple pole on ∂ Σ, it satisfiesDirichlet conditions away from the pole, and it is positive in the interior of Σ.Notice that h ( w, w ) has a singularity at w = 0, on the first boundary. This pole can beshifted to any point on ∂ Σ by a real translation, so that h ( w − x, w − x ) has a singularity at16 = x . Instead, to obtain the harmonic functions with singularities at ∂ Σ one needs to define, w (cid:48) ≡ τ − w. (3.25)Then the function h ( w (cid:48) + y, w (cid:48) + y ) has a pole at w (cid:48) = − y , on the second boundary, for a real y . In other words the pole is localised at w = y + it/ A + A , B + B , H and K are expressed as linearcombinations of h harmonic functions with poles on both boundaries, A + A = M A (cid:88) (cid:96) A =1 r (cid:96) A h ( w − x (cid:96) A , w − x (cid:96) A ) + M (cid:48) A (cid:88) j A =1 r (cid:48) j A h ( w (cid:48) + y j A , w (cid:48) + y j A ) ,B + B = M B (cid:88) (cid:96) B =1 r (cid:96) B h ( w − x (cid:96) B , w − x (cid:96) B ) + M (cid:48) B (cid:88) j B =1 r (cid:48) j B h ( w (cid:48) + y j B , w (cid:48) + y j B ) ,H = M H (cid:88) (cid:96) H =1 r (cid:96) H h ( w − x (cid:96) H , w − x (cid:96) H ) + M (cid:48) H (cid:88) j H =1 r (cid:48) j H h ( w (cid:48) + y j H , w (cid:48) + y j H ) ,K = M K (cid:88) (cid:96) K =1 r (cid:96) K h ( w − x (cid:96) K , w − x (cid:96) K ) + M (cid:48) K (cid:88) j K =1 r (cid:48) j K h ( w (cid:48) + y j K , w (cid:48) + y j K ) . (3.26)Each harmonic function is taken to have M i poles x (cid:96) i with (cid:96) i = 1 , ..., M i on ∂ Σ , and M (cid:48) i poles y j i with j i = 1 , ..., M (cid:48) i on ∂ Σ . The corresponding residues are r (cid:96) i and r j i .In addition to the regularity conditions given in subsection 3.1, the harmonic functions(3.26) satisfy an extra condition coming from the requirement that e >
0. Namely, ( A + A ) K − ( B + B ) > r A r K = r B . In this subsection we show that it is possible to recover well-defined global solutions withsource branes at the ends of the space from the general analysis above for the annulus. Thesesolutions do not satisfy most of the regularity conditions imposed in [34, 35], and, moreover,contain not only D3 but also D7-brane sources at the ends of the space. Still, we will be ableto recover them in a particular limit from the formalism in [35].As we have already stressed, the choice of constants in the general (cid:98) h and h functions definedby equations (2.6) and (2.7) allows for discontinuities in the RR sector of our backgrounds ateach ρ = 2 πk value, with k = 1 , . . . , P . The discontinuities in (cid:98) h (cid:48) are interpreted as generatedby D3-brane sources, while the discontinuities in h (cid:48) are interpreted as generated by D7-branes.Both types of branes are smeared in the ψ direction. The space is terminated in the ρ directionby imposing that (cid:98) h = h = 0 at ρ = 0 , π ( P + 1). When u = constant the closure of the space17y setting (cid:98) h = h = 0 generates D3 and D7 sources, in the boundary of the space, as explainedaround eq.(2.10).Instead, in the general discussion for the annulus in [35] the D3-branes occur as poles ofa basic harmonic function at its two boundaries. The basic harmonic function must howeverbe regular in the interior. Therefore, in order to fit in the discussion for the annulus we needcontinuous (cid:98) h (cid:48) and h (cid:48) functions. This is imposed taking β k ≡ β, ν k ≡ ν, for k = 0 , , . . . , P, (3.27)in (2.6), (2.7), which implies α k = k β, µ k = k ν, for k = 0 , . . . , P. (3.28)The solutions are then defined by the functions (cid:98) h = β π ρ, h = ν π ρ (3.29)at all ρ -intervals. Yet, the closure of the space at ρ = 2 π ( P +1) requires that ( P +1) β D3-branesand ( P + 1) ν D7-branes are present at the end of the space. Instead of closing the space byintroducing sources as we did with the choice of (cid:98) h and h functions given by (2.6) and (2.7),these branes will be automatically present at the end of the space in the annulus construction.Let us now see how these solutions arise from the general formalism in [35]. We take theannulus in (3.22) as defined from, w = z π = (cid:101) ψ + i (cid:101) ρ, with (cid:101) ψ = ψ π , (cid:101) ρ = ρ π . (3.30)Then (cid:101) ψ ∈ [0 ,
