F-theory and AdS_3/CFT_2 (2,0)
FF-theory and AdS / CFT (2 , Christopher Couzens , Dario Martelli , and Sakura Sch¨afer-Nameki Department of Mathematics, King’s College London,The Strand, London, WC2R 2LS, UK Mathematical Institute, University of OxfordWoodstock Road, Oxford, OX2 6GG, UK kcl.ac.uk: christopher.couzens, dario.martelligmail: sakura.schafer.nameki
We continue to develop the program initiated in [1] of studying supersymmetric AdS back-grounds of F-theory and their holographic dual 2d superconformal field theories, which aredimensional reductions of theories with varying coupling. Imposing 2d N = (0 ,
2) supersym-metry, we derive the general conditions on the geometry for Type IIB AdS solutions withvarying axio-dilaton and five-form flux. Locally the compact part of spacetime takes the formof a circle fibration over an eight-fold Y τ , which is elliptically fibered over a base (cid:102) M . Weconstruct two classes of solutions given in terms of a product ansatz (cid:102) M = Σ × M , whereΣ is a complex curve and M is locally a K¨ahler surface. In the first class M is globallya K¨ahler surface and we take the elliptic fibration to vary non-trivially over either of thesetwo factors, where in both cases the metrics on the total space of the elliptic fibrations arenot Ricci-flat. In the second class the metric on the total space of the elliptic fibration overeither curve or surface are Ricci-flat. This results in solutions of the type AdS × K × M τ ,dual to 2d (0 ,
2) SCFTs, and AdS × S / Γ × CY , dual to 2d (0 ,
4) SCFTs, respectively.In all cases we compute the charges for the dual field theories with varying coupling andfind 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 geometryto be non-compact leads to the most general class of Type IIB AdS solutions with varyingaxio-dilaton, i.e. F-theoretic solutions, that are dual to 4d N = 1 SCFTs. a r X i v : . [ h e p - t h ] M a r ontents Solutions in F-theory dual to 2d N = (0 , Ansatz and (0 ,
2) Supersymmetry 92.3 Constraints on the Geometry 102.4 Sufficiency of the Conditions 132.4.1 Equations of Motion 142.4.2 Supersymmetry 152.5 Summary of Conditions 162.6 F-theoretic Formulation 17 with 2d N = (2 , and AdS with Varying τ τ : AdS Duals to N = (2 ,
2) 193.3 Varying τ : AdS Duals to 4d N = 1 20 N = (0 , Solutions with Varying τ ,
4) Solutions 304.2.2 Baryonic Twist Solutions 31 Y p , q Case 43 N = 4 SYM 456.2 Twisted N = 1 Field Theories 482.2.1 Universal Twist 506.2.2 Baryonic Twist 516.3 Duality Twisted N = 1 Field Theories 526.3.1 Universal Duality Twist: Elliptic Surface S τ Case 546.3.2 Universal Twist: Elliptic Three-fold T τ Case 566.3.3 Baryonic Duality Twist 57 Solutions 59 Solutions in M-theory 597.2 Flux Quantisation and Central Charges 607.3 AdS Solutions with Elliptic Surface Factor 617.4 AdS Solutions with Elliptic Three-fold Factor 627.5 AdS × Y p , q × K U (1) D in Type IIB 67 A.1 Duality U (1) D B Details for the Derivations in Section 2 70
B.1 Torsion conditions 70B.2 Derivation of the “Master Equation” 71
C Derivation of the N = (2 , Solutions 72
C.1 Torsion Conditions 74C.2 Reducing the Conditions 75C.3 Recovering known (2 ,
2) Solutions 78C.4 Case 1: K¨ahler–Einstein 79C.5 Case 2: Product of Riemann Surfaces 80C.6 Further Generalisations 84
D AdS to AdS D.1 AdS Solutions with (2 ,
2) and Varying τ Solution with varying τ E Details for the Baryonic Twist Solution 90
E.1 Polynomial Solution to the “Master Equation” 90E.2 The local F-theory geometry 92E.3 Regularity 94E.4 Toric Geometry of Y p , q F Summary of the 4d Y p,q Field Theories 103
Twenty years after holography was uncovered in string theory, it still provides us with sur-prising and deep results about strongly coupled superconformal field theories (SCFTs) andquantum gravity in anti-de Sitter (AdS) spacetimes. Progress is as far ranging as finding newsupergravity solutions, matching with dual field theory observables, as well as performing pre-cision tests of the duality in particular regimes. In the present paper we expand this AdS/CFTdictionary towards theories with spacetime varying coupling constant, a program initiated in[1]. The main goal is the construction of Type IIB solutions, where the axio-dilaton τ variesover parts of the spacetime, including monodromies in the SL Z duality group of Type IIB.In this sense these are AdS solutions in F-theory [2]. In a brane realisation, the non-trivialmonodromies arise through the presence of non-perturbative ( p, q ) 7-branes, which contributea new sector to the field theory duals.There are a multitude of motivations for studying field theories with varying coupling, e.g.field theories arising within F-theory such as D3-branes and duality defects in SCFTs. Oftenthe field theory side is somewhat difficult to study due to the genuinely non-perturbativeeffects. For example, in F-theory D3-branes wrapped on cycles inside the compactificationgeometry give rise to a varying complexified coupling τ . A field theoretic description of theseis available for abelian theories [3–6], but remains elusive for the non-abelian generalization.Some special cases of S-duality twists can be studied along the lines of [7], but do not corre-spond to varying axio-dilaton configurations. In this context the holographic dual can shedsome light upon some of the physical properties of these theories. In this paper we determineseveral new classes of solutions in Type IIB supergravity, which have a varying axio-dilaton4im SUSY dτ (cid:54) = 0 Type IIB/F-theory Field Theory Sect.2 (0 , (cid:88) AdS × S × Y τ D3s on C ⊂ Y τ [1]2 (2 , × AdS × M Example in [8] 3.24 1 (cid:88)
AdS × ( S → T τ ) D3s at conical singularity 3.32 (0 , (cid:88) AdS × ( S → ( S τ × M )) Section 6.3.1 4.1.12 (0 , (cid:88) AdS × ( S → ( T τ × Σ)) Section 6.3.2 4.1.22 (0 , (cid:88) AdS × K τ × Y p , q Section 6.3.3 4.2
Table 1 : Summary of various AdS and AdS solutions in Type IIB/F-theory with five-form flux, including supersymmetries, the geometry of the solution and the dual field theory.The spaces with a superscript τ are elliptically fibered, however only in the case of Y τ and K τ they have Ricci-flat metrics. In general they are elliptic fibrations with non-trivial Riccicurvature.profile, dual to both 2d and 4d SCFTs. A summary of the solutions is given in Table 1. Inall cases we will explore both the solutions, as well as the holographic dual field theories andcorroborate the duality by comparing central charges and other characteristics. Furthermore,we determine dual M-theory solutions, which support some of the assumptions made in theF-theory setting.In [1] this approach was initiated by studying the F-theory solutions dual to 2d (0 , × S / Γ × Y τ where Y τ is an ellipticallyfibered Calabi–Yau three-fold, and the complex structure of the elliptic fiber E τ is identifiedwith the axio-dilaton τ IIB . This provides a generalization of the known solutions with (4 , Y = Y × E τ with Y = K3 or T 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 descrip-tion in terms of D3-branes wrapped on a curve inside the base K¨ahler B of the ellipticallyfibered Calabi–Yau three-fold: C ⊂ B ⊂ Y τ . The varying axio-dilaton induces a varyingcoupling of the 4d N = 4 Super-Yang Mills theory on the D3-brane, along the curve C . Theresulting theory is supersymmetric when a particular topological twist, the so-called topo-logical duality twist [3, 4], is applied along the curve C [5, 6, 10]. The dual M-theory setupis the MSW string wrapped on an elliptic surface, and a dual M-theory solution confirmsthe F-theoretic results in [1], including the holographic comparison of the central chargesto leading and sub-leading orders. In this context this comparison is non-trivial due to the5resence of duality defects, i.e. 7-branes, which in the M-theory picture have a geometricorigin in resolution cycles of the singular elliptic fibers.A richer class of theories in 2d preserves only (0 ,
2) supersymmetry, where the centralcharge is not determined by the UV spectrum of the field theory, but due to mixing ofthe U (1) R symmetry with global symmetries along the RG flow, one needs to invoke c -extremization to compute the central charges [11] (see also [12]).Holographically the constant axio-dilaton case supported by only five-form flux was stud-ied in [13], where it was shown that the internal space locally admits a circle fibration, realisingthe U (1) R symmetry in the dual field theory. A related analysis appeared in [14], which againhas trivial τ but allows for a particular three-form flux on the internal manifold M . Ex-amples of solutions were obtained in [15, 16], again for constant τ , where starting with thegeneral framework of [13], the 6d K¨ahler base is assumed to be a direct product C g ×M , with C g a genus- g constant curvature Riemann surface, and M a locally K¨ahler space equippedwith a metric admitting an SU (2) × U (1) isometry.In the present paper we generalise these results to allow for varying axio-dilaton τ , anddetermine the geometric constraints on the supergravity solutions preserving (0 ,
2) in thedual 2d SCFT. Locally the F-theory solution takes the form AdS × ( S → Y τ ), where Y τ is elliptically fibered and K¨ahler, however not Calabi–Yau . The U (1) isometry of the S corresponds to the R-symmetry of the dual (0 ,
2) SCFTs and the curvature of Y τ has tosatisfy (2.63). This is of course a formidable equation to solve and in this paper we will focuson two classes of solutions to this equation, which result from specialisations of Y τ . Let usdenote the base of the elliptic fibration by (cid:102) M , which is (locally) a K¨ahler three-fold.The first class of solutions, which will be referred to as universal twist solutions , arisefrom making a product ansatz for the base in terms of a complex curve and surface E τ (cid:44) → Y τ ↓ (cid:102) M = Σ × M where (cid:40) Y τ = ( E τ → Σ) × M ≡ S τ × M Y τ = ( E τ → M ) × Σ ≡ T τ × Σ (1.1)Thus Y τ , either has an elliptic surface S τ or an elliptic three-fold T τ as a factor. The keyis that none of the factors appearing in Y τ are Ricci-flat. This is to be contrasted with thestandard flat-space analysis, which results in an elliptic Calabi-Yau compactification. Here,due to the fluxes, the supersymmetry implies a non-trivial condition on the curvature of thefactors in Y τ , which is not simply Ricci-flatness. These solutions will be discussed in section4.1. Their field theory duals have a characterization in terms of a topological duality twisted6eduction of 4d N = 1 SCFTs.The second class of solutions, that are the subject of section 4.2 can be summarised interms of an ansatz Y τ = ( E τ → P ) × M = K τ × M , (1.2)whereby the base factors as (cid:102) M = P × M and the elliptic fibration over the P is Ricci-flat .The dual field theories are obtained by turning on baryonic flux, and we shall refer to theseas the baryonic flux solutions . We find that this yields a class of geometries that are closelyrelated to the Y p,q Sasaki–Einstein manifolds [17] as well as the constant axio-dilaton AdS solutions obtained in [14, 15] and interpreted in [16] as gravity duals to certain topologicallytwisted compactification of the Y p,q quiver theories [18]. The F-theory solution, which wefind, including the geometrized axio-dilaton, is AdS × K τ × Y p , q , or as a Type IIB solutionAdS × P × Y p , q , where – similarly to [17] – the latter is a circle fibration over the Hirzebruchsurface F = P × P , equipped with a non-direct product metric. The constant τ precursorof the baryonic flux solutions were conjectured in [16] to be gravity dual to the Y p,q fieldtheories compactified on a T , with a twisting by the baryonic symmetry of these theories.Upon a careful analysis of the geometry that we include in Appendix E, we have uncoveredsome puzzling features of this duality, that deserve a separate investigation.Another interesting class of SCFTs in 2d have (2 ,
2) supersymmetry, which are amenableto localization computations [19, 20]. Recent progress on constructions of dual pairs, usingorbifolds of the NS-NS-flux supported AdS × S × T solutions with 2d (4 ,
4) supersymmetrywere obtained in [21, 22]. We find that by turning on the axio-dilaton as well as requiring2d N = (2 ,
2) supersymmetry in the dual SCFT, no compact solutions exist. However, wedetermine the general constraints for these Type IIB solutions with constant axio-dilaton,showing in particular that they realise geometrically the U (1) R × U (1) R R -symmetry of thedual SCFTs. We will recover some known solutions, but we have refrained from exploringthe general conditions further. Interestingly, if we relax the assumption that the internalspace is compact, the axio-dilaton can vary and we find AdS × M τ , where S (cid:44) → M τ → T τ with T τ the same elliptic three-fold geometries that appeared in the AdS solutions in (1.1).As we will discuss in the paper this motivates the conjecture that the 2d SCFTs dual tothe second type of universal twist solutions are the compactification of the 4d SCFTs dualto this class of AdS solutions. In fact it is not too difficult to show that this is the most Likewise we can consider a factorization Y τ = CY τ × Σ, with a Calabi-Yau three-fold factor, and as weshall show in section 4.2.1, this gives rise to the already known solutions in [1] with (0 ,
4) supersymmetry in2d. solution, dual to SCFTs preserving N = 1 in 4d, supported with only five-formflux. This makes contact with the results in [23], however, we shall provide an intrinsically F-theoretic interpretation in terms of elliptically fibered Calabi–Yau four-fold compactifications.This surprising appearance of duals to 4d SCFTs, generalising known constant but non-perturbative τ solutions of [24–28], is welcome and will be studied elsewhere in more detail[29].The plan of the paper is as follows: in section 2 we begin by developing the generalsetup for solutions in Type IIB with five-form flux and varying axio-dilaton, requiring only(0 ,
2) supersymmetry in 2d. The “master equation” derived in this section will underliemost of the remaining part of the paper. We then analyse in section 3 the special case of(2 ,
2) supersymmetry and find, as already alluded to, that there are no varying axio-dilatonsolutions, unless the internal space decompactifies. The resulting solutions have AdS factorsand varying τ , as explained in section 3.3, and a proof that these are the most general suchAdS solutions dual to 4d N = 1 is provided in appendix D. In section 4 we derive two newclasses of (0 ,
2) F-theory solutions: the universal twist and baryonic twist solutions. Theholographic central charges are computed in section 5. The dual field theories are discussedin section 6, where a brief review of the duality twist is included. Dual M-theory solutions arediscussed in section 7, before we conclude with a summary and outlook in section 8. Severalappendices supplement the content of the main body of the paper. Solutions in F-theory dual to 2d N = (0 , The starting point of our analysis is a comprehensive exploration of the conditions of TypeIIB/F-theory supergravity which yield AdS solutions with at least 2d (0 ,
2) supersymmetryand vanishing three-form fluxes. The main difference to earlier results in [13] is that weallow the axio-dilaton τ to have a non-trivial dependence on spacetime. This generalises ourearlier work in [1], where we found the most general solutions in F-theory with only five-formflux dual to 2d (0 ,
4) SCFTs. The requirement for 2d (0 ,
2) supersymmetry leads us to bothrecover the earlier (0 ,
4) results, as well as new classes of solutions dual to (0 ,
2) SCFTs. The“master equation” (2.41), which constrains the internal geometry, yields potentially moresolutions, whose exploration we leave for the future. This equation has a reformulation interms of an F-theoretic setting, where the axio-dilaton becomes part of the compactificationgeometry. 8 .1 Killing Spinor Equations
The starting point is the 10d Type IIB supergravity Killing spinor equations , i.e. the re-quirement that the supersymmetry transformations of the fermions vanish identically δψ M = D M (cid:15) + i192 Γ P ...P F MP ...P (cid:15) − (cid:16) Γ P ..P M G P ..P − P P G MP P (cid:17) (cid:15) c = 0 , (2.1) δλ = iΓ M P M (cid:15) c + i24 Γ P ..P G P ...P (cid:15) = 0 . (2.2)The covariant derivative D is with respect to both Lorentz transformations and local U (1) D transformations, where the duality U (1) D has gauge field depending on the variation of theaxio-dilaton τ = τ + iτ Q = − τ d τ . (2.3)In addition we define the combination P = i2 τ d τ . (2.4)The Killing spinors have U (1) D charge 1 / P has charge 2 and G has charge 1. This U (1) D symmetry will play a key role in the following as it encodes the varying axio-dilaton. Insupergravity the gauge symmetry itself is well-known and we summarise some of the salientpoints in appendix A. We will return to this in section 6 in the field theory analysis.The equations of motion consist of the Einstein equation R MN = 2 P ( M P ∗ N ) + 196 F MP ..P F P ..P N + 18 (cid:18) G P P ( M G ∗ N ) P P − g MN G P ..P G ∗ P ..P (cid:19) (2.5)and the flux equations of motion and Bianchi identities D ∗ G = P ∧ ∗ G ∗ − i F ∧ G ,
D ∗ P = − G ∧ ∗ G , F = ∗ F , D P = 0 , D G = − P ∧ G ∗ , d F = i2 G ∧ G ∗ . (2.6) Ansatz and (0 , Supersymmetry
In this paper we consider the most general class of bosonic, minimally supersymmetric TypeIIB supergravity solutions with SO (2 ,
2) symmetry and vanishing three-form flux G = 0. Asin [1, 13] the 10d metric will be taken in Einstein frame to be a warped product of the form d s = e (cid:0) d s (AdS ) + d s ( M ) (cid:1) , (2.7) Our conventions will be those of [1, 30]. For the entirety of the paper, we will indicate by subscripts of spaces always the real dimension. s (AdS ) is the metric on AdS , with Ricci tensor R µν = − m g µν and d s ( M ) is themetric on an arbitrary internal seven-dimensional manifold M . We take ∆ ∈ Ω (0) ( M , R ) , P ∈ Ω (1) ( M , C ) , τ ∈ Ω (0) ( M , C ) and the five-form flux to be of the form F (5) = (1 + ∗ )dvol(AdS ) ∧ F (2) , (2.8)with F (2) ∈ Ω (2) ( M , R ) in order to preserve the SO (2 ,
2) symmetry of AdS . The Bianchiidentity for F (5) implies d F (2) = 0 , dˆ ∗ F (2) = 0 , (2.9)where ˆ ∗ is the hodge star on the unwarped metric d s ( M ). We use the spinor ansatzdeveloped in appendix A of [1] (cid:15) = ψ ⊗ e ∆ / ξ ⊗ θ + ψ ⊗ e ∆ / ξ ⊗ θ , (2.10)where ψ i are Majorana Killing spinors on AdS and satisfy ∇ α ψ i = α i m ρ α ψ i . (2.11)The chirality of the spinor of the dual SCFT is determined by the choice of α i = ±
1. Thespinors ψ i are taken to be independent Killing spinors on AdS , whilst the ξ i are Diracspinors on M . Each independent Dirac spinor ξ / will give 2 (anti-) chiral supercharges onthe boundary SCFT. To preserve (0 ,
2) supersymmetry we take ξ to vanish. We shall alsobe interested in preserving (2 ,
2) supersymmetry in section 3 in which case both spinors arekept, but with opposite values for α .The reduced supersymmetry equations for the spinors on M are, as in [1], obtained byinserting the ansatz (2.10) into the 10 d supersymmetry equations, (2.1) and (2.2), γ µ P µ ξ cj = 0 , (2.12) (cid:18) ∂ µ ∆ γ µ − i α j m − /F (2) (cid:19) ξ j = 0 , (2.13) (cid:18) D µ + i α j m γ µ − e − F (2) ν ν γ ν ν µ (cid:19) ξ j = 0 . (2.14) In this section we investigate the torsion conditions arising from imposing the minimal amountof supersymmetry, namely N = (0 ,
2) in 2d. This amount of supersymmetry is preserved bythe existence of a single Dirac spinor on M , and signifies that the internal 7d space admitsan SU (3) structure. In 7d an SU (3) structure implies the existence of a real vector which10oliates the space with the transverse 6d space admitting a canonical SU (3) structure. In thefollowing we show that the transverse 6d space is conformally K¨ahler and the existence of asupersymmetric solution is determined by a single partial differential equation, similar to theequation found in [13], for the K¨ahler metric on the 6d space. The remaining geometry isfixed by the choice of this K¨ahler metric.The torsion conditions for preserving N = (0 ,
4) supersymmetry were computed in ap-pendix C of [1] and may be specialised to preserve N = (0 ,
2) by setting, without loss ofgenerality, ξ = 0. We present the non-trivial torsion conditions in the present case below and the general equations in appendix B.1. Supersymmetry implies both differential and alge-braic constraints on the fluxes and bilinears. The independent differential conditions satisfiedby the bilinears are d S = 0 (2.15)e − d (cid:0) e K (cid:1) = − mU − e − F (2) , (2.16)d (cid:0) e U (cid:1) = 0 , (2.17)e − D (cid:0) e Y (cid:1) = 2 m ∗ Y , (2.18)e − D (cid:0) e ∗ Y (cid:1) = 0 , (2.19)4d∆ ∧ ∗ Y = − ie − F (2) ∧ Y , (2.20)e − d (cid:0) e ∗ U (cid:1) = 2i m ∗ K . (2.21)Again, as in [1] the scalar S can be set to 1 by a constant rescaling of the Killing spinor.To proceed we introduce an orthonormal frame for the metric and by a suitable framerotation we may set K to be parallel to the vielbein e . In this frame the remaining bilinearsbecome K = − e , (2.22) U = − i( e + e + e ) , (2.23) X = U ∧ K , (2.24) Y = ( e − i e ) ∧ ( e − i e ) ∧ ( e − i e ) . (2.25) We refine the notation of [1] for ease of reading. By setting ξ = 0 the bilinears with a ‘2’ index are setto zero, and it therefore becomes superfluous to keep the ‘11’ subscript on the non-zero bilinears; apart fromremoving this labelling the names of the bilinears are otherwise kept the same. We also set the parameter α in [1] to 1 without loss of generality in the following. As explained previously this parameter takes values ±
11 2d SCFT with N = (0 ,
2) supersymmetry has a U (1) R R-symmetry, which by the AdS/CFTdictionary is dual on the gravity side to a Killing vector generating a U (1) isometry of thefull solution. From the torsion conditions it follows that K defines such a Killing vector andthus is identified with the R-symmetry of the putative dual SCFT. It is useful to introducecoordinates adapted to this Killing vector (and dual one-form) K = 2 m∂ ψ , K = 12 m (d ψ + ρ ) , (2.26)so that the 7d metric can be written as followsd s = 14 m (d ψ + ρ ) + d s ( M ) . (2.27)Observe from (2.17) that the bilinear U is conformally closed and this motivates us to definethe following conformally rescaled forms J = i m e U , ¯Ω = m e Y . (2.28)These new forms define a canonical SU (3) structure on (cid:102) M whose metric is conformallyrelated to M by d s ( M ) = e − m d s ( (cid:102) M ) . (2.29)They satisfy the SU (3) structure algebraic conditions J ∧ Ω = 0 , Ω ∧ ¯Ω = − J ∧ J ∧ J = −
8i dvol( (cid:102) M ) (2.30)and in addition the differential conditions¯ D Ω = − mK ∧ Ω , d J = 0 , (2.31)which imply integrability of the complex structure defined by Ω and that (cid:102) M is K¨ahler.Finally, we should extract the conditions of the varying axio-dilaton on the metric. From thesupersymmetry equation (2.12) J µν P µ = i P µ , P µ K µ = 0 , (2.32)i.e. P varies holomorphically on (cid:102) M and the Killing vector K is a symmetry of τ , L K τ = 0.Due to the foliation of the space by the Killing vector we may decompose the exteriorderivative as d = d ψ ∧ ∂ ψ + d . (2.33)12ith this splitting of the exterior derivative (2.31) becomes ∂ ψ Ω = − iΩ , (2.34)d Ω = − i( Q + ρ ) ∧ Ω . (2.35)Equation (2.34) may be solved by extracting a suitable ψ dependent phase from Ω. Thisphase will play no role in the following analysis and will be assumed to have been extracted.Subsequently, (2.35) implies R = − (d Q + d ρ ) , (2.36)where R is the Ricci form on (cid:102) M . The Ricci tensor on (cid:102) M is given in terms of the Ricciform as R µν = − J ρµ R ρν . (2.37)The flux is fixed by equation (2.16) to be mF (2) = − J −
12 d(e (d ψ + ρ )) . (2.38)Notice that the flux has legs along the Killing direction and may be decomposed such thatˆ F (2) = F (2) + de ∧ K , (2.39)has no legs along the Killing direction . By contracting the indices of the Ricci-form with thecomplex structure one finds the Ricci scalar for (cid:102) M to be R = 2 | P | + 8e − . (2.40)By imposing equations (2.16), (2.31), (2.36) it follows that equations (2.19)-(2.21) are imme-diately satisfied. So far supersymmetry has implied that the solution satisfies (2.31), (2.32), (2.36), (2.38) and(2.40). We show in this section that this set of equations in addition to imposing the equationof motion for F (2) are both necessary and sufficient conditions for a bosonic supersymmetricsolution. As we show, the equation of motion for F (2) may be rephrased as a differentialcondition on the K¨ahler metric of the 6d space. We proceed by first considering the equationsof motion before proving that there exists a globally defined Killing spinor satsifying theKilling spinor equations (2.12)-(2.14). The explicit K factor cancels out with that in F (2) . In deriving this result it is necessary to use the algebraic equation F (2) µν J µν = ˆ F (2) µν J µν = − m , which isobtained from the supersymmetry equation (2.12). .4.1 Equations of Motion Recall that the equation of motion for the five-form flux is equivalent to the two equationsin (2.9) for the two-form F (2) . Using equation (2.38) as the definition of F (2) it is clearafter using (2.31) that it is closed. Supersymmetry does not however impose the equation ofmotion for the flux, d ∗ F (2) = 0, which must be imposed in addition. One may understandthis equation as giving a “master equation” for the K¨ahler base which generalises the onefound in [13] to include varying axio-dilaton, (cid:3) ( R − | P | ) − R + R µν R µν + 2 | P | R − R µν P µ P ∗ ν = 0 . (2.41)A discussion of its derivation is given in appendix B.2. We conclude that both the equation ofmotion for the five-form flux and the self-duality constraint are satisfied. The Bianchi identityfor P is implied by construction whilst its equation of motion reduces to τ being harmonicon the K¨ahler manifold. As τ is holomorphic it follows that it is also harmonic and thereforethe flux equations of motion and Bianchi identities are satisfied.By using the analysis of [30] and some case dependent algebra we may show that theEinstein equation is satisfied. Integrability of the Killing spinor equations and use of the fluxequations of motion and Bianchi identities implies E MN Γ N (cid:15) = 0 (2.42)where E MN = 0 is equivalent to Einstein’s equation and (cid:15) is the 10d Killing spinor. One mayconstruct a null vector bilinear, (cid:98) K ≡ ¯ (cid:15) Γ (1) (cid:15) , which implies that the metric admits a framesuch that it takes the form d s = 2 e + e − + e a e a , (2.43)with (cid:98) K = e + and a = 1 , ..,
