Warped AdS_6\times S^2 in Type IIB supergravity III: Global solutions with seven-branes
22017 May 25
Warped
AdS × S in Type IIB supergravity III Global solutions with seven-branes
Eric D’Hoker a,b , Michael Gutperle a and Christoph F. Uhlemann a a Mani L. Bhaumik Institute for Theoretical PhysicsDepartment of Physics and AstronomyUniversity of California, Los Angeles, CA 90095, USA b Kavli Institute for Theoretical PhysicsUniversity of California Santa Barbara, CA 93106, USA [email protected]; [email protected]; [email protected]
Abstract
We extend our previous construction of global solutions to Type IIB supergrav-ity that are invariant under the superalgebra F (4) and are realized on a spacetime ofthe form AdS × S warped over a Riemann surface Σ by allowing the supergravityfields to have non-trivial SL (2 , R ) monodromy at isolated punctures on Σ. We obtainexplicit solutions for the case where Σ is a disc, and the monodromy generators areparabolic elements of SL (2 , R ) physically corresponding to the monodromy allowed inType IIB string theory. On the boundary of Σ the solutions exhibit singularities atisolated points which correspond to semi-infinite five-branes, as is familiar from theglobal solutions without monodromy. In the interior of Σ, the solutions are everywhereregular, except at the punctures where SL (2 , R ) monodromy resides and which physi-cally correspond to the locations of [ p, q ] seven-branes. The solutions have a compellingphysical interpretation corresponding to fully localized five-brane intersections with ad-ditional seven-branes, and provide candidate holographic duals to the five-dimensionalsuperconformal field theories realized on such intersections. a r X i v : . [ h e p - t h ] J un ontents SU (1 ,
1) transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 Identification with 5-brane intersections . . . . . . . . . . . . . . . . . . . . . 9 ∂ w A ± . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3 The functions A ± . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.4 Regularity conditions for G . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.5 Summary of solutions and regularity conditions . . . . . . . . . . . . . . . . 193.6 Counting free parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.7 Identification of punctures with [ p, q ] 7-branes . . . . . . . . . . . . . . . . . 22 ,
0] branch point . . . . . . . . . . . . . . . . . . . . . 284.3 3-pole solution with [0 ,
1] branch point . . . . . . . . . . . . . . . . . . . . . 31 G on ∂ Σ Introduction
Five-dimensional superconformal field theories (SCFTs) exhibit many intriguing and exoticproperties, including the uniqueness of the exceptional superconformal symmetry algebra F (4), the possibility for exceptional global symmetries, the absence of a useful Lagrangiandescription, and many non-trivial dualities and relations to theories in other dimensions.In the absence of a conventional Lagrangian description, the theories have been accessedindirectly, for example as non-trivial UV fixed-points of five-dimensional gauge theories con-sidered on the Coulomb branch or as low-energy description of certain brane configurationsin string theory or M-theory on Calabi-Yau manifolds [1, 2]. A very fruitful approach hasbeen to engineer these theories using brane constructions in Type IIB string theory. Five-dimensional gauge theories can be realized on the world-volume of D5-branes that are sus-pended between semi-infinite external ( p, q ) 5-branes [3, 4]. In the limit where these branewebs collapse to a fully localized intersections of ( p, q ) 5-branes one recovers the SCFTs atthe origin of their moduli spaces. While the string theory constructions provide access tomany features of the 5d SCFTs and have led to many insights, the corresponding supergrav-ity solutions in Type IIB supergravity are the prerequisite for utilizing AdS/CFT as toolfor comprehensive quantitative analyses. In recent work we have constructed large classesof warped AdS solutions in Type IIB supergravity that are in direct correspondence withfully localized 5-brane intersections in Type IIB string theory [5, 6, 7]. They allow for quan-titative analyses of the theories realized on intersections of 5-branes, including as a first stepthe study of free energies and entanglement entropies [8].The 5-brane web constructions in Type IIB string theory can be generalized considerablyby including 7-branes [9, 10]. External 5-branes are allowed to terminate on 7-branes and7-branes may be added into the open faces of the web. The Hanany-Witten brane creationeffect [11] provides a way to relate certain webs with 7-branes to webs without 7-branes.Many recent insights are based on manipulations involving 7-branes, including new dualitiesbetween 5d theories from branch cut moves, the construction of gauge theory descriptionsfor 5d uplifts of 4d class S theories, the realization of theories violating the flavor boundsof [1, 2] and connections to 6d SCFTs [14, 15, 16, 17, 18, 19]. These observations providea clear motivation for the construction of warped AdS solutions in Type IIB supergravitycorresponding to 5-brane intersections which include 7-branes. 7-branes placed inside thefaces of a 5-brane web, for example, should be directly accessible via supergravity solutionscorresponding to the conformal limit of the web. In the present paper we will constructwarped AdS solutions to Type IIB supergravity which include 7-branes.The geometry of the solutions constructed in [5, 6, 7] takes the form of AdS × S warped over a two-dimensional Riemann surface Σ with boundary, and the solutions realize Recent attempts towards a more complete classification can be found in [12, 13]. F (4) superconformal algebra in 5d [20, 21]. The solutions arespecified in terms of two locally holomorphic functions A ± on Σ and a crucial feature isthat the differentials of these functions have poles on the boundary of Σ. At these poles thegeometry approaches that of ( p, q ) 5-branes with the charges given by the residues, and thisallows for a clear mapping between 5-brane intersections and supergravity solutions. For theglobal solutions constructed explicitly so far Σ was taken to be a disc [6, 7]. These solutionswill provide the basis for the construction of solutions with 7-branes.The distinct feature of 7-branes, amongst the brane solutions in Type IIB supergravity,is the defect they create in the space transverse to their world-volume, and the non-trivialmonodromy the axion and dilaton fields exhibit around this defect. The duality group ofType IIB supergravity is SL (2 , R ) and, mathematically, the axion-dilaton field and the three-form field strengths might have monodromy with arbitrary values in SL (2 , R ). Physically,however, we are interested in supergravity solutions which embed into Type IIB string theory.The duality group of Type IIB string theory is SL (2 , Z ), and string theory solutions onlyallow for SL (2 , Z )-valued monodromy. For example the monodromy of a D7-brane leavesthe dilaton invariant and shifts the axion field by 1, corresponding to a parabolic elementof SL (2 , Z ). Just as strings and 5-branes, 7-branes transform non-trivially under SL (2 , Z )so that a general 7-brane carries a charge labeled by a pair of integers [ p, q ] which specifythe monodromy around the 7-brane [22, 23, 24]. In supergravity, SL (2 , Z ) is replaced by SL (2 , R ), p and q become real numbers, and the monodromy can be a generic parabolicelement of SL (2 , R ).The supersymmetry conditions on branes allow for the preservation of the full F (4)superalgebra in the presence of both 5-branes and 7-branes, and we shall henceforth restrictto solutions with this full symmetry. Preserving the full F (4) requires the 7-branes to belocated at isolated points or punctures in the interior of the surface Σ, around which thesupergravity fields have non-trivial monodromy given by a parabolic element of SL (2 , R ).We will show that the monodromy of the supergravity fields around the punctures can berealized by suitable monodromies of the locally holomorphic functions A ± which parametrizethe solutions, and we will explicitly construct such A ± and the corresponding supergravitysolutions. We will allow for an arbitrary number of punctures with mutually commutingmonodromies. These are the appropriate monodromies for an arbitrary number of mutuallylocal 7-branes, and we will show that the asymptotic behavior of the solutions near thepunctures indeed approaches the form expected on physical grounds.The remainder of the paper is organized as follows. In sec. 2 we will review the globalsolutions constructed in [5, 6, 7] and highlight the points that will be relevant for the con-struction of solutions with 7-brane monodromy. The actual construction will be carried outin sec. 3, where we explicitly set up the holomorphic data for solutions with monodromyand derive the regularity conditions constraining the parameters. We will also show that thesupergravity fields close to the punctures match to the expected form for [ p, q ] 7-branes. In4ec. 4 we will solve the regularity conditions and present explicit example solutions, showingthat the solutions indeed have the desired properties. The connection to 5-brane webs with7-branes will be discussed in more detail in sec. 5 and we close with a discussion in sec. 6.5 Review of solutions without monodromy