1] and the parameter t in the definition of the annulus is t = 2( P + 1). As recalledin section 2, our class of solutions is valid when P is large. This allows us to approximate theJacobi theta function introduced in (3.23) by its asymptotic expansion when t → ∞ , θ ( πw | τ ) | t →∞ ≈ e − π t sin πw ≈ ie − π t e − iπw . (3.31)This approximation will be key in showing the matching with our solutions. Indeed, in thisapproximation it is easy to see that the basic harmonic function defined by (3.24) reads, h ( w, w ) ≈ π + iπP + 1 ( w − w ) . (3.32)This gives, at the two boundaries ∂ Σ and ∂ Σ , h ( w − x (cid:96) i , w − x (cid:96) i ) ≈ π + iπP + 1 ( w − w ) ,h ( w (cid:48) + y j i , w (cid:48) + y j i ) ≈ − iπP + 1 ( w − w ) , (3.33)respectively, where for the second boundary we have used the relation (3.25). These expressionsare thus independent of the positions of the poles at both boundaries. This is in agreement with18he fact that our D3/D7 branes are smeared in the ψ -direction. We then get for the harmonicfunctions in eq.(3.26), A + A =2 π M A (cid:88) (cid:96) A =1 r (cid:96) A − iπP + 1 ( w − w ) M (cid:48) A (cid:88) j A =1 r (cid:48) j A − M A (cid:88) (cid:96) A =1 r (cid:96) A ,B + B =2 π M B (cid:88) (cid:96) B =1 r (cid:96) B − iπP + 1 ( w − w ) M (cid:48) B (cid:88) j B =1 r (cid:48) j B − M B (cid:88) (cid:96) B =1 r (cid:96) B ,H =2 π M H (cid:88) (cid:96) H =1 r (cid:96) H − iπP + 1 ( w − w ) M (cid:48) H (cid:88) j H =1 r (cid:48) j H − M H (cid:88) (cid:96) H =1 r (cid:96) H ,K =2 π M K (cid:88) (cid:96) K =1 r (cid:96) K − iπP + 1 ( w − w ) M (cid:48) K (cid:88) j K =1 r (cid:48) j K − M K (cid:88) (cid:96) K =1 r (cid:96) K . (3.34)In order to match these expressions with the expressions for A + A and K given in eq.(3.15)we take into account that w = z/ (2 π ), and we obtain, M A (cid:88) (cid:96) A =1 r (cid:96) A = 0 and M (cid:48) A (cid:88) j A =1 r (cid:48) j A = ( P + 1) ν π , (3.35)for the matching of A + A , and M K (cid:88) (cid:96) K =1 r (cid:96) K = 0 and M (cid:48) K (cid:88) j K =1 r (cid:48) j K = ( P + 1) βπ , (3.36)for the matching of K . Rescaling the residues as r (cid:48) j A → r (cid:48) j A , r (cid:48) j K → r (cid:48) j K . (3.37)and replacing the sums by M (cid:48) A (cid:88) j A =1 r (cid:48) j A → π (cid:90) π dr (cid:48) j A , (3.38)as implied by the smearing of the branes in the ψ -direction, we can finally interpret the residuesas the charge-densities of D7 and D3 brane sources at both boundaries of the annulus. We wouldlike to stress that even if the general formalism in [35] does not account for D7-branes at theboundaries of the annulus, we have associated these to the (smeared) poles of the basic harmonicfunction for A + A . The analysis goes in complete parallelism to the analysis of the residues andpoles of the K function, associated to the D3-brane sources at both boundaries of the annulus.It is unclear to us the precise reason why this seems to work in the presence of D7-branes. Note that a rescaling is also necessary in order to interpret the residues of the solutions in [48] as chargesof ( p, q ) 5-branes. B + B and H functions we find M B (cid:88) (cid:96) B =1 r (cid:96) B = M (cid:48) B (cid:88) j B =1 r (cid:48) j B = 0 and M H (cid:88) (cid:96) H =1 r (cid:96) H = M (cid:48) H (cid:88) j H =1 r (cid:48) j H = u π . (3.39)These expressions do not seem to have however a direct interpretation in terms of charges ofour solutions.The previous analysis holds true as well for the LNRS backgrounds discussed in [1]. Thematching of the A + A , H and K functions is valid for both solutions, while the harmonicfunction B + B vanishes. Again, there are smeared D3 and D7-branes at the end of the spacewith the same relations between residues and charges. × S × CY × Σ solutions with Σ aninfinite strip In this section we construct a new class of AdS solutions to Type IIB supergravity with 8supercharges by acting with non-Abelian T-duality (NATD) on the AdS × S × CY × I ρ solutionsobtained in [23]. The non-Abelian T-duality transformation is performed with respect to afreely acting SL(2 , R ) isometry group of the AdS subspace. This transformation gives rise toa new class of solutions in which the AdS subspace is replaced by AdS times an interval.These solutions fit in the classification of [34] for a Riemann surface with a single boundary,equivalent to an infinite strip. × S × CY solutions The study of NATD as a solution generating technique of supergravity was initiated in [49].Further works include [50, 51, 52, 53]. In all these examples the dualisation took place withrespect to a freely acting SU(2) subgroup of the entire symmetry group of the solutions. Instead,in this section we perform the non-Abelian T-duality transformation with respect to one of thefreely acting SL(2 , R ) isometry groups of the AdS subspace.In order to perform the dualisation with respect to the SL(2 , R ) isometry group we follow thederivation in [54]. We take the sl (2 , R ) generators analytically continuing the su (2) generators,as t a = τ a / √