8. The argument of [31] shows that the only component of E MN which may be non-zero is E ++ . For this class of solutions E ++ lies along AdS and by explicitcomputation one finds that the Ricci-tensor on the warped AdS satisfies R µν = (cid:0) − m + 8 ∇ µ ∆ ∇ µ ∆ − (cid:3) ∆ (cid:1) g µν . (2.44)It follows that E ++ ∝ g ++ which therefore vanishes and we conclude supersymmetry im-plies the Einstein equation. We determine that all the equations of motion are satisfied bysupersymmetry and equation (2.41). 14 .4.2 Supersymmetry We now show that any solution satisfying the necessary conditions presented above admits aglobally defined Killing spinor satisfying (2.1) and (2.2). By construction it follows that anyglobal solution to the 7d Killing spinor equations (2.12)-(2.14) may be uplifted to a globalKilling spinor in 10d satisfying both (2.1) and (2.2). Preserving supersymmetry is thereforeequivalent to proving that equations (2.12)-(2.14) admit a globally defined Killing spinor. Weshall construct such a spinor by making use of the canonical spin c structure that every K¨ahlermanifold admits.We begin by defining the notation and vielbein we shall be using in the following. Recallthat the metric takes the formd s ( M ) = e − m d s ( (cid:102) M ) + 14 m (d ψ + ρ ) = d s ( (cid:102) M ) + ( e ) , (2.45)where, in keeping with the frame in section 2.3, e = − m (d ψ + ρ ). The flat index forthe vielbein on (cid:102) M will be taken from the middle of the Latin alphabet, i, j, k and run from1 , . . . , (cid:102) M will be from the middle of the Greek alphabet, µ, ν, σ ,finally the seven-dimensional indices will be from the beginning of the respective alphabets.The K¨ahler two-form on (cid:102) M , written in terms of the vielbeine, is j = e + e + e , whichin general is only conformally closed, whilst the closed two-form on (cid:102) M is denoted J .On any K¨ahler manifold there exists a spin c structure that admits a section η satisfyingthe spin c Killing spinor equation (cid:18) (cid:101) ∇ µ + i2 (cid:98) P µ (cid:19) η = 0 , (2.46)where (cid:98) P is the one-form Ricci potential of the K¨ahler metric. For a 6d space, if one takes thespinor η to satisfy the projection conditions γ η = γ η = γ η = − i η , (2.47)it is easy to see that the term arising from the spin-connection precisely cancels the contribu-tion from (cid:98) P and therefore any constant section η , subject to the projection conditions, solves(2.46). Clearly this spinor is globally defined on (cid:102) M , and we may use it to to construct aKilling spinor satisfying the 7d supersymmetry equations. On (cid:102) M , equation (2.46) reads (cid:18) (cid:101) D µ − i2 ρ µ (cid:19) η = 0 . (2.48)The spin connection on M is found to be ω jk = (cid:101) ω jk − ∂ k ∆ e j − ∂ j ∆ e k ) − m (cid:104) R jk − i( P j P ∗ k − P ∗ j P k ) (cid:105) e , (2.49)15 j = 14 m (cid:104) R jk e k − i( P j P ∗ − P ∗ j P ) (cid:105) , (2.50)and the flux is mF (2) = − m e j + e R + d Q ) + 4 m e d∆ ∧ e . (2.51)By inserting the above spin connection, (2.36), (2.38) and (2.40) into (2.14), and computingalong the Killing direction and along (cid:102) M , respectively, yields0 = (cid:18) ∇ ψ + i m − e − F ab γ abψ (cid:19) ξ = (cid:18) ∂ ψ − i2 (cid:19) ξ (cid:18) D µ + i m γ µ − e − F ab γ abµ (cid:19) ξ = (cid:18) (cid:101) D µ − i2 ρ µ (cid:19) ξ . (2.52)We may solve both equations by taking the Killing spinor to be ξ = e i2 ψ η . (2.53)Notice that the functional dependence on ψ is consistent with (2.34).It remains to show that the algebraic conditions (2.12) and (2.13) are satisfied. Using theholomorphicity of P one finds that the dilatino equation, (2.12), vanishes upon applicationof the projection conditions (2.47). The algebraic gravitino equation becomes0 = (cid:18) ∂ µ ∆ γ µ − i α j m − /F (2) (cid:19) ξ = (cid:18) i m m (cid:18) R ij − i P i P ∗ j (cid:19) γ ij (cid:19) ξ , (2.54)which vanishes after some gamma matrix algebra and application of (2.40) and (2.47). Weconclude that supersymmetry is preserved if we satisfy (2.31), (2.32), (2.36), (2.38), (2.40)and (2.41). Let us summarise the necessary and sufficient conditions for a supersymmetric solution withat least N = (0 ,
2) supersymmetry, metric of the form (2.7), arbitrary five-form flux, F andvarying axio-dilaton, τ all preserving the isometries of AdS . We have shown that the metricof the solution takes the formd s = e (cid:20) d s (AdS ) + 1 m (cid:18)
14 (d ψ + ρ ) + e − d s ( (cid:102) M ) (cid:19)(cid:21) , (2.55)where d s ( (cid:102) M ) is a K¨ahler metric satisfying the “master equation” (2.41). The remaininggeometry is determined in terms of the metric on d s ( (cid:102) M ) to bee − = 18 ( R − | P | ) , (2.56)16 ρ = − (d Q + R ) , (2.57)and the flux is given by F = (1 + ∗ )dvol(AdS ) ∧ F (2) mF (2) = − J −
12 d(e (d ψ + ρ )) . (2.58)The axio-dilaton τ is a holomorphic function on (cid:102) M , and when it is constant, the aboveconditions consistently reduce to those in [13]. As shown in the previous subsection theseconditions are necessary and sufficient for the existence of a supersymmetric solution. The condition on the curvature and axio-dilaton (2.41) has again a nice geometrised formwhich will allow a re-interpretation of the Type IIB supergravity equations with varying τ interms of an F-theory model, where the axio-dilaton τ is identified with the complex structureof an elliptic curve. The varying of the complex structure, which is compatible with the SL Z duality group action on Type IIB string theory, is then encoded in a geometric ellipticfibration in a putative 12d space.The geometry that incorporates the axio-dilaton in terms of an elliptic fibration over theType IIB spacetime (cid:102) M is a K¨ahler four-fold, with metricd s ( Y τ ) = 1 τ (cid:0) (d x + τ d y ) + τ d y (cid:1) + d s ( (cid:102) M ) , (2.59)whose Ricci-form is written in terms of that of (cid:102) M , R ( (cid:102) M ) , as R ( Y ) = R ( (cid:102) M ) − i P ∧ P ∗ . (2.60)It is clear from this expression that the Ricci-form has legs only along (cid:102) M and therefore R ( Y ) µν = R ( (cid:102) M ) µν − P ( µ P ∗ ν ) , (2.61) R ( Y ) = R ( (cid:102) M ) − | P | . (2.62)Using the above expressions in (2.41) and that the coordinates of the auxiliary elliptic fibrationgenerate Killing directions of the full solution we find0 = (cid:3) (cid:102) M ( R ( (cid:102) M ) − | P | ) −
12 ( R ( (cid:102) M ) ) + R ( (cid:102) M ) µν R ( (cid:102) M ) µν + 2 | P | R ( (cid:102) M ) − R ( (cid:102) M ) µν P µ P ∗ ν = (cid:3) Y R ( Y ) −
12 ( R ( Y ) ) + R ( Y ) ij R ( Y ) ij . (2.63)17his is the “master equation” presented in [13] in two more dimensions. Solving (2.41) isequivalent to solving (2.63) and imposing that the 8d K¨ahler metric for Y τ is ellipticallyfibered. The condition is thus not that this space is Calabi-Yau, but a more refined condition,which only in special cases will be shown to reduce to containing Ricci-flat elliptic fibrations.Alternatively, the geometry may also be specified in terms of the metric on Y τ using (2.56)and (2.57) as R ( Y ) = 8e − , d ρ = − R ( Y ) . (2.64)Note that solutions to this equation will also automatically give rise to supersymmetric so-lutions of eleven dimensional supergravity of the form AdS × M [32], where M is locallya circle fibration over Y τ . We thus obtain a 1–1 correspondence of F-theory AdS solutionsand elliptically fibered M-theory AdS solutions. We shall discuss this point later in section7. with 2d N = (2 , and AdS with Varying τ Before entering an extensive analysis of new solutions with N = (0 ,
2) supersymmetry, it isworthwhile singling out the special case of N = (2 ,
2) supersymmetry. Again we consider onlyfive-form flux in the present setup, and analyse the general torsion conditions on the geometry.There are two main conclusions: the first is that there are no AdS solutions which preserve(2 ,
2) and have non-constant axio-dilaton. We again provide the constraints on the geometry,and show how known solutions for constant axio-dilaton τ such as AdS × S × CY arerecovered from this in appendix C.3. Interestingly, requiring the axio-dilaton to vary, impliesthat the solution is non-compact, and in fact becomes AdS . These solutions are in fact themost general solutions of this type, and we conclude this section by showing the classificationof the constraints on Type IIB AdS solutions with varying axio-dilaton and vanishing three-form fluxes, preserving N = 1 supersymmetry in the dual 4d gauge theory. The torsion conditions for N = (2 ,
2) supersymmetry may be extracted from [1] by specialisingthe α parameters to be α = − α = 1 for the two spinors ξ i . The existence of two non-vanishing Dirac Killing spinors on M implies that M supports an SU (2) structure. In 7dan SU (2) structure is determined by three independent vectors which specify a dreibein fora 3d space M and a transverse 4d space M , admitting a canonical SU (2) structure M = M (cid:111) M . (3.1)18n 4d an SU (2) structure is determined by the existence of a real two-form, j of maximalrank and a holomorphic two-form ω , satisfying the algebraic conditions j ∧ ω = 0 , (3.2) ω ∧ ¯ ω = 2 j ∧ j . (3.3)From (2.12) and (2.13) we may find various algebraic conditions that the bilinears mustsatisfy. From (2.13) we have( α i + α j ) A ij = 0 ⇒ A = A = 0 . (3.4)From (2.12) we find A ∗ ij P = 0 , (3.5)and therefore for τ to vary we require A = 0. We may then split the cases into those withvarying τ and those where τ is fixed to be constant or equivalently to the cases of vanishing A or non-trivial A respectively. Using the results of [1] we see that both K and K areKilling vectors. In addition one finds from (2.13) the two equations i K ij d∆ = − i m α i − α j ) S ij , (3.6) S ij d∆ + i m α i − α j ) K ij = e − i K ij F , (3.7)which may be used to show that the vectors K and K are also symmetries of both the warpfactor and flux. They correspond to the left and right moving R-current in the putative dualSCFT. In all cases the scalars S and S are constant and the spinors may be normalisedsuch that both of these scalars are unity. This concludes the general analysis, and we must nowspecialise to one of the two cases. Requiring that the solution space transverse to the AdS is compact implies that τ is constant. This case will be discussed in section 3.2, supplmentedwith details in appendix C. If we relax the compactness condition then we find AdS withvarying τ . This allows us to classify all F-theoretic AdS solutions in Type IIB with five-formflux in section 3.3. τ : AdS Duals to N = (2 , ,
2) supersymmetry, further discussion can be found in appendixC where the conditions are derived and known solutions in the literature are recovered. Theanalysis of the torsion conditions shows that for a compact internal space constant τ is a19ecessary condition. In this section we consider the case where the scalar bilinear A isnon-trivial. The conditions for the existence of a solution are reminiscent of the conditionsfound in [33] for AdS solutions in M-theory. Locally the internal metric takes the form m d s ( M ) = (1 − y e − )(d ψ + σ ) + y e − d ψ + e − y (1 − y e − ) d y +e − g (4) ( y, x ) ij d x i d x j , (3.8)where both ψ and ψ are Killing vectors and generate the expected U (1) × U (1) symmetrythat is dual to the R-symmetry on the field theory side. For fixed y the metric g (4) is K¨ahlerwith K¨ahler form J satisfying ∂ y J = 12 d σ , ∂ y log √ g = − y e − − y e − ∂ y ∆ , (3.9)where σ = − (cid:98) P + 2 y e − − y e − d c ∆ mF (2) = − (cid:16) (e − y )d σ + 2 J + 4e d∆ ∧ (d ψ + σ ) (cid:17) . (3.10) (cid:98) P is the Ricci-form potential for the metric g (4) . The complex structure J ji is independentof y and this allows us to rewrite (3.9) as(d σ ) + = − y e − − y e − ∂ y ∆ J , (3.11)where (d σ ) + is the self-dual part of the two-form d σ .Examples of geometries of this type were found in [8, 13] but is is likely that there existother interesting solutions. Since the case of constant axio-dilaton is not the focus of thispaper, we leave a further analysis of these solutions and their duals for the future. Weshall proceed in the next section with the analysis of non-trivially varying τ by relaxing thecondition of compactness of the internal 7d manifold. τ : AdS Duals to 4d N = 1In the previous section the requirement for 2d N = (2 ,
2) supersymmetry and compactnessof the solution led to the result that τ is constant. We shall now relax the latter conditionand find that there are non-trivial varying τ solutions which are AdS duals to N = 1 in 4d.In the following we will provide the derivation of this starting from the present setup of AdS solutions. We supplement this with an analysis from a direct AdS × M τ ansatz which showsthat these are in fact all such varying τ AdS solutions with five-form flux. As explained in appendix C the Killing directions dual to the left and right moving R-symmetries arelinear combinations of these two Killing vectors, they correspond to the diagonal and anti-diagonal. s = e (cid:18) d s (AdS ) + e − m (1 − e − ) d∆ + 4(1 − e − ) m d ϕ (cid:19) + 1 m (cid:104) (d ψ + σ ) + d s ( (cid:102) M ) (cid:105) = d s (AdS ) + 1 m (cid:104) (d ψ + σ ) + d s ( (cid:102) M ) (cid:105) , (3.12)where (cid:102) M is a K¨ahler surface. The axio-dilaton varies holomorphically over (cid:102) M and obeysthe following curvature condition R = 6 J − d Q . (3.13)In particular, to find solutions we should solve this equation. Notice that for constant τ thisreduces to the K¨ahler–Einstein condition.As an F-theory background, this can written asd s = d s (AdS ) + d s ( M τ )= d s (AdS ) + 1 m (d ψ + σ ) + d s ( (cid:102) M ) + 1 τ (d x + τ d y ) + τ d y , (3.14)where S (cid:44) → M τ → T τ , with the elliptically fibered three-fold E τ (cid:44) → T τ → (cid:102) M , which is notCalabi-Yau . There is however a nice reformulation of these solutions in term of an ellipticallyfibered Calabi–Yau four-fold. The compact part of the geometry has an obvious relationwith the metrics on Sasaki–Einstein solutions and in fact may be shown to be the link of theconical base of an elliptically fibered Calabi–Yau four-fold. Specifically, that the metricd s ( Y ) = 1 τ (d x + τ d y ) + τ d y + d r + r (cid:16) (d ψ + σ ) + d s ( (cid:102) M ) (cid:17) (3.15)is both Ricci-flat and K¨ahler, where the elliptic fiber varies over the K¨ahler manifold (cid:102) M .For constant τ the fibration is trivial and we reduce to the usual Sasaki–Einstein solutions,which can be written as the link of a Calabi–Yau three-fold cone. Including varying τ thesolution remains Sasakian, however the Calabi–Yau condition of the 6d cone is now relaxed.In fact as we briefly show in appendix D, this set of solutions is the most general with N = 1supersymmetry in the dual 4d theory, with five-form flux and vanishing three-form fluxes.A more detailed study of these solutions and their holographic interpretation will appear in[29].The starting point in appendix D is the ansatz AdS ×M for the geometry, now allowing τ to vary. We rename the K¨ahler form on the transverse space (cid:102) M to be J T , and theholomorphic two-form Ω T . Furthermore we introduce both a K¨ahler two-form and maximal21olomorphic form in complex dimension 3 and 4 and we shall distinguish them by labelingwith their complex dimension. Let us first construct the conical metric C ( M τ ) = Y d s ( Y ) = d r + r (cid:16) (d ψ + σ ) + d s ( (cid:102) M ) (cid:17) . (3.16)The K¨ahler two-form is J = r d r ∧ (d ψ + σ ) + r J T , (3.17)and from (D.41) it follows that it is closed d J = 0. We have proven our first assertion thatthe solution is Sasakian. We next check the Ricci tensor of the solution. As the metric iscomplex this is most easily achieved by computing the exterior derivative of the holomorphicthree-form of the cone, Ω = r (d r + i rK ) ∧ Ω T . (3.18)Using (D.42) results in dΩ = − i Q ∧ Ω , from which we may abstract the Ricci form of theK¨ahler cone to be R Y = − d Q . (3.19)Recall that τ varies holomorphically over (cid:102) M , so that R Y can be reinterpreted as the Ricciform for the base Y of an elliptically fibered Calabi–Yau four-fold, X F ,d s ( X F ) = 1 τ (cid:0) (d x + τ d y ) + τ d y (cid:1) + d r + r (cid:16) (d ψ + σ ) + d s ( (cid:102) M ) (cid:17) . (3.20)For trivial τ this reduces to the Sasaki–Einstein case as necessary. Notice firstly that theCalabi–Yau is non-compact and that the elliptic fibration depends only on (cid:102) M . Moreoverthe Reeb vector is fibered over (cid:102) M . The full solution isd s = d s (AdS ) + 1 m (cid:16) (d ψ + σ ) + d s ( (cid:102) M ) (cid:17) = 1 m (cid:18) r d s ( R , ) + 1 r d s ( Y ) (cid:19) (3.21)with self-dual five-form flux F = 4 m (dvol(AdS ) + dvol( M τ )) , (3.22)where d s ( Y ) = d r + r (cid:16) (d ψ + σ ) + d s ( (cid:102) M ) (cid:17) (3.23)22s the metric on the K¨ahler three-fold which is the base of an elliptically fibered Calabi-Yaufour-fold with elliptic fiber varying over the K¨ahler manifold (cid:102) M ⊂ X .Of course the five-form flux needs to be quantized through the unique five-cycle in theten-dimensional geometry i.e. we must impose1(2 π(cid:96) s ) g s (cid:90) M τ F ∈ Z , (3.24)which with the above form for the flux results in1(2 π(cid:96) s ) g s (cid:90) M τ m dvol( M τ ) = 4vol( M τ )(2 πm(cid:96) s ) g s = N . (3.25)The integer N is interpreted as the number of D3-branes as usual. From this it followsstraightforwardly that the leading order holographic central charge of the dual 4d SCFT isgiven by a d = π G (10) N (cid:90) M τ e dvol( M τ ) = N π M τ ) , (3.26)exactly as in the constant τ , Sasaki–Einstein case.However, due to the relation (3.13), the volume receives corrections with respect to theconstant τ case. In particular, using 6 J = d Q + R , we havevol( M τ ) = (cid:90) M τ (d ψ + σ ) ∧ J ∧ J π (cid:96) (cid:90) M c ( M ) − c ( M ) ∧ c ( L D )+ c ( L D ) , (3.27)where the first term is the result of the volume for quasi-regular Sasaki–Einstein manifolds.These solutions and their dual field theories will be further investigated in [29]. In thepresent paper they will make a re-appearance in the so-called universal twist solutions insection 4.1, which are holographic dual of topologically twisted compactifications of the 4d N = 1 SCFTs dual to the above AdS solutions.We conclude that that there are no compact AdS solutions with varying axio-dilatondual to (2 ,
2) SCFTs, and this is supported by the absence of any non-chiral field theoriesfrom wrapped D3-branes in F-theory [6]. Nevertheless this analysis has led us to the excitingdirection of AdS solutions in F-theory. N = (0 , Solutions with Varying τ In this section we turn to exploring solutions to the “master equation” (2.41) or equivalently(2.63), for duals to 2d (0,2) SCFTS, which incorporates a varying Type IIB axio-dilaton in23erms of an elliptically fibered local K¨ahler four-fold Y τ . We will study two new classes ofsolutions, which result from different specialisations of the K¨ahler four-fold.The first type of solution is a specialisation of the F-theoretic reformulation in section2.6, where (cid:102) M is a direct product of a complex curve and a complex surface, (cid:102) M = Σ × M , (4.1)such that the elliptic fibration is only non-trivial over one of these subspaces, i.e. there aretwo cases Elliptic Surface: Y τ = ( E τ → Σ) × M = S τ × M Elliptic Three-fold: Y τ = Σ × ( E τ → M ) = Σ × T τ . (4.2)where none of the factors has a Ricci-flat metric. This class will correspond in the dualfield theory to “universal twist solutions”, which generalise to varying τ the universal twistsolutions in [16], that were originally found in [34]. We will see that they are dimensionalreductions with topological duality twist of 4d N = 1 SCFTs with rational R charges, withvarying coupling. In this class of solutions we do not assume that the elliptic fibration over Σor M are Ricci-flat. In fact the “master equation” implies that they are not. These solutionswill be studied in section 4.1.Another class of solutions can be obtained by a similar splitting, however we now requirethat the factor with the non-trivial elliptic fibration is Ricci-flat, i.e. has a Calabi-Yau (4 − s )-fold factor Y τ − s ) Y τ = Y τ − s ) × M s . (4.3)Clearly M s has to be K¨ahler as well and only the values s = 1 , . Insertingthe direct product metric into (2.63) one immediately finds that the K¨ahler metric on M s must again obey the same equation originally found in [13], namely (cid:3) M s R ( M s ) − R ( M s )2 + R ( M s ) ij R ( M s ) ij = 0 . (4.4)We shall first consider the case when s = 1 where Y τ is the direct product of an ellipticallyfibered Calabi–Yau three-fold and a Riemann surface before considering the s = 2 case. Aswe shall show the former recovers the (0 ,
4) solutions determined in [1] whilst the latter givesrise to a new class of strictly (0 ,
2) supersymmetic solutions. These solutions will be thesubject of section 4.2. Note that s = 0 is ruled out because the Ricci scalar of Y τ , and therefore also the warp factor e − ,vanishes in this case. s = 3 corresponds to Y = T × M , which has constant axio-dilaton [13]. .1 Universal Twist Solutions In this section we begin with the product ansatz in (4.1)d s ( (cid:102) M ) = d s (Σ) + d s ( M ) , (4.5)where Σ is a complex curve and M a K¨ahler surface. It is most convenient to express ouransatz in the reformulation of section 2.6. The Ricci-form of the 8d space Y τ , which is theelliptic fibration over (cid:102) M is R Y = k J M + k J Σ , (4.6)where k and k are constants. We consider the two cases oulined in (4.2): τ varies non-trivially only over the curve Σ giving an elliptic surface, or τ varies non-trivially only over M giving an elliptic three-fold. Though the supergravity solutions are distinct much of theanalysis will be similar, and therefore it will be useful to keep the discussion as general aspossible. Inserting the above ansatz into the ‘master equation’ (2.63) the necessary conditionis k ( k + 2 k ) = 0 and R Y = 4 k + 2 k . (4.7)Clearly to solve (4.7) either k = 0 or k = − k . The former recovers the (0 ,
4) solutiondiscussed in [1] . We therefore consider the latter solution in the remainder of this section.Evaluated on such a solution the Ricci scalar is R Y = − k and thus the positivity constraintof the Ricci scalar implies that k <
0. The overall scale of the K¨ahler metric on (cid:102) M maybe removed by a coordinate change, thus without loss of generality we may set k to be anynegative value, for convenience we choose k = −
3. The 10d solution in Einstein frame isd s = 23 (cid:18) d s (AdS ) + 94 (cid:18)
19 (d χ + ρ ) + d s ( M ) + d s (Σ) (cid:19)(cid:19) , (4.8)e − = 94 , (4.9) F (2) = −
23 (4 J Σ + J M ) , (4.10) F = −
34 (d χ + ρ ) ∧ J M ∧ (2 J M + J Σ ) −
23 dvol(AdS ) ∧ (4 J Σ + J M ) , (4.11) ρ = − A M + 3 A Σ , (4.12)d A i = J i . (4.13) We will recover these (0,4) solutions in a slightly different way in section 4.2. Y τ becomes R Y = 6 J M − J Σ , in matrix block form this is R Y = − J Σ J M . (4.14)In the above we have not specified over which factor in (cid:102) M , τ varies non-trivially. In thefollowing we shall consider the two cases in which τ varies non-trivially only over Σ, giving anelliptic surface, or over M , giving an elliptic three-fold. We are not aware of any existenceresults for metrics on either the elliptic surface or the elliptic three-fold with the specificconditions imposed on the curvatures (in particular let us re-emphasise that these are notRicci-flat). We will assume that such metrics exist on these spaces with the Ricci-form givenas above. It would indeed be of great interest to develop the mathematics that shows theexistence of such metrics. Of course the bases of these elliptic fibrations will have singularitiesat points where τ becomes singular, but by assumption they will be otherwise smooth. Inthe following sections we analyse the two distinct types of solutions discussed above. Theconsistency of the holographic computations using these solutions with the proposed fieldtheory duals corroborates our conjecture that these metrics exist. Let us first consider the case where τ varies non-trivially only over Σ. We require the metricon Y τ to factorise as d s ( Y τ ) = d s ( S τ ) + d s ( M ) , (4.15)where E τ (cid:44) → S τ → Σ is an elliptic surface with section, over Σ. The Ricci curvature thenfactorises into two 4 × (cid:32) R S τ R M (cid:33) = − J Σ J M . (4.16)To solve this equation we therefore have that the metric on M is K¨ahler–Einstein withRicci-form R M = 6 J M , and we require the existence of a metric on the elliptic surface S τ to satisfy R S τ = − J Σ ⇐⇒ R Σ + d Q = − J Σ . (4.17)Notice that the K¨ahler–Einstein metric on M has the normalisation of the base of a Sasaki–Einstein manifold. In fact the one form dual to the Reeb vector field of the Sasaki–Einstein26anifold is given by − (d χ + ρ ) at fixed coordinate on Σ. We conclude that at fixed coordinateon Σ the U (1) fibration over M is a (quasi-regular) Sasaki–Einstein manifold.Solutions of this form, where Σ is the constant curvature Riemann surface H have beenstudied in [34], however there are some differences once τ is allowed to vary non-trivially overΣ. Topologically the 7d internal space is a U (1) fibration over M × Σ. Such fibrations areregular if the first Chern class of the bundle is integral over all two-cycles in H ( M × Σ , Z ).Let the period of χ be 2 π(cid:96) , then for a regular U (1) fibration we require12 π (cid:96) d ρ = 12 π(cid:96) ( − J M + 3 J Σ ) ∈ H ( M × Σ , Z ) . (4.18)This may be rephrased in terms of the elliptic surface S τ with base Σ as c ( U (1)) = − (cid:96) ( c ( M ) + c ( S τ ) | Σ ) ∈ H ( M × Σ , Z ) . (4.19)Notice that the non-trivial elliptic fibration implies that the quantisation condition differs tothat in [34]. Concretely we have used the first Chern class of the elliptic surface S τ to rewritethe condition on c ( U (1)). A convenient basis for H ( (cid:102) M ) is furnished by the set { Σ , Σ α } where { Σ α } is a basis of H ( M , Z ). Then c ( U (1)) being integer implies12 π (cid:90) Σ c ( U (1)) = − (cid:96) (2( g −