In this section we will briefly review the local solutions to Type IIB supergravity with 16supersymmetries and metric of the form
AdS × S warped over a Riemann surface Σ asconstructed in [5], the regularity conditions they have to satisfy and the global solutionswithout monodromy constructed in [7]. The global solutions without monodromy will bethe starting point for the construction of solutions with monodromy in the next section. The general local solution with 16 supersymmetries and SO (2 , × SO (3) isometry can beexpressed in terms of two locally holomorphic functions A ± defined on the Riemann surface Σwith so far arbitrary topology. The symmetry requirement restricts the metric and two-formfield strength to take the form ds = f ds AdS + f ds S + 4 ρ | dw | F (3) = d C ∧ vol S (2.1)with f , f ρ real functions on Σ while C is an in general complex function on Σ. The four-form field vanishes. The functions appearing in the ansatz can be conveniently expressed interms of A ± by using the composite objects κ = −| ∂ w A + | + | ∂ w A − | G = |A + | − |A − | + B + ¯ B ∂ w B = A + ∂ w A − − A − ∂ w A + R + 1 R = 2 + 6 κ G| ∂ w G| (2.2)where B is defined up to an integration constant. The metric functions then take the form f = c √ G (cid:18) R − R (cid:19) / f = c √ G (cid:18) − R R (cid:19) / ρ = κ √ G (cid:18) R − R (cid:19) / (2.3)The remaining fields are the axion-dilaton scalar B , which is given by B = ∂ w A + ∂ ¯ w G −
R ∂ ¯ w ¯ A − ∂ w G R ∂ ¯ w ¯ A + ∂ w G − ∂ w A − ∂ ¯ w G (2.4)and the complex function C parametrizing the two-form gauge field, which reads C = 4 ic (cid:18) ∂ ¯ w ¯ A − ∂ w G κ − R ∂ w G ∂ ¯ w ¯ A − + ∂ ¯ w G ∂ w A + ( R + 1) κ − ¯ A − − A + (cid:19) (2.5)6 .2 Regularity conditions and global solutions For physically sensible solutions additional regularity conditions are required, and these canbe expressed concisely as conditions on the composite quantities κ and G defined in (2.2).To have the metric functions f , f , ρ positive in the interior of Σ it is sufficient to require κ > G > S on the boundary ∂ Σ of Σ, which amounts to the additional conditions κ (cid:12)(cid:12)(cid:12) ∂ Σ = 0 G (cid:12)(cid:12)(cid:12) ∂ Σ = 0 (2.7)This finishes the general discussion of the regularity conditions. As shown in [7] they can besatisfied by choosing Σ to be the upper half plane and the locally holomorphic functions as A ± ( w ) = A ± + L (cid:88) (cid:96) =1 Z (cid:96) ± ln( w − p (cid:96) ) (2.8)where p (cid:96) for (cid:96) = 1 , · · · , L denote the L poles of the differentials ∂ w A ± ; they lie on the realline which is the boundary ∂ Σ. The residues of ∂ w A ± at these poles, Z (cid:96) ± , are expressed interms of L − s n , n = 1 , · · · , L − Z (cid:96) + = σ L − (cid:89) n =1 ( p (cid:96) − s n ) L (cid:89) k (cid:54) = (cid:96) p (cid:96) − p k (2.9)with an overall complex normalization parametrized by σ , and Z (cid:96) − = − Z (cid:96) + . The locallyholomorphic functions constructed this way satisfy the regularity conditions on κ , produce G constant along each boundary component free of poles and G > G = 0 on the boundary. The only condition left to satisfy therefore is G = 0 on the boundary,which constrains the parameters to satisfy A Z k − + ¯ A Z k + + (cid:88) (cid:96) (cid:54) = k Z [ (cid:96)k ] ln | p (cid:96) − p k | = 0 (2.10)for k = 1 , · · · , L , where we have defined 2 A = A − ¯ A − and Z [ (cid:96)k ] = Z (cid:96) + Z k − − Z k + Z (cid:96) − .Regularity of the string-frame geometry near the poles furthermore requires c = 1.7 .3 SU (1 , transformations The SL (2 , R ) ∼ SU (1 ,
1) duality symmetry transformations of Type IIB supergravity havebeen realized on the locally holomorphic data A ± and on the composite quantities κ and G in [5]. Parametrizing a generic SU (1 ,
1) transformation by u, v ∈ C with | u | − | v | = 1, thelocally holomorphic functions A ± transform as follows A + → A (cid:48) + = + u A + − v A − + a + A − → A (cid:48)− = − ¯ v A + + ¯ u A − + a − (2.11)where a ± are complex constants parametrizing a shift in addition to a pure SU (1 ,
1) trans-formation. On the differentials ∂ w A ± this induces a pure SU (1 ,
1) transformation ∂ w A + → ∂ w A (cid:48) + = + u∂ w A + − v∂ w A − ∂ w A − → ∂ w A (cid:48)− = − ¯ v∂ w A + + ¯ u∂ w A − (2.12)and implies that κ and its complex conjugate are invariant under SU (1 , B is definedonly up to a constant by (2.2), the transformation of A ± determines the transformation of B only up to a further constant shift. As discussed in [5], however, for the transformation of thelocally holomorphic data to induce the correct SU (1 ,
1) transformations on the supergravityfields, this shift has to vanish and we in addition have to require a − = ¯ a + (2.13)This condition is itself SU (1 ,
1) invariant and it implies that G is invariant under (2.11) aswell. As a result, the metric functions f , f , ρ are invariant, as expected for the metric inEinstein frame, and the axion-dilaton scalar B and gauge field C transform as B → B (cid:48) = uB + v ¯ vB + ¯ u (2.14) C → C (cid:48) = u C + v ¯ C + C (2.15)Note that the transformation of C includes a shift by a constant C which can be compensatedby a gauge transformation. SL (2 , R )To translate the SU (1 ,
1) transformation of B to the corresponding SL (2 , R ) transformationof τ , we note that B and τ are related by B = τ − i − τ − i = U ( τ ) U = 1 √− i (cid:18) − i − − i (cid:19) (2.16)8he normalization factor in U has been chosen such that det U = 1. The SU (1 ,
1) transfor-mation in (2.14) can be written as B → B (cid:48) = uB + v ¯ vB + ¯ u = V ( B ) V = (cid:18) u v ¯ v ¯ u (cid:19) (2.17)with | u | − | v | = 1, while the SL (2 , R ) transformation of τ is given by τ → τ (cid:48) = aτ + bcτ + d = M ( τ ) M = (cid:18) a bc d (cid:19) (2.18)with a, b, c, d ∈ R and ad − bc = 1. The two transformations are related to one another by V ( B ) = B (cid:48) = U ( τ (cid:48) ) = U M ( τ ) = U M U − ( B ) (2.19)from which the identification of SU (1 ,
1) and SL (2 , R ) parameters can be read off as u = 12 ( a + ib − ic + d ) v = 12 ( − a + ib + ic + d ) (2.20)Given the relation between B and ∂ w A ± in (2.4) we find that the transformation of B canbe realized if the differentials ∂ w A ± are transformed as follows (cid:18) ∂ w A + ∂ w A − (cid:19) → (cid:18) ∂ w A (cid:48) + ∂ w A (cid:48)− (cid:19) = ( V † ) − (cid:18) ∂ w A + ∂ w A − (cid:19) ( V † ) − = (cid:18) u − v − ¯ v ¯ u (cid:19) (2.21) As discussed in detail in [7], the geometry of the supergravity solution close to a pole p (cid:96) precisely matches the near-brane limit of the 5-brane solutions constructed in [25]. In thenotation of [25], the charges of the 5-brane, ( q , q ) Q , are identified with the residue of ∂ w A + at the pole p m , given by Z m + , via ( q − iq ) Q = 83 c Z m + (2.22)We note that in the convention of [25] q corresponds to NS5 charge and q to D5 charge, andcorrespondingly Im ( Z m + ) translates to D5 charge while Re ( Z m + ) translates to NS5 charge.9 Solutions with monodromy on the disc
In this section we will start from the global solutions without monodromy on the disc reviewedin the previous section and use them to construct physically regular solutions on a disc withpunctures and non-trivial monodromy. We will allow for an arbitrary number of puncturesand for generic parabolic SL (2 , R ) monodromies, as appropriate for the inclusion of 7-branes,but restrict the monodromies to be mutually commuting. In sec. 3.1 - 3.4 we will detail theconstruction and derive the regularity conditions. The results will be summarized in sec. 3.5.In sec. 3.6 and 3.7 we will count the free parameters labeling distinct solutions and identifythe punctures with [ p, q ] 7-branes. Before discussing the construction of solutions with general monodromies in the upper halfplane, we will outline the basic strategy for a simple example where we take Σ to be a disc.A general parabolic element of SL (2 , R ) can be parametrized by two real numbers p, q as M [ p,q ] = (cid:18) − pq p − q pq (cid:19) (3.1)and we will use this parametrization in the following. The parameters of the corresponding SU (1 ,