2, with τ = (cid:18) ii (cid:19) , τ = (cid:18) − ii (cid:19) , τ = (cid:18) i − i (cid:19) . (4.1)These generators satisfy,Tr( t a t b ) = ( − a δ ab , [ t , t ] = i √ t , [ t , t ] = i √ t , [ t , t ] = − i √ t . (4.2)20 group element in the Euler parametrisation is given by, g = e i φτ e i θτ e i ψτ , where 0 ≤ θ ≤ π, ≤ ψ < ∞ , ≤ φ < ∞ , (4.3)from which we write the left invariant one forms, L a = − i Tr( t a g − d g ), in the following fashion, L = sinh ψ d θ − cosh ψ sin θ d φL = cosh ψ d θ − sinh ψ sin θ d φL = − cos θ d φ − d ψ. (4.4)The backgrounds in [23] support an SL(2 , R ) isometry such that the metric, the Kalb-Ramond field and the dilaton can be written as , ds = 14 g µν ( x ) L µ L ν + G ij ( x ) dx i dx j , B = B ij ( x ) dx i ∧ dx j , φ = φ ( x ) , (4.5)where x i are the coordinates in the internal manifold, for i, j = 1 , , ...,
7, and L µ are the formsgiven by (4.4). All the coordinate dependence on the SL(2 , R ) group is contained in theseforms. The subsequent details on how to technically compute the NATD transformation havebeen developed extensively in the literature [49, 53] (see these reference for more details).The geometries obtained through NATD with respect to a freely acting SL(2 , R ) group onthe AdS of the solutions in [23] are given by,d s st = u (cid:113)(cid:98) h h r (cid:98) h h − u r d s + (cid:115)(cid:98) h h d s + u (cid:113)(cid:98) h h (cid:98) h h + u (cid:48) d s + (cid:113)(cid:98) h h u (d ρ + dr ) ,e − φ = (cid:16) r (cid:98) h h − u (cid:17) (cid:16) (cid:98) h h + u (cid:48) (cid:17) (cid:98) h ,B = − r (cid:98) h h r (cid:98) h h − u vol AdS − ρ (cid:98) h h − u (cid:48) ( u − ρu (cid:48) )2 (cid:16) (cid:98) h h + u (cid:48) (cid:17) vol S . (4.6) We have taken g µν = − Tr( t µ t ν ) to have signature (+ , − , +). F = − rh (cid:48) r + 12 (cid:18) h + ∂ ρ (cid:20) uu (cid:48) (cid:98) h (cid:21)(cid:19) d ρ,F = h (cid:16) (cid:98) h h + u (cid:48) (cid:17) (cid:32)(cid:98) h (cid:48) u (cid:98) h d ρ + rh (cid:18) (cid:98) h + ∂ ρ (cid:20) uu (cid:48) h (cid:21)(cid:19) d r (cid:33) ∧ vol S + r h (cid:16) r (cid:98) h h − u (cid:17) (cid:18) h (cid:48) u h d r − r (cid:98) h (cid:18) h + ∂ ρ (cid:20) uu (cid:48) (cid:98) h (cid:21)(cid:19) d ρ (cid:19) ∧ vol AdS ,F = 12 (cid:18) r (cid:98) h (cid:48) d r − (cid:18) (cid:98) h + ∂ ρ (cid:20) uu (cid:48) h (cid:21)(cid:19) d ρ (cid:19) ∧ vol CY − r u h (cid:16) r (cid:98) h h − u (cid:17) (cid:16) (cid:98) h h + u (cid:48) (cid:17) (cid:18) r (cid:98) h (cid:48) d ρ + (cid:18) (cid:98) h + ∂ ρ (cid:20) uu (cid:48) h (cid:21)(cid:19) d r (cid:19) ∧ vol AdS ∧ vol S ,F = − (cid:98) h (cid:16) (cid:98) h h + u (cid:48) (cid:17) (cid:18) h (cid:48) u h d ρ + r (cid:98) h (cid:18) h + ∂ ρ (cid:20) uu (cid:48) (cid:98) h (cid:21)(cid:19) d r (cid:19) ∧ vol S ∧ vol CY − r (cid:98) h (cid:16) r (cid:98) h h − u (cid:17) (cid:32)(cid:98) h (cid:48) u (cid:98) h d r − rh (cid:18) (cid:98) h + ∂ ρ (cid:20) uu (cid:48) h (cid:21)(cid:19) d ρ (cid:33) ∧ vol AdS ∧ vol CY ,F = r u (cid:98) h (cid:16) r (cid:98) h h − u (cid:17) (cid:16) (cid:98) h h + u (cid:48) (cid:17) (cid:18) rh (cid:48) d ρ + 14 (cid:18) h + ∂ ρ (cid:20) uu (cid:48) (cid:98) h (cid:21)(cid:19) d r (cid:19) ∧ vol AdS ∧ vol CY ∧ vol S . (4.7)The previous background is a solution to the Type IIB supergravity EOM whenever 4 r (cid:98) h h − u >
0. Namely we get a well-defined geometry for r > r = u (cid:113)(cid:98) h h . (4.8)In the next section we show that a subset of the solutions defined by (4.6) and (4.7) fit inthe general classification of AdS × S × CY × Σ geometries given in [34] with Σ an infinite strip.22 = 0 r = u √ ˆ h h ρ = 2 π ( P + 1) z ρr Figure 4: Infinite strip associated to the NATD solution.