1) + deg( L D )) ∈ Z π (cid:90) Σ a c ( U (1)) = − ˜ mn α (cid:96) ∈ Z , (4.20)where ˜ m is the Fano-index of M , see appendix B of [34] for a review of properties of4d K¨ahler–Einstein spaces, and n α are relatively prime integers. The period (cid:96) of χ mustbe a divisor of both ˜ m and (2( g −
1) + deg( L D )) and consequently it has maximal value (cid:96) = gcd { ˜ m, (2( g −
1) + deg( L D )) } . Recall that this construction only works for the regularand quasi-regular Sasaki–Einstein metrics. Flux Quantisation
The cycles of interest are the compact five-cycles of the geometry, of which there are twoclasses. The first is the five-cycle given at fixed Σ coordinates, which is a Sasaki–Einstein(SE) manifold. The second class of five-cycles, which we denote D α , are obtained as U (1)fibrations over Σ α × Σ, where Σ α ∈ H ( M , Z ). For the former we find N (SE ) = 1(2 π(cid:96) s ) g s (cid:90) SE F = 9(2 π(cid:96) s m ) g s vol( SE ) , (4.21)27here the volumes are computed with the canonical Sasaki–Einstein metrics, which haveRicci-tensor satisfying R µν = 4 g µν . As it is necessary for the fibration to be quasi-regular wemay rewrite this quantisation condition as N ( SE ) = (cid:96)M πm (cid:96) s4 g s ∈ Z , (4.22)where the integer M is the topological invariant M = (cid:90) M c ( M ) ∧ c ( M ) . (4.23)For the five-cycles D α the condition is N ( D α ) = − (cid:96) ˜ mn α (2( g −
1) + deg( L D ))2 πm (cid:96) s4 g s ∈ Z . (4.24)Quantisation of the flux such that the above integers are minimal implies n = ˜ m(cid:96)h πm (cid:96) s4 g s , h = gcd (cid:18) M ˜ m , g −
1) + deg( L D ) (cid:19) , (4.25)from which we obtain N ( SE ) ≡ N = nM ˜ mh , N ( D α ) = nn α h (2( g −
1) + deg( L D )) . (4.26)In comparing with the field theory results we shall identify the integer N as the number ofD3-branes in the setup. Notice that the above analysis is a generalisation to that performedin [34], corresponding to deg( L D ) = 0. Consider now the case where τ varies non-trivially only over M , so that the metric on Y τ factorises as d s ( Y τ ) = d s (Σ) + d s ( T τ ) , (4.27)where E τ (cid:44) → T τ → M is the elliptic three-fold. The Ricci curvature of this metric nowfactorises in one 2 × × (cid:32) R Σ R T τ (cid:33) = − J Σ J M . (4.28)The upper block of this equation implies that the metric on the Riemann surface has constantcurvature R Σ = − J Σ . We then require the existence of a metric on the elliptic three-fold T τ to satisfy R T τ = 6 J M ⇐⇒ R M + d Q = 6 J M . (4.29)28n fact, the elliptic three-fold T τ is precisely that appeared in section 3.3. At fixed coordinateson Σ, the solutions can be obtained in the same way as the AdS solutions discussed in section3.3 and will be studied in more detail in [29]. We nevertheless give a brief discussion on globalproperties of the solutions following the above. Topologically the solution is again a U (1)fibration over a K¨ahler base. Giving χ period 2 π(cid:96) as before the first Chern class of the U (1)bundle is c ( U (1)) = − (cid:96) ( c (Σ) + c ( T τ ) | M ) . (4.30)Using the same basis as previously we require12 π (cid:90) Σ α c ( U (1)) = 1 (cid:96) ( c ( T τ ) · Σ α ) ∈ Z , π (cid:90) Σ c ( U (1)) = χ (Σ) (cid:96) ∈ Z . (4.31)Here χ (Σ) is the Euler number of the Riemann surface Σ. The period (cid:96) must divide both χ (Σ) and c ( T τ ) · Σ α for all α . Flux Quantisation
Recall that at fixed coordinates on the constant curvature Riemann surface Σ, the metric isno longer Einstein, though it remains Sasakian. We will refer to this space as M τ as it willbe related to the M τ of section 3.3. The possible five-cycles are as before and we keep thesame notation as in the previous quantisation condition. Then the quantisation condition is N ( M τ ) = 1(2 π(cid:96) s ) g s (cid:90) M τ F = 9(2 πm(cid:96) s ) g s vol( M τ ) , (4.32)which has the same form as for the the first class of solutions. We may rewrite the volume of M τ asvol( M τ ) = 12 (cid:90) M τ
13 d χ ∧ J M ∧ J M = π (cid:96) (cid:90) M (cid:0) c ( M ) − c ( M ) c ( L D ) + c ( L D ) (cid:1) , (4.33)where the integral on the right-hand side is an integer, given by the sum of three topologicalnumbers, whose value we denote by (cid:102) M . Then N ( M τ ) = (cid:96) (cid:102) M · π(cid:96) s4 m g s . (4.34)The quantisation over the remaining five-cycles gives N ( D α ) = χ (Σ) (cid:96) · πm (cid:96) s4 g s (cid:18) ˜ mn α − (cid:90) Σ α c ( L D ) (cid:19) . (4.35)29s before we impose that the fluxes are minimal integers through all integral cycles whichimplies the quantisation of the length scale m as n = (cid:96)h · πm (cid:96) s4 g s , h = gcd (cid:20) ˜ M , χ (Σ) gcd (cid:18)(cid:26) ˜ mn α − (cid:90) Σ α c ( L D ) (cid:27) α (cid:19)(cid:21) . (4.36)We have N ( M τ ) ≡ N = (cid:102) M nh , N ( D α ) = χ (Σ) nh (cid:18) ˜ mn α − (cid:90) Σ α c ( L D ) (cid:19) . (4.37) We now consider the ansatz (4.3), with one of the factors in Y τ an elliptically fibered Calabi-Yau. (0 , Solutions
The case s = 1, i.e. Y τ = Y × Σ, where Y is an elliptically fibered Calabi–Yau three-foldand Σ is a complex curve recovers the classifiaction of N = (0 ,
4) theories that we presentedin [1]. The metric is d s ( Y τ ) = d s ( Y ) + d s (Σ) , (4.38)and as any Riemann surface is conformally flat we may write the metric on M asd s (Σ) = e − f ( x,y ) (d x + d y ) . (4.39)A Riemann surface trivially satisfies R = 2 R µν R µν and therefore (4.4) reduces to (cid:3) Σ R Σ = 0 . (4.40)On any smooth compact manifold any bounded harmonic function is constant and it followsthat for a smooth and compact internal manifold we must have that R Σ is constant and there-fore the Riemann surface is of constant curvature . For positive curvature, as is necessaryby (2.64), the only possibility is a round two-sphere and it follows that the only solutions areof the form d s = d s (AdS ) + d s ( S / Γ) + d s ( B ) (4.41)where B is the base of Y , the elliptically fibered Calabi–Yau introduced above. This preciselyreproduces the solutions discussed in [1], where they were shown to be the unique (0 , Removing the smoothness assumption, there could exist further (0,2) solutions where Σ has singularities.However, in [1] we did not make any global assumption and therefore those are indeed the most generalsolutions preserving (0,4) supersymmetry. .2.2 Baryonic Twist Solutions A new class of solutions with exactly (0 ,
2) supersymmetry can be obtained for s = 2 in theansatz (4.3), i.e. where the geometry consists of an elliptic K3 surface Y and a local K¨ahlersurface M as factors d s ( Y τ ) = d s ( Y ) + d s ( M ) . (4.42)Any solution to the “master equation” (4.4) for the metric on M will furnish a solutionwith varying axio-dilaton. In fact, solutions have been found previously in the literature for M , [14, 15] and in the following section we shall discuss a particular example. We begin bywriting the full local solution with varying axio-dilaton, and subsequently we will investigateits global regularity, including quantisation of the fluxes. The computations are very similarto those presented in [14, 15] for the solutions with constant axio-dilaton. However, we includethe details below and in appendix E.1 to be self-contained and highlight some subtle features,which were not emphasised before.The solutions bear an uncanny resemblance to the five-dimensional Y p,q Sasaki–Einsteinmanifolds [17]. Following the ideas in [16], this connection will be sharpened by a dual fieldtheory discussion in section 6, where we will propose that the dual 2d SCFTs are obtainedfrom a particular twisted compactification of the Y p,q theories on a curve, with a varyingcoupling.The local metric in string frame isd s IIB ( SF ) = 1 √ axτ (cid:20) d s (AdS ) + 14 m (cid:18) w [d ψ + g ( x ) Dφ ] + 4 ax d s ( B )+ a (cid:18) d x x U + Uw Dφ + d θ + sin θ d χ (cid:19)(cid:19)(cid:21) , (4.43)with RR five-form flux F = − m dvol(AdS ) ∧ (cid:18) ax ( Dψ − g ( x ) Dφ ) ∧ d x + 2dvol( B ) + 12 dvol( S ) (cid:19) + a m Dψ ∧ Dφ ∧ d x ∧ (cid:18) dvol( B ) + 14 x dvol( S ) (cid:19) + a m dvol( B ) ∧ dvol( S ) ∧ (cid:18) xDψ − U ( x ) w ( x ) Dφ (cid:19) . (4.44)The axio-dilaton varies holomorphically over B = P , such that the total space of the ellipticfibration Y , E τ (cid:44) → Y → B , where the axio-dilaton parametrises the complex structure ofthe fiber, is a K3 surface. The warp factor is e − x ) = ax and in the above expressions we31ave used the following definitions U ( x ) = 1 − a (1 − x ) ,w ( x ) = 1 + a (2 x − ,g ( x ) = − axw ( x ) ,Dφ = d φ + cos θ d χ ,Dψ = d ψ + g ( x ) Dφ , (4.45)with a an integration constant. After performing the global regularity analysis, that weinclude in Appendix E, one discovers that a takes rational values, given in terms of twointegers p , q . The resulting Type IIB solution takes the formAdS × P × Y p , q , Y p , q = ( S → F ) , (4.46)where Y p , q is a circle fibration over F = S × S , with Chern numbers p and q , respectively,that are related to the parameter a as a = q p . (4.47)Of course the K¨ahler metric on this F is not the Einstein, direct-product metric on S × S .Regularity of the metric requires that a >
1, which implies that the integers p , q obey0 < p < q . (4.48)This notation is closely related to the one in [17], and a further discussion of the relation tothe standard Y p,q is provided in appendix E. Flux Quantisation
Finally, we need to check that the flux of the solution is properly quantized, i.e. N ( D ) = 1(2 π(cid:96) s ) g s (cid:90) D F ∈ Z (4.49)for any five-cycle D ∈ H ( M ; Z ). There are two independent five-cycles in M = P × Y p , q ,namely Y p , q at a point on the base B of the elliptic K3, and E × B = E × P , where the E is the unique generator of H ( Y p , q ; Z ). The flux as given in (E.33) is mF (2) = − (cid:18) ax ( Dα − g ( x ) Dφ ) ∧ d x + 2 J B + 12 sin θ d θ ∧ d χ (cid:19) . (4.50)32ue to the self-duality of the five-form flux, it is the Hodge star of the above two-form thatneeds to be quantised. An explicit computation reveals that m ∗ F (2) = a Dα ∧ Dφ ∧ d x ∧ (cid:18) dvol( B ) + 14 x dvol( S ) (cid:19) + a B ) ∧ dvol( S ) ∧ (cid:18) xDα − U ( x ) w ( x ) Dφ (cid:19) . (4.51)The flux through the cycles Y p , q is (cid:90) Y p , q ∗ F (2) = 1 m (cid:90) Y p , q a x sin θ d x ∧ d θ ∧ d χ ∧ Dα ∧ Dφ = − (2 π ) m (cid:18) q p ( p − q ) (cid:19) . (4.52)Which implies the quantisation condition1(2 π(cid:96) s ) g s m = N π p ( p − q ) q , (4.53)where π(cid:96) s ) g s (cid:90) Y p , q F = − N , N ∈ N . (4.54)The integer N is interpreted as the number of D3-branes along R , × P .To perform the quantisation over the other five-cycle, we must first identify the correctgenerator for H ( Y p , q ; Z ). It is not simple to identify this three-cycle in the metric as itis not a product metric, however there are four easily identifiable three-cycles at each ofthe degeneration surfaces, further discussion of these degeneration surfaces is provided inappendix E.4. Let the generator of H ( Y p , q ; Z ) be denoted E , and the three-cycles at eachof the degeneration surfaces be E a where a ∈ { + , − , , π } . The closed three-form dual to thegenerator E is ω = p − q (4 π ) (cid:20) Dα ∧ Dφ ∧ d x + (cid:18) xDα − U ( x ) w ( x ) Dφ (cid:19) ∧ dvol( S ) (cid:21) (4.55)and satisfies (cid:90) E ω = 1 . (4.56)One may use the above three-form to verify that the following homology relations E + = ( p + q ) E , E − = ( p − q ) E , E = E π = − p E , (4.57) We chose this sign to ensure that
N > E × B gives (cid:90) E × B ∗ F (2) = 1 m (cid:90) E × B a B ) ∧ (cid:18) Dα ∧ Dφ ∧ d x + (cid:18) xDα − U ( x ) w ( x ) Dφ (cid:19) ∧ dvol( S ) (cid:19) = 4 aπ ( p − q ) m (cid:90) E × B dvol( B ) ∧ ω = 4 π m q p ( p − q ) vol( B ) . (4.58)Flux quantisation imposes 1(2 π(cid:96) s ) g s (cid:90) E × B F = − M , M ∈ N , (4.59)which may be interpreted as quantisation of the volume of B vol( B ) = M πN pq q − p . (4.60)This concludes the discussion of the new AdS solutions in F-theory dual to (0 ,
2) SCFTs.In the following we will use these to test the duality by comparing holographic charges withthe dual field theory observables.
To compare physical observables with the dual SCFTs, we now turn to computing holograph-ically the central charge as well as the R-charges and baryonic charges of baryonic operators,which will be compared to the dual field theories in section 6. At leading order in N , theresults of the holographic computations presented in this section also apply, with minor mod-ifications, to the holographic duals with constant axio-dilaton [16]. The leading order central charge is computed using the standard Brown-Henneaux prescrip-tion [35], relating it to Newton’s constant G N in 3d as c sugra = 32 mG (3) N . (5.1)This can be extracted from the solution by computing the volume of the compact part ofthe spacetime M . We remark that in all the solutions presented above the bases of theelliptic surfaces and three-folds considered above are singular, however the volumes of thesespaces can be computed indirectly either by using flux quantisation or relating it to varioustopological quantities. Here we furthermore assume that the fibration is a smooth Weierstrassmodel, i.e. with only I fibers. This will allow us to circumnavigate having to resolve any34dditional singularities, in passing to an M-theory picture. A similar logic was employed in[1], and cross-checked against a smooth M-theory dual, field theory and anomalies. Using theconventions in appendix D of [1] we have c sugra = 32 mG (10) N (cid:90) M e ∆ dvol( M ) , (5.2)where G (10) N = 2 π (cid:96) s8 is the 10d Newton’s constant.The subleading contribution to the central charge can be determined by anomaly inflowon the 7-branes as in [1], which follows an argument presented in [28]. Starting with a singleD7-brane whose world-volume is extended along W , the Wess–Zumino term in the effectiveaction of the D7-brane induces a 3d CS coupling by S CS = µ π (cid:96) s4 (cid:90) W C (4) ∧ Tr(
R ∧ R ) , (5.3)with µ = ((2 π ) (cid:96) s8 g s ) − . The results of [36] allow one to extract the subleading contributionfrom the coefficient of the Chern-Simons term S CS (AdS ) = c L − c R π (cid:90) AdS ω CS (AdS ) . (5.4)One should then sum over all the 7-branes in the solution.The number (and type of) 7-branes in the background are encoded in the elliptic fibration.In the simplest case of an elliptic surface E τ (cid:44) → S τ → Σ the number of 7-branes, assumingonly I fibers, is given by 12deg( L D ). The canonical bundle of the total space of an ellipticsurface is K S τ = π ∗ K Σ + | ∆ | (cid:88) i =1 a i P i , (5.5)where i is summed over the components of the discriminant ∆ of the elliptic fibration and a i are coefficients determined by the type of the singular fibers and π is the projection to thebase. For I fibers as considered here a i = . In order to satisfy R S τ = − J Σ = − K S τ , and R Σ = − J Σ − d Q = − K Σ , (5.6)one obtains that the number of I fibers is | ∆ | = 12deg( L D ) . (5.7)Notice that for an elliptically fibered K3 surface, whose base is necessarily a P , deg( L D ) = c ( P ) = 2 implies the well-known result of 24 7-branes.35e will also compare R-charges and baryonic charges in the holographic duality. Re-call that in the Sasaki–Einstein setup one may compute these by evaluating the volumes ofcertain supersymmetric three cycles { Σ i } . Below we present a version of this computationin the context of the AdS solutions of interest. We assume that, similarly to their AdS counterparts, D3-branes wrapped on Σ i give rise to BPS particles moving in AdS , which weconjeture to be dual to some baryonic-type operator in the CFT . These are BPS objects,and in 2d their conformal dimension equals their R-charge. Denoting by B Σ i the operatorsin the dual field theory associated to the three-cycle Σ i , the conformal dimension is R [ B Σ i ] = ∆[ B Σ i ] = M [ B Σ i ] m , (5.8)where M [ B Σ i ] is the mass of the wrapped D3-brane. As our solutions include a warp factorfor AdS depending on the internal manifold, the mass of the D3-brane wrapped on thethree-cycle Σ i , is given by M [ B Σ i ] = T (cid:90) Σ i e ∆ m dvol(Σ i ) , T = 18 π (cid:96) s4 g s , (5.9)where T is the D3-brane tension. The factor of e ∆ m is precisely the warp factor due to thewarping of the time coordinate. In summary R [ B Σ i ] = 2 πN (cid:82) Σ i e d (cid:99) vol(Σ i ) (cid:82) M F . (5.10)The volume form with a hat is defined to be the volume form of the unwarped dimensionlessmetric obtained from the bracketed expression in (2.55). Notice the similarity with theformulas for geometric R -charge in warped AdS backgrounds [37, 38].The supersymmetric cycles are divisors in the complex cone over M , which implies thatthey are calibrated with respect to the four-form e r J cone ∧ J cone , with J cone the K¨ahler formon the 8d metric cone d s = d r + r d s ( M ). Recall that unlike in the Sasaki–Einsteincase, the cone is neither Ricci-flat nor K¨ahler, however as follows from [39] we haved( r − e J cone ∧ J cone ∧ J cone ) = 0 . (5.11)In fact for all the solutions presented above a stronger condition holds. In each of the solutionspresented above there is a distinguished Riemann surface. Define ˜ J to be the K¨ahler form atfixed coordinate on the cone, then we haved( r − e ˜ J ∧ ˜ J ) = 0 . (5.12)36his implies that the R-charges, (5.10), are topological quantities and independent of thecoordinates chosen.The final holographic charges that we can compute are the baryonic charges, (a summaryof the related computation for the Sasaki–Einstein case is given in appendix F). In particular,we shall use the observation that the integral of a harmonic three-form over each of thethree-cycles gives the baryonic charges of each of the baryons dual to that cycle in the fieldtheory up to some overall normalisation which is fixed by requiring the results are integer.We note that as this result is a topological invariant we are free to multiply the metric byan arbitrary bounded and non-vanishing warp factor and perform the computation using thewarped metric. We shall make use of this freedom later. Consider first the universal twist solutions, where Y τ has an elliptic surface factorAdS × S → ( M × S τ ) , E τ (cid:44) → S τ → Σ . (5.13)Recall also that for a fixed coordinate on Σ the transverse space is a Sasaki–Einstein manifold SE = ( S → M ). Central Charges
We first consider the holographic charges of the universal twist solution with τ varying overΣ. From (5.2) we have c sugra = 2 · π ((2 πm(cid:96) s ) g s ) vol(SE )vol(Σ) = 2 π N vol(Σ)vol( SE )= 36 n M ˜ m h (cid:96) (2( g −
1) + deg( L D )) , (5.14)where we have used the quantisation conditions in (4.26). In the final step we have re-expressed the volume of Σ, by using the fact that the Ricci form on Σ satisfies (4.17), asvol(Σ) = (cid:90) Σ J Σ = − (cid:18)(cid:90) Σ R Σ + d Q (cid:19) = − (cid:18) π (1 − g ) + (cid:90) Σ d Q (cid:19) , (5.15)and using − π (cid:90) Σ d Q = (cid:90) Σ c ( L D ) = deg( L D ) , (5.16)we have vol(Σ) = 13 (4 π ( g −
1) + 2 π deg( L D )) . (5.17)37oreover it follows that the central charge is integer for any K¨ahler–Einstein base and anysurface Σ. To make contact with the field theory this can be related to the “ a ” central chargeof the 4d quiver theory dual to the Sasaki–Einstein solution (with constant τ ) as c sugra = 8 a d π vol(Σ) , where a d = N π SE ) . (5.18)we conclude that at leading order in N the central charge is integral and given by c sugra = a d (cid:20) g − L D ) (cid:21) . (5.19)The first term is precisely the result one obtains for the constant τ solution. Notice thateven at leading order there is a correction to the central charge due to the varying axio-dilaton τ , proportional to the first Chern class of the U (1) D duality bundle.We note that this central charge is integer, independent of the choice of K¨ahler-Einsteinbase and curve Σ. To see this one should consider the last expression in (5.14). There arethree possible choices for K¨ahler-Einstein base; CP with ( M, ˜ m ) = (9 , S × S with( M, ˜ m ) = (8 ,
2) and d P k for k = 3 , .., M, ˜ m ) = (9 − k, c L ) sugra − ( c R ) sugra ) = N . (5.20)Therefore the total contribution from the 7-branes is given by( c L ) sugra − ( c R ) sugra = (number of 7-branes) · N N deg( L D ) . (5.21) R-charges
Recall that at fixed coordinates on Σ the U (1)-fibration over the K¨ahler–Einstein space M is a Sasaki–Einstein manifold, therefore the three-cycles which are dual to baryonic operatorsin 2d are the same as those in 4d . From (5.10) the R-charges are R [ B Σ i ] = 2 πN (cid:82) Σ i e (cid:0) (cid:1) d (cid:102) vol(Σ i )9 (cid:102) vol( SE ) = N π (cid:82) Σ i d (cid:102) vol(Σ i ) (cid:102) vol( SE ) = R d [ B Σ i ] , (5.22)where we have used [40, 41] to compare with the corresponding 4d R-charge. The Sasaki–Einstein metric (at fixed Σ coordinates) appearing in the AdS solution has a constant rescalingin comparison with the AdS metric and therefore the volume form on the any three-cycle differs by a factorof e (cid:0) (cid:1) / in comparison with the AdS normalised metric. We shall write all volume forms with respectto the canonically normalised metric on the Sasaki–Einstein manifold with a tilde. aryonic Charges During the discussion on baryonic charges we noted that the result is independent of a rescal-ing of the metric. Clearly this implies that the baryonic charges for these solutions will beidentical to the original AdS computation and therefore we shall not present it.Holographic charge Result c sugra 32( g − a d + a d deg( L D )( c L ) sugra − ( c R ) sugra N deg( L D )R-charges R (2 d ) [ B Σ i ] = R (4 d ) [ B Σ i ]Baryonic charges B (2 d ) [ B Σ i ] = B (4 d ) [ B Σ i ] Table 2 : Holographic charges for the universal twist solution with elliptic surface S τ . Here, a d is the 4d central charge (5.18) associated to the dual of the AdS × SE solutions. Consider now the universal twist solution where Y τ has a factor given by an elliptic three-fold.It will be instructive to compare these solutions to the AdS solutions in section 3.3 in ananalogous manner to the way in which the discussion in the previous section referenced theSasaki–Einstein solutions. Central Charges
The leading order central charge is easily found to be c = 2 · π ((2 πm(cid:96) s ) g s ) vol( M τ )vol(Σ) = 8 π ( g − N M τ ) = 32( g − a dτ , (5.23)where a dτ is the central charge of the τ dependent 4d field theory dual to the solutionsdiscussed in section 3.3.As in the previous cases, the subleading contribution to the difference of central chargescan be determined by anomaly inflow on the 7-branes, from the Wess–Zumino term (5.3) inthe effective action of a single D7-brane. In contrast to the first case, the discriminant locusof the elliptic fibration is now a curve in M . We consider only I singular fibers and thusonly single 7-branes are wrapped on curves C x in the discriminant locus . ∆. Imposing thatthe elliptic fibration satisfies (4.29) implies[∆] = (cid:88) x ω x = 12 c ( L D ) , (5.24) With a slight abuse of notation we denote simply as ∆ the locus { ∆ = 0 } . ω x are the two-forms dual to the curves C x , which are wrapped by the single 7-branes.Each 7-brane is extended along AdS × ( U (1) χ → Σ × C x ), where χ is the angular coordinatewith period 2 π(cid:96) along the R-symmetry direction. The total contribution to the WZ term isobtained by summing over all the single 7-branes, so that the effective world-volume can bewritten as W = AdS × ( M → Σ), where M = U (1) χ → ∆ is a three-cycle in M τ . Thethree-dimensional Chern-Simons term arising from the Wess-Zumino action then reads S CS = − ˜ µ (cid:90) W F ∧ ω CS (AdS )= − µ m πχ (Σ) vol( M ) (cid:90) AdS ω CS (AdS ) , (5.25)wherevol( M ) = (cid:90) M