1) transformation are given via (2.20) by u [ p,q ] = 1 + i p + q ) v [ p,q ] = i p − iq ) (3.2)We will now consider the special case of a single puncture at the center of the disc, whereonly the axion has non-trivial monodromy and shifts by 1. The corresponding SL (2 , R )matrix is M [1 , = (cid:18) (cid:19) (3.3)The entries of the corresponding SU (1 ,
1) transformation matrix V in (2.17) are given by u = u [1 , , v = v [1 , with (3.2). The differentials correspondingly have to transform via (2.21)as ∂ w A ± → ∂ w A (cid:48)± = ∂ w A ± + i ∂ w A + − ∂ w A − ) (3.4)To realize this monodromy around the point at the center of the disc, we introduce a coor-dinate such that w = 0 corresponds to the center of the disc and | w | = 1 to the boundary.We may then realize the above monodromy by considering the logarithmic function, whichhas the appropriate monodromy as we wrap around the center by w → e πi w . Let ∂ w A (0) ±
10e the differentials for a solution without monodromy on the disc, which are single-valuedand meromorphic. We then set ∂ w A ± = ∂ w A (0) ± + f (cid:16) ∂ w A (0)+ − ∂ w A (0) − (cid:17) f ( w ) = 14 π ln w (3.5)The function f is locally holomorphic on the disc, and this produces locally holomorphicdifferentials with the desired monodromy. What we have left to verify is that they satisfy theregularity conditions on κ reviewed in sec. 2.2. A straightforward calculation shows that κ = −| ∂ w A (0)+ | + | ∂ w A (0) − | − ( f + ¯ f ) (cid:12)(cid:12)(cid:12) ∂ w A (0)+ − ∂ w A (0) − (cid:12)(cid:12)(cid:12) (3.6)The first term is positive in the interior of Σ and vanishes on the boundary, since thedifferentials ∂ w A (0) ± were assumed to correspond to a regular solution. For the second termwe note that − ( f + ¯ f )( w ) = − π ln | w | (3.7)is positive in the interior of the disc and vanishes on the boundary. The second term in (3.6)therefore is non-negative in the interior of the disc and zero on the boundary, such that κ satisfies the regularity conditions in (2.6) and (2.7). ∂ w A ± We will now generalize the strategy outlined in the previous subsection to construct the dif-ferentials for an arbitrary number of punctures with commuting monodromies of the generalform in (3.1). Instead of working with the disc, we will map to the upper half plane, so wecan directly use the solutions of [6, 7] reviewed in sec. 2.The first step is to generalize the locally holomorphic function f to the case with multiplepunctures with commuting monodromies at points w i , i = 1 , · · · , I in the upper half plane.This is straightforward and yields f ( w ) = I (cid:88) i =1 n i π ln (cid:18) γ i w − w i w − ¯ w i (cid:19) (3.8)where n i ∈ R and | γ i | = 1 for i = 1 , · · · I . We note the following properties of the function f • f is locally holomorphic in the upper half plane, with branch points at w i around which f ( w i + e πi ( w − w i )) = f ( w ) + i n i (3.9)11 the branch cuts associated with w i extend in a direction determined by γ i and can beparametrized as w = w i + c ¯ w i − w i c + γ i c ∈ [0 ,
1] (3.10)in particular, γ i = +1 and γ i = − • − ( f + ¯ f ) is positive in the interior of Σ and vanishes on the boundary ∂ Σ.Using the function f defined in (3.8) and the differentials ∂ w A (0) ± for a solution withoutmonodromy in the upper half plane, as given in (2.8), we can now construct the differentialsfor a solution with axion monodromy in the upper half plane, by setting ∂ w A ax ± = ∂ w A (0) ± + f (cid:16) ∂ w A (0)+ − ∂ w A (0) − (cid:17) (3.11)The monodromy of these differentials around w i is given by the SU (1 ,
1) transformation in(2.21) with u = u [ n i , , v = v [ n i , and (3.2). This corresponds to the SL (2 , R ) transformation M [ n i , = (cid:18) n i (cid:19) (3.12)thus realizing axion monodromies as desired.To generalize the construction to general parabolic SL (2 , R ) monodromies of the form(3.1), we note that the transformation given in (3.1) can be generated from M [1 , given in(3.3) by conjugating with an SL (2 , R ) matrix Q as follows M [ p,q ] = QM [1 , Q − Q = (cid:18) p − q/ ( p + q ) q p/ ( p + q ) (cid:19) (3.13)To realize the transformation by Q on the differentials in (3.11) we translate it to an SU (1 , u Q = 1 + η + η − η − v Q = 1 − η + η − η − η ± = p ∓ iq (3.14a)Transforming the differentials (3.11) according to (2.12) then yields differentials realizing thedesired monodromies, and with ∂ w A ± ≡ ( ∂ w A ax ± ) (cid:48) we find ∂ w A + = + u Q ∂ w A (0)+ − v Q ∂ w A (0) − + η + f (cid:16) ∂ w A (0)+ − ∂ w A (0) − (cid:17) ∂ w A − = − ¯ v Q ∂ w A (0)+ + ¯ u Q ∂ w A (0) − + η − f (cid:16) ∂ w A (0)+ − ∂ w A (0) − (cid:17) (3.14b)12his completes the construction of the differentials. The expressions in (3.14) realize SL (2 , R )monodromies M [ n i p,n i q ] = (cid:18) − n i pq n i p − n i q n i pq (cid:19) (3.15)around the points w i in the upper half plane, as desired. Moreover, since κ is SU (1 , SU (1 ,
1) transformation from thosein (3.11), we have κ = −| ∂ w A ax+ | + | ∂ w A ax − | = −| ∂ w A (0)+ | + | ∂ w A (0) − | − ( f + ¯ f ) (cid:12)(cid:12)(cid:12) ∂ w A (0)+ − ∂ w A (0) − (cid:12)(cid:12)(cid:12) (3.16)Due to the properties of f collected above, the differentials in (3.14) therefore produce κ thatis positive in the interior of the upper half plane and zero on its boundary, thus satisfying theregularity conditions in (2.6), (2.7). For any choice of global solution without monodromy,we therefore get suitable differentials for a solution with monodromy.To facilitate the computations and arguments in the following sections, we will introducea more convenient notation. Namely, we split ∂ w A ± = ∂ w A s ± + η ± F (3.17)where ∂ w A s ± denotes the single-valued part of the differentials and the logarithmic part isdenoted by F . In terms of the seed solution without monodromy we have ∂ w A s + = + u Q ∂ w A (0)+ − v Q ∂ w A (0) − ∂ w A s − = − ¯ v Q ∂ w A (0)+ + ¯ u Q ∂ w A (0) − (3.18)for the single-valued part and the logarithmic part is given by F = f (cid:16) ∂ w A (0)+ − ∂ w A (0) − (cid:17) (3.19)This can be spelled out more explicitly as ∂ w A s + ( w ) = L (cid:88) (cid:96) =1 Y (cid:96) ± w − p (cid:96) F ( w ) = f ( w ) (cid:96) (cid:88) (cid:96) =1 Y (cid:96) w − p (cid:96) (3.20)where we have defined convenient combinations of the residues as Y (cid:96) + = + u Q Z (cid:96) + − v Q Z (cid:96) − Y (cid:96) = Z (cid:96) + − Z (cid:96) − Y (cid:96) − = − ¯ v Q Z (cid:96) + + ¯ u Q Z (cid:96) − (3.21)Due to the conjugation properties of Z (cid:96) ± we have Y (cid:96) ± = − Y (cid:96) ∓ and that Y (cid:96) is real. Moreover,since the Z (cid:96) ± sum to zero the same holds for Y (cid:96) ± and we have (cid:80) (cid:96) Y (cid:96) ± = 0.13 • w i • w j • w k • • p (cid:96) (cid:48) p (cid:96) • w Figure 1: Branch cuts for F are drawn as black dashed lines and do not intersect each otheror poles on the real line. The cuts shown correspond to γ i = − γ j = 1 and γ k = e iπ/ . Anintegration contour for I , which does not intersect any of the branch cuts, is shown in red. A ± In this subsection we will construct the locally holomorphic functions A ± from the dif-ferentials. This in particular involves realizing a monodromy of the form (2.11) with theconstant shifts related as in (2.13). The differentials for solutions without monodromy couldbe integrated straightforwardly to obtain the locally holomorphic functions A ± . For thedifferentials constructed in the previous section this is still possible, but due to the presenceof the logarithms in F , their integrals involve dilogarithms. We find it more convenient towork with the integrals explicitly, and introduce the following notation A ± = A s ± + η ± I (3.22)We have once again separated the single-valued part A s ± from the part resulting from thelogarithmic terms in the differentials η ± I . The expressions for A s ± are A s + = + u Q A (0)+ − v Q A (0) − A s − = − ¯ v Q A (0)+ + ¯ u Q A (0) − (3.23)with A (0) ± the holomorphic functions for the seed solution without monodromy as given in(2.8). More explicitly, we may write this as A s ± ( w ) = A ± + L (cid:88) (cid:96) =1 Y (cid:96) ± ln( w − p (cid:96) ) (3.24)with the Y (cid:96) ± given in (3.21) and integration constants A ± , which are appropriate combinationsof the