In this section we discuss how the solutions given by (4.6)-(4.7) fit in the class of CGK. Goingto Einstein frame we get the warp factors of the metric, dilaton and axion, f = ur √ h (4 (cid:98) h h + u (cid:48) ) / (cid:98) h h r − u ) / , f = u √ h (4 (cid:98) h h r − u ) / (cid:98) h h + u (cid:48) ) / ,f = (4 (cid:98) h h r − u ) / (4 (cid:98) h h + u (cid:48) ) / √ h , e = e − φ = (cid:113) (4 (cid:98) h h r − u )(4 (cid:98) h h + u (cid:48) )2 (cid:98) h ,χ = 12 (cid:18) ν ( ρ − r ) + 4 µρ + uu (cid:48) (cid:98) h (cid:19) , (cid:101) ρ = √ h (4 (cid:98) h h r − u ) / (4 (cid:98) h h + u (cid:48) ) / u . (4.9)In χ , the axion field, we have taken h = µ + νρ , with µ , ν constants. This choice correspondsto backgrounds without D7-branes, as those constructed in [34].The 2d Riemann surface associated to the solutions is the strip depicted in Figure 4,parametrised as, z = ρ + i r where ρ ∈ [0 , π ( P + 1)] and r ∈ [ r , ∞ ] , (4.10)where the value of r is determined in eq.(4.8).Taking the ’plus-solution’ defined by equation (3.9) we obtain the A , B , H and K functionsin terms of the defining functions of our backgrounds, (cid:98) h , h and u , A = 12 (cid:18) µ ( r − iρ ) + 2 iν ( r − iρ ) + u (cid:48) (cid:98) h ( ru (cid:48) − iu ) (cid:19) , B = 12 (cid:113) (cid:98) h h + u (cid:48) √ u + r u (cid:48) (cid:113)(cid:98) h h ,H = ru , K = r (4 (cid:98) h h + u (cid:48) )2 h . (4.11)We anticipate these functions are neither harmonic nor holomorphic. In order to ensure har-monicity -in H and K - and holomorphicity -in A and B - we need to choose u (cid:48) = 0. In that23ase we obtain, A = 4 µ ( r − iρ ) + 2 iν ( r − iρ ) = − iz µ + zν , B = u = u ,H = ru = − i u ( z − z ) , K = r (cid:98) h = − i ( z − z )2 ( β ( z + z ) + 2 α ) , (4.12)where we have used (3.14). The harmonic function conjugated to K is, (cid:101) K = − ( β ( z + z ) + 2 α ( z + z )) , with C CY = 12 ( β ( r − ρ ) + 2 αρ ) . (4.13)Note that we have taken (cid:98) h = α + βρ , with α , β constants, which corresponds to backgroundswithout D3-branes, as those constructed in [34].The functions associated to the complex three-form are, h = − i z − z ) , (cid:101) h = −
14 ( z + z ) ,h = − (cid:16) µ ( z + z ) + ν z + z ) (cid:17) , (cid:101) h = i (cid:16) µ ( z − z ) + ν z − z ) (cid:17) . (4.14)Notice that h i and (cid:101) h i are harmonic functions conjugate to each other. The potentials given in(3.7) are, b (1) = − i u ( z − z )( z − z ) ( β ( z + z ) + 2 α )( ν ( z + z ) + 2 µ ) + 4 u − h , b (2) = −
14 ( z + z ) ,c (1) = − i u ( z − z )(2 µ ( z + z ) + ν ( z + z ))16(( z − z ) ( β ( z + z ) + 2 α )( ν ( z + z ) + 2 µ ) + 4 u ) + (cid:101) h ,c (2) = − u β ( z + z ) + 2 α ) + h , (4.15)which agree with the expressions (4.6) and (4.7) for u (cid:48) = 0.The previous analysis shows that the new class of solutions constructed through non-AbelianT-duality provide an explicit example of CGK geometries where the Riemann surface is aninfinite strip. We will provide a more detailed global study of these solutions in a futurepublication. In this section we extend two results discussed in [1, 39] to our new infinite family of AdS solutions.The first result is a relation between the holographic central charge in eq.(2.24) and an integral24f the product of the electric and magnetic fluxes of the Dp-branes present in the background.This relates the holographic central charge in Section 2.2, computed purely in terms of theNS-NS sector of the background, with a calculation purely in terms of the Ramond-Ramondsector.Furthermore, in section 5.2, we explore this relation from a geometrical point of view. Wedefine a quantity in terms of geometric forms in our geometries and through an extremisationprinciple relate it to the holographic central charge in eq. (2.24). In summary, in this sectionwe present a connection between the holographic central charge, the product of the electric andmagnetic charges and an extremised functional. We provide a relation between the holographic central charge found in eq.(2.24) and the fluxesof the Ramond-Ramond sector in eq (2.2). Consider a Dp brane and the associated electric (cid:98) F p +2 and magnetic (cid:98) F − p Page field strengths. We define the “density of electric and magneticcharges”, ρ e Dp and ρ m Dp , as follows, ρ e Dp = 1(2 π ) p (cid:98) F p +2 , ρ m Dp = 1(2 π ) − p (cid:98) F − p . (5.1)From these we construct the quantity, (cid:90) (cid:88) p =1 , , , ρ e D p ρ m D p == 1 π Vol
AdS (cid:18) Vol CY π (cid:19) (cid:90) d ρ (cid:34) (cid:98) h h − ( u (cid:48) ) ∂ ρ (cid:18) u ( h h ) (cid:48) h h (cid:19) − u (cid:32)(cid:98) h (cid:48)(cid:48) (cid:98) h + h (cid:48)(cid:48) h (cid:33)(cid:35) . (5.2)In the absence of sources (cid:98) h (cid:48)(cid:48) = h (cid:48)(cid:48) = 0 and, up to a boundary term, this is proportional tothe expression for the holographic central charge in equation (2.24). We explore below thecontribution of the sources to this expression. Notice that eq.(5.2) links the holographic centralcharge in eq.(2.24)—a calculation purely in terms of the NS-NS sector—with one purely interms of the Ramond-Ramond sector. Following the ideas of [55, 56] and the lead of the works [1, 39], we construct a functional interms of an integral of forms defined in the internal space. Once such functional is extremisedthe holographic central charge in eq.(2.24) is recovered, up to a boundary term.We define forms J i and F i (for i = 1 , , ,
7) on the internal space X =[S , CY , S ψ , I ρ ].These forms are inherited from the Page fluxes (2.15) . As explained in [1, 39], they are the The same result can be obtained considering the Maxwell fluxes in (2.2). estriction of the fluxes to the internal space. Writing the Page fluxes in eqs.(2.15) in terms offorms J i and F i as, (cid:98) F = J , (cid:98) F = F ∧ vol AdS + J , (cid:98) F = F ∧ vol AdS + J , (cid:98) F = F ∧ vol AdS + J , (cid:98) F = F ∧ vol AdS . (5.3)The forms J i and F i are, J = h (cid:48) d ψ , J = 14 (cid:18) h + u (cid:48) ( u (cid:98) h (cid:48) − (cid:98) h u (cid:48) )2 (cid:98) h (cid:19) vol S ∧ d ρ, J = − (cid:98) h (cid:48) vol CY ∧ d ψJ = − (cid:18) (cid:98) h + u (cid:48) ( uh (cid:48) − h u (cid:48) )2 h (cid:19) vol CY ∧ vol S ∧ d ρ, F = 12 ( h (cid:48) ( ρ − πk ) − h ) d ψ , F = 14 (cid:32) ( ρ − πk ) h − ( u − ( ρ − πk ) u (cid:48) ) ( u (cid:98) h (cid:48) − (cid:98) h u (cid:48) )4 (cid:98) h (cid:33) vol S ∧ d ρ, F = 12 ( (cid:98) h − ( ρ − πk ) (cid:98) h (cid:48) ) vol CY ∧ d ψ , F = − (cid:18) ( ρ − πk ) (cid:98) h − ( u − ( ρ − πk ) u (cid:48) ) ( uh (cid:48) − h u (cid:48) )4 h (cid:19) vol CY ∧ vol S ∧ d ρ. (5.4)With the forms in eqs.(5.4), we construct the functional, C = (cid:90) X F ∧ J + F ∧ J − ( J ∧ F + J ∧ F )= (cid:90) X (cid:32) (cid:98) h h − ( u (cid:48) ) − u (cid:32)(cid:98) h (cid:48) (cid:98) h + h (cid:48) h (cid:33) + uu (cid:48) (cid:32)(cid:98) h (cid:48) (cid:98) h + h (cid:48) h (cid:33)(cid:33) vol CY ∧ vol S ∧ d ψ ∧ d ρ, (5.5)We minimise the functional C by imposing the Euler-Lagrange equation for u ( ρ ),2 u (cid:48)(cid:48) = u (cid:32)(cid:98) h (cid:48)(cid:48) (cid:98) h + h (cid:48)(cid:48) h (cid:33) . (5.6)This equation of motion is solved if, h (cid:48)(cid:48) = 0 , (cid:98) h (cid:48)(cid:48) = 0 , u (cid:48)(cid:48) = 0 , (5.7)the first two are Bianchi identities for the background and the last is a BPS equation. Thefunctional in eq.(5.5) can be rewritten as, C = 18 (cid:90) X (cid:32) (cid:98) h h − ( u (cid:48) ) + ∂ ρ (cid:34) u (cid:32)(cid:98) h (cid:48) (cid:98) h + h (cid:48) h (cid:33)(cid:35) − u (cid:32)(cid:98) h (cid:48)(cid:48) (cid:98) h + h (cid:48)(cid:48) h (cid:33)(cid:33) vol CY ∧ vol S ∧ d ψ ∧ d ρ. (5.8)The last term (that would vanish in the absence of sources), is proportional to the quotientof the number of flavours by the number of colours in each node. Using the condition that26he flavours are sparse, as explained below eq.(2.17), we see that its contribution is subleadingin front of the other terms. Furthermore, the boundary term gives a divergent contribution.Indeed, for the case u = u and (cid:98) h , h in eqs.(2.6)-(2.7) the boundary term reads, (cid:90) π ( P +1)0 ∂ ρ (cid:34) u (cid:32)(cid:98) h (cid:48) (cid:98) h + h (cid:48) h (cid:33)(cid:35) vol CY ∧ vol S ∧ d ψ ∧ d ρ = − lim (cid:15) → πu (cid:15) ( α P + µ P + β + ν )Vol CY = − lim (cid:15) → πu (cid:15) ( Q total D3 + Q total D7 )Vol CY , (5.9)where we regularised (cid:98) h (0) = h (0) = (cid:98) h (2 π ( P + 1)) = h (2 π ( P + 1)) = (cid:15) . The divergence ineq.(5.9) is associated with the presence of sources in the background as was found in [1, 39].In summary, the functional in eq.(5.5) is proportional to the holographic central chargeof eq.(2.24), plus a subleading contribution and a boundary term. For our infinite family ofbackgrounds, we have linked a calculation purely in terms of the NS-NS sector—eq.(2.24), witha calculation purely in terms of the Ramond-Ramond sector—eq.(5.2), with the extremisationof a functional constructed as a restriction of the Ramond-Ramond forms to the internal space—eq.(5.5). We believe that this may be a generic feature, worth exploring in backgrounds dualto various SCFTs in different dimensions. We close this paper by presenting a short summary of the contents of this work and proposingfuture lines of investigation.This work presents two new infinite families of backgrounds with an AdS factor. Thepresentation focuses mostly on geometrical aspects of the new solutions. The new family ofbackgrounds in Section 2 can be obtained by analytically continuing the backgrounds of [1] orvia T-duality, on the Hopf-fibre of the S , from the solutions in [39]. These connections aresummarised in Figure 1. A precise brane set-up was proposed for these backgrounds and theholographic central charge was calculated. We used the brane set-up to argue for a precisequiver. The IR dynamics of such quivers should be the SCQMs dual to our backgrounds.The family of AdS backgrounds in Section 2 and that in the paper [1] have been shown tobe connected to the solutions of [34, 35]. In fact, under certain circumstances they extend thisclass of solutions. The connection between these qualitatively different backgrounds requires ofa subtle zoom-in procedure that we explained in detail in Section 3.A second family of new backgrounds is presented in Section 4. These interesting solutionsdepend explicitly on two coordinates (labelled as ρ and r in Section 4) and were obtainedby the application of non-Abelian T-duality on the AdS factor of the backgrounds in [23].We leave for future work to discuss the associated brane set-up, though it seems clear thatthe ideas described in [40, 41, 42, 43, 14] will play an essential role in the global-definition of27hese solutions. By the same token, it would be interesting to study the integrability (or not)of the backgrounds presented here, as well as those in [1, 39]. Integrable string backgroundsdual to field theories described by linear quivers in dimensions d = 2 , ,
6, have been found in[57, 58, 59, 60]. Similar techniques should probably apply for the d = 1 case.Finally, in Section 5 the holographic central charge defined in Section 2—a quantity com-puted solely in terms of the NS-NS sector of the backgrounds, has been connected with acalculation purely in terms of the Ramond-Ramond sector of our solutions. A functional whoseextremisation yields the holographic central charge was also discussed. It should be interestingto find out if a similar structure occurs generically for other AdS d +1 backgrounds. Acknowledgements
We would like to thank Niall Macpherson and Salomon Zacarias for very useful discussions.The work of CN is supported by STFC grant ST/T000813/1. Y.L. and A.R. are partiallysupported by the Spanish government grant PGC2018-096894-B-100. AR is supported byCONACyT-Mexico.
References [1] Y. Lozano, C. Nunez, A. Ramirez, and S. Speziali, “New AdS backgrounds and N = 4Conformal Quantum Mechanics,” arXiv:2011.00005 [hep-th] .[2] J. M. Maldacena, “The Large N limit of superconformal field theories and supergravity,” Int. J. Theor. Phys. (1999) 1113–1133, arXiv:hep-th/9711200 [hep-th] . [Adv.Theor. Math. Phys.2,231(1998)].[3] A. Hanany and E. Witten, “Type IIB superstrings, BPS monopoles, andthree-dimensional gauge dynamics,” Nucl. Phys. B (1997) 152–190, arXiv:hep-th/9611230 .[4] F. Apruzzi, M. Fazzi, D. Rosa, and A. Tomasiello, “All AdS solutions of type IIsupergravity,” JHEP (2014) 064, arXiv:1309.2949 [hep-th] .[5] D. Gaiotto and A. Tomasiello, “Holography for (1,0) theories in six dimensions,” JHEP (2014) 003, arXiv:1404.0711 [hep-th] .[6] S. Cremonesi and A. Tomasiello, “6d holographic anomaly match as a continuum limit,” JHEP (2016) 031, arXiv:1512.02225 [hep-th] .[7] C. Nunez, J. M. Penin, D. Roychowdhury, and J. Van Gorsel, “The non-Integrability ofStrings in Massive Type IIA and their Holographic duals,” JHEP (2018) 078, arXiv:1802.04269 [hep-th] . 288] A. Brandhuber and Y. Oz, “The D-4 - D-8 brane system and five-dimensional fixedpoints,” Phys. Lett. B (1999) 307–312, arXiv:hep-th/9905148 .[9] O. Bergman and D. Rodriguez-Gomez, “5d quivers and their AdS(6) duals,”
JHEP (2012) 171, arXiv:1206.3503 [hep-th] .[10] Y. Lozano, E. ´O Colg´ain, D. Rodriguez-G´omez, and K. Sfetsos, “Supersymmetric AdS via T Duality,” Phys. Rev. Lett. no. 23, (2013) 231601, arXiv:1212.1043 [hep-th] .[11] E. D’Hoker, M. Gutperle, A. Karch, and C. F. Uhlemann, “Warped
AdS × S in TypeIIB supergravity I: Local solutions,” JHEP (2016) 046, arXiv:1606.01254 [hep-th] .[12] E. D’Hoker, M. Gutperle, and C. F. Uhlemann, “Holographic duals for five-dimensionalsuperconformal quantum field theories,” Phys. Rev. Lett. no. 10, (2017) 101601, arXiv:1611.09411 [hep-th] .[13] E. D’Hoker, M. Gutperle, and C. F. Uhlemann, “Warped
AdS × S in Type IIBsupergravity III: Global solutions with seven-branes,” JHEP (2017) 200, arXiv:1706.00433 [hep-th] .[14] Y. Lozano, N. T. Macpherson, and J. Montero, “AdS T-duals and type IIB AdS × S geometries with 7-branes,” JHEP (2019) 116, arXiv:1810.08093 [hep-th] .[15] D. Gaiotto and J. Maldacena, “The Gravity duals of N=2 superconformal field theories,” JHEP (2012) 189, arXiv:0904.4466 [hep-th] .[16] R. Reid-Edwards and j. Stefanski, B., “On Type IIA geometries dual to N = 2 SCFTs,” Nucl. Phys. B (2011) 549–572, arXiv:1011.0216 [hep-th] .[17] O. Aharony, L. Berdichevsky, and M. Berkooz, “4d N=2 superconformal linear quiverswith type IIA duals,”
JHEP (2012) 131, arXiv:1206.5916 [hep-th] .[18] C. Nunez, D. Roychowdhury, S. Speziali, and S. Zacarias, “Holographic aspects of fourdimensional N = 2 SCFTs and their marginal deformations,” Nucl. Phys. B (2019)114617, arXiv:1901.02888 [hep-th] .[19] E. D’Hoker, J. Estes, and M. Gutperle, “Exact half-BPS Type IIB interface solutions. I.Local solution and supersymmetric Janus,”
JHEP (2007) 021, arXiv:0705.0022[hep-th] .[20] E. D’Hoker, J. Estes, M. Gutperle, and D. Krym, “Exact Half-BPS Flux Solutions inM-theory. I: Local Solutions,” JHEP (2008) 028, arXiv:0806.0605 [hep-th] .[21] B. Assel, C. Bachas, J. Estes, and J. Gomis, “Holographic Duals of D=3 N=4Superconformal Field Theories,” JHEP (2011) 087, arXiv:1106.4253 [hep-th] .[22] C. Couzens, C. Lawrie, D. Martelli, S. Schafer-Nameki, and J.-M. Wong, “F-theory andAdS /CFT ,” JHEP (2017) 043, arXiv:1705.04679 [hep-th] .2923] Y. Lozano, N. T. Macpherson, C. Nunez, and A. Ramirez, “AdS solutions in MassiveIIA with small N = (4 ,
0) supersymmetry,”
JHEP (2020) 129, arXiv:1908.09851[hep-th] .[24] Y. Lozano, N. T. Macpherson, C. Nunez, and A. Ramirez, “1/4 BPS solutions and theAdS /CFT correspondence,” Phys. Rev. D no. 2, (2020) 026014, arXiv:1909.09636 [hep-th] .[25] Y. Lozano, N. T. Macpherson, C. Nunez, and A. Ramirez, “Two dimensional N = (0 , solutions in massive IIA,” JHEP (2020) 140, arXiv:1909.10510[hep-th] .[26] Y. Lozano, N. T. Macpherson, C. Nunez, and A. Ramirez, “AdS solutions in massiveIIA, defect CFTs and T-duality,” JHEP (2019) 013, arXiv:1909.11669 [hep-th] .[27] Y. Lozano, C. Nunez, A. Ramirez, and S. Speziali, “ M -strings and AdS solutions toM-theory with small N = (0 ,
4) supersymmetry,”
JHEP (2020) 118, arXiv:2005.06561 [hep-th] .[28] F. Faedo, Y. Lozano, and N. Petri, “Searching for surface defect CFTs within AdS ,” JHEP (2020) 052, arXiv:2007.16167 [hep-th] .[29] F. Faedo, Y. Lozano, and N. Petri, “New N = (0 ,
4) AdS near-horizons in Type IIB,” arXiv:2012.07148 [hep-th] .[30] G. Dibitetto and N. Petri, “AdS from M-branes at conical singularities,” arXiv:2010.12323 [hep-th] .[31] G. Dibitetto and N. Petri, “AdS solutions and their massive IIA origin,” JHEP (2019) 107, arXiv:1811.11572 [hep-th] .[32] J. P. Gauntlett, N. Kim, and D. Waldram, “Supersymmetric AdS(3), AdS(2) and BubbleSolutions,” JHEP (2007) 005, arXiv:hep-th/0612253 .[33] N. Kim, “Comments on AdS solutions from M2-branes on complex curves and thebackreacted K¨ahler geometry,” Eur. Phys. J. C no. 2, (2014) 2778, arXiv:1311.7372[hep-th] .[34] M. Chiodaroli, M. Gutperle, and D. Krym, “Half-BPS Solutions locally asymptotic toAdS(3) x S**3 and interface conformal field theories,” JHEP (2010) 066, arXiv:0910.0466 [hep-th] .[35] M. Chiodaroli, E. D’Hoker, and M. Gutperle, “Open Worldsheets for HolographicInterfaces,” JHEP (2010) 060, arXiv:0912.4679 [hep-th] .[36] D. Corbino, E. D’Hoker, J. Kaidi, and C. F. Uhlemann, “Global half-BPS AdS × S solutions in Type IIB,” JHEP (2019) 039, arXiv:1812.10206 [hep-th] .[37] D. Corbino, “Warped AdS and SU (1 , |
4) symmetry in Type IIB,” arXiv:2004.12613[hep-th] . 3038] G. Dibitetto, Y. Lozano, N. Petri, and A. Ramirez, “Holographic description of M-branesvia AdS ,” JHEP (2020) 037, arXiv:1912.09932 [hep-th] .[39] Y. Lozano, C. Nunez, A. Ramirez, and S. Speziali, “AdS duals to ADHM quivers withWilson lines,” arXiv:2011.13932 [hep-th] .[40] Y. Lozano and C. N´u˜nez, “Field theory aspects of non-Abelian T-duality and N = 2linear quivers,” JHEP (2016) 107, arXiv:1603.04440 [hep-th] .[41] Y. Lozano, N. T. Macpherson, J. Montero, and C. Nunez, “Three-dimensional N = 4linear quivers and non-Abelian T-duals,” JHEP (2016) 133, arXiv:1609.09061[hep-th] .[42] Y. Lozano, C. Nunez, and S. Zacarias, “BMN Vacua, Superstars and Non-AbelianT-duality,” JHEP (2017) 008, arXiv:1703.00417 [hep-th] .[43] G. Itsios, Y. Lozano, J. Montero, and C. Nunez, “The AdS non-Abelian T-dual ofKlebanov-Witten as a N = 1 linear quiver from M5-branes,” JHEP (2017) 038, arXiv:1705.09661 [hep-th] .[44] N. T. Macpherson, C. Nunez, L. A. Pando Zayas, V. G. J. Rodgers, and C. A. Whiting,“Type IIB supergravity solutions with AdS from Abelian and non-Abelian T dualities,” JHEP (2015) 040, arXiv:1410.2650 [hep-th] .[45] Y. Bea, J. D. Edelstein, G. Itsios, K. S. Kooner, C. Nunez, D. Schofield, and J. A.Sierra-Garcia, “Compactifications of the Klebanov-Witten CFT and new AdS backgrounds,” JHEP (2015) 062, arXiv:1503.07527 [hep-th] .[46] B. Assel and A. Sciarappa, “On monopole bubbling contributions to ’t Hooft loops,” JHEP (2019) 180, arXiv:1903.00376 [hep-th] .[47] E. D’Hoker, J. Estes, and M. Gutperle, “Gravity duals of half-BPS Wilson loops,” JHEP (2007) 063, arXiv:0705.1004 [hep-th] .[48] E. D’Hoker, M. Gutperle, and C. F. Uhlemann, “Warped AdS × S in Type IIBsupergravity II: Global solutions and five-brane webs,” JHEP (2017) 131, arXiv:1703.08186 [hep-th] .[49] K. Sfetsos and D. C. Thompson, “On non-abelian T-dual geometries with Ramondfluxes,” Nucl. Phys. B (2011) 21–42, arXiv:1012.1320 [hep-th] .[50] Y. Lozano, E. O Colgain, K. Sfetsos, and D. C. Thompson, “Non-abelian T-duality,Ramond Fields and Coset Geometries,”
JHEP (2011) 106, arXiv:1104.5196[hep-th] .[51] G. Itsios, Y. Lozano, E. O Colgain, and K. Sfetsos, “Non-Abelian T-duality andconsistent truncations in type-II supergravity,” JHEP (2012) 132, arXiv:1205.2274[hep-th] .[52] G. Itsios, C. Nunez, K. Sfetsos, and D. C. Thompson, “On Non-Abelian T-Duality andnew N=1 backgrounds,” Phys. Lett. B (2013) 342–346, arXiv:1212.4840 [hep-th] .3153] G. Itsios, C. Nunez, K. Sfetsos, and D. C. Thompson, “Non-Abelian T-duality and theAdS/CFT correspondence:new N=1 backgrounds,”
Nucl. Phys. B (2013) 1–64, arXiv:1301.6755 [hep-th] .[54] E. Alvarez, L. Alvarez-Gaume, J. Barbon, and Y. Lozano, “Some global aspects ofduality in string theory,”
Nucl. Phys. B (1994) 71–100, arXiv:hep-th/9309039 .[55] C. Couzens, J. P. Gauntlett, D. Martelli, and J. Sparks, “A geometric dual of c -extremization,” JHEP (2019) 212, arXiv:1810.11026 [hep-th] .[56] J. P. Gauntlett, D. Martelli, and J. Sparks, “Toric geometry and the dual of c -extremization,” JHEP (2019) 204, arXiv:1812.05597 [hep-th] .[57] K. Filippas, “Non-integrability on AdS supergravity backgrounds,” JHEP (2020)027, arXiv:1910.12981 [hep-th] .[58] K. S. Rigatos, “Non-integrability in AdS vacua,” arXiv:2011.08224 [hep-th] .[59] C. Nunez, D. Roychowdhury, and D. C. Thompson, “Integrability and non-integrabilityin N = 2 SCFTs and their holographic backgrounds,” JHEP (2018) 044, arXiv:1804.08621 [hep-th] .[60] K. Filippas, C. Nunez, and J. Van Gorsel, “Integrability and holographic aspects ofsix-dimensional N = (1 ,
0) superconformal field theories,”
JHEP (2019) 069, arXiv:1901.08598 [hep-th]arXiv:1901.08598 [hep-th]