13 (d χ + ρ ) ∧ J M = 2 π(cid:96) (cid:90) ∆ J M = 8 π(cid:96) (cid:90) M J M ∧ c ( L D ) = 8 π (cid:96) (cid:90) M (cid:0) c ( M ) ∧ c ( L D ) − c ( L D ) (cid:1) . (5.26)The gravitational anomaly, by using (5.4), is therefore found to be c L − c R = − π ( g − µ vol( M ) m , (5.27)where notice that vol( M ) is essentially an intersection number, providing the effective num-ber of 7-branes, as in [1].However, as for the leading order central charges, we will relate c L − c R in the dual 2dSCFT to a corresponding holographic quantity in the parent 4d SCFT, therefore vol( M ) willdrop out form the equation. Later we will show that this relationship is reproduced exactlyby a field theoretic calculation, although we will not attempt to calculate the precise valuesof the 4d central charges in specific examples.Concretely, we wish to identify the above result with the linear ’t Hooft anomaly k R inthe 4d theory, and therefore with the difference of 4d central charges c d − a d = k R , whererecall that a d = 332 (3 k RRR − k R ) , c d = 132 (9 k RRR − k R ) . (5.28)For any 4d N = 1 SCFT with an R -symmetry, the R -symmetry current R µ satisfies theanomalous conservation equation [42–44] ∂ µ (cid:104)√ g R µ (cid:105) = k R π (cid:15) µνρσ R µνκτ R ρσκτ + k RRR π (cid:15) µνρσ F µν F ρσ , (5.29) In the following discussion the overall constant of the Wess-Zumino term in equation (5.3) will cancel inthe computation and therefore for simplicity we define the new constant ˜ µ = µ π (cid:96) s4 . F is the field strength of the background gauge field A sourcing the R-symmetrycurrent.Consider the AdS solutions of section 3.3. Recall that for the universal twist solution tobe well-defined the manifold M τ is required to be quasi-regular. As such we may write themetric on M τ as a U (1) fibration over a K¨ahler base M asd s ( M τ ) = 19 (d χ + 3 σ ) + d s ( M ) , (5.30)with d σ = 2 J M . As we consider only the quasi-regular cases we may fix the period of χ tobe 2 π(cid:96) . By changing coordinates as χ = (cid:96) ˜ χ we define a new 2 π periodic coordinate. As theReeb vector field is dual to the R-symmetry direction it is natural, as explained in [40], thata shift in the coordinate ˜ χ induces a gauge transformation of the R-symmetry gauge field A , that is ˜ χ → ˜ χ + α Λ , A → A + dΛ . (5.31)The identification of the constant α is fixed by using the fact that the holomorphic 3-formon the cone is associated to the superpotential and therefore has R-charge 2. The functionaldependence of the holomorphic three-form on ˜ χ may be read off from (D.42) which fixes α = (cid:96) . We may include A in the usual Kaluza-Klein ansatz by deforming the internal metricas d s ( M τ ) → (cid:18) (cid:96) (cid:19) (cid:18) d ˜ χ + 3 (cid:96) σ + 2 (cid:96) A (cid:19) + d s ( M ) . (5.32)Moreover, for consistency, the five-form flux must be deformed as F → (1 + ∗ ) 2 (cid:96) m (cid:18)(cid:18) d ˜ χ + 3 (cid:96) σ + 2 (cid:96) A (cid:19) ∧ J M ∧ J M −
13 d
A ∧ (cid:18) d ˜ χ + 3 (cid:96) σ (cid:19) ∧ J M (cid:19) , (5.33)which by construction is closed upon using the equation of motion for the new gauge fieldd ∗ d A = 0. The term of the four-form potential of interest is C ⊃ − m A ∧ (cid:18) (cid:96) (cid:18) d ˜ χ + 3 (cid:96) σ (cid:19)(cid:19) ∧ J M . (5.34)In this configuration, the world-volume of each 7-brane is AdS × ( U (1) χ → C x ), thereforethe total contribution from all the 7-branes is obtained by integrating on the world-volume W = AdS × M where M = U (1) χ → ∆ is the same three-cycle in M τ that appears in More precisely, here A is a gauge field in AdS , whose boundary value is identified with the backgroundR-symmetry gauge field A in the four dimensional SCFT. The length scale associated to the AdS will be denoted as m in the following. It will be shown to beproportional to the length scale m in the AdS solutions. solution. We may use this to extract from the Wess-Zumino term a contribution tothe gravitational action in AdS given by S CS = ˜ µ (cid:90) W C ∧ Tr[
R ∧ R ] = − µ m vol( M ) (cid:90) AdS A ∧
Tr[
R ∧ R ] . (5.35)According to the gauge/gravity duality master formula, the generating functional for(connected) current correlators in the boundary theory, i W [ A ] = log Z [ A ], equates the on-shellgravitational action, W [ A ] = S AdS [ A ], and therefore as explained in [45] the non-invarianceunder gauge transformations of the latter corresponds to the anomaly in the dual field theory.Specifically, a gauge transformation of the boundary gauge field A induces a transformationof the Chern-Simons term δ Λ W [ A ] = δ Λ S CS = − µ m vol( M ) (cid:90) AdS dΛ ∧ Tr[
R ∧ R ]= − µ m vol( M ) (cid:90) ∂ AdS Λ Tr[
R ∧ R ] , (5.36)implying that on the boundary we have (cid:90) ∂ AdS Λ ∂ µ (cid:104)√ g R µ (cid:105) dvol( ∂ AdS ) = 2˜ µ m vol( M ) (cid:90) ∂ AdS Λ Tr[
R ∧ R ] , (5.37)where Tr[ R ∧ R ] = − (cid:15) µνρσ R µνκτ R ρσκτ dvol( ∂ AdS ) . (5.38)In conclusion we find k R = − N π vol( M )12vol( M τ ) = − π ˜ µ vol( M ) m , (5.39)and inserting this into (5.27) we obtain c L − c R = 94 m m ( g − k R . (5.40)We may relate the different length scales of the two solutions by comparing the quantisationcondition used to obtain the integer N . In both cases this gives the number of D3-branesin the solution and should therefore be fixed in flowing from the AdS solution to the AdS solution, by comparing (3.25) and (4.32) we find 9 m = 4 m and therefore we conclude that c L − c R = ( g − k R . (5.41) It would be interesting to match this formula, using (4.33) and (5.26), to a purely field theoretic compu-tation in the 4d SCFT. -charges In a similar manner to the previous section, at fixed coordinate on Σ, which is now H / Γwith Γ a subgroup of SL Z , equipped with the constant curvature metric, one finds that themetric on M τ is the same (up to an overall constant factor) as the metrics discussed in section3.3. Again we have that the three-cycles of the two solutions agree and therefore the dualbaryonic operators in 2d and 4d are identified. Clearly by the same arguments as presentedin section 5.2 the R-charges of the baryonic operators in 2d and 4d coincide. Baryonic Charges
As above the metrics agree up to a numerical factor. The topological nature of this compu-tation implies that the baryonic charges of the 2d theory and the 4d theory agree.Holographic charge Result c sugra 32( g − a dτ ( c L ) sugra − ( c R ) sugra g − c d − a d )R-charges R (2 d ) [ B Σ i ] = R (4 d ) [ B Σ i ]Baryonic Charges B (2 d ) [ B Σ i ] = B (4 d ) [ B Σ i ] Table 3 : Holographic charges for the universal twist with elliptic threefold T τ . Here a dτ isthe central charge of the dual to the solutions in section 3.3. Y p , q Case
In this final section we shall consider the baryonic twist solutions using Y p , q as the example.We expect the computations to extend to other solutions with baryonic twists in a similarmanner. Some such solutions will be discussed in [46]. Central Charges
From (5.2) we compute c sugra = 6 N M p ( q − p ) q . (5.42)Notice that despite the differences of our solution with respect to the constant τ versiondiscussed in [14], the value of the holographic central charge (5.42) agrees exactly with thevalue obtained in eq. (4.18) of [14]. We anticipate that this is a general property of thebaryonic twist solutions, that does not depend on the details of the M τ geometry. Moreprecisely, for any solution of the type AdS × T × M τ and constant axio-dilaton, we canconstruct a solution of the type AdS × P × M τ with axio-dilaton varying holomorphically43n P , such that the F-theory lift has an elliptic K3 factor. These two solutions will haveequal holographic central charges, at leading order in N .The subleading contribution may also be simply computed from the geometry. Moreoverit can be seen that the result is independent of the choice of M τ , one obtains the universalcontribution of N for a single 7-brane. For a K3 surface the number of 7-branes for a consistentgeometry is 24 and therefore the subleading contribution is( c L ) sugra − ( c R ) sugra = 24 · N N . (5.43)Notice that although at leading order the central charges of the solution with constant andvarying τ agree, the sub-leading contribution (5.43) is clearly zero in the model with constant τ , as there are no seven-branes. In the next section we will argue that in the dual field theoryside this result is exactly reproduced combining contributions that come both from the bulkmodes (3-3 strings) as well as 3-7 strings. Note that there are O ( N ) terms in the bulk for thetheory with varying coupling, that are due to the duality twisting. R-charges
The three-cycles in the geometry that are calibrations are the four three-cycles located at thefour degeneration surfaces of the metric. Recall that at each degeneration surface a Killingvector has zero norm, which determines a codimension two locus (namely a three-manifold)in the geometry. By explicit computation one can see that the volume form on the three-manifolds will be equal to the pullback of this closed four-form and hence these cycles arecalibrated. We find R [ B Σ + ] = R [ B Σ − ] = N q − p q ,R [ B Σ ] = R [ B Σ π ] = N p q . (5.44)Observe that the sum of the normalised volumes of submanifolds satisfies exactly the samerelation to their Sasaki–Einstein counterparts, namely R [ B Σ + ] + R [ B Σ − ] + R [ B Σ ] + R [ B Σ π ] = 2 N . (5.45)It would be nice to obtain a general proof of this formula, analogous to that in [47].
Baryonic Charges
As discussed in the introduction of this section we are free to multiply the metric by anarbitrary warp factor so long as the warp factor is bounded and non-vanishing. We shall44ake use of this freedom to find such a harmonic form. As dim[ H ( Y p , q )] = 1 there is aunique closed three-form representative which may be extracted from (4.51) and is given by ω = k (cid:18) Dα ∧ Dφ ∧ d x + dvol( S ) ∧ (cid:18) xDα − U ( x ) w ( x ) Dφ (cid:19)(cid:19) . (5.46)Observe that for the warped metric on Y p , q d s = e − d s ( Y p , q ) (5.47)this three-form is both closed and co-closed and therefore harmonic. The constant k is fixed byrequiring that the results are integer, to be k = − q − p . Integrating this over the three-cycleswe find (cid:90) Σ π ω = (cid:90) Σ ω = p , (cid:90) Σ − ω = q − p , (cid:90) Σ + ω = − ( q + p ) (5.48)which gives the baryonic charges of the fields and agrees with the result in (E.70) for thewould-be GLSM charges.Holographic charge Result c sugra 6 NM p ( q − p ) q ( c L ) sugra − ( c R ) sugra N R-charges R (2 d ) [ B Σ ] = R (2 d ) [ B Σ π ] = N q − p q R (2 d ) [ B Σ ] = R (2 d ) [ B Σ π ] = N p q Baryonic charges B (2 d ) [ B Σ ] = B (4 d ) [ B Σ π ] = p B (2 d ) [ B Σ ] = q − p B (2 d ) [ B Σ ] = − ( q + p ) Table 4 : Holographic charges of the Baryonic twist for Y p , q . Field theories with spacetime varying coupling are not a new concept in itself. Howeverthe inclusion of S-duality monodromies specifically in 4d N = 4 SYM and, as we will see,generalizations to N = 1, have only received recent attention in [4, 6]. N = 4 SYM
For 4d N = 4 the question arose in the context of D3-branes in F-theory, which naturallyimplements the varying complexified coupling τ in terms of a complex structure of an elliptic45urve. Field theoretically the τ variation along a curved manifold, e.g. a complex curve orsurface, together with retaining some supersymmetry, implies that a particular new topo-logical twist needs to be applied to the field theory. This topological duality twist was firstintroduced for abelian theories in [3], and a proposal for the non-abelian generalization wasput forward based on a realization in terms of M5-branes in [4]. For D3-branes wrapped oncurves along which the coupling varies, the duality twist was implemented in [5, 6].The key point about the topological duality twist is that fields and supercharges transformas sections of a duality bundle L D with connection given in terms of τ = τ + iτ by Q in(2.3). The transformation of the supercharges is such that they have charge ± / U (1) D : Q α → e − iα ( γ ) Q α ˜ Q ˙ α → e + iα ( γ ) ˜ Q ˙ α , (6.1)where e iα ( γ ) = ( cτ + d ) / | cτ + d | for γ = (cid:0) a bc d (cid:1) ∈ SL Z . The remaining fields of the N = 4 SYMtheory are charged q Φ = 0 (scalars), q F ± = ∓ F ± = √ τ ( F ± (cid:63)F ) /
2) and q λ = − , q ˜ λ = (fermions). To offset this transformation the duality twist redefines the U (1) D withan R-symmetry transformation. More generally for spacetimes of the form M = R , × C the twist can involve U (1) C , U (1) D and an R-symmetry U (1) R , as discussed in [6].One of the classes of solutions that we will encounter is the compactification of a 4d N = 1theory on a curve C = P , which is the base of an elliptic K3. This has many similaritiesto the elliptic K3 compactifications of F-theory as discussed in appendix D of [6]. Briefly, inthis case the twist only requires one to combine U (1) twist : T C twist = 12 ( T C − T D ) , (6.2)without an R-symmetry twist. The resulting theory has 2d (0 ,
8) supersymmetry. The fieldsare counted by cohomologies h i,j ( C ), depending on the twist charges q twist = − , , +1 core-sponding to ( i, j ) = (1 , , (0 , , (0 , (cid:98) C ,which geometrizes the axio-dilaton variation in terms of the elliptic fiber. Studying these the-ories has so far not been done, but some progress for D3-branes in CY three-folds in F-theorywas put forward in [1] and will appear in [48], using anomaly arguments.For N = 1 theories in 4d, similar compactifications with spacetime dependent couplingscan be defined. Although not every such theory has a duality group, whenever there is a46olographic dual setup, and an embedding into Type IIB (or F-theory), the theory shouldhave an induced U (1) D symmetry. One way to argue for this is presented in [49] by Intriligator,where the so-called bonus- U (1), which for the abelian theories was identified with U (1) D in[4]. Again there is a question as to how to generalise this to non-abelian theories, where thereis no manifest way to define this duality symmetry. We should remark that this symmetryfor the abelian theory is a symmetry only of the equations of motion, not of the action. Fromconsiderations in [49], the bonus symmetry is an approximate symmetry only for certainobservables in a particular limit, namely when both stringy and D-stringy corrections aresuppressed, but then should also be a feature of 4d N = 1 theories.Here we will consider well-known quiver gauge theories with Type IIB Sasaki–Einsteinduals, for which we will discuss generalisations of the “universal twist” and “baryonic twist”[16]. The first class of theories is characterised by having rational R-charges in four dimen-sions, and otherwise unspecified global symmetries; examples include N = 4 SYM and theKlebanov-Witten model, but more generally the theories discussed in section 3.3, which arethe most general F-theory solutions with AdS factors dual to 4d N = 1 theories. The sec-ond class of theories is characterised by having a global baryonic global symmetry, and mayhave rational or irrational R-charges in four dimensions; our main example will be the Y p,q quivers [50]. In all cases, the R-charges of the 2d SCFTs will be rational.In the gauge theories each node of the quiver has a complex coupling constant τ i and thediagonal combination τ = (cid:88) i τ i (6.3)is identified in the dual supergravity solution with the axio-dilaton of Type IIB. Unlike thecase of N = 4 there is no direct way to identify the charges, however we will argue that thefermions are all charged in the same way, exactly as in N = 4 SYM. The argument to supportthis uses the duality with AdS : although the bonus U (1) is not an actual symmetry of thetheory, it is a symmetry for large N and for short operators. In the holographic dual thesecorrespond to Kaluza-Klein modes on the compact part of the supergravity solution. As thelatter have definite charges under U (1) D , the expectation is that the dual states will also havea well-defined charge. The state of the art of the KK-spectrum on Sasaki–Einstein manifoldswas obtained in [51].We begin with 4d N = 1 with supercharges Q = ( , ) and ˜ Q = ( , ) under SO (1 , L In order to be self-contained, we give a mini-review of the 4d Y p,q quiver theories in Appendix F. CSO (1 , L → SO (1 , L × U (1) C ( , ) → ++ ⊕ −− ( , ) → + − ⊕ − + . (6.4)The duality charges are conjecturally q Q = − q ˜ Q = +1. Then performing the topologicaltwist as in (6.2) results in two scalar supercharges of negative chirality (i.e. −− and − + inthe above equation). For abelian N = 1 theories the multiplets are such that the scalars areuncharged under the U (1) D and the fermions carry all the same charge, which agrees withthat of the supercharges. This is much alike the charges in the N = 4 SYM case. For thenon-abelian theory, we propose to study the theory in a mesonic or Coulomb branch, whereusing anomalies we can determine the central charges [48]. N = 1 Field Theories
Before addressing the dual field theory interpretation of the solutions we discussed in section4 we review some aspects of the dualities proposed in [16] for the solutions with constant τ [33, 34]. We will follow the notation of these references, except, when we discuss the baryonictwist of the Y p,q theories where we will be careful in distinguishing the parameters p, q in thefield theories from the parameters p , q in the gravity solution [33, 34]. As we have alreadymentioned, although these parameters can be formally identified, they turn out to be definedin disjoint domains.A 4d N = 1 field theory can be compactified on a Riemann surface C g of genus g byperforming a topological twist that preserves N = (0 ,
2) supersymmetry in two dimensions.Although the details of these two-dimensional theories may be complicated to work out,if these flow to (0 ,
2) SCFTs then many of their properties can be inferred by employingthe method of c -extremization [8]. In particular, this method allows one to determine the 2dcentral charge c R of these theories, starting from the ’t Hooft anomalies of their “parent” four-dimensional theories. The most reliable method to implement this is to consider the anomalypolynomial I of the N = 1 4d theory, that can be usually computed exactly starting fromthe fermionic field content of the 4d theory. This is given by I = 16 k IJK c ( F I ) ∧ c ( F J ) ∧ c ( F K ) − k I c ( F I ) ∧ p ( T ) , (6.5)where the index I runs over all the U (1) global symmetries of the theory. Here c ( F I ) arethe first Chern classes of the different U (1) I bundles and p ( T ) is the first Pontryagin class48f the manifold the theory is placed upon. The constants K IJK and K I are the cubic andlinear ‘t Hooft anomalies which can be determined from the charges of the fermions in thetheory, namely K IJK = Tr[ U (1) I U (1) J U (1) K ] = (cid:88) i Q iI Q iJ Q iK , K I = Tr[ U (1) I ] = (cid:88) i Q iI , (6.6)where Q iI denotes the charge of the i -th fermion under U (1) I . This can be reduced to theanomaly polynomial I of the 2d theory by integrating it over C g , which in a similar notation,reads I = 12 k IJ c ( F I ) ∧ c ( F J ) − k p ( T ) . (6.7)In the (0 ,
2) SCFT we can then read off the central charges c R and the gravitational anomalyas c R = 3 k RR , c R − c L = k . (6.8)In general, to compute the k IJ and k one requires information on the spectrum of fermions ofthe 2d theory, but for theories coming from a parent 4d theory with known ’t Hooft anomalies,these can be extracted simply from I = (cid:90) C g I . (6.9)The 2d superconformal U (1) R symmetry is determined by extremizing the trial k RR .The topological twist can be performed by switching on background gauge fields for thevarious global symmetries of the 4d theory, with quantised fluxes through C g . Consider aquiver gauge theory for which the global symmetries are( U (1) F ) n F × ( U (1) B ) n B × U (1) dR , (6.10)where F stands for flavour and B stands for baryonic symmetries, respectively. The super-script on the R-symmetry-factor emphasises that this is the exact superconformal R-symmetryof the interacting 4d SCFT, determined by a -maximization.In the notation of [16], the topological twist can be generically performed along T twist = n F (cid:88) I b I T I + n B (cid:88) I B I T B I + κ T dR , (6.11) Twisted compatifications of various four-dimensional quiver gauge theories were studied in [52] and furtherexamples of dual supergravity solutions will be discussed in [46]. T twist refers to the combination of symmetry generators that are used to twist the local Lorentz symmetryalong the curve. T I , T B I , T dR are the generators of the respective global symmetries and κ = 1 , , − g = 0 ,
1, or g >
1, respectively . Here b I , B I are suitably quantised parameters, andthe factor κ is determined by requiring that the Killing spinors on C g become constants, asusual. Notice that as the Killing spinors are not charged under the other global symmetries,this particular way of preserving supersymmetry does not fix the parameters b I , B I .Note that when κ (cid:54) = 0 the twisting (6.11) makes sense only when U (1) dR is a compactsymmetry. In particular, for the Y p,q theories this is true iff z ≡ (cid:112) p − q is an integer andthe 4d R-charges (F.1) are rational numbers. This implies that generically the 2d R-chargeswill be rational numbers. When κ = 0 (namely for C g =1 = T ) there is no twist by the 4dR-symmetry and therefore one can start from 4d field theories with irrational R -charges. Inthe next section we will explain a variant of this twisting, in which we can again start from a4d field theory with irrational R-charges, and nevertheless compactify this on a C g =0 = P .The R-symmetry U (1) dR of the (0 ,
2) theory can in general mix with all the global sym-metries of the 4d theory , namely in terms of generators we have T d trial = n F (cid:88) I (cid:15) I T I + n B (cid:88) I (cid:15) B I T B I + T dR , (6.12)where (cid:15) I , (cid:15) B I are a priori real numbers that will be determined my extremizing the trial2d central charge as a function of these parameters. This calculation was performed in[16] for various examples, using the index theorem to count the fermionic zero modes in 2d[53]. However, as we discussed above, the computation using the reduction of the anomalypolynomial of the 4d theory is more robust, as there is no need to assume that the theoryis weakly coupled (which is not a correct assumption for most N = 1 theories with Sasaki–Einstein duals). This twist can be applied to any theory provided the 4d R-charges are rational, and consistsin taking T twist = κ T dR , (6.13) In this equation it is assumed that C g has constant curvature. A priori, there can be global symmetries that emerge in the 2d theory. In this case c -extremization (like a -maximization) cannot be used effectively to determine the R symmetry in the IR. κ = −
1. Assuming a general parameterization as in (6.12) the outcome of the extrem-ization procedure is that (cid:15) I = (cid:15) B I = 0, so that the 2d and 4d R-symmetries are identified ,namely R d = R d .At leading order in N , this yields the universal relation c R = c L = 323 ( g − a d . (6.14)Recalling that (at leading order in N ) in 4d theories a d = 932 (cid:88) i ( R di − , (6.15)one sees that (6.14) is indeed equivalent to R d = R d and c R = − · g − (cid:88) i (cid:0) − (cid:1) ( R d − R d − , (6.16)where − ( R d −
1) is the net number of 2d fermion zero modes associated to each 4d fermion.The results of section 5.2 may be used to compare with the constant τ version presentedhere by setting deg( L D ) = 0. We see that, as noted in [16], the central charges match exactly.Moreover we see that the holographic computations for constant τ implies that c L − c R = O (1)as follows from the field theory computation. Finally the results for the R-charges as presentedin section 5.2 are in agreement with the results from the field theory computation. Let us now consider theories that possess at least one baryonic symmetry with generator T B ,so that we can twist as T twist = BT B + κ T dR , (6.17)and in particular the theories can now be compactified on a torus, C = T , with κ = 0.This twist is purely baryonic and for concreteness we focus on the Y p,q theories, which have n B = 1 and n F = 2. One finds that extremizing k RR gives (cid:15) = 0 , (cid:15) = p + z q , (cid:15) B = p − z q , (6.18)and c R = c L = − Bp ( p − q ) q N . (6.19) This holds if the 4d ’t Hooft anomaly coefficients obey k RRF = k F = 0 and k RRB = k B = 0, which is truefor all quiver gauge theories with toric Sasaki–Einstein duals [54]. B <
0. As remarked in [16], from (6.18) we see that the 2d superconformal R-symmetry involves mixing the 4d one with the baryonic symmetry. Moreover, notice thatdespite the mixing coefficients (cid:15) , (cid:15) B and the ’t Hooft anomalies being irrational numbers,this irrationality drops out of the final expression for c R .This result matches that of the holographic computation [14] ( c.f. (5.42)) upon makingthe following identifications [16]: p = p , q = q , M = BN . (6.20)However, some comments are now in order. First of all, we note that since p < q and p > q ,strictly speaking this identification is contradictory . This issue was overlooked in the literatureand certainly deserves further scrutiny in the future. Here we will not attempt to resolve it,but we will make a number of checks that confirms the plausibility of these identifications.So far the only assumptions we made on the 2d field theories are that they are (0 , Y, Z, U α , V α can be computed from (6.12), namely using R d [ X d ] = (cid:15) Q F [ X d ] + (cid:15) B Q B [ X d ] + R d [ X d ] . (6.21)Plugging in (6.18) and the values of the 4d charges gathered from Table 5 we obtain R d [ Y d ] = R d [ Z d ] = q − p q ,R d [ U d ] = p q , R d [ V d ] = 1 , (6.22)in agreement with the results (5.44) for the normalised volumes of calibrated submanifolds.However, in the field theory the R -charges associated to the fields Y and Z are negative ,indicating that a better understanding of the duality proposed in [16] would be desirable. N = 1 Field Theories
In this subsection we shall extend the above computations to compactifications of the four-dimensional theories on a Riemann surface C g , with τ varying (holomorphically) over this.In particular, we shall promote the U (1) D symmetry obtained for varying τ to be a bundleover the Riemann surface C g , with curvature two-form d Q . This implies we must introduceadditional terms to the 4d anomaly polynomial for this bundle. By introducing the additional52 (1) D bundle the 4d anomaly polynomial I is modified by the inclusion of additional termsas I τ = I + 12 k DIJ c ( F D ) ∧ c ( F I ) ∧ c ( F J ) + k DDI c ( F D ) ∧ c ( F D ) ∧ c ( F I )+ k DDD c ( F D ) ∧ c ( F D ) ∧ c ( F D ) − k D c ( F D ) ∧ p ( T ) , (6.23)where I ∈ { R, B I , F I } as before. The anomaly polynomial for the 2d theory, I τ is againcomputed by integrating I τ over C g .With the introduction of the additional U (1) bundle in the anomaly polynomial we mustcompute the additional ‘t Hooft cubic and linear anomalies involving U (1) D . We shall arguethat the cubic and linear anomalies involving the duality bundle will scale as N and bymaking a plausible assumption we will be able to compute sub-leading contributions to the2d anomalies, obtaining perfect agreement with the holographic computations.Let us consider for example the linear trace k D ≡ Tr[ U (1) D ] = (cid:88) i q iD , (6.24)where the sum is over all the fermions (of the 4d theory) and q iD are their charges under U (1) D . However, exactly as for N = 4 SYM, in the non-abelian theories the bonus U (1) D is not a symmetry [49] and therefore these charges are not meaningful.To circumvent this problem, it is expedient to Higgs the N = 1 quiver theories withgauge group G = SU ( N ) χ to an abelian theory, at a generic point of the (mesonic) vacuummoduli space. In the low energy limit this theory has gauge group U (1) N − and contains N − N chiral multiplets, parameterising the flat directions of themesonic moduli space, that is the symmetric product of N copies of the related Calabi–Yauthree-fold conical singularity X = C ( Y ), Sym N X . See [55] for some discussion in the case ofthe Klebanov-Witten model with G = SU ( N ) , and [56] for an explicit analysis in the Y p,q theories. This is an abelian theory for which U (1) D is now a symmetry of the equations ofmotion, and we can infer the charges of the fields under U (1) D from the supergravity analysis.As we recall in Appendix A, in our conventions the supergravity Killing spinors havecharge q D = − /
2. In the boundary (abelian) field theory this translates to the fact thatthe scalar field φ and the fermion field ψ in a chiral multiplet have U (1) D charges satisfying q D [ φ ] − q D [ ψ ] = − /
2. The U (1) D charge of the scalar bifundamental fields can be fixed by anextension of the arguments in [49], by noting that mesonic gauge-invariants operators (closedloops in the quiver) correspond to scalar harmonics on the Sasaki–Einstein manifold Y that53re in 1–1 correspondence with holomorphic functions on the cone [57]. In particular, thesemodes are fluctuations of a mixture of the metric and the RR four-form potential [58]. Sincethese are both inert under SL R transformations, it follows that an infinite tower of dual scalaroperators is uncharged under U (1) D . In N = 4 SYM these operators are Tr X I X I · · · X I k [45] and correspond to a KK tower on S , uncharged under U (1) D [59]. This clearly impliesthat the scalar bifundamental fields themselves must be uncharged and therefore the fermionsin the chiral mutliplets have q D [ ψ ] = 1 /
2. The U (1) D charge of the gauginos is fixed by the(abelian) supersymmetry transformations to be q D [ λ ] = 1 /
2. Putting all together, we obtain k D = 12 3 N + 12 ( N −
1) = 2 N −
12 (at a generic point on the Higgs branch) , (6.25)It remains to justify the assumption that, differently from other symmetries, for U (1) D thereare no other contributions on the Higgs branch, arising from integrating out the massive off-diagonal modes [60, 61]. This is plausible, as at the origin of the Higgs branch U (1) D ceasesto be a symmetry. Moreover, this scaling with N is fully consistent with the results for the(0 ,
4) theories that we discussed in [1]. We will return to this elsewhere [48].Using this prescription it is straightforward to compute the mixed cubic anomaly coef-ficients involving one D index, k DIJ . However, the result of this computation will provideus the sub-leading term of c R , which we have not attempted to compute holographically,and therefore we do not present the results here. It would be interesting to compute thisperforming a KK analysis of the U (1) R isometry in the geometry, along the lines of [36].Below we will discuss the matching with the holographic computations of c R , c L at leadingorder in N , and of c R − c L at sub-leading order. S τ Case
Let us now consider the field theory dual to the solutions discussed in section 4.1.1 andcompare with the results of section 5.2. Like the universal twist solutions revised above weshall compensate for the curvature of the base by coupling the 4d R-symmetry to a backgroundfield. This however is not sufficient to cancel off all of the curvature of C g and we must alsotwist with U (1) D . As before we allow the 2d R-symmetry to mix with the flavour and baryonicflavour symmetries, however we do not allow it to mix with U (1) D , as implied by the analysisin the gravity side. The topological twist ensures that the Killing spinor equation on Σ admitsa constant spinor solution. To achieve this, couple to two background fields A i (unlike theconstant τ cases) as ( ∇ Σ + i A T D + i A T R ) (cid:15) = 0 (6.26) There is no contribution to c R from the 37 sector, therefore this is the full contribution. ω = R = − J − d Q . (6.27)Requiring that τ is holomorphic on Σ implies that the spinor on Σ satisfies the projectioncondition γ (cid:15) = − i (cid:15) and therefore requiring that a constant spinor satisfies (6.26) impliesthe topological twist d A = − d Q , d A = 3 J , (6.28)which is precisely like the topological duality twists in [1, 6] and results in the twisted U (1) T twist = T D − T dR , (6.29)whilst the trial R-charge is given by (6.12). Concretely the twisting induces the followingidentifications of the curvatures of the various bundles F dR → F dR − J Σ , F dF I → F dF I + (cid:15) I F dR , F dB I → F dB I + (cid:15) B I F dR , F dD → πc ( L D ) , (6.30)where F are the flavour symmetries and B the baryonic symmetries. Upon extracting the k RR coefficient and extremizing with respect to the (cid:15) ’s one finds (cid:15) I = 0 = (cid:15) B I , (6.31)and therefore there is no mixing in 2d of the exact R-symmetry and the flavour and baryonicsymmetries. This is true at leading order but may be corrected at subleading order due tocubic ‘t Hooft anomalies involving U (1) D . The central charge is given by c R = 3 k RR and isobtained from reducing the I on the base of the elliptic surface, Σ, as c R = 163 (2( g −
1) + deg( L D )) a (4 d ) , (6.32)which is in perfect agreement with (5.19). An important point to note here is that the centralcharge has at leading order already a τ -dependence through L D .By extracting k from the I τ anomaly polynomial we find the subleading contribution tobe ( c L − c R ) bulk = − k D (cid:90) Σ c ( L D ) = − k D deg( L D ) . (6.33)55he subscript indicates that this contribution arises from the dimensional reduction of the4d theory, ignoring the defect modes from the 7-branes. Furthermore, assuming that thecontributions of the 7-branes to the spectrum are again Fermi multiplets as in [6], we canconjecture that the 3-7 defect modes gives an additional contribution( c L − c R ) defect = 8 N deg( L D ) . (6.34)From the discussion at the begining of this section we have k D = 2 N − / c L − c R = 8 N deg( L D ) − N deg( L D ) = 6 N deg( L D ) , (6.35)which agrees with the result given in (5.21).One may also compute the R-charges of the fields from the anomaly polynomial. As theextremization forces all the (cid:15) I to vanish one finds that the R-charges of the 2d fields are thesame as the R-charges of the 4d fields, in agreement with the conclusion reached in section5.2. T τ Case
The field theory duals to the universal twists with an elliptic three-fold factor are obtainedby a twisted reduction of the 4d N = 1 SCFTs in section 3.3, whose duals are F-theoreticAdS solutions. The field theory is reduced along a curve with constant τ , so that thestandard universal twist of [16] can be implemented as in (6.13), with the trial R-symmetrygiven as usual. In the following we shall assume that the 4d ’t Hooft coefficients still obey k RRF = k F = 0 and k RRB = k B = 0 as in the toric Sasaki–Einstein case, we make norestriction on k RRD and k D . This starting point implies that the 2d R-symmetry to leadingorder is given exactly by the 4d one, and we have (cid:15) I = (cid:15) B i = 0. In particular the centralcharge is c R = 32( g Σ − a (4 d ) τ , (6.36)which agrees with the holographic result. Moreover the subleading contribution is given by c L − c R = ( g − k τR . (6.37)Since this corresponds to the twisted reduction on H / Γ above which the theory has novarying coupling this is the exact result to this order. In the constant τ field theory one hasto subleading order k R = 0 and therefore c L = c R at subleading order. As discussed in section5.3, k τR is non-zero at sub-leading order in the varying τ field theory, and therefore non-trivial56 not only modifies the leading order central charge of the theory it also implies that the leftand right moving central charges differ at subleading order.As a final check of our results in section 5.3 the identification of the 2d R-symmetry withthe 4d one implies that the R-charges of the fields in 2d and 4d are identical, which agreeswith the results presented in the holographic setup. We now discuss theories with varying coupling, which have a baryonic symmetry. We cancompactify on a complex curve C g of genus g (cid:54) = 1 and preserve supersymmetry by twistingwith U (1) D , as explained in section 6.1. As the supercharges are uncharged under the baryonic(and flavour) symmetries we are free to twist with these as well. In particular, we take C = P , with curvature given by − d Q , which is also the connection of the duality linebundle L D . Concretely, the topological twist we take is T twist = BT B + T D . (6.38)We again assume that the R-symmetry does not mix with U (1) D and therefore we take astrial R-charge T trial = (cid:15) T + (cid:15) B T B + T dR . (6.39)Under the twisting the curvatures of the various bundles become F dR → F dR , F dF → F dF F dF → F dF + (cid:15) F dR , F dB → F dB + (cid:15) B F dR − Bt g , F dD → c ( P ) . (6.40)The last line is fixed as the compactification geometry is an elliptic K3-surface. The anomalypolynomial for the 2d theory, I τ is computed by integrating I τ in (6.23) over the base P ofthe elliptic K τ (cid:90) P I τ = I τ ⊃ − ( B ( k R B + k B (cid:15) + k BB (cid:15) B ) + 2( k R D + k D (cid:15) + k BD (cid:15) B )) c ( F ) ∧ c ( F R ) − ( B ( k R B + k B (cid:15) + k BB (cid:15) B ) + 2( k R D + k D (cid:15) + k BD (cid:15) B )) c ( F ) ∧ c ( F R ) Note that there is a minus sign difference in the F B term with that in equation (2.47) of [16]. We fixedthis by first recovering the results for constant τ on a T via the anomaly polynomial. As the expression one finds for I is unwieldy we present only the salient terms. ( B ( k RBB + k BB (cid:15) + k BBB (cid:15) B ) + 2( k RBD + k BD (cid:15) + k BBD (cid:15) B )) c ( F B ) ∧ c ( F R ) −
12 [ B { k RRB + (cid:15) B (2 k RBB + k BBB (cid:15) B ) + (cid:15) (2 k RRB + k B (cid:15) + 2 k BD (cid:15) B ) } +2 { k RRD + (cid:15) B (2 k RBD + k BBD (cid:15) B ) + (cid:15) (2 k R D + k D (cid:15) + 2 k BD (cid:15) B ) } ] c ( F R ) + 124 ( Bk B + 2 k D ) p ( T ) (6.41)Comparing this with the general structure of the I polynomial (6.7) and (6.8) yields c R = 3 k RR = − B { k RRB + (cid:15) B (2 k RBB + k BBB (cid:15) B ) + (cid:15) (2 k RRB + k B (cid:15) + 2 k BD (cid:15) B ) } +2 { k RRD + (cid:15) B (2 k RBD + k BBD (cid:15) B ) + (cid:15) (2 k R D + k D (cid:15) + 2 k BD (cid:15) B ) } ] ,c L − c R = − Bk B − k D . (6.42)The exact central charge is obtained by extremizing c R with respect to (cid:15) B , (cid:15) , the ex-pression one obtains is prohibitorily large and so we do not present it here. The key is tonote how the various ’t Hooft anomalies scale with N [48]. Those not involving the dualitysymmetry, U (1) D will be unaffected by its inclusion and scale as N , on the other hand anyterm involving U (1) D will scale as N and therefore it will be sub-leading. Observe that inthe universal twist solutions presented previously a non-trivial variation induces a shift in thecentral charge at leading order, not just at subleading order as is the present case.Note that so far we have not specified a theory, and therefore the conclusion that theleading order central charge is unchanged with respect to the value of the same theory,compactified on C g =1 = T , and twisted by T twist = BT B is quite general. Specializing to the Y p,q quivers, we of course recover the result (6.19) c R = − Bp (cid:0) p − q (cid:1) q N + O ( N ) . (6.43)This is in agreement with our observations from gravity that the corrections due to τ aresub-leading in N . Using k B = 0 we also obtain( c L − c R ) bulk = − N + 1 , (6.44)where again the subscript indicates that this contribution arises from the dimensional reduc-tion of the 4d theory, ignoring the defect modes from the 7-branes. For an elliptic K3 the 3-7defect modes gives an additional contribution( c L − c R ) defect = 16 N , (6.45)so that the total contribution at order O ( N ) is precisely 12 N as in (5.43). We thank Craig Lawrie for discussions on this point and collaboration in [48]. Dual M-theory AdS Solutions
In all F-theory solutions, where the axio-dilaton varies non-trivially and thus there existsingular fibers, the metric on the base of the elliptic fibration, which is contained in the TypeIIB spacetime, is necessarily singular. This is the case even if we assume a smooth Weierstrassmodel, which only has I singular fibers. In light of this, it is advisable to also consider thedual M-theory solutions, where the total space of the elliptic fibration becomes part of thefull spacetime, and thus statements about the existence of smooth metrics on the total spacecan be used.For the (0 ,
4) solutions in [1] a dual AdS solution existed and the elliptic Calabi-Yauthree-fold, was a factor of this solution, and various statements in F-theory could be substan-tiated in this way. In the present cases of duals to (0 ,
2) SCFTs, one might therefore also wishto consider the dual M-theory AdS solution obtained by T-dualising along one of the internalnon R-symmetry Killing directions and uplift. However we find that in all cases (universaltwist and baryonic twist), the resulting spacetime has a key difference to the (0 ,
4) setup: thetotal space of the elliptic fibration does become part of the M-theory compactification space,however there is a warp-factor which only affects the base of the elliptic fibration. As it isfar from clear, whether there is a smooth metric on this total space of warping plus ellipticfibration, we will here dualise to AdS solutions in M-theory, which do not have this problem.This provides M-theoretic duals to the entire class of F-theory geometries that we studied inthis paper, albeit one which admits 1d dual SCFTs. It would be very interesting to explorefurther the connection between these and the 2d SCFTs that are prominent in the F-theoryduality frame. Solutions in M-theory
The “master equation” (2.63) is structurally the same as the equation in [32] governing AdS solutions in 11d supergravity with only electric flux. As we shall see the solutions presentedin this paper are dual to a subclass of those solutions when the (real) 8d K¨ahler base is takento be elliptically fibered. To perform this duality chain we must write the AdS metric as afoliation by AdS [15], that is we use the metricd s (AdS ) = 14 m (cid:18) − r d t + d r r + (2d ϕ + r d t ) (cid:19) . (7.1)We have normalised the metric such that the Ricci-tensor satisfies R µν = − m g µν . One maynow perform a T-duality along the azimuthal coordinate ϕ to obtain the metric on AdS × S SO (1 ,
1) isometry group of AdS preserved. Performing the T-duality on thegeneral Type IIB solution given in (2.55) and (2.58) results in the string frame Type IIAsolution m d s ( M IIA ) = e √ τ (cid:18) (cid:18) − r d t + d r r (cid:19) + 14 (d χ + ρ ) + e − d s ( M ) (cid:19) + √ τ e − d ϕ ,F A = 14 m dvol(AdS ) ∧ F (2) ,F A = 1 m d τ ∧ d ϕ , (7.2) H = 12 d ϕ ∧ dvol(AdS ) , e − IIA = τ e , which uplifts to 11d supergravity as an AdS × M solution m d s ( M ) = e (cid:18) m s (AdS ) + 14 (d χ + ρ ) + e − (cid:18) d s ( M ) + τ d ϕ + 1 τ (d ψ + τ d ϕ ) (cid:19)(cid:19) ,G = 14 m dvol(AdS ) ∧ (cid:20) − J −
12 d(e (d χ + ρ )) (cid:21) , (7.3)which agrees with the general form presented in [32] upon making the identifications A KP = 4∆3 , B KP = 12 ρ , ψ KP = χ . (7.4)Observe that the elliptically fibered space Y τ , that underlies the F-theory solutions of section2.6, now appears in the solution explicitly as part of the geometry. In the following we shallanalyse this map for the previously discussed solutions. As in the Type IIB solutions presented above it is necessary to quantise the flux through allcompact integral cycles in the geometry. To quantise the flux we impose that over all integralseven-cycles in the geometry, { A i } ∈ H ( M ; Z ) and integral four-cycles D i ∈ H ( M ; Z )[62] π(cid:96) p ) (cid:90) A i ∗ G ∈ Z , (cid:90) D i p ∈ Z . (7.5)The leading order holographic central charge of the dual 1d SCFTs may be extracted from[63] c d = 34 πG (2) N , G (2) N = 1 G (11) N (cid:90) M dvol( M ) , (7.6) One should also quantise G through all four-cycles however as the legs always lie along AdS no quanti-sation is necessary. G (2) N is the 2d Newton’s constant . Solutions with Elliptic Surface Factor
From the general form of the AdS solutions we haved s ( M ) = e (cid:20) d s (AdS ) + 9 m (cid:18)
19 (d χ + ρ ) + d s ( M ) + d s ( S τ ) (cid:19)(cid:21) , (7.7) G = − m dvol(AdS ) ∧ (cid:20) J S τ + 13 ( J M + J Σ ) (cid:21) , (7.8)e − = 94 , (7.9) ρ = − A M + 3 A Σ , (7.10)where as before d A i = J i . Note that the factors in the metric have conspired such that themetric on Σ and the elliptic fibration combine into the smooth metric on the elliptic surface S τ . Flux Quantisation
First consider the condition on the Pontryagin class following (7.5). There are two four-cyclesto consider, M and S τ . First note that p ( M ) = p ( M ) + p ( S τ ) . (7.11)We see that both the manifold S τ and M must have first Pontryagin class divisible by four.For the base of Y p,q for example this has vanishing first Pontryagin class.There are three distinct types of seven-cycle in the geometry; the seven cycle, D Σ givenby the five-cycle M τ fibered over Σ, the five-cycle M τ with the elliptic fiber which we call D E and finally the seven-cycles D α consisting of a three-cycle in M τ fibered over Σ alongwith the elliptic fibration. From the rules of T-duality and uplift one finds the relations (cid:96) p3 = m(cid:96) s4 β , vol( E τ ) = (2 π ) (cid:96) p6 m (cid:96) s4 g s , (7.12)where β is the period of the U (1) isometry dualised along. We find n ( D E ) = 1(2 π(cid:96) p ) (cid:90) M τ × E τ ∗ G = 1(2 π(cid:96) p ) (cid:90) M τ × E τ (cid:18) − m (cid:19) Dχ ∧ J ∧ J ∧ J E τ = 9vol( SE )(2 πm(cid:96) s ) g s ≡ N , (7.13) Notice that in the evaluation of the 3d Newton’s constant, (5.2), there is a factor of e ∆ coming fromthe warping of AdS , this is not the case for G (2) N . Recall that reduced Newton’s constant is computed byextracting the coefficient of the reduction of (cid:82) M √ gR . Under a conformal transformation g → e g wehave √ gR → e ( d − √ gR . The factors of the warp factor appearing (or not appearing) in the formula for thedimensional reduction of Newton’s constant is therefore clear. n ( D α ) = 1(2 π(cid:96) p ) (cid:90) Σ α ×T τ − m Dχ ∧ J ∧ J Σ ∧ J E τ = (2 π ) (cid:96) ˜ mn α (2 πm(cid:96) s ) g s (2( g −
1) + deg( L D )) . (7.14)Over the final type of seven-cycle we have1(2 π(cid:96) p ) (cid:90) D Σ ∗ G = 2 β a d g −
1) + deg( L D )) . (7.15)Flux quantization implies that β needs to take particular values, depending on a d and thegeometric data, and only in these cases do we expect the M-theory solution to be consistent . Central Charge
Using the formula for the central charge in equation (7.6) we find c d = 34 π π (cid:96) p9 (cid:18) m (cid:19) e vol( SE )vol(Σ)vol( E τ )= 16 a d g −
1) + deg( L D ) . (7.16)We find agreement with the central charge obtained from the Type IIB solution as expected. Solutions with Elliptic Three-fold Factor
From the general form of the AdS solutions we haved s ( M ) = e (cid:20) d s (AdS ) + 9 m (cid:18)
19 (d χ + ρ ) + d s ( H ) + d s ( T τ ) (cid:19)(cid:21) , (7.17) G = − m dvol(AdS ) ∧ (cid:20) J E τ + 13 ( J M + 4 J Σ ) (cid:21) , (7.18)e − = 94 , (7.19) ρ = − A M + 3 A Σ , (7.20)where as before d A i = J i . By assumption the metric on T τ is smooth. This expression is in fact proportional to the central charge in the AdS dual, and it would be interestingto understand what the physical interpretation of this quantization is. lux Quantisation As before we begin by considering the condition on the first Pontryagin class. There are twotypes of four-cycles to consider; the base M and a two-cycle in M with the elliptic fibrationover it and p ( M ) = p ( T τ ) . (7.21)We must impose the topological restriction that the first Pontryagin class of T τ is divisibleby 4.As in the previous class of solution there are three distinct types of seven-cycle in thegeometry; the seven cycle, D Σ given by the five-cycle M τ fibered over Σ, the U (1) fibrationover T τ which we call D E and finally the seven-cycles D α consisting of the elliptic fibrationover a three-cycle in M τ fibered over Σ. The same relations between (cid:96) s , (cid:96) p and vol( E τ ) holdas in the previous case. The first quantization condition is n ( D E ) = 1(2 π(cid:96) p ) (cid:90) D E ∗ G = 9vol( M τ )(2 πm(cid:96) s ) g s ≡ N , (7.22)which is the same as in Type IIB. Consider next the second type of seven-cycle containingthe circle fibration over the two-cycle in M with the elliptic fibration over it all fibered overthe curve, which we assume to be simply H then n ( D α ) = 1(2 π(cid:96) p ) (cid:90) D α − m Dχ ∧ J ∧ J Σ ∧ J E τ = π χ (Σ) (cid:96) πm(cid:96) s ) g s (cid:18) ˜ mn α − (cid:90) Σ α c ( L D ) (cid:19) . (7.23)Over the final type of seven-cycle we have1(2 π(cid:96) p ) (cid:90) D Σ ∗ G = 2 β a dτ g − . (7.24)As in the previous case we find that this is proportional to the central charge in the AdS case as can be seen by using (5.23). We may again tune the period of the U (1) dualised togive an integer result as necessary. Central Charge
Using the formula for the central charge in equation (7.6) we find c d = 34 π π (cid:96) p9 (cid:18) m (cid:19) e vol( M τ )vol( H )vol( E τ )= 32( g − a dτ c d . (7.25)As in the previous case we find agreement with the central charge obtained from the TypeIIB solution as expected. 63 .5 AdS × Y p , q × K Solutions
Following the discussion in the previous subsection there exists an AdS solution of 11dsupergravity which is M/F dual to the baryonic twist solutions in F-theory, of the formd s = e (cid:18) d s (AdS ) + w [d ψ + g ( x ) Dφ ] + a (cid:18) d x x U + Uw Dφ + d θ + sin θ d χ (cid:19) + 4e − d s ( K (cid:17) , (7.26) G = − m dvol(AdS ) ∧ (cid:18) ax ( Dψ − g ( x ) Dφ ) ∧ d x + 2 J K + 12 sin θ d θ ∧ d χ (cid:19) , (7.27)where K3 denotes an elliptically fibered K3 surface, d s ( K
3) is the smooth Ricci-flat K¨ahlermetric and J K is the K¨ahler two-form on the K3. By assumption we only have I singularfibers, and so the K3 surface is smooth – recall that the metric induced on the base B = P of the fibration, which is part of the Type IIB spacetime, has singularities. If we allowedfor general singular fibers, then in this M-theory setup, this would allow us to resolve thesesingularities. In either case we are considering a smooth K3 surface. Furthermore, theprevious regularity analysis for the metric on Y p , q shows that the full 11d solution is perfectlysmooth. Flux Quantisation
Let us first consider the Pontryagin class in (7.5). There are two four-cycles in the geometrythe first is the base Z of the U (1) fibration of Y p , q and the second is the K3 surface. As themetric is topologically AdS × Y p , q × K p ( M ) = p ( Z ) + p ( K Z is zero which is athird of the first Pontryagin class for a four-manifold. Similarly σ ( K
3) = −
16 and thereforeno additional constraint is imposed. Instead consider the quatisation of the flux over compactintegral seven-cycles. There are two distinguished classes of seven-cycles in the geometry, theK3 surface with the unique three-cycle generator E ∈ H ( Y p , q ) which we call Σ E and theseven-cycles arising from a two-cycle, { C i } ∈ H ( K Z ) and Y p , q . The flux may be writtenas ∗ G = dvol( E τ ) ∧ ˆ ∗ F (2) + 2 m e dvol( M ) , (7.29)64here ˆ ∗ is the Hodge star on the unwarped internal manifold in the Type IIB solution.Consider first the seven-cycle Σ E π(cid:96) p ) (cid:90) Σ E ∗ G = 1(2 π(cid:96) p m ) vol( B )vol( E τ ) 4 π q p ( p − q )= vol( B )(2 π(cid:96) s m ) g s q π p ( p − q ) ≡ M , (7.30)where we have used vol( E τ ) = (2 π ) (cid:96) p6 m (cid:96) s4 g s (7.31)as follows from correctly identifying the periods in the duality chain. Consider instead theseven-cycle Y p , q × E τ , we find1(2 π(cid:96) p ) (cid:90) Y p , q × E τ ∗ G = 4 π q (2 π(cid:96) s m ) p ( p − q ) ≡ N . (7.32)Observe that the integers that we have introduced are the same as those appearing in theType IIB analysis.
Central Charge
For the central charge we obtain c d = 3 . π (2 π(cid:96) p m ) (cid:90) M e dvol( Y p , q ) ∧ dvol( B ) ∧ dvol( E τ ) = c d , (7.33)where use has been made of the relation (cid:96) p3 = (cid:96) s4 m as follows from the T-duality and uplift. The aim of this paper was to extend the AdS /CFT dictionary in the F-theory context to 2dsuperconformal field theories with (0 ,
2) supersymmetry. There are at least two motivationsfor pursuing this program: on the one hand, it is interesting to explore the relatively unchartedterritory of holography in the context of F-theory. On the other hand, we believe that it isworthwhile broadening the set of field theories, which have varying coupling, such as was donein the case of 4d N = 4 SYM with topological duality twist [3, 4].The starting point of our analysis is the derivation of the general constraints for AdS solutions in Type IIB, with varying axio-dilaton, that follow from the Killing spinor equationson the internal geometry (2.41). We then investigated various solutions to these equations.A summary of all solutions obtained in this paper can be found in Table 1.65he first class of solutions have enhanced supersymmetry to (2 , is allowed to become a slicing of anAdS solution, τ can vary, and the solutions are the most general F-theory solutions dualto 4d N = 1 theories. The brane-setup is given in terms of D3-branes probing F-theorygeometries that are elliptic Calabi-Yau four-folds. A further analysis of these will appear in[64].For duals to 2d (0 ,
2) SCFTs there are two classes of solutions discussed in this paper,which are all based on the general form of the F-theory solution (i.e. including the axio-dilatoninto the geometric description in terms of the elliptic fibrations) given byAdS × ( S → Y τ ) . (8.1)Here Y τ is elliptically fibered. The base of this elliptic fibration (cid:102) M is a K¨ahler three-fold. Thefirst class of solutions are of the type (cid:102) M = Σ × M , i.e. a product of a curve and a surface.This gives rise to the universal twist solutions, where the elliptic fibration is non-trivial onlyover one of the two factors. The key characteristic of these universal twist solutions in F-theory is that they do not have any Calabi-Yau factors, i.e. the elliptic fibration restrictedto Σ and M , respectively, cannot be Ricci flat! The second class of solutions is obtained byimposing that there is explicitly a Ricci-flat factor in the direct product Y τ = M × K τ . Theresulting solutions are of the type AdS × K τ × Y p , q , or as Type IIB solution AdS × P × Y p , q ,where Y p , q are circle-fibrations over F . These are the baryonic twist solutions. In each casewe determined the holographic central charges and matched them to dual field theory, wherethe central charge is obtained using c-extremization applied in the context of 4d N = 1 fieldtheories with varying coupling. Key to our analysis are various topological twists of the 4dtheories that involve the U (1) D “bonus” symmetry inherited from Type IIB supergravity. Inparticular, we have demonstrated in several examples how this twisting affects the F-theorygeometry as well as the dual field theories, through an analysis based on an U (1) D -augmentedanomaly polynomial of these theories.For the baryonic twist solutions, based on the Y p , q geometries, we have uncovered somepuzzling aspects (see Section (6.2.2)) of the proposed duality with the Y p,q quiver gaugetheories [16], already present in the solutions with constant τ . It is clearly an interestingquestion to resolve these puzzles, and we hope to return to this in the near future.Let us mention some other directions following from this work. Enhancement to (2 , τ constant. Clearly one of the extensions ofthis work is to find more general solutions to (2.41) in tandem with the dual field theories,66oth for (0 ,
2) theories with varying coupling, as well as the (2 ,
2) theories, with constant τ .Extensions to the holographic dictionary in F-theory to higher dimensions could build alsoon the work [65–67], which could be put into a more F-theoretic setting.In Section 7 we have discussed M-theory duals to the entire class of F-theory AdS solutions, and argued that these are more naturally represented as AdS solutions in elevendimensions. In particular, in the baryonic twist solutions the K τ factor is geometrizedas in the standard M/F duality. However, the universal twist solutions result in M-theorygeometries, where the elliptically fibered part of the space is not a Calabi–Yau. This hints ata universal relation between 2d and 1d SCFTs. We have shown that for all these solutionsthe leading order holographic central charge agrees precisely with the F-theory result. Howto extract the sub-leading contributions, remains an interesting open question.Finally, in this paper we have shown that a simple extension of the anomaly polynomial tothe “bonus” U (1) D symmetry provides a powerful tool for studying field theories with varyingcouplings. It will be interesting to make more rigorous the arguments that we employed inSection 6 to deduce the contribution of the seven-brane modes to the central charges of thetwo-dimensional theories. We anticipate that doing this will improve our understanding ofthe still elusive field theories with varying couplings, including the case of non-abelian N = 4SYM and N = 1 , Acknowledgments
We especially thank Craig Lawrie for discussions and collaboration on a related work, andJenny Wong for collaboration at an earlier stage of this paper. We also thank BenjaminAssel, Francesco Benini, Nikolay Bobev, Heeyeon Kim, Yolanda Lozano, Dave Morrison,Sameer Murthy, Itamar Shamir, and James Sparks for discussions. CC is supported bySTFC studentships under the STFC rolling grant ST/N504361/1. DM is supported by theERC Starting Grant 304806 “The gauge/gravity duality and geometry in string theory”. SSNis supported by the ERC Consolidator Grant 682608 “Higgs bundles: Supersymmetric GaugeTheories and Geometry (HIGGSBNDL)”.
A S-duality and U (1) D in Type IIB In this appendix we collect some basic facts about the transformation of Killing spinors inType IIB supergravity under the SL R duality, as well as the relevance of the U (1) D symmetrywith connection Q . 67 .1 Duality U (1) D Let γ be an element of the Type IIB self-duality group SL R , which acts on the axio-dilaton τ as γ.τ = aτ + bcτ + d , ad − bc = 1 . (A.1)Due to quantum corrections the true duality group is SL Z . Define α ( γ ) in terms of thephase e i α ( γ ) = cτ + d | cτ + d | . (A.2)We define the U (1) D gauge field Q by Q = − τ d τ , (A.3)which transforms under SL R as Q → Q − d α ( γ ) , d α ( γ ) = c ( d + cτ )d τ − cτ d τ | cτ + d | . (A.4)Furthermore, the Killing spinors transform under the duality transformation as (cid:15) → e i qα ( γ ) , (cid:15) (A.5)where q is the charge under the U (1) D . We see that the τ dependent part of the Killingspinor equation transforms as D (cid:15) = (cid:18) ∇ − i2 Q (cid:19) (cid:15) → (cid:18) ∇ − i2 Q (cid:48) (cid:19) e i qα ( γ ) (cid:15) = e i qα ( γ ) (cid:18) i q (( ∇ α ( γ )) (cid:15) + i2 (d α ( γ )) (cid:15) + D (cid:15) (cid:19) = e i qα ( γ ) D (cid:15) (A.6)which holds with q = − .To determine how the Killing spinor equations with a general background transformunder the U (1) D , recall that the following combination of supergravity fields transform underthe duality group G = i √ τ (cid:16) τ d B − d C (2) (cid:17) , P = i τ d τ (A.7)as G → | cτ + d | cτ + d G = e − i α ( γ ) GP → c ¯ τ + dcτ + d P = e − α ( γ ) P . (A.8)68he five-form field F is of course invariant. The gravitini and diliatini Killing spinor equationsthen transform as follows δ Ψ M → e − i2 α ( γ ) δ Ψ M δλ → e − α ( γ ) δλ , (A.9)where we have used that (cid:15) c has charge − q where q is the charge of the Killing spinor (cid:15) .Let us now consider the action. Of course for Type IIB supergravity there is no actionthat imposes the correct equations of motion and the self-duality of the five-form. One mayhowever take as action S = 12 κ (cid:90) (cid:18) R − τ ∂ µ τ ∂ µ ¯ τ − | G | − | F | (cid:19) ∗ κ (cid:90) C (4) ∧ ¯ G ∧ G (A.10)to derive the equations of motion and impose the self-duality of F afterwards. From theabove transformations it follows that the action is invariant under SL R . A.2 Gauge theory couplings from Supergravity
In this section we recall how the gauge couplings in the field theory are identified in theholographic dual.We shall begin with describing how this works in the Klebanov-Witten theory [68]. Recallthat the Klebanov-Witten theory is a quiver theory with two nodes. At each of these nodesthere is an associated complexified coupling constant which we shall denote by ˜ τ i . Thecomplex coupling constant is ˜ τ i = θ i π + 4 π i g i . (A.11)From [68] we know that there are two complex moduli in the theory and these correspond tothe sum and difference of the two gauge coupling constants. The sum of the two is identifiedon the gravity side with the τ , see [69] for an early example and [70] for a later use, τ = ˜ τ + ˜ τ . (A.12)The second marginal coupling corresponds to the integration over the unique two-cycle of thetwo-form τ B − C (2) , (A.13)where C (2) and B are the RR and NS-NS two form potentials respectively. One has [71] (seealso [72]) ˜ τ − ˜ τ = 12 π (cid:20)(cid:90) S (cid:16) τ B − C (2) (cid:17) − π (cid:21) mod2 π . (A.14)69ith these expressions we can compute how the two combinations of couplings transformunder SL R . We have ˜ τ + ˜ τ = τ → a (˜ τ + ˜ τ ) + bc (˜ τ + ˜ τ ) + d (A.15)whilst ˜ τ − ˜ τ → c (˜ τ + ˜ τ ) + d (˜ τ − ˜ τ ) . (A.16)As remarked in [70] these formulae are derived for the N = 2 orbifold theory and in theliterature are assumed to hold for the conifold also. The difference between the two couplingshas the interpretation as the distance between two NS5 branes in the T-dualised theory whichis associated to the non-anomalous baryonic symmetry. B Details for the Derivations in Section 2
B.1 Torsion conditions
This appendix summarises the torsion conditions relevant for section 2. They are the sameas computed in [1], and we refer the reader there for further details.
Scalar differential equations d S ij = i m α i − α j ) K ij , (B.1)e − D (e A ij ) = − i m α i − α j ) B ij . (B.2) One-form differential equations e − d (cid:0) e K ij (cid:1) = − i m ( α i + α j ) U ij − S ij e − F (2) (B.3) D (e B ij ) = 0 (B.4) Two-form differential equations e − d(e U ij ) = − i m α i − α j ) X ij , (B.5)e − D (cid:0) e V ij (cid:1) = − m α i − α j ) Y ij + e − F (2) ∧ B ij (B.6) Three-form differential equations e − d (cid:0) e X ij (cid:1) = 2 m ( α i + α j ) ∗ X ij − e − F (2) ∧ U ij , (B.7)e − D (cid:0) e Y ij (cid:1) = m ( α i + α j ) ∗ Y ij (B.8)70 our-form differential equations e − d (cid:0) e ∗ X ij (cid:1) = − m α i − α j ) ∗ U ij (B.9)e − D (cid:0) e ∗ Y ij (cid:1) = − m α i − α j ) ∗ V ij − ie − F (2) ∧ Y ij , (B.10)e − D (cid:0) e ∗ Y ij (cid:1) = − i m α i − α j ) ∗ V ij − e − A ij ∗ F (2) (B.11) Five-form differential equations e − d (cid:0) e ∗ U ij (cid:1) = i m ( α i + α j ) ∗ K ij , (B.12)e − D (cid:0) e ∗ V ij (cid:1) = i m ( α i + α j ) ∗ B ij (B.13) Six-form differential equations e − d (cid:0) e ∗ K ij (cid:1) = i m ( α i − α j ) S ij Vol( M ) , (B.14)e − D (cid:0) e ∗ B ij (cid:1) = − m α i − α j ) A ij Vol( M ) . (B.15) B.2 Derivation of the “Master Equation”
In this appendix we provide an extensive discussion on the derivation of the “master equation”(2.41). Supersymmetry implies that a solution satisfies the Einstein equation and the Bianchiidentity for F (2) but not the equation of motion for F (2) . In this appendix we show that theequation of motion for F (2) is equivalent to (2.41). In [13] the F (2) equation of motion isshown to be equivalent to the differential equation (cid:3) R − R + R µν R µν = 0 (B.16)on the K¨ahler base. We shall find that a similar equation governs the existence of a solutionwhen τ becomes non-trivial.In the main text it was shown that the internal space is a U (1)-fibration over a warpedsix-dimensional K¨ahler base. In the following it will be necessary to reduce along the Killingdirection and to express everything in terms of the K¨ahler metric rather than the warpedone, as such it is necessary to first clarify the notation we shall be using. We denote by ∗ the Hodge dual operator on the internal space, ∗ is the Hodge dual operator on the base ofthe U (1) fibration and ˆ ∗ the Hodge dual operator on the K¨ahler metric. The Ricci tensor,Ricci scalar and Ricci-form appearing are that of the K¨ahler metric and the K¨ahler two formis denoted by J . 71upersymmetry implies that the flux satisfies m ∗ F (2) = ∗ (cid:18) − J − m e d∆ ∧ K −
12 e d ρ (cid:19) = e − m K ∧ J ∧ J − m ˆ ∗ de − − m ˆ ∗ ( R + d Q ) ∧ K . (B.17)Making use of the identities (which are easily proven)ˆ ∗ R = R J ∧ J − R ∧ J , (B.18)ˆ ∗ P ∧ P ∗ = − i | P | J ∧ J − P ∧ P ∗ ∧ J (B.19)we have m ∗ F (2) = − m ˆ ∗ d( R − | P | ) + 12 m ( R ∧ J − i P ∧ P ∗ ∧ J ) ∧ K (B.20)Imposing (2.9) is then equivalent to0 = dˆ ∗ d( R − | P | ) + 2 R ∧ R ∧ J + 4i R ∧ P ∧ P ∗ ∧ J . (B.21)Taking the Hodge dual of the above and using the identitiesˆ ∗ R ∧ R ∧ J = 14 R − R µν R µν , (B.22)ˆ ∗ R ∧ P ∧ P ∗ ∧ J = − i (cid:18) R | P | − R µν P µ P ∗ ν (cid:19) (B.23)one obtains ˆ (cid:3) ( R − | P | ) = 12 R − R µν R µν − | P | R + 4 R µν P µ P ∗ ν , (B.24)where ˆ (cid:3) = ˆ ∗ dˆ ∗ d . (B.25)Equation (2.41) determines the K¨ahler metric from which the remaining geometry may berecovered. Notice that for constant axio-dilaton one recovers the equation of [13] as expected. C Derivation of the N = (2 , Solutions
In this appendix we derive the results of section 3.2 and show that the classic AdS × S × T or K M-theory solutions. We conclude with a shortdiscussion about a more general ansatz.Recall from section 3.2 that the scalar bilinear A was non-trivial and therefore τ wasconstant. The analysis may be split into two further subcases, the first when the scalarbilinear S is non-constant and the second being the constant case. In the first case ofnon-constant S one can show that it is not possible to satisfy the torsion conditions andtherefore we restrict to the case of constant S in the following. In fact integrability of thetorsion conditions forces S = 0 and the two spinors are orthogonal.The SU (2) structure specifies a basis of three vectors which may be chosen to be thebilinears { K , K , Im[ A ∗ B ] } . (C.1)Both K and K are Killing vectors and are dual to the right and left moving U (1) R-symmetries in the field theory. This defines a foliation of the 7d metric and the metric on thespace defined by the three vectors is found to bed s ( M ) = 14 | A | (1 − | A | ) (cid:0) | A | ( K + K ) + (1 − | A | ) ( K − K ) + 4 Im[ A ∗ B ] (cid:1) . (C.2)The canonical SU (2) structure two-forms may be expressed in terms of the bilinears as U = − i (cid:18) j − | A | (1 − | A | ) ( K + (2 | A | − K ) ∧ Im[ A ∗ B ] (cid:19) , (C.3) U = − i (cid:112) − | A | ω , (C.4)Here j and ω are the two SU (2) two-forms and after putting a vielbein on M take thecanonical form j = e + e , ω = ( e + i e ) ∧ ( e + i e ) . (C.5)All other forms may be expressed in terms of these two two-forms, the three one-forms in(C.1) and A . We use this to reduce the torsion conditions to the minimal set acting on thisbasis of bilinears. The torsion conditions on the bilinears, that are not in the basis, must allbe either automatically satisfied or they impose additional algebraic relations. We find thatintegrability will constrain the warp factor and flux F (2) to take specific forms. For ease of notation we shall drop the ‘12’ subscript on A and B in this appendix. .1 Torsion Conditions From (B.2) we find that (B.4) is automatically satisfied. Moreover one finds that Im[ A ∗ B ] isconformally closed, d(e | A | ) = 2 m e Im[ A ∗ B ] , (C.6)and this allows us to introduce a coordinate for Im[ A ∗ B ]. Using the differential equations for U , U , K , K and A one finds that all the other torsion conditions are satisfied. From(B.10) and (B.11) one may find an expression for ∗ F which reads ∗ F (2) = e A (cid:16) ∧ ∗ Y + 4i m ∗ V + ie − F (2) ∧ Y (cid:17) , (C.7)and it follows that d ∗ F (2) = 0 is satisfied. Contrast this with the (0 ,
2) case here and in [13]where it is necessary to impose the “master equation” for the metric in order to satisfy (2.9).A similar expression for the flux F (2) as in the (0 ,
2) analysis is possible and will be givenlater.The torsion conditions for the SU (2) structure two-forms imply that M is complex andthat the flux F (2) and warp factor give an obstruction to M being K¨ahler,e − d(e j ) = − − | A | (cid:20) − | A | d∆ ∧ ( K + K ) + 2 mj + e − F (2) (cid:21) ∧ Im[ A ∗ B ] , (C.8)e − d(e ω ) = − − | A | ) (cid:2) | A | d∆ + i m ( K + K ) (cid:3) ∧ ω . (C.9)In light of the above equations it is natural to make the following redefinitions m e j → J , m e ω → Ω , (C.10) S = K + K , T = K − K . (C.11)The conditions that we must impose to preserve (2 ,
2) supersymmetry becomee − d(e S ) = 2 m − | A | S ∧ Im[ A ∗ B ] − − F (2) − − m J , (C.12)e − d(e T ) = − m | A | T ∧ Im[ A ∗ B ] , (C.13)d(e | A | ) = 2 m e Im[ A ∗ B ] , (C.14) The first line of the redefinitions extracts out a conformal factor from the metric on M . In the secondwe mix the two Killing vectors for later simplicity. This implies that the Killing vectors S and T are now dualto a combination of the left and right moving R-currents. J = − − | A | (cid:20) m e − | A | d∆ ∧ S + 2 mJ + m F (2) (cid:21) ∧ Im[ A ∗ B ] , (C.15)dΩ = − − | A | (cid:20) i m S + 2 | A | d∆ (cid:21) ∧ Ω . (C.16)In the next section we introduce local coordinates which leads to a simplification of theseequations. The 7d metric takes the formd s ( M ) = 14 | A | (1 − | A | ) (cid:0) | A | S + (1 − | A | ) T + 4 Im[ A ∗ B ] (cid:1) + e − m d s ( (cid:102) M ) , (C.17)where (cid:102) M is a 4d space with SU (2) structure defined by the two two-forms J and Ω . C.2 Reducing the Conditions
To proceed we introduce coordinates for each of the three vectors. We may introduce localcoordinates adapted to each of the Killing vectors S and T . Moreover we have seen thatIm[ A ∗ B ] is conformally closed and we may therefore introduce a further local coordinate forthis direction. In light of equation (C.6) we introduce the coordinate y , via the equation y e − ≡ | A | (C.18)such that e − m d y = Im[ A ∗ B ] . (C.19)It can be shown that this defines an integrable almost product structure. Explicit computation by Fierz identities (or equivalently via an orthonormal frame com-putation) gives the conditions S µ T µ = 0 , S µ S µ = 4(1 − y e − ) , T µ T µ = 4 y e − , (C.22)the first signifies that the two Killing vectors are orthogonal. Introducing local coordinates, ψ and ψ for these Killing directions we have S = 2 m ∂∂ψ , S = 2(1 − y e − ) m (d ψ + σ ) , (C.23) If one defines the unit norm form Π = 1 | A | (cid:112) − | A | Im[ A ∗ B ] (C.20)then the almost product structure defined by J νµ = Π µ Π ν − δ νµ (C.21)is integrable. This implies that the remaining metric may have y dependence however there are no d y termsappearing in the metric other than the one in Im[ A ∗ B ], this will become pertinent soon. = 2 m ∂∂ψ , T = 2 y e − m (d ψ + σ ) . (C.24)The integrable almost product structure implies that the σ i have no d y term but may other-wise depend on y non-trivially.Using the local coordinates defined above the one-form equations (C.12) and (C.13)become mF (2) = − (cid:16) (e − y )d σ + 2 J + 4e d∆ ∧ (d ψ + σ ) (cid:17) , (C.25)d σ = 0 . (C.26)Equation (C.25) will be used as the defining equation for the flux F (2) . As σ is closed andtherefore locally exact, it may, through a local change of coordinates, be set to zero. TheKilling vector T is then unfibered. The metric on M in local coordinates isd s ( M ) = 1 m (cid:18) (1 − e − y )(d ψ + σ ) + y e − d ψ + 14 y (e − y ) d y (cid:19) . (C.27)The two two-form equations (C.15) and (C.16) becomed J = 12 d σ ∧ d y , (C.28)dΩ = − (cid:18) i(d ψ + σ ) + 2 y e − − y e − d∆ (cid:19) ∧ Ω . (C.29)Having introduced three local coordinates the exterior derivative splits asd = d + d y ∧ ∂∂y + d ψ ∧ ∂∂ψ + d ψ ∧ ∂∂ψ , (C.30)and decompose the remaining torsion conditions. Let the coordinates on (cid:102) M be denoted by x i and let the metric on (cid:102) M be g (4) ij ( x, y ) where non-trivial y dependence of the metric ispermitted. The decomposition of (C.29) gives ∂ ψ Ω = − iΩ , (C.31) ∂ ψ Ω = 0 , (C.32) ∂ y Ω = − y e − − y e − ∂ y ∆Ω , (C.33)d Ω = − (cid:18) i σ + 2 y e − − y e − d ∆ (cid:19) ∧ Ω . (C.34)From the decomposition of (C.28) we find ∂ ψ J = ∂ ψ J = 0 , (C.35)76 y J = 12 d σ , (C.36)d J = 0 . . (C.37)A well known fact of complex geometry is that if the maximal holomorphic form Ω satisfiesthe differential equation dΩ = i (cid:98) P ∧ Ω for some one-form (cid:98) P then the almost complex structuredefined by Ω is integrable and the manifold is complex of real dimension 2 n , in fact it followsthat the one form (cid:98) P satisfies R = d (cid:98) P as has been used previously in the paper. Equation(C.34) shows that (cid:102) M is a complex manifold, furthermore as discussed in [33] this impliesthat the complex structure J ji is independent of ψ , ψ and y . One may solve (C.31) byextracting out a suitable ψ dependent phase from Ω . Equation (C.33) fixes the y variationof the volume of g (4) . From Ω ∧ ¯Ω = 4vol( (cid:102) M ) we find ∂∂y log √ g = − y e − − y e − ∂ y ∆ . (C.38)Finally (C.34) implies σ = − (cid:98) P + 2 y e − − y e − d c ∆ , (C.39)where d c = i( ¯ ∂ − ∂ ) with ∂ , ¯ ∂ the Dolbeault operators on (cid:102) M . From (C.37) we seethat J is closed on (cid:102) M and therefore g (4) locally defines a family of K¨ahler metrics on (cid:102) M parametrised by y .For a supersymmetric solution we must solve the two differential equations ∂ y J = 12 d σ , (C.40) ∂ y log √ g = − y e − − y e − ∂ y ∆ , (C.41)where σ and the flux F (2) are given by σ = − (cid:98) P + 2 y e − − y e − d c ∆ , (C.42) mF (2) = − (cid:16) (e − y )d σ + 2 J + 4e d∆ ∧ (d ψ + σ ) (cid:17) . (C.43)The seven-dimensional metric is m d s ( M ) = (1 − y e − )(d ψ + σ ) + y e − d ψ + e − y (1 − y e − ) d y +e − g (4) ( y, x ) ij d x i d x j (C.44) (cid:98) P should not be confused with the one-form P which depends on the axio-dilaton and is vanishing in thiscase. g (4) K¨ahler at fixed y . Following the arguments in [33], we define the self-dual andanti-self-dual combinations of an arbitrary two-form, ζ on (cid:102) M to be ζ ± = ζ ± ∗ ζ , then wehave the identity ( ∂ y J ) + = 12 ∂ y log √ gJ (C.45)which is valid when the complex structure J j i is independent of y . We may use this torewrite the equation for the volume as(d σ ) + = − y e − − y e − ∂ y ∆ J . (C.46)The necessary conditions to solve are (C.40) and (C.46) along with the definitions (C.42) and(C.43). C.3 Recovering known (2 , Solutions
In this subsection we show that the classic AdS × S × T or K
3, obtained from purelyD3-branes and the solutions discussed in [73] fit into this classification. We shall use a co-homogeneity one ansatz inspired by [33]. The distinguished coordinate in this co-homogeneityone ansatz is y . We shall take the warp factor to satisfy ∆ = ∆( y ) and moreover we impose ∂ y σ = 0. With these assumptions the Ricci-form on (cid:102) M is R = − d σ , (C.47)and therefore R = − ∂ y J , (C.48) R + = 4 y e − − y e − ∂ y ∆ J . (C.49)We see that R + is pointwise proportional to J and that the proportionality factor is inde-pendent of the coordinates on (cid:102) M . As the metric is K¨ahler and the proportionality factorconstant on (cid:102) M the Ricci scalar must also be constant on (cid:102) M , that isd R = 0 . (C.50)From the assumption ∂ y σ = 0 we see that ∂ y R ij = 0 and therefore as the complex structureis independent of y we have ∂ y R ij = 0 . (C.51)As the complex structure is y independent, equation (C.48) is equivalent to R ij = − ∂ y g (4) ij ⇒ R ij R ij = 2 ∂ y R , (C.52)78nd therefore d R ij R ij = 0 . (C.53)On a K¨ahler manifold the eigenvalues of the Ricci tensor come in pairs and therefore for a4d metric there are at most two distinct eigenvalues with degeneracy at least two. Equations(C.50) and (C.53) show that both the sum of the eigenvalues and sum of the squares of theeigenvalues are constant over the base. This implies that the two pairs of eigenvalues of theRicci tensor are constant on (cid:102) M . Using [74], which assumes that the Goldberg conjecture istrue, we find that the K¨ahler surface is the sum of two complex curves each with constantcurvature. In the case where the two pairs of eigenvalues are the same, by definition themanifold is K¨ahler–Einstein. There are then two natural classes to consider, either (cid:102) M isK¨ahler–Einstein or it is the product of two Riemann surfaces. C.4 Case 1: K¨ahler–Einstein
When (cid:102) M is K¨ahler–Einstein, the Ricci-form is R = κF ( y ) J , (C.54)where κ ∈ { , ± } and F ( y ) >
0, the last inequality follows from requiring the metric to bepositive definite. For κ = 0 (cid:102) M is Ricci-flat and K¨ahler and therefore is Calabi–Yau. Thewarp factor in this case is found to be constant and we recover the classic AdS × S × T ( K κ (cid:54) = 0 in the remainder of this section. From (C.48) F ( y ) = c − κ y (C.55)where c is an integration constant. Note that in [33] the analogous function was found to bequadratic rather than linear. We may then solve (C.49) for the warp factor to obtaine − = c + 2 y − c κ log y ( yκ − c ) . (C.56)We must fix the range of y . To do so we look at the values of y at which the metric degenerates.For smoothness y must be prevented from attaining the value 0 due to the logarithmic termand therefore we require that 1 − y e − has two roots. We were unable to find two such roots.One may be tempted to set the constant c = 0, this implies κ = −
1. One then finds thatthere is only one solution for the root of 1 − y e − . Note that as the warp factor is singular at y = 0 we must avoid this value of y here and it follows that the metric will be non-compact.Instead let us consider the case of a product of Riemann surfaces.79 .5 Case 2: Product of Riemann Surfaces The second case to consider is when for fixed y , (cid:102) M = Σ × Σ is a product of two Riemannsurfaces of constant curvature, i.e.d s ( (cid:102) M ) = d s (Σ ) + d s (Σ ) . (C.57)Denoting K¨ahler forms for each curve as J i the Ricci-forms are given by R = κ G ( y ) J , R = κ G ( y ) J , (C.58)where as before κ i ∈ { , ± } and G i ( y ) > y within its domain. On (cid:102) M TheK¨ahler form and Ricci-form factor as J = J + J , R = R + R . (C.59)We may solve for the functions G i ( y ) by using equation (C.48) G i ( y ) = c i − κ i y . (C.60)Using (C.49) the warp factor is found to bee − = 2 κ κ y + k − κ c + κ c ) log y (2 c − κ y )(2 c − κ y ) , (C.61)where k is an integration factor. We analyse the regularity of these solutions in the remainderof this section.Σ = T First consider the case where one of the Riemann surfaces is a T , without loss of generalitylet us set κ = 0. Observe that for κ = 0 we return to the Calabi–Yau case which wasdiscussed previously, therefore restrict to κ (cid:54) = 1. The warp factor becomese − = k − κ c log y c (2 c − κ y ) (C.62)As y appears in the metric we require that it is strictly positive. We wish to find the range of y such that the metric is both compact and smooth. For both κ = ± × H The metric in this case is m d s ( M ) =4(1 − y e − )(d ψ + σ ) + 4 y e − d ψ + e − y (1 − y e − ) d y + e − (cid:0) G ( y )d s (Σ ) + G ( y )d s (Σ ) (cid:1) , (C.63) σ = −
12 ( ˆ P + ˆ P ) (C.64)where (cid:98) P i satisfy d (cid:98) P i = R i . Due to the Logarithmic term appearing in the warp factor (C.61)we shall choose the constants such that the term c κ + c κ vanishes. Clearly for this casewe must take c = c = c and we have G ( y ) = c − y , G ( y ) = c + y , (C.65)e − = k − y (2 c − y )(2 c + y ) = k − y G ( y ) G ( y ) . (C.66)There are two constants however we may remove one of these constants by a rescaling of the y coordinate. Observe that under the rescaling y → qy we have G i ( y ) → q (cid:18) cq ± y (cid:19) = q ˜ G i ( y ) , (C.67)e − → kq − y q ˜ G ( y ) ˜ G ( y ) = e − , (C.68)and the full metric is seen to be rescaled by an irrelevant constant factor. We may then usethis coordinate transformation to set without loss of generality k = 4.For a smooth compact metric we require that both e − and G i ( y ) are strictly positivefor all values of y . Moreover as is clear from the explicit metric in (3.8) we require y ≥ c > ≤ y ≤ min { , c } . Computing the scalar invariants of the metric one finds that the Ricciscalar vanishes as expected from the equation of motion, whilst the contraction of the Riccitensor into itself has poles when G i ( y ) = 0 or y = 2. This latter singularity may be rephrasedin terms of the divergence of the warp factor at these points. We must therefore require thatthe warp factor does not vanish or degenerate for all y . This implies that the manifold istopologically the direct product of AdS with M and regularity of the solution is equivalentto regularity of M .The y coordinate is fixed by finding points where the metric degenerates and making achoice of coordinate period such that it is smooth. The metric on M degenerates at y = 081nd a real positive root of 1 − y e − . As y is strictly positive G ( y ) is strictly positive for allvalues of positive y . On the other hand G (2 c ) = 0 and unless c = 1 the metric is singular.We are then restricted to have 0 ≤ y ≤ y ∗ where y ∗ is a positive root of 1 − y e − which issmaller than 2. The roots are y ± = 2(1 ± (cid:112) − c ) (C.69)and therefore to allow real roots we fix 0 ≤ c ≤ y to be0 ≤ y ≤ y = 2(1 − (cid:112) − c ) . (C.70)We first analyse the metric at the two endpoints for 0 ≤ c <
1, returning to the special caseof c = 1 after. Expanding the metric around y = 0 we find m d s = 1 c (cid:18) y d ψ + d y y (cid:19) + 4(d ψ + σ ) + 1 c (d s ( S ) + d s ( H )) . (C.71)The bracketed term is R if ψ has period π . The remaining part of the metric is a U (1)fibration over S × H and is regular. We choose to redefine the coordinate ψ so that it hasthe canonical 2 π period. Performing the expansion around y ∗ the ψ coordinate is fixed tohave period π , and again we perform a redefintion of the coordinate to give the canonical 2 π period. The metric is then smooth for all 0 ≤ c < S × S × H .We return to the special case of c = 1. The analysis around y = 0 is identical and so weshall not repeat it here. Around y = y ∗ = 2 the metric becomes m d s = d ˜ ψ + d s ( H ) + 2 − y (cid:0) (d ψ + cos θ d φ ) + d θ + sin θ d φ (cid:1) + 14(2 − y ) d y , (C.72)which after a coordinate transformation can be seen to be S × H × R if ψ has period 4 π .Again the metric is smooth for c = 1 but is topologically different to the 0 ≤ c < ψ = χ + z , ψ = χ − z (C.73)which puts the metric into the form m d s ( M ) = (cid:0) d χ + 2(1 − y e − ) σ + (1 − y e − )d z (cid:1) + e − d s ( (cid:102) M ) (C.74)d s ( (cid:102) M ) = d y y (1 − y e − ) + 4 y (1 − y e − )(d z + σ ) + G ( y )d s ( S ) + G d s ( H ) This value of c will be studied separately later (cid:102) M is K¨ahler and satisfies the defining equation (cid:3) R = 12 R − R µν R µν . (C.75)We have checked that all the remaining conditions in [13] are satisfied and the flux given in(C.43) can be repackaged into the form given in [13].We observe that for c = 1 this is the solution explicitly presented in [73], (11)-(12), aftera change of coordinates. The form of the solution presented in [73] isd s = (cid:18) d z − cos θ θ (d ϕ + cos θ d φ − x d x ) (cid:19) + e − (cid:32) θ )d θ + 2 sin θ cos θ θ (cid:18) d ϕ + cos θ d φ − x d x (cid:19) + cos θ (d θ + sin θ d φ ) + (1 + sin θ ) (cid:18) d x + d x x (cid:19)(cid:19) , (C.76)e − = 11 + sin θ . (C.77)Performing the change of coordinates y = 2 sin θ , z = − ψ , ϕ = − ( ψ + ψ ) (C.78)puts the metric (C.76) in the form (C.63). In later pages of [73] a solution in differentdimensions is presented with this additional parameter included. S × S Again the strategy to obtain a smooth solution is to eliminate the logarithmic term appearingin (C.61). We impose that c = − c . By definition the coordinate y satisfies y ≥ G ( y ) and G ( y ) to be simultaneously positive. Thereforeif we tune the parameters such that the logarithmic term vanishes then no solution exists. H × H Eliminating the logarithmic term appearing in (C.61) we impose c = − c ≡ c . Due to thesymmetry in c we may take c ≥ S × S caseboth G ( y ) and G ( y ) can be made positive if y ≥ c . However if we compute the roots of1 − y e − we find that they are both negative and therefore there is no way to make the spaceclose and have a positive definite metric. We conclude that there are no smooth solutions ofthis form. We correct some terms that are missing from both (11) and (12) in [73]. .6 Further Generalisations In the preceding sections we have recovered known (2 ,
2) solutions in the literature. A naturalansatz to use to find new solutions is that presented in [75]. The ansatz imposes a U (1) isom-etry for the base (cid:102) M and is the most general complex metric in 4d with a U (1) isometry. The7d internal metric will admit three commuting U (1) Killing vectors. A natural interpretationof this ansatz is that this is the near horizon of D R , × CY where the CY is non-compact and is decomposed as the sum of three line bundles over aRiemann surface. D AdS to AdS In this appendix we provide some of the computational derivations for section 3.3. We lookat the AdS solutions with N = (2 ,
2) and varying τ by relaxing the compactness conditionof the internal space. We find the only solutions of this problem decompactify to an AdS solution. In fact the resulting AdS varying τ solutions of IIB supergravity are the mostgeneral of this kind, which we show in section D.2. In [23] AdS solutions with five-form fluxand varying axio-dilaton were considered. We recover the analysis presented there and givean F-theoretic interpretation in terms of an elliptically fibered Calabi–Yau four-fold. D.1 AdS Solutions with (2 , and Varying τ D.1.1 Torsion Conditions
The starting point for this analysis is (3.5) where for P to be non-zero and thus τ varying,we need A = 0. It is easy to see that by setting S to be constant it must in fact vanish.Moreover it is trivial to see that it is impossible to satisfy the torsion conditions if both ofthese scalars simultaneously vanish. We shall therefore restrict to the case when S is non-constant in the remainder of this subsection. As before we find that both S and S areconstant and therefore we may normalise the spinors such that they are both unity.Recall that M admits an SU (2) structure which implies there is a 3 + 4 splitting, suchthat the ’3’ part, (cid:102) M has a vielbein given by the three vectors of the SU(2) structure. Onemay take as a basis for the three independent vectors { K , K , Im[ S ∗ K ] } (D.1)in terms of which we may write the metric on M asd s ( M ) = 14 | S | (1 − | S | ) (cid:0) K + K + 2(1 − | S | ) K ⊗ K + 4 Im[ S ∗ K ] (cid:1) . (D.2)84he canonical SU (2) structure two-forms are written in terms of the bilinears as j = i U − | S | (1 − | S | ) (cid:0) K + (1 − | S | K (cid:1) ∧ Im[ S ∗ K ] , (D.3) ω = 1(1 − | S | ) V ∗ . (D.4)We may construct a basis of independent bilinears consisting of the scalar S , the three one-forms in (D.1) and the two canonical SU (2) two-forms in (D.3) and (D.4). All other bilinearsmay be obtained from wedge products of these bilinears. The torsion conditions of the non-basis elements should then be either automatically satisfied by imposing the equations for thebasis forms or impose additional algebraic constraints.Integrability of the torsion conditions (3.6) imply the warp factor satisfies∆ = −
12 log[1 − | S | ] . (D.5)We may use it as a coordinate for Im[ S ∗ K ]. Moreover integrability of the torsion conditionsimplies that the flux F (2) is fixed to be F (2) = − | S | d∆ ∧ ( K + K ) , (D.6)which is easily shown to be both closed and co-closed and therefore F (2) satisfies both itsBianchi identity and its equation of motion. The torsion conditions for K and K implyfor the K¨ahler form on M that j = − e − m d (cid:0) e ( K − K ) (cid:1) , (D.7)which is conformally closed. In light of this we define the rescaled real and complex two forms J = m e j , Ω = m e V ∗ , (D.8)for the resulting four-fold as (cid:102) M , which satisfyd J = 0 , ¯ D Ω = − m ( K − K ) ∧ Ω . (D.9)From (2.12) we see that P is holomorphic with respect to the induced complex structuredefined by J . The metric after this redefinition takes the formd s ( M ) = e K − K ) + 14(1 − e − ) ( K + K ) + e − m (1 − e − ) d∆ + e − m d s ( (cid:102) M )(D.10)85here d s ( (cid:102) M ) is K¨ahler. As K and K are Killing vectors so are the linear combinations K = K − K , L = K + K , (D.11)and they satisfy the algebraic conditions || K || = 4e − , || L || = 4(1 − e − ) , K µ L µ = 0 (D.12)and the differential equationsd(e K ) = − m J , d (cid:18) − e − L (cid:19) = 0 . (D.13) D.1.2 Decompactification to AdS We may introduce local coordinates adapted to these two Killing directions as K = m ∂∂ψ , L = m ∂∂ϕ , (D.14)with dual one-forms K = 4 m e − (cid:18) d ψ + 12 ρ (cid:19) , L = 4 m (1 − e − )(d ϕ + σ ) . (D.15)The one-forms ρ and σ are both independent of ψ and ϕ . From (D.13) we see that σ is closedand therefore locally exact and may be set to zero by a local change of coordinates. Themetric takes the form m d s ( M ) = e − − e − d∆ +4(1 − e − )d ϕ +e − (cid:32) (cid:18) d ψ + 12 ρ (cid:19) + d s ( (cid:102) M ) (cid:33) . (D.16)These explicit coordinates induce a splitting of the exterior derivative asd → d ϕ ∂∂ϕ + d∆ ∂∂ ∆ + d ψ ∂∂ψ + d . (D.17)With this splitting equation (D.9) decomposes as ∂ ϕ Ω = ∂ ∆ Ω = 0 , (D.18) ∂ ψ Ω = − , (D.19)¯ D Ω = − ρ ∧ Ω . (D.20)Equation (D.19) may be solved by extracting a phase from Ω. Equation (D.20) implies thatthe Ricci form on (cid:102) M is R = 6 J − d Q . (D.21)86ombining these terms, the full 10 d metric isd s =e (cid:18) d s (AdS ) + e − m (1 − e − ) d∆ + 4(1 − e − ) m d ϕ (cid:19) + 1 m (cid:104) (2d ψ + ρ ) + d s ( (cid:102) M ) (cid:105) =d s (AdS ) + 1 m (cid:104) (2d ψ + ρ ) + d s ( (cid:102) M ) (cid:105) . (D.22)The first term in the brackets with the warp factor included is in fact the metric on AdS with Ricci-tensor satisfying R µν = − m g µν . D.2 Generality of the AdS Solution with varying τ In this section we show briefly how the AdS solution obtained in the last section, are in factthe most general solutions with varying axio-dilaton, and dual N = 1 supersymmetry. Weperform the general analysis of these solutions directly from an AdS ansatz. These AdS solutions with varying τ were originally studied in [23], though the F-theoretic Calabi–Yaufour-fold interpretation given in this paper was not noticed there. A more in depth analysisof the solutions and their holographic duals will be relegated to [29]. D.2.1 Conditions for Supersymmetry
We make an ansatz for an AdS -solutiond s = e (cid:0) d s (AdS ) + d s ( M τ ) (cid:1) F = f (vol(AdS ) + vol( M τ )) , (D.23)with ∆ ∈ Ω ( M τ , R ), τ ∈ Ω ( M τ , C ) and f a constant. The metric on AdS is such that theRicci tensor satisfies R µν = − m g µν and d s ( M τ ) is an arbitrary five-dimensional metric onthe internal space M τ . This allows us to use the supersymmetry equations in d = 5 directlyfrom [30] by setting G = 0 and we obtain0 = D m ξ + i4 ( f e − − m ) γ m ξ , (D.24)0 = ¯ D m ξ − i4 ( f e − + 2 m ) γ m ξ , (D.25)0 = γ m ∂ m ∆ ξ − i4 ( f e − − m ) ξ , (D.26)0 = γ m ∂ m ∆ ξ + i4 ( f e − + 4 m ) ξ , (D.27)0 = P m γ m ξ , (D.28)0 = P ∗ m γ m ξ . (D.29)87he first implication from (D.26) and (D.27) is that ∂ m ∆ = 0. Substituting this back into(D.26) and (D.27) it is necessary to set one of ξ or ξ to be zero and to set f = ± m e depending on which Killing spinor we keep. Without loss of generality we set ξ = 0 and set∆ = 0 so that f = 4 m . The SUSY equations reduce to0 = D m ξ + i m γ m ξ , (D.30)0 = P ∗ m γ m ξ . (D.31)Computing the integrability conditions for the Killing spinor equation implies that both theEinstein equation and P equation of motion are satisfied. D.2.2 Torsion Conditions
To compute the necessary and sufficient conditions for the existence of bosonic supersym-metric solutions in this class we shall use again G -structure techniques. For a single Killingspinor in five dimensions the solution must admit an SU (2) structure, which in 5d is specifiedby the existence of a real one-form, which defines a foliation of the space with a transverse 4dspace admitting an SU (2)-structure. The latter consists of a real two-form of maximal rankand a holomorphic two-from. First define the spinor bilinears A = ¯ ξ ξ , K = ¯ ξ γ (1) ξ , j = i ¯ ξ γ (2) ξ , ω = ¯ ξ c γ (2) ξ . (D.32)From Fierz identities it follows that the one-form satisfies K (cid:121) j = 0 , K (cid:121) ω = 0 , (D.33)and therefore the space transverse to the one-from admits indeed an SU (2)-structure. From(D.30) and (D.31) follow the torsion conditionsd A = 0 , d K = 2 m j , d j = 0 , Dω = − m ∗ ω = − m K ∧ ω , (D.34)and the algebraic conditions J µν P ν = i P µ , i K P = 0 . (D.35)Equation (D.35) implies that the complex field P is holomorphic with respect to the complexstructure. From the torsion conditions it follows that the vector K is not only a Killingvector but a symmetry of the full solution, corresponding in the dual field theory to the U (1)88-symmetry. We shall use this as customary to be the Reeb vector field, or simply Reeb, inthe following. To proceed we introduce coordinates adapted to the Killing direction. Define K = m ∂∂ψ , K = 1 m (d ψ + σ ) , (D.36)where the factor of m has been introduced for later convenience and we have used the factthat the Reeb has unit norm. The one-from σ is independent of ψ . It is also convenient atthis point to extract out a dimensionful parameter from the metric on the transverse space,and we extract out a factor of m . The metric on M τ is m d s ( M τ ) = (d ψ + σ ) + d s ( (cid:102) M ) , (D.37)where (cid:102) M is a K¨ahler surface, as follows from (D.34), and we shall refer to it as the ‘transverse’space. Introducing a vielbein on the transverse space yields j = 1 m J = 1 m ( e + e ) , (D.38) ω = 1 m ¯Ω = 1 m ( e − i e ) ∧ ( e − i e ) . (D.39)Here J and Ω are the canonical K¨ahler-form and holomorphic two-form respectively. Havingintroduced a coordinate along the Killing direction we may perform a splitting of the exteriorderivative as d = d ψ ∧ ∂∂ψ + d . (D.40)With this decomposition (D.34) becomesd σ = 2 J , d J = 0 , (D.41)and ∂ ψ Ω = 3iΩ , d Ω = i(3 σ − Q ) ∧ Ω . (D.42)The first equation is easily solved by extracting out a ψ dependent phase, which we shallimplicitly do in the following and by an abuse of notation keep the notation Ω. The secondequation determines the Ricci form on (cid:102) M to be R = 6 J − d Q . (D.43)To find a solution we should solve this final equation (D.43). Notice that for constant τ thisreduces to the case of Sasaki–Einstein. This may be written as a fourth order equation forthe K¨ahler potential of M , however there is a nice interpretation of this geometry arisingfrom the base of an elliptically fibered Calabi-Yau four-fold as is explained in section 3.3.This classifies all possible N = 1 AdS solutions with G = 0 and varying τ .89 Details for the Baryonic Twist Solution
In this appendix we provide some more details on the baryonic twist solution of section 4.2.2.The solution to (4.4) that we shall use was found in [15] for general values of s , here we areinterested in the s = 2 case . This was later discussed in [14], where it was interpreted asa Type IIB solution of the form AdS × T × M τ , with the regularity analysis performedtherein. As we show here, the same solution to (4.4) yields an F-theory geometry of theform AdS × K × M τ , with the same manifold (in particular the same metric) M τ . Afterreviewing the derivation of the local form of the solution, for completeness, we shall performa similar analysis of the regularity and global properties, with some minor changes from [14].The starting point is a cohomegenity one ansatz for the K¨ahler metric on M ,d s ( M ) = d r U ( r ) + U ( r ) r (cid:18) d ϕ + 12 cos θ d χ (cid:19) + r θ + sin θ d χ ) , (E.1)for which, after changing variable to x = 1 /r , one can find the explicit solution to (2.41) as U ( x ) = 1 − a ( x − (E.2)depending on one integration constant a . This is reviewed below. E.1 Polynomial Solution to the “Master Equation”
Let us denote C = cos θ d χ and KE = S the round two-sphere. The associated K¨ahlerform reads J = − (cid:0) r d r ∧ (d ϕ + C ) + r J KE (cid:1) (E.3)where J KE is the K¨ahler form on KE . Thend J = r d r ∧ d C − r d r ∧ J KE (E.4)and d J = 0 implies d C = 2 J KE . (E.5)The holomorphic 2-form is given byΩ = e ϕ r (cid:32) (cid:112) U ( r ) d r + i (cid:112) U ( x ) r (d ϕ + C ) (cid:33) ∧ Ω KE , (E.6) In the notation of [15] s = n + 1. The authors of [15] were mainly interested in the cases s = 3 and s = 4. KE is the holomorphic one-form on S and satisfiesd Ω KE = 2i C ∧ Ω KE . (E.7)Then dΩ = i (cid:18) − U ( r )) − r U ( r )d r (cid:19) (d ϕ + C ) ∧ Ω , (E.8)and defining f ( r ) ≡ − U ( r )) − r U ( r )d r (E.9)we have that the Ricci form is R = d( f (d ϕ + C ))= d f d r d r ∧ (d ϕ + C ) + 2 f J KE . (E.10)The Ricci scalar is R = 4 fr + 2 r d f d r . (E.11)It is convenient to make the change of coordinates x = 1 /r , under which the metric becomesd s = 1 x (cid:18) d x x U + U (d ϕ + C ) + d s ( KE ) (cid:19) , (E.12)and we have f ( x ) = 2(1 − U ( x )) + x d U ( x )d x , (E.13) R = 4 f x − x d f d x , (E.14) R = d f d x d x ∧ (d ϕ + C ) + 2 f J KE . (E.15)Using (cid:3) R = 2 x ∂ x (2 U ∂ x ) R , (E.16)(4.4) reduces to the following ODEdd x (cid:18) U d R d x + 2 f (cid:19) = 0 , (E.17)which is immediately integrated once to U d R d x + 2 f = 8 c , (E.18)91here c is an arbitrary constant. Inserting (E.13) and (E.14) we obtain the third ordernon-linear equation − U ( x ) + 8 xU (cid:48) ( x ) + x (2 U (cid:48) ( x ) − U ( x ) U (cid:48)(cid:48) ( x )) − x U ( x ) U (cid:48)(cid:48)(cid:48) ( x ) = 8 c , (E.19)where a is an arbitrary constant. This may be rewritten asdd x g ( x ) − x g ( x ) = 8 c , (E.20)where g ( x ) ≡ U ( x ) x + x (2 U (cid:48) ( x ) − U ( x ) U (cid:48)(cid:48) ( x )) . (E.21)It is simple to find the most general solution to (E.20) g ( x ) = x c − c x , (E.22)where c is another integration constant. Thus finally we obtain a non-linear second orderequation for U ( x ) that reads8 U ( x ) + x (2 U (cid:48) ( x ) − U ( x ) U (cid:48)(cid:48) ( x )) − x c + 8 c = 0 . (E.23)The most general solution to this equation remains unknown, but we one may find two simplesolutions. Setting c = 1, and only for c = 1, we find the solution U ( x ) = 4 √ c √ √ x − , (E.24)however this solution is problematic for furnishing a smooth metric as it possesses only oneroot. Instead we find the quadratic solution U ( x ) = − c + c x + c c − x , (E.25)which has in principle two arbitrary parameters. By making the redefinition c = a − x → ac x we see that in fact only one of the parameters is physical andthat U ( x ) takes the form U ( x ) = 1 − a ( x − . (E.26) E.2 The local F-theory geometry
The full geometry can then be reconstructed as follows. The Ricci form of the K¨ahler base (cid:102) M = P × M is R = − (d ρ + d Q ) = − d ρ + R B = R + R B , (E.27)92ith R = d (cid:18) f ( x ) (cid:18) d ϕ + 12 cos θ d χ (cid:19)(cid:19) f ( x ) = 2(1 − U ( x )) + x d U ( x )d x = 2 a (1 − x ) . (E.28)Thus we take ρ = − f ( x ) (cid:18) d ϕ + 12 cos θ d χ (cid:19) . (E.29)Furthermore the Ricci scalar of (cid:102) M = P × M is R = 2 | P | + 8e − = R B + R . (E.30)From which we may identify the warp factor ase − = R (cid:18) xf − x d f d x (cid:19) = ax . (E.31)The 5d part of the metric takes the form m d s ( M ) = 14 (d ψ − a ( x − Dφ ) + a (cid:20) d x x U + U Dϕ + 14 (cid:0) d θ + sin θ d χ ) (cid:1) + x d s ( B ) (cid:21) , (E.32)where Dϕ = d ϕ + cos θ d χ . Using (2.58) we can read off the expression for the flux mF (2) = − ax d ψ ∧ d x − B ) −
12 dvol( S ) . (E.33)The regularity analysis performed in the next subsection shows that the base of this local U (1) fibration is itself not a manifold , instead a change of coordinates is useful to describethe global geometry and results in the solution in the form presented in (4.43) and (4.44).The profile of the axio-dilaton is determined (implicitly) by the condition that the metricon Y is a Ricci-flat metric, and thus Y is an elliptically fibered K
3, with base B = P .Note that we will not determine explicitly the metric on P and in particular this cannotbe the Einstein metric, but the stringy-cosmic string metric of [76], induced by the ellipticfibration. In particular, the metric will have singularities at the discriminant loci. We thuscontinue to distinguish the two two-spheres in the geometry by referring to them as S and P , respectively.At this stage the background depends on two arbitrary constants m , a and we nowdetermine which values of these allow for a globally defined solution. A similar situation occurs with the Sasaki–Einstein Y p,q manifolds. The K¨ahler base of Y p,q in thecanonical Sasaki–Einstein coordinates is not in general a manifold. .3 Regularity We first consider regularity of the metric and later address the quantisation of the flux (similardiscussions have appeared in [14, 17]). The metric on M isd s ( M ) = 14 (cid:18) d ψ − a (1 − x ) (cid:18) d ϕ + 12 cos θ d χ (cid:19)(cid:19) (E.34)+ a (cid:32) d x x U + U (cid:18) d ϕ + 12 cos θ d χ (cid:19) + 14 (d θ + sin θ d χ ) + x d s ( B ) (cid:33) . We require that the warp factor does not vanish and therefore the range of the coordinate x cannot include x = 0. This implies that the 7d geometry is topologically M τ × P , with M τ the five-dimensional space defined by x = constant, and therefore we need only analyse theregularity of M τ , subject to x avoiding x = 0. The range of x is fixed to lie between the tworoots of U ( x ) x ± = 1 ± √ a . (E.35)Clearly to avoid x = 0 it is necessary to have x − >
0, so that a >
1, and it follows that U ( x )is positive between the two roots for all values a > The Base Z Let us first consider the four-dimensional part of the metric, namely the K¨ahler base M . Theround S appearing in M has coordinates θ and χ with the canonical coordinate periodicities θ ∈ [0 , π ] and χ ∈ [0 , π ]. Near to the zeroes of U at x = x ± , the degenerating part of themetric is 1 x ± U (cid:48) ( x ± ) (cid:18) d ρ + ( x ± U (cid:48) ( x ± )) ρ d ϕ (cid:19) , (E.36)where ρ = 2 √ x − x ± , respectively. For this to be locally R at both end-points it is necessarythat x ± U (cid:48) ( x ± ) has the same value at both roots; a trivial calculation shows this is not thecase and therefore there is no choice of periodicity of ϕ that gives a smooth metric.As in [17], the way to proceed is to show that one can still view the five-dimensional spaceas a circle fibration over a base Z , albeit one with metric different from the local K¨ahlermetric on M . Changing coordinates from ( ψ, ϕ ) to ( α, φ ) by α = ψ , φ = 2 ϕ + ψ , the 5dmetric takes the formd s ( M ) = w ( x )4 (d α + g ( x )(d φ + cos θ d χ )) + a (cid:20) d x x U + Uw (d φ + cos θ d χ ) + d θ + sin θ d χ (cid:21) . (E.37)94ith this change of coordinates we may avoid potential conical singularities at the endpointsof x if φ has period 2 π due to the remarkable fact that( U (cid:48) ( x ± ) x ± ) w ( x ± ) = 1 . (E.38)As in [17] we may introduce a new angular coordinate defined bycos ζ = − a ( x − √ w , sin ζ = √ aU √ w , (E.39)with ζ ∈ [0 , π ]. However, performing this change of coordinates is not particularly useful andso we shall keep the x coordinates in the following.At fixed x between the two roots the base Z with metric given in the second line of(E.37), is a circle bundle over the round two-sphere, where the U (1) fiber coordinate is φ .The Chern number of this bundle is obtained by computing the integral of the curvaturetwo-form of the connection on the U (1) and gives12 π (cid:90) S d( − cos θ d χ ) = 2 . (E.40)This identifies the three-dimensional space at fixed x to be S / Z . Furthermore it followsthat the four-dimensional base Z has topology S × S . For the following it is useful to havean explicit basis for the homology group H ( Z ; Z ) = Z ⊕ Z . The two natural two-cycles arethe two S ’s, whose cycles we denote by C , C in keeping with the notation in [17]. Since themetric on Z is not a product metric the location of the two S ’s is not clear, however we maytake C to be the fiber S at fixed θ, χ on the round two-sphere. There are two two-cycleswhich are visible in the geometry; namely the two S ’s at the south and north poles of thefiber S (at ζ = 0 , π respectively or equivalently x = x − , x + ), let us call them S and S .Then the two-cycles C i are 2 C = S − S , C = S + S , (E.41)with dual cohomology elements ω = − π (sin ζ d ζ ∧ (d φ + cos θ d χ ) − cos ζ sin θ d θ ∧ d χ ) ,ω = 14 π sin θ d θ ∧ d χ , (E.42)satisfying (cid:90) C i ω j = δ ij . (E.43)95s we wish to be precise in comparing the geometry here with that of the Y p,q manifolds, wewill perform some additional checks on the base Z .The Euler characteristic of a four-manifold M may be computed by using the Chern-Gauss-Bonnet theorem χ ( M ) = 132 π (cid:90) M √ g (cid:32) | W | − (cid:12)(cid:12)(cid:12)(cid:12) Ric − R g (cid:12)(cid:12)(cid:12)(cid:12) + 16 R (cid:33) , (E.44)where norms are computed using the metric, W denotes the Weyl tensor and Ric the Riccitensor. Computing this for our metric we find χ ( Z ) = 4 . (E.45)We may compute the signature using the Hirzebruch signature theorem σ ( M ) = 148 π (cid:90) M √ g ( | W + | − | W − | ) (E.46)and indeed we find σ ( Z ) = 0 . (E.47)Let us also check that Z is a complex manifold. To do so we compute the exteriorderivative of the associated (0 ,
2) two-form. As is well-known the exterior derivative of theholomorphic n -form on a complex manifold of complex dimension n satisfiesdΩ = i (cid:98) P ∧ Ω , (E.48)where (cid:98) P is a one form potential for the Ricci-form R , that is, d (cid:98) P = R . For the metric on Z we have Ω = ax (cid:32) x √ U d x + i √ U √ w (d φ + cos θ d χ ) (cid:33) ∧ (d θ + i sin θ d χ ) (E.49)and upon taking the exterior derivative one finds that the manifold is complex, with theone-form Ricci potential given by (cid:98) P = (cid:18) a (2(2 x − − a ( x − x ( x − w / (cid:19) (d φ + cos θ d χ ) . (E.50) In general, for the product of two Riemann surfaces Σ × Σ of genus g , g , respectively, we have χ (Σ × Σ ) = 4(1 − g )(1 − g ). The signature of the product of two Riemann surfaces Σ × Σ vanishes by Rohlin’s theorem, as this isthe boundary of its handlebody. R = d (cid:98) P is proportional to the first Chern class of the tangentbundle of the manifold, and may be integrated over the two two-cycles discussed above. Wefind 12 π (cid:90) S R = 0 , π (cid:90) S R = 4 , (E.51)which implies c ( C ) = c ( C ) = 2 . (E.52)This discussion establishes that in fact Z is complex-diffeomorphic to the Hirzebruch surface F = S × S , exactly as for the 4d base that appeared in the Y p,q construction in [17] . The Circle Fibration
We now turn to the circle fibration. The norm of the Killing vector ∂/∂α is w ( x ) / U ( x ). In order to get a compactfive-dimensional manifold we need the coordinate α to describe an S bundle over Z . Wethen take it to have period 0 ≤ α ≤ π(cid:96) , (E.53)where (cid:96) parametrises the arbitrariness of the period of α . We may then rescale α by (cid:96) − which implies that (cid:96) − A = (cid:96) − g (d φ + cos θ d χ ) (E.54)should be a connection on a U (1) bundle over Z (cid:39) S × S . In general such U (1) bundlesare completely specified topologically by the gluing on the equators of the two S cycles C and C . These are measured by the corresponding Chern numbers in H ( S ; Z ) = Z whichwe label p and q . These are given by the integrals of the U (1)-curvature two-form d A/ π overthe two two-cycles which form the basis of H ( Z ; Z ) = Z ⊕ Z . We may choose (cid:96) such that p and q are coprime, ( p , q ) = 1. We first check that d A is a globally defined two-form. At fixed x between the two roots x − , x + we see that d A is proportional to the “global angular form”on the U (1) bundle with fibre φ and is a globally well-defined one-form, therefore so is d A Recall that for a Hirzebruch surface F n , in a basis of H ( F n ; Z ) with intersection matrix (cid:32) − n
11 0 (cid:33) we have the following invariants χ ( F n ) = 4 , σ ( F n ) = 0 , c ( C ) = − n + 2 , c ( C ) = 2. We have checked thatthese invariants identify the base manifold B in [17] as B (cid:39) F . We have also checked that computingexplicitly these for the metric on F found in [41], gives correctly χ ( F ) = 4 , σ ( F ) = 0 , c ( C ) = 1 , c ( C ) = 2,where C = H − E, C = E .
97n a fixed x slice of Z . We must also check how the curvature two-form behaves near to thezeroes of U . We find that the only piece that may be troublesome is the term proportional tod x ∧ d α near the poles, however the true radial coordinate is r = ( x − x i ) / and so this termis proportional to the volume form near the fibre poles and thus is well-defined. Consequentlyd A is a globally well-defined smooth two-form on Z .Let us now calculate the periods P i ≡ π (cid:90) C i d A . (E.55)The corresponding integrals of (cid:96) − d A/ π give the Chern numbers p , q , so that we have P = (cid:96) p and P = (cid:96) q . These are most easily found by first computing the integrals of d A over the twocycles S i , namely 12 π (cid:90) S i d A = 2 g ( x i ) , (E.56)from which we find P = 2 √ a − a , P = 2 a − a ⇒ P P = 1 √ a = pq (E.57)which implies that a = q p , (cid:96) = 2 qq − p , x ± = 1 ± pq . (E.58)Recall that the regularity of the metric required that a >
1, which implies that the integers p , q obey 0 < p < q , (E.59)for which there is clearly an infinite number of solutions. We have deliberately used a notationas close as possible to [17], and found that topologically the base Z and the circle fibrationare formally identical. More precisely, Z here and the base B in [17] are diffeomorphic ascomplex manifolds, and the two corresponding circle bundles are characterised by a pair ofcoprime integers. However, the regularity of the metric here, implies that the Chern numbers p , q characterising the fibration obey an inequality that is opposite to those obeyed by theintegers p , q in the Y p,q Sasaki–Einstein manifolds, which was p > q >
0! We denote thecorresponding five-dimensional manifolds as M = Y p , q .To summarise, the geometry of the full Type IIB solution isAdS × P × Y p , q , Y p , q = S → F , (E.60)98here Y p , q is a circle fibration over F = S × S . Of course the K¨ahler metric on this F isnot the Einstein, direct-product metric on S × S .As already mentioned, the same M = Y p , q geometry enters in the solutions with constant τ presented in [14]. Indeed, one can show that the global analysis conducted in [14] matchesthat presented above . E.4 Toric Geometry of Y p , q The fact that the manifolds Y p , q are not Sasaki–Einstein leads to the cones constructedover these, C ( Y p , q ), not being Calabi–Yau. In fact, the cone over these M τ geometries admitan integrable complex structure, but not a symplectic structure. In particular they are notK¨ahler [39]. However, both the five-dimensional manifolds Y p , q and their cones C ( Y p , q ) admitan isometric (and holomorhic) T (cid:39) U (1) action. Therefore, on the one hand, the methodsfrom toric symplectic geometry employed in [41] cannot be applied here. In particular, we donot have moment maps whose images would determine the convex polyhedral cones underlyingseveral properties of the toric Sasaki–Einstein geometries [41, 47]. On the other hand, we stillhave a T action and one may attempt to understand these geometries from a complex toricgeometry viewpoint [77]. Below we will use the example of the Y p , q solution to illustrate somefeatures of these geometries, that we expect to persist more generally.A key property of toric Calabi-Yau singularities is that the image of the moment mapassociated to the T action is a convex polyhedral cone. The primitive normals to the facets ofthis cone can be projected to a plane, where they provide the toric diagram of the singularity.Equivalently, these normals correspond to the vanishing of different (Killing) vectors in T ,and thus define co-dimension two loci that are toric divisors in the Calabi–Yau cone, orequivalently calibrated three-manifolds in the Sasaki–Einstein base. These vectors may beextracted from an analysis of the explicit metric, and written in a basis for T they yieldthe toric diagram [78, 79]. Following these references, below we will employ this methodfor obtaining a toric diagram assocated to the Y p , q geometries, albeit one that will not beconvex. As we will explain, this diagram is formally in 1–1 correspondence with that of the Y p,q geometries.The analysis below will follow closely the discussion in [78, 79] for the regularity ofthe five-dimensional L a,b,c toric Sasaki–Einstein metrics. This gives an alternative method toperforming the regularity analysis of the metric, and in particular to determine the constraint Denoting the integers p, q in [14] as p DGK , q
DGK , one has the following identifications p = q DGK and q = p DGK + q DGK . In fact, they are neither Einstein nor Sasakian. They are not even contact manifolds [77]. < q . The starting point is the local five-dimensional metric (E.32) depending on theparameter a . There are four codimension two fixed point sets, where the metric degenerates;these are at x = x + , x = x − , θ = 0 and θ = π . At each of these points a Killing vectorhas vanishing norm. We may introduce a 2 π periodic coordinate for each of these angulardirections at the degeneration loci by normalising the associated Killing vector such that itssurface gravity, defined as κ = ∂ µ | V | ∂ µ | V | | V | , (E.61)is unity on the degeneration surface. With this choice of periodicity the Killing vector degen-erates smoothly onto the degeneration surface.The most general Killing vector one can construct is V = S∂ α + T ∂ φ + W ∂ χ , (E.62)where S, T, W are three constants. This has norm | V | = w ( x )4 ( S + g ( x )( T + cos θW )) + a (cid:18) U ( x ) w ( x ) ( T + cos θW ) + sin θW (cid:19) . (E.63)The norm is a sum of three positive terms and therefore for it to vanish each of these termsmust independently be zero. We find that the Killing vectors after being suitably normalisedare k + = 1 x + (cid:0) ∂ α + x + ∂ φ (cid:1) , k − = 1 x − (cid:0) ∂ α + x − ∂ φ (cid:1) ,k = ∂ φ − ∂ χ , k π = ∂ φ + ∂ χ , (E.64)where the superscript denotes the associated degeneration point. Clearly these four Killingvectors are not linearly independent as they span a three-dimensional space and thereforethey must satisfy Hk + + J k − + Kk + Lk π = 0 , (E.65)for some constant coefficients. As explained in [78] the constant coefficients must be integers.This follows because each of the Killing vectors generate 2 π periodic translations, and there-fore the coefficients must be rational. Then by taking integer combinations of translationsaround these circles one generates a translation which would identify arbitrarily close points.To prevent this from occurring one must take the coefficients to be integers which may beassumed to be coprime. One finds that the integers satisfy H + J + K + L = 0 , K = L , (E.66)100nd Hx + + Jx − = 0 ⇒ √ a = H − JH + J . (E.67)Taking into account the constraints above, we may redefine the integers H and J as H − J = 2 q , H + J = 2 p (E.68)for consistency with the previous section’s notation, i.e. (E.58). Then from the constraintthat a > p < q . Moreover, rewriting the linear relation between thevectors in terms of these two integers we find( p + q ) k + + ( p − q ) k − − p k − p k π = 0 . (E.69)From this we can read off what in the GLSM language is called the “charge matrix” (up toan overall sign) to be ( p , p , − p + q , − p − q ) . (E.70)Notice that this is formally identical to the charge matrix of Y p,q singularities, in particularthe sum of all these charges vanishes. However, due to the different sign of p − q , here thereare three positive charges and one negative for Y p , q , in contrast to the two positive and twonegative for Y p,q . In the Calabi–Yau context, these charges can be used to reconstruct thesingularity (and all its resolutions) from the K¨ahler quotient C //U (1). Then two charges ofthe same sign give rise to toric non-orbifold singularities, whereas three charges with the samesign produce a C / Z n orbifold. However the cone over Y p , q is not an orbifold singularity, asit follows from the preceding analysis, this is not in contradiction because the cone is notK¨ahler.To extract a toric diagram from the previous analysis we need to write the four vectorsabove in an effectively acting basis of T . Locally the T action is generated by the vectorfields ∂ α , ∂ φ and ∂ χ , however they do not give an effectively acting basis. Let an effectivelyacting basis of these Killing vectors be the set { e , e , e } , which are linear combinations of ∂ α , ∂ φ , ∂ χ and are taken to be suitably normalised such that all have period 2 π . Any SL Z transformation of this basis will also generate the effective T action. Writing the degeneratingKilling vectors as a linear combination of the e i and applying SL Z transformations to bring For a similar analysis in L a,b,c and conventions see [50, 80]. k + k − k k π = A B C D E F e e e . (E.71)Consider the degeneration surface defined by x = x + with degenerating Killing vector k + = e .The T fibration reduces smoothly to a T fibration over this surface which is spanned by { e , e } . At the intersection of this degeneration surface with the degeneration surfaces locatedat θ = 0 and θ = π we have an additional degenerating Killing vector. Recall from previousarguments that this vector must be 2 π periodic for the degeneration to be smooth. At θ = 0we have the Killing vector Ce + De degenerating on the surface. For this to be 2 π periodicit is necessary that C and D are relatively prime, gcd( C, D ) = 1. A similar argument followsfor the degeneration surface at θ = π and so we also have gcd( E, F ) = 1. Notice that there isno condition on
A, B as the degeneration surface at x = x − does not intersect with the oneat x = x + . As gcd( C, D ) = 1 there exist integer solutions to RC + SD = 1 and therefore byan SL Z ⊂ SL Z transformation we may set C = 1 , D = 0. We find k + k − k k π = A B
E F e e e . (E.72)Next, using the linear relation between the four Killing vectors we find( p − q ) B − p F = 0 , ( p − q ) A − p − p E = 0 . (E.73)These can be solved by B = p , G = p − q , A = 0 , E = − , (E.74)and we obtain k + k − k k π = p − p − q e e e . (E.75)We may now introduce three 2 π periodic coordinates ψ i for each of the three e i , the changeof coordinates from the original set is α = 1 x + ( ψ − ψ ) + (cid:18) µ − νx + − νx − (cid:19) ψ , φ = ψ , χ = − ψ + µψ , (E.76)102 p , p )( p - q -1, p - q ) (0,0) Figure 1 . Toric diagram for Y p , q . Noticethat this is not convex, however the externallines do not intersect. The figure is represen-tative of the choice p = 3, q = 4. (1,0) ( p, p )( p - q -1, p - q )(0,0) Figure 2 . Toric diagram for Y p,q . The figureis representative of the choice p = 4 and q = 3. where the integers µ and ν satisfy µ ( p − q ) − p ν = 1 and are guaranteed to exist by the factgcd( p , p − q ) = 1. With these coordinates the T action acts effectively.Finally, we may read off the toric data from the matrix Λ: the four vertices are given bythe rows of Λ, namely , p , , − p − q . (E.77)By an additional SL Z transformation the vectors take the form , pp , , p − q − p − q , (E.78)which agree formally with the ones for the Y p,q Calabi–Yau singularity [18]. Notice howeverthat because q > p , this no longer defines a convex polytope. For comparison, we contrasttwo examples of toric diagrams in the two cases in the figures 1 and 2. F Summary of the 4d Y p,q Field Theories
In the paper we use the N = 1 four-dimensional theories [18] compactified on a Riemannsurface as an example of AdS / CFT duality for which we can obtain an F-theory embedding.103 Figure 3 . The Y , quiver diagram. The quiver diagram for Y , . The fields have been colour-codedas follows: the Y fields are shown in blue, Z fields in red, U i fields are purple and V i fields green. Fields Multiplicity U (1) U (1) U (1) B U (1) R Y p + q − p − q R Y Z p − q p + q R Z U p − p R U U p − − p R U V q q R V V q − q R V λ p Table 5 : The charges of the various fields in the 4d Y p,q theories.For clarify of notation, below we present a summary of useful facts about the 4d field theoriesand the related AdS / CFT duality, before compactifying on a Riemann surface. These arean infinite family of quiver gauge theories specified by the gauge group G = SU ( N ) p andby a set of 4 p + 2 q chiral multiplets, transforming in bi-fundamental representations of pairsof SU ( N ) factors. Here p and q are two positive integers satisfying p > q . The preciserepresentation can be conveniently encoded in a quiver diagram, as in Figure 3.The bi-fundamentals are grouped in four types of fields, denoted Y, Z, U α , V α , each withdifferent global charges. There is a superpotential W , whose detailed form we will not needhere. The theories have global symmetries SU (2) × U (1) × U (1) R × U (1) B , but for manypurposes it is convenient to consider the charges with respect to the Cartan generator U (1) ⊂ SU (2) . There are two categories of global symmetries, referred to as flavour and baryonicsymmetries, respectively. For the convenience of the reader, the charges of the fields, togetherwith their multiplicities, are summarised in Table F. The R X in the last column denote the R -104harges of the (scalar) fields ( X = Y, Z, U α , V α ) under the true R -symmetry of the SCFTs attheir IR fixed point. The R -charges of the fermions in the chiral multiplets are given by R X − p gauginos λ for reference. The superconformal R -symmetry in theIR can mix with the abelian global symmetries of the theory and can be determined uniquelyby employing a -maximization [43]. In fact, it turns out that the baryonic symmetry U (1) B does not participate to this mixing, and the result of the extremization [18, 81] provides theR-charges of the theory, which read R Y = 3 q + 2 pq − p + (2 p − q ) z q , R U = 2 p (2 p − z )3 q ,R Z = 3 q − pq − p + (2 p + q ) z q , R V = 3 q − p + z q , (F.1)where we have defined z = (cid:112) p − q . For generic values of the parameters p and q thesenumbers are famously irrational, which corresponds to the fact that the R -symmetry is notcompact.There are two related ways to think about the gravity duals of these field theories. Onone hand, one can show that there is a branch of the mesonic vacuum moduli space of thesetheories that contains a copy of a Calabi–Yau three-fold singularity, denoted C ( Y p,q ) [41].It follows that the field theories may be thought of as arising from N D3 branes transverseto this conical singularity. On the other hand, in the large N limit, the Type IIB geometrynear the branes (“near horizon”) is AdS × Y p,q , where Y p,q are the five-dimensional Sasaki–Einstein manifolds [17]. The integers p, q characterising these manifolds can be consistentlyidentified with the p, q characterising the field theories.There are several checks that can be performed on this conjectured duality. The mostbasic check consists in matching the central charges on the two sides. Moreover, one canalso compare successfully the charges under the global symmetries on the two sides. To thisend, it is useful to consider certain baryonic operators B X , which correspond to particlesmoving in AdS , arising from D3-branes wrapping supersymmetric three-manifolds in theSasaki–Einstein manifold [40]. This establishes a map between baryonic operators B X andsupersymmetric three-manifolds Σ, in particular the R -charges of the former may be computedin terms of volumes of the latter by using the formula [40] R [ X ] = N π Y p,q ) . (F.2)The baryonic charges of the fields can also be inferred from the string duals of the baryonicoperators. Recall that the RR four-form potential C gives rise upon Kaluza-Klein reduction105o five dimensions to the background gauge field A B associated to the baryonic symmetrythrough the ansatz C = H ∧ A B , where H is a (harmonic) representative of H ( Y p,q ; Z ) (cid:39) Z .From this it follows that the baryonic charges of the baryonic operators can be computed inthe supergravity solution by integrating H on the various sub-manifolds Σ [50], so that Q B ( X ) = (cid:90) Σ X H . (F.3)Indeed this was explicitly done in [82], obtaining agreement with the field theory U (1) B charges written in Table F. Finally, the multiplicities of the fields written in the secondcolumn of F can be reproduced by calculating π (Σ X ) in the geometry Σ [50].Notice that all the comparisons that we have recalled here can be done without usingtechniques of toric geometry. 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