integration constants of the seed solution without monodromy. For the logarithmicpart I we have to discuss the choice of integration contour. We will assume that all γ i and w i are chosen such that the resulting branch cuts do not intersect a pole on the real line,14nd moreover that the branch cuts do not intersect each other in the interior of Σ. Theintegration contour for I with starting point at + ∞ + i + can then be chosen such that itdoes not intersect any of the branch cuts in F . The contour is illustrated in fig. 1. Theexpression for I becomes I ( w ) = (cid:90) w ∞ dz F ( z ) (3.25) A ± across branch cuts We can now evaluate the behavior of the holomorphic functions A ± across the branch cut,for each cut individually. Let w be a point on the branch cut associated with a particularbranch point w i . We can then evaluate the shift in the holomorphic functions by integratingaround the branch cut as follows, A ± ( w + (cid:15) ) − A ± ( w − (cid:15) ) = (cid:90) C dz ∂ z A ± = (cid:90) C dz (cid:104) ∂ z A s ± + η ± f (cid:16) ∂ z A (0)+ − ∂ z A (0) − (cid:17)(cid:105) (3.26)where the contour C is illustrated in fig. 2 and the second equality follows using (3.17),(3.19). Since A s ± are holomorphic in the interior of Σ, the first term in square bracketscancels between the segments C and C as (cid:15) →
0. Moreover, again due to holomorphicityof ∂ w A (0) ± , we can write the remaining part as A ± ( w + (cid:15) ) − A ± ( w − (cid:15) ) = (cid:90) C dz η ± (∆ f )( ∂ z A (0)+ − ∂ z A (0) − ) (3.27)where ∆ f is the shift in f across the branch cut. Using ∆ f = in i /
2, we then find A ± ( w + (cid:15) ) − A ± ( w − (cid:15) ) = η ± in i (cid:104) A (0)+ ( w ) − A (0) − ( w ) − A (0)+ ( w i ) + A (0) − ( w i ) (cid:105) (3.28)The logarithmic singularity in the differentials ∂ w A ± is integrable, and the functions A ± therefore finite in the upper half plane. But they shift across the branch cut as given above.The shift in eq. (3.28) can be written as SU (1 ,
1) transformation supplemented by anadditional complex shift as follows, A + ( w + (cid:15) ) = + u [ n i p, n i q ] A + ( w − (cid:15) ) − v [ n i p, n i q ] A − ( w − (cid:15) ) + a + A − ( w + (cid:15) ) = − ¯ u [ n i p, n i q ] A + ( w − (cid:15) ) + ¯ v [ n i p, n i q ] A − ( w − (cid:15) ) + a − (3.29)with the parameters u , v as defined in (3.2). The complex constants a ± are given by a ± = − η ± in i (cid:104) A (0)+ ( w i ) − A (0) − ( w i ) (cid:105) (3.30)15 • w i • w j • w k • • p (cid:96) (cid:48) p (cid:96) w − (cid:15) • • w + (cid:15)C C Figure 2: Integration contour C = C ∪ C , where C denotes the left half of the contourshown in red and C the right half.The SU (1 ,
1) parameters in (3.29) correspond to the SL (2 , R ) transformation in (3.15) andthis is precisely the desired monodromy. To guarantee single-valued G we only have to impose(2.13) on the shift parameter a ± . Since we have η − = ¯ η + , this condition amounts to thedifference A (0)+ ( w i ) − A (0) − ( w i ) being imaginary A (0)+ ( w i ) − A (0) − ( w i ) + c . c . = 0 (3.31)This can be expressed as a condition on the single-valued part of the differentials by notingthat the residues are related by η − Y k + − η + Y k − = Y k (3.32)This yields the relation A (0)+ − A (0) − = η − A s + − η + A s − for the locally holomorphic functionsand we can express the condition in (3.31) as η − A s + ( w i ) − η + A s − ( w i ) + c . c . = 0 (3.33)We have thus constructed the holomorphic functions A ± for a solution with monodromy,and find that the location of the branch points is constrained by (3.33). G We have constructed the differentials and the locally holomorphic functions A ± and imple-mented the regularity conditions on κ . It remains to implement the regularity conditionson G , which we will do in this section. The positivity condition in the interior of Σ in (2.6)is automatically satisfied if we implement the condition G = 0 on the boundary in (2.7), forthe same reasons as discussed in sec. 2.3 of [7]. Implementing G = 0 on ∂ Σ proceeds in twosteps. The first is to ensure that G is piecewise constant along each boundary segment freeof poles. The second is to then ensure that G is also constant across poles. The remainingfree integration constant in G (recalling its definition in terms of B in (2.2) and that B isfixed only up to a constant) can then be used to set it to zero.16 .4.1 Piecewise constant G on ∂ ΣPiecewise constant G on the boundary ∂ Σ = R can be implemented by realizing a reflectionsymmetry across ∂ Σ on the locally holomorphic functions A ± due to the fact that ∂ w G + ∂ ¯ w G = (cid:16) A + ( w ) − A − ( w ) (cid:17)(cid:16) ∂ w A − ( w ) − ∂ ¯ w A − ( ¯ w ) (cid:17) − (cid:16) A − ( w ) − A + ( w ) (cid:17)(cid:16) ∂ w A + ( w ) − ∂ ¯ w A + ( ¯ w ) (cid:17) (3.34)It is therefore sufficient to establish the conjugation property A ± ( ¯ w ) = −A ∓ ( w ) (3.35)which guarantees that ∂ w G + ∂ ¯ w G = 0 on ∂ Σ and hence that G is piecewise constant.To implement this conjugation property we start with the weaker condition on the deriva-tives. The solution without monodromy is assumed to obey ∂ ¯ w A (0) ± ( ¯ w ) = − ∂ w A (0) ∓ ( w ), asdiscussed in [7], and from the explicit expressions for ∂ w A s ± in (3.18) we see that the sameis true for the single-valued part of the differentials. From (3.17) we therefore have ∂ ¯ w A ± ( ¯ w ) = ∂ ¯ w A s ± ( ¯ w ) + η ± F ( ¯ w )= − ∂ w A s ∓ ( w ) + η ∓ f ( ¯ w ) (cid:16) − ∂ w A (0) − ( w ) + ∂ w A (0)+ ( w ) (cid:17) (3.36)To realize differentials with the desired conjugation property we therefore have to impose f ( ¯ w ) = − f ( w ) (3.37)With the symmetric distribution of the points w i and ¯ w i under complex conjugation and thefact that the γ i are pure phases, we indeed find from the definition of f in (3.8) that f ( ¯ w ) = I (cid:88) i =1 n i π ln (cid:18) γ i ¯ w − w i ¯ w − ¯ w i (cid:19) = I (cid:88) i =1 n i π ln (cid:18) γ i w − ¯ w i w − w i (cid:19) = − f ( w ) (3.38)if the branch cut of the logarithm ln is chosen symmetrically with respect to complex con-jugation. With (3.36) this yields the desired conjugation condition for the differentials ∂ ¯ w A ± ( ¯ w ) = − ∂ w A ∓ ( w ) (3.39)Lifting this relation to the holomorphic functions A ± now simply amounts to choosing theintegration constants A ± such that A ± = −A ∓ (3.40)With the symmetric choice of branch cuts and the contour for I ± in (3.22), this suffices toensure the conjugation property for the locally holomorphic functions A ± in (3.35), and thusconstant G along each boundary component free of poles or branch cuts ( G also does notshift across branch cuts if the conditions (3.33) are satisfied).17 .4.2 Vanishing G on ∂ ΣImplementing the vanishing of G amounts to realizing vanishing monodromy of G aroundeach pole. Since we assumed that the branch cuts do not intersect the poles on the real axis,they will play a role only at the very end. For the evaluation of the monodromy of G aroundthe pole p k , ∆ k G , we note that, with a small (cid:15) ∈ R + ,∆ k G = |A + ( p k − ε ) | − |A + ( p k + ε ) | − |A − ( p k − ε ) | + |A − ( p k + ε ) | + ∆ k B + ∆ k ¯ B (3.41)With C k a half circle contour of radius (cid:15) centered on p k with counter-clockwise orientation,the shift in B is given by ∆ k B = (cid:90) C k dz (cid:16) A + ∂ z A − − A − ∂ z A + (cid:17) (3.42)To evaluate the first line in (3.41) explicitly, we note that we can evaluate the shift in A ± across the pole by integrating the differentials along C k , which yields A ± ( p k − (cid:15) ) − A ± ( p k + (cid:15) ) = (cid:90) C k dw ∂ w A ± = iπ (cid:2) Y k ± + η ± f ( p k ) Y k (cid:3) (3.43)This also directly gives the residues of the differentials in the new solution at the poles onthe real line. Using that ¯ η ± = η ∓ and that f is imaginary on ∂ Σ, we find∆ k G = iπ (cid:2) Y k + + η + f ( p k ) Y k (cid:3) (cid:16) A + ( p k + (cid:15) ) − A − ( p k + (cid:15) ) (cid:17) + ∆ k B + c . c . (3.44)Explicitly evaluating ∆ k B with its conjugate shows that it precisely reproduces the firstterm. We give the details of this calculation in app. A. Evaluating the first term in (3.44)explicitly and using that f ( p k ) is imaginary, the resulting expression for the shift reads∆ k G πi = 2 ¯ A Y k + + 2 A Y k − + (cid:88) (cid:96) (cid:54) = k Y [ (cid:96),k ] ln | p (cid:96) − p k | + Y k (cid:16) f ( p k ) (cid:104) η − A s + ( p k + ε ) − η + A s − ( p k + ε ) (cid:105) − I ( p k + ε ) − c . c . (cid:17) (3.45)where 2 A = A − ¯ A − and Y [ (cid:96),k ] = Y (cid:96) + Y k − − Y k + Y (cid:96) − . The individual terms in the second line aredivergent as (cid:15) →
0, but their combination is finite. To make this manifest, it is convenientto perform an integration by parts in the expression for I . We relegate the details again toapp. A. In the resulting expression we can then take (cid:15) →
0, as desired, and find∆ k G πi = 2 ¯ A Y k + + 2 A Y k − + (cid:88) (cid:96) (cid:54) = k Y [ (cid:96),k ] ln | p (cid:96) − p k | + 2 f ( p k ) Y k ( η − A − η + A − ) + Y k (cid:32)(cid:90) p k ∞ dw L (cid:88) (cid:96) =1 Y (cid:96) ln( w − p (cid:96) ) ∂ w f − c . c . (cid:33) (3.46)18s discussed in sec. 3.3 the integration contour has to be chosen in such a way that it doesnot cross any of the branch cuts, and this is a natural form of the regularity conditions. Wecan simplify the choice of contour by noting that ∂ w f is meromorphic with simple poles inthe upper half plane at the w i , such that the integrand in the second line is holomorphicexcept for at the poles of ∂ w f . We can therefore also move the contour to the real line.When deforming the integration contour shown in fig. 1 to approach the real line, we willonly pick up the residues for the poles in ∂ w f at those w i that are crossed. This yields (cid:90) p k ∞ dw ln( w − p (cid:96) ) ∂ w f − c . c . = (cid:90) p k ∞ dx ln | x − p (cid:96) | f (cid:48) ( x ) + (cid:88) i ∈S k in i | w i − p (cid:96) | (3.47)where S k ⊂ { , · · · , I } is the set of branch points for which the associated branch cutintersects the real line in the interval ( p k , ∞ ) and the integral over x is along the real line.The explicit expression for f (cid:48) ( x ) reads f (cid:48) ( x ) = I (cid:88) i =1 in i π Im ( w i ) | x − w i | (3.48)The final form for ∆ k G can then be written as∆ k G πi = 2 ¯ A Y k + + 2 A Y k − + (cid:88) (cid:96) (cid:54) = k Y [ (cid:96),k ] ln | p (cid:96) − p k | + 2 f ( p k ) Y k ( η − A − η + A − )+ Y k L (cid:88) (cid:96) =1 Y (cid:96) (cid:34) (cid:90) p k ∞ dx f (cid:48) ( x ) ln | x − p (cid:96) | + (cid:88) i ∈S k in i | w i − p (cid:96) | (cid:35) (3.49)To ensure that G = 0 on the entire boundary of Σ we have to enforce ∆ k G = 0 for all k = 1 , · · · , L . We will now give a self-contained summary of the construction of solutions with monodromyand of the regularity conditions, and discuss some additional points. The data feeding intothe construction are L ≥ p (cid:96) on the real line, L − s n in the upper half planeand an overall complex normalization σ . From those one constructs Z (cid:96) ± via Z (cid:96) + = σ L − (cid:89) n =1 ( p (cid:96) − s n ) L (cid:89) k (cid:54) = (cid:96) p (cid:96) − p k Z (cid:96) − = − Z (cid:96) + (3.50)The additional data for the monodromies is given by a pair of real numbers p, q and I punctures w i in the upper half plane, with a real number n i for each puncture and a complex19hase γ i fixing the direction of the branch cut. This data fixes a function f ( w ) = I (cid:88) i =1 n i π ln (cid:18) γ i w − w i w − ¯ w i (cid:19) (3.51)which encodes the branch points and additional branch cut structure. Moreover, with u Q = 1 + η + η − η − v Q = 1 − η + η − η − η ± = p ∓ iq (3.52)we define convenient shorthands for linear combinations of the Z (cid:96) ± as Y (cid:96) + = + u Q Z (cid:96) + − v Q Z (cid:96) − Y (cid:96) − = − Y (cid:96) + Y (cid:96) = Z (cid:96) + − Z (cid:96) − (3.53)The locally holomorphic functions for a solution with monodromy are then given by A ± = A ± + L (cid:88) (cid:96) =1 Y (cid:96) ± ln( w − p (cid:96) ) + η ± (cid:90) w ∞ dz f ( z ) L (cid:88) (cid:96) =1 Y (cid:96) z − p (cid:96) (3.54)where A ± are integration constants that are constrained by ¯ A ± = −A ∓ . The contour forthe integral is chosen inside the upper half plane in such a way that it does not cross any ofthe branch cuts in f , as illustrated in fig. 1.The supergravity fields for these solutions have SL (2 , R ) monodromies given by (3.15)around the points w i , as desired. The residues of the differentials at the poles on the realline played a crucial role in the solutions without monodromy, for the identification withexternal 5-branes. The differentials corresponding to A ± in (3.54) are given by ∂ w A ± = L (cid:88) (cid:96) =1 Y (cid:96) ± + η ± f ( w ) Y (cid:96) w − p (cid:96) (3.55)where we note that the numerators are non-trivial functions of w . The residues of thesedifferentials at the poles on the real line appeared already in (3.43), and are given by Y (cid:96) ± = Y (cid:96) ± + η ± f ( p (cid:96) ) Y (cid:96) (3.56)It is these residues that correspond to the charges of the external 5-branes via the identi-fication reviewed in sec. 2.4: since the match of the geometry close to a pole to a 5-branesolution only uses the local form of the solution around the pole, this match carries over tothe solution with monodromy straightforwardly. We will therefore use the Y (cid:96) ± as shorthandfor the combination in (3.21) whenever convenient.The parameters introduced above are constrained by regularity requirements and there-fore not all independent. The construction already ensures that the regularity conditions on20 are satisfied, but to have single-valued G which vanishes on the boundary the parametersin addition have to be chosen such that eqs. (3.33) and (3.49) are satisfied. Using the con-jugation condition ¯ A ± = −A ∓ and the relation (3.32), we can write these conditions moreexplicitly as0 = 2 η − A − η + A − + L (cid:88) (cid:96) =1 Y (cid:96) ln | w i − p (cid:96) | i = 1 , · · · , I (3.57)0 = 2 A Y k − − A − Y k + + (cid:88) (cid:96) (cid:54) = k Y [ (cid:96),k ] ln | p (cid:96) − p k | + Y k J k k = 1 , · · · , L (3.58)where Y [ (cid:96),k ] = Y (cid:96) + Y k − − Y k + Y (cid:96) − . With S k ⊂ { , · · · , I } denoting the set of branch points forwhich the associated branch cut intersects the real line in the interval ( p k , ∞ ), J k is givenby J k = L (cid:88) (cid:96) =1 Y (cid:96) (cid:34) (cid:90) p k ∞ dxf (cid:48) ( x ) ln | x − p (cid:96) | + (cid:88) i ∈S k in i | w i − p (cid:96) | (cid:35) (3.59)where the integral over x is along the real line. The conditions in (3.58) ensure that the shiftin G across the pole p k , ∆ k G , vanishes, while those in (3.57) ensure that G is continuousacross the branch cuts associated with w i . In contrast to the case without monodromy, thesum over the regularity conditions in (3.58) does not manifestly vanish, and we therefore ingeneral have L independent conditions. However, satisfying the branch point conditions in(3.57) does imply that (cid:80) k ∆ G = 0, and consequently that the sum over the conditions in(3.58) vanishes: By the arguments of sec. 3.4.1, G is constant along each boundary segmentfree of poles. Therefore, since (cid:80) k ∆ G gives the total change in G across all poles, it mustequal the shift in G across all branch cuts. Satisfying (3.57) for each branch point impliesthat G is continuous across all branch cuts and therefore (cid:80) k ∆ G = 0. Having gathered the parameters and the constraints on the parameters for general solutionsto be regular with monodromy in a convenient form, we can now count the moduli. Theparameters associated with the Z (cid:96) ± are s n L − σ p (cid:96) L real parameters (3.60)The remaining parameters are the integration constants A ± , which are related by the conju-gation condition (3.40) and therefore correspond to only 2 real parameters, and the param-21ters directly associated with the punctures and monodromies. Namely, A ± p, q ω i I real parameters n i I real parameters γ i I real parameters (3.61)Altogether, (3.60) and (3.61) are 3 L + 4 I + 2 real degrees of freedom. Those have to satisfythe L + I − SL (2 , R ) automorphisms of the upper half plane, which map toequivalent solutions and can be used to fix, e.g., the position of three of the poles at will.Moreover, one of the parameters in ( p, q, n i ) is redundant, since an overall rescaling of p and q can be compensated by rescaling the n i . We are thus left with2 L − I (3.62)free real parameters. As discussed in [7], the general L -pole solution without monodromyhas 2 L − SL (2 , R ) monodromy that isfixed by ( p, q, n i ). With the n i unconstrained we can take it e.g. as the the phase of p − iq .For I = 0 the dependence on that extra parameter becomes trivial, and the parameter counttherefore reduces to the expected number for a solution without monodromy. [ p, q ] In this section we will discuss the identification of the punctures w i with the location of [ p, q ]7-branes. The monodromies around the punctures in (3.15) are precisely those expectedfor a [ p, q ] 7-brane [22], which certainly suggests this identification. We will discuss this inmore detail by explicitly working out the form of all supergravity fields near the w i . It willbe sufficient to fix [ p, q ]= [1 ,
0] and discuss the relation of the punctures in the resultingsolution to D7-branes. Since the solutions with general [ p, q ] monodromies were obtainedfrom those with [1 ,
0] monodromies by an SL (2 , R ) transformation, and the [ p, q ] 7-branesare related to [1 ,
0] 7-branes by the same SL (2 , R ) transformation, the identification directlyextends to general [ p, q ] once it is established for [1 , w i for [ p, q ]= [1 ,
0] and in sec. 3.7.2we will compare to the expected behavior for D7-branes.22 .7.1 Asymptotic behavior near [1 , branch points For [p,q] = [1 ,
0] we have u Q = 1, v Q = 0, η ± = 1, and the expressions simplify correspond-ingly. In particular, ∂ w A s ± = ∂ w A (0) ± . To analyze the metric factors near w i we need thebehavior of κ , G and ∂ w G . We introduce a coordinate ξ centered on the branch point ξ = γ i w − w i w − ¯ w i (3.63)and will assume | ξ | (cid:28) κ we then find κ = − n i π | c | ln | ξ | + O (1) c = ∂ w A (0) − − ∂ w A (0)+ (cid:12)(cid:12)(cid:12) w = w i (3.64)For the expansion of G it is convenient to separate off the contribution purely from the A (0) ± as ∂ w G (0) , so we have ∂ w G = ∂ w G (0) + c ( I − ¯ I ) + F ( ¯ A (0)+ − A (0) − + A (0)+ − ¯ A (0) − ) (3.65)Due to (3.33), the term multiplying F is O ( ξ ). Moreover, due to the same condition ∂ w G (0) = c ( A (0)+ − ¯ A (0) − ) + O ( ξ ). Therefore, ∂ w G = g + O ( ξ ln | ξ | ) g = c ( A + − ¯ A − ) (cid:12)(cid:12)(cid:12) w = w i (3.66)Note that I and thus A ± are finite at w i , so ∂ w G is finite as w → w i as well. Upon integratingthe same applies for G , and to conveniently collect the O (1) terms we use G = g + O ( ξ ) (3.67)Due to the regularity condition G > g ∈ R + , while g is not constrained, g ∈ C . Using the definition of R yields R = 4 π | g | n i g | c | ( − ln | ξ | ) + O (cid:0) (ln | ξ | ) − (cid:1) (3.68)and therefore R → + as w → w i . With the expression for the metric factors in (2.3), wethen find, to leading order in the ξ -expansion, f ≈ c (cid:112) g f ≈ c (cid:112) g ρ ≈ n i π | c | √ g ( − ln | ξ | ) (3.69)That is, in Einstein frame the radii of AdS and S are finite, while ρ diverges logarithmi-cally. Note that the sign ensures that ρ is positive. With | dw | ≈ | w i − ¯ w i | | dξ | and (2.1),the complete metric takes the form ds ≈ c (cid:112) g (cid:18) ds AdS + 19 ds S (cid:19) + n i | c | π √ g ( − ln | ξ | ) | w i − ¯ w i | | dξ | (3.70)23o derive the expansion of B it is convenient to rewrite (2.4) as B = − ∂ w A + − ∂ w A − ) ∂ ¯ w G + R ( ∂ ¯ w ¯ A + − ∂ ¯ w ¯ A − ) ∂ w G R∂ ¯ w ¯ A + ∂ w G − ∂ w A − ∂ ¯ w G (3.71)Since ∂ w A + − ∂ w A − = ∂ w A (0)+ − ∂ w A (0) − , the numerator in the second term is O (1), whilethe denominator is O (ln | ξ | ). The explicit expansion reads B = − c F + O (cid:0) (ln | ξ | ) − (cid:1) (3.72)The expansion for τ is conveniently derived using τ = − i + 2 i/ (1 + B ), which yields τ = − in i π ln ξ + τ (3.73)where τ is finite at ξ = 0 and single-valued up to terms of O (1 / ln | ξ | ). With τ = χ + ie − φ we find the explicit expressions for axion and dilaton, to leading order near the branch point, χ ≈ − in i π (cid:0) ln ξ − ln ξ (cid:1) + χ e − φ ≈ − n i π ln | ξ | (3.74)where χ is finite at ξ = 0 and single-valued up to terms of O (1 / ln | ξ | ). We therefore findthe expected axion monodromy χ → χ + n i when encircling w i counterclockwise at an in-finitesimal radius. Moreover, we see that the exponentiated dilaton diverges logarithmically.To derive the form of C near the branch point we start from the expression in (2.5). With(3.66) and (3.33), one finds that ∂ w G ∂ ¯ w ¯ A − + ∂ ¯ w G ∂ w A + = O (1). The second term in thebracket of (2.5) therefore is O ((ln | ξ | ) − ). Up to terms of O ((ln | ξ | ) − ), the behavior of C near the branch point is thus given by C ≈ ic (cid:18) − ¯ A − − A + + ln ξ ln | ξ | ( A + − ¯ A − ) (cid:19) (3.75)The two-form potential is therefore finite at the branch point but not necessarily single-valuedacross the branch cut. We note that, due to (3.33), A + − ¯ A − is imaginary at w i . Since themonodromy of ln ξ is imaginary as well, we find that the real part of C is single-valued andonly the imaginary part is in general not.In general, C (2) and correspondingly C transform non-trivially under SL (2 , R ), as givenin (2.15). For the monodromy considered here we would expect the imaginary part of C to receive a shift proportional to the real part of C . However, since ln ξ/ ln | ξ | → C vanisheswhen encircling the branch point at an infinitesimal radius. This reveals the constant gaugetransformation in (2.15) as C = − in i (cid:0) C ( w i ) + ¯ C ( w i ) (cid:1) (3.76)24 .7.2 Matching to 7-branes With the asymptotic behavior of the solution with [1 ,
0] monodromy near the branch pointin hand, we can now attempt a physical interpretation. The form of the monodromy clearlysuggests that the branch points correspond to D7-branes, and we will now extend the discus-sion to include all supergravity fields. The D7-brane solution has been worked out alreadyin [26], but we will take it in the form given in [27]. To match to [27], we rewrite the metricnear the branch point, as given in (3.70), as ds ≈ c (cid:112) g (cid:18) ds AdS + 19 ds S (cid:19) + Im ( H ) | dz | H = − in i π ln z (3.77)where we changed coordinates to z = c | w − w i | ξ . The axion-dilaton τ near the branchpoint, as given in (3.73), then takes the form τ ≈ H + ˜ τ , where ˜ τ is finite at z = 0 andsingle-valued up to terms of O (1 / ln | z | ).The metric of the transverse space parametrized by z immediately matches the form ofthe flat-space D7-brane solution given in (19.74), (19.75) of [27], taking into account thedifference in conventions for the spacetime signature. The (trivial) scaling of the remainingpart of the metric with z in Einstein frame also agrees with the flat-space D7-brane solution,but with AdS × S replacing R , . The axion-dilaton τ matches up to the finite offset ˜ τ ,and for n i = 1 we find the same monodromy. The two-form gauge field is generically non-vanishing at w i , which is another difference to the flat-space D7-brane solution. We thereforefind a D7-brane in a non-trivial background, where the axion-dilaton and the two-form fieldshave non-trivial background values and the D7-brane wraps AdS × S .The stronger background dependence exhibited by a D7-brane compared to the virtualbackground independence observed near any of the semi-infinite 5-branes (as discussed in[7]) can be understood from the behavior of ρ . Close to the poles on the real line, where thesemi-infinite 5-branes reside, the metric factor ρ behaves as O ( r − / | ln r | − / ). Therefore,the metric distance of any interior point of Σ to the location of the pole is infinite. This offersthe possibility to move out on each of the semi-infinite 5-branes of the web and decouplefrom the intersection. Close to the branch point, however, ρ only diverges logarithmically,so the proper distance to other points in Σ remains finite. We can not move away from theintersection to a point where the 5-branes decouple. We will expand on the interpretationin the context of 5-brane webs in sec. 5.The asymptotic behavior of the supergravity fields for a branch point with genericparabolic [ p, q ] monodromy can be obtained by an SU (1 ,
1) transformation with parame-ters given in (3.14a) from the results in sec. 3.7.1. The Einstein-frame metric is invariantwhile the axion-dilaton B and the gauge field C transform as in (2.14), (2.15). Since [ p, q ] 7-branes are obtained from D7-branes precisely by the SL (2 , R ) transformation correspondingto this element of SU (1 , p, q . 25 Example solutions with monodromy
In this section we will explicitly construct example solutions with monodromy and illustratethat the regularity conditions derived in the previous sections are indeed sufficient to guar-antee smooth supergravity solutions with the desired monodromies. We will also explicitlyexhibit the real degree of freedom in choosing the position of the 7-branes.The simplest case to consider are 3-pole solutions. 3-pole solutions without 7-branes areall SL (2 , R ) dual to each other up to an overall rescaling of the charges, as discussed in [7].This is to be expected already from the parameter count: For solutions without monodromythere are 4 independent parameters after taking into account the redundancy due to the SL (2 , R ) automorphisms of the upper half plane. These parameters are further reduced bythe SL (2 , R ) duality transformations of Type IIB supergravity to a single parameter corre-sponding to the overall scale of the residues. For solutions with monodromy, however, this isnot true anymore. For solutions with I ≥ I free parametersaccording to the counting in sec. 3.6 and 2+3 I after taking into account the SL (2 , R ) dualitytransformations of Type IIB supergravity. So the 3-pole solutions already yield families ofinequivalent solutions and we will discuss some of the features in the following. We will start with a simple example where the regularity conditions can be solved straight-forwardly in closed form, to illustrate the procedure and discuss some general points. Wewill consider the case where a solution with D7-brane monodromy is constructed from a3-pole solution where one of the poles corresponds to D5-branes. Recalling the discussion insec. 2.4 that means the corresponding residue Z (cid:96) + is purely imaginary. By SL (2 , R ) dualitythe discussion extends straightforwardly to the case where generic 7-brane charges coincidewith the charges of one of the 5-branes, but to keep the expressions simple we fix them ascorresponding to D7 and D5-branes. In that case we have v Q = 0 and u Q = η ± = 1.For 3-pole solutions the SL (2 , R ) automorphisms of the upper half plane can be used tofix the location of all poles, and we will use p = 1 p = 0 p = − p to correspond to a stack of D5-branes. Since the residues in theseed solutions sum to zero, this constrains the real parts of the other two residues to sum tozero, and we have Z = iN Y = 0 Y = − Y (4.2)with N ∈ R \ { } . This simplifies the regularity conditions (3.57), (3.58) considerably. 2 + I of these conditions are independent and have to be solved. The condition in (3.58) for k = 126xes the real part of the integration constants A ± as A − A − = − Y ln 2 (4.3)Recall that the integration constants are related by the conjugation condition (3.40). Withthat real part fixed we can solve the branch point conditions (3.57), which imply w i = 13 (cid:0) e iα i (cid:1) ≤ α i ≤ π (4.4)The location of the branch points is thus constrained to a half circle of radius 2 / /
3. It intersects the real line at the location of the pole p , corresponding toD5-branes, and at − /
3, in between the other two poles. The remaining regularity conditionsare the conditions in (3.58) for k = 2 ,
3. Since we solved the branch point conditions in (3.57),these remaining conditions are not linearly independent and solving one of them implies theother one. We therefore find only one more real constraint, fixing the imaginary part of A ± which was left unconstrained by (4.3). This yields A = 12 J − Y Y Y ln 2 (4.5)The combination of this A with w i in (4.4) solves all the regularity conditions (3.57), (3.58).The regularity conditions do not fix n i and γ i , and the curve on which the branch pointscan be placed is independent of both parameters. In addition we have one real parameter α i for each puncture, specifying the position of the branch point on the curve in Σ. Thisclearly exhibits the 3 extra parameters introduced by each branch point, in line with thediscussion in sec. 3.6. The additional parameters associated with the branch points do affectthe residues Y (cid:96) ± of the differentials at the poles on the real line, as given in (3.56). With(4.2) they explicitly read Y = Z Y = Z + f ( p ) Y Y = Z − f ( p ) Y (4.6)Since f is imaginary on the real line, the residue at each pole changes by an imaginaryamount proportional to the real part of the residue. That is, the D5 charge of the 5-branechanges by an amount proportional to its NS5 charge. In particular, the residue at p ,corresponding to the D5 charge there, is unaffected by the addition of the D7-branes. Thetotal charge non-conservation is given by (cid:80) (cid:96) Y (cid:96) + = ( f ( p ) − f ( p )) Y . It is independent ofthe choice of γ i , but varies with n i and α i .Regarding the choice of orientation for each branch cut, one can realize “topologically”different configurations, by choosing different pairs of adjacent poles between which thebranch cut intersects the real line. These different configurations have an immediate inter-pretation from the brane intersection picture, namely as the choice of semi-infinite external27ranes between which the branch cut is located. The phases γ i fixing the orientation ofthe branch cuts, however, can be varied continuously. Indeed, fixing all other parametersand varying one of the γ i such that the associated branch cut varies without crossing anyof the poles, we find a linear dependence of the residues Y (cid:96) + on arg( γ i ). The change in theresidue Y k + as the branch cut associated with w i crosses the pole p k is discrete and given by∆ Y k + = i n i Y k . We will come back to the interpretation of the continuous moduli in thebrane web picture in sec. 5. As a last point, we note that the solution without monodromycan be recovered if the branch cuts are chosen e.g. along the negative imaginary directionand the branch points are moved along the allowed curve in (4.4) to approach the real lineat − /
3. At the real line the w i “annihilate” with their mirror points in the lower half plane,leading back to a solution without monodromy. [1 , branch point To illustrate that the constructions outlined in sec. 2 indeed yield solutions with the desiredmonodromies and regularity properties, we will now show explicit plots for a generic solutionwith three poles and one puncture corresponding to a D7-brane. We fix the poles again asin (4.1). As an explicit example we start from the 3-pole solution discussed in sec. 4.1 of [7],for which the zero in the upper half plane and σ were chosen as s = 12 + 2 i σ = i (4.7)Plots of the solution without punctures were shown in sec. 4 of [7]. Adding 7-branes intro-duces additional parameters ( w i , n i , γ i ) as well as the charges p , q . We add a single D7-branewith [p,q] = [1 , u Q = η ± = 1, v Q = 0, and fix n = 1 γ = − k = 1 , A ± straightforwardly, andas the remaining independent constraint we can then take the condition associated with thebranch point in (3.57). That constrains the location of the D7-brane. For the particularsolution (4.7), the resulting curve to which w is restricted is shown in fig. 3. It is not a halfcircle as in the previous example but of similar form. The curve starts and ends on the realline, between the poles p , p and p , p , respectively. For any value of w along the curve,with A ± as described above, all regularity conditions in (3.57) and (3.58) are solved. Wenote that there is no direction along which the branch point could be moved out of Σ alongits branch cut for this choice of γ . The puncture is “trapped” inside Σ in that sense. As inthe previous example, the real parts of the residues are constant along the curve, and givenby Re ( Y ) = 1 Re ( Y ) = − Y ) = 1 (4.9)28 - Re H w L Im H w L Π Π Π Π arg H w L - - Im H Y + { L Figure 3: On the left hand side the allowed locations for the branch point w in the upperhalf plane, for the solution (4.7) with a single puncture corresponding to a D7-brane and n = 1. On the right hand side the imaginary part of the charges along the curve shown onthe left. At arg( w ) = π the curves are, from top to bottom, Im ( Y ), Im ( Y ) and Im ( Y ).But the imaginary parts vary, as shown on the right had side in fig. 3.To explicitly construct the supergravity fields for a set of parameters that solve theregularity conditions as above, we now have to construct the locally holomorphic functions A ± and the composite quantities κ , G explicitly. We do this numerically as follows. Oncethe regularity conditions are solved it is straightforward to construct the differentials ∂ w A ± via (3.55). Constructing the locally holomorphic functions A ± themselves, however, alreadyrequires a more non-trivial integration than in the case without monodromy, as is evidentfrom the expression in (3.54). From the functions A ± we then have to construct the locallyholomorphic function B defined in (2.2) by a further integration. With these functions inhand one can then construct G and R in (2.2) and from those the metric functions via (2.3),the axion-dilaton scalar B via (2.4) and the gauge field via (2.5). To explicitly constructthe supergravity fields we implement a two-step numerical integration procedure. In a firststep we construct I defined in (3.25) and from that the locally holomorphic functions A ± on a dense grid in the upper half plane. Since the A ± feed into the construction of B viaa further integration, they are needed with higher precision than the desired precision forthe supergravity fields. To accurately capture the rapidly varying behavior of A ± aroundthe poles on the real line and around the branch cuts, the grid in particular contains a largenumber of points around the poles and also a large number of points closely tracing thebranch cuts. The freedom in choosing the integration contour in (3.54) (illustrated in fig. 1)can be exploited to avoid rapidly varying regions for all other points. In a second step wethen determine B by a further numerical integration. The grid can be chosen less dense butagain contains a large number of points around the poles and branch cuts, to accuratelycapture the behavior there. Once these functions are determined it is then straightforwardto compute the supergravity fields. 29igure 4: The metric factors f , f and ρ , the real and imaginary parts of the two-formpotential C and axion and dilaton for the 3-pole solution with [1 ,
0] branch point.30or the sake of presenting explicit plots of a solution, we pick a generic point on the curveshown in fig. 3, namely w = 0 . e iπ/ (4.10)Plots of the supergravity fields for the resulting solution are shown in fig. 4. They showthat the branch cut indeed starts at w and from there extends in the positive imaginarydirection. The plots also show that the metric functions are smooth and single-valued, withonly ρ diverging at the position of the D7-brane, as desired. The dilaton blows up at thelocation of the D7-brane but is otherwise smooth, as expected, and the axion has non-trivialmonodromy around w , realizing precisely the shift expected for a D7-brane. The real partof the two-form field is smooth, and also the imaginary part behaves precisely as discussedin sec. 3.7.1. Namely, C transforms by the appropriate SU (1 ,
1) transformation combinedwith a constant gauge transformation such that the limit of C as w → w is well defined.The imaginary part of C also reflects the fact that the imaginary parts of the residues inthe presence of a D7-brane do not have to sum to zero: after crossing all three poles, theboundary value of Im ( C ) does not return to its original value. The discrepancy in the valueof Im ( C ) on the boundary to the left of all poles and to the right of all poles is given by thediscontinuity of Im ( C ) across the branch cut at infinity. The real parts of the residues, onthe other hand, still sum to zero and correspondingly the value of Re ( C ) on ∂ Σ does returnto its original value after crossing all three poles. The behavior of all fields at the poles onthe real line is as expected for an identification of the poles with 5-branes, in the same wayas discussed in more detail in [7]. [0 , branch point As a second explicit example we will consider a case with a different choice of the chargesand a different orientation of the branch cut, to illustrate the features of the solutions in thatcase. We start again from the 3-pole solution (4.1) with (4.7), and add a branch point with[p,q] = [0 ,
1] monodromy, corresponding to the S-dual of a D7-brane. Choosing [p,q] = [0 , η + = − η − = u Q = − i and v Q = 0, and we fix n = 1 γ = 1 (4.11)Solving the regularity conditions proceeds in the same way as outlined for the previousexample, and the location of the branch point is once again restricted to a curve in Σ whichcan be parametrized by arg( w ). From the expression for the residues at the poles on the realline in (3.56) we now see that their imaginary part is unaffected by the addition of the branchpoint, but their real parts change. The conserved linear combination of the ( p, q ) 5-branecharges therefore is the D5-charge, corresponding to the imaginary parts of the residues. The31S5-charge, corresponding to the real parts of the residues, is modified and in general notconserved. To show explicit solutions we again pick a generic point on the curve, namely w = 0 . e iπ/ (4.12)The residues for this particular choice for the location of the branch point are given by Y = 0 . − i Y = 1 . i Y = − . − i (4.13)where the imaginary parts are exact and the real parts evidently do not sum to zero. Plots ofthe metric functions, the two-form gauge field and the axion and dilaton for that solution areshown in fig. 5. The behavior of the metric functions is qualitatively similar to the examplewith [1 ,
0] monodromy: the radii of
AdS and S are finite at the branch point while ρ diverges, as expected. For the two-form gauge field, on the other hand, the imaginary partis now continuous across the branch cut, while the real part is not. Their roles are thusswitched compared to the previous example, as expected. The non-conservation of the realpart of the residues at the poles is reflected in the values of Re ( C ) on the boundary as well:since there is no pole or branch cut at infinity, the boundary value of Re ( C ) to the left of allpoles equals its boundary value to the right of all poles, but the non-conservation is manifestin the discontinuity at the point where the branch cut intersects the real line. Axion anddilaton now both behave non-trivially when crossing the branch cut, reflecting the expectedbehavior for a [1 ,
0] monodromy. Moreover, the exponentiated dilaton e − φ is finite at thebranch point, instead of diverging as previously for the branch point corresponding to aD7-brane. This is the expected behavior after performing an S-duality transformation andcompletes the discussion of all the non-trivial supergravity fields. In summary, we find asolution that satisfies the physical regularity conditions and realizes the desired monodromy.The behavior of the supergravity fields for generic [ p, q ] 7-brane charges is qualitativelysimilar and shows a combination of the features seen for the specific examples we discussedin detail. In general, the real and imaginary parts of C both have a discontinuity across thebranch cut, corresponding to the fact that the conserved linear combination of the chargesdoes not simply reduce to the real or imaginary part of the residues. Likewise, as seen alreadyfor the [0 ,
1] example, axion and dilaton both transform non-trivially. The exponentiateddilaton e − φ is finite at the branch point when q (cid:54) = 0 and diverges if q = 0. The generalizationto multiple branch points with commuting monodromies is likewise straightforward, the plotsbecome more busy but the regularity conditions derived in sec. 3 again guarantee smoothmetric functions and that the two-form gauge field and the axion-dilaton scalar show thedesired behavior across the branch cuts. 32igure 5: The metric factors f , f and ρ , the real and imaginary parts of the two-formpotential C and axion and dilaton for the 3-pole solution with [0 ,
1] monodromy.33
Connection to 5-brane webs with 7-branes
In this section we will discuss the connection of the supergravity solutions constructed insec. 3 to 5-brane webs with additional 7-branes in more detail. We will first revisit the iden-tification with 5-brane intersections and then turn to the punctures and their identificationwith additional 7-branes.As argued in [7], the solutions without monodromy have a compelling interpretation assupergravity descriptions for fully localized intersections of 5-branes, as obtained by takingthe conformal limit of 5-brane webs describing 5d gauge theories. The arguments werebased on having the correct symmetries and parameter count, and in particular on theidentification of the poles on the real line with the external 5-branes defining the intersection.This identification directly carries over to the solutions with monodromy, since it only usesthe leading behavior of the holomorphic data close to the poles and the differentials for thesolutions with monodromy again have simple poles on the real line. By direct extension ofthe identification in sec. 2.4, we therefore find that the poles p (cid:96) on the real line correspondto 5-branes with charges determined by the residues Y (cid:96) ± in (3.56). Analogously to (2.22),the identification with the charge vector ( q , q ) Q in the conventions of [25] is given by( q − iq ) Q = 83 c Y (cid:96) + (5.1)with the real part of Y (cid:96) + corresponding to NS5 charge and the imaginary part correspondingto D5 charge. Compared to the Z (cid:96) ± which determined the charges in the solutions withoutmonodromy, however, the residues Y (cid:96) ± are less constrained. For solutions with D7-branes,only the real parts of the Y (cid:96) ± have to sum to zero: Since f ( p (cid:96) ) is imaginary, η ± = 1 and Y (cid:96) real, eq. (3.56) shows that the real parts of Y (cid:96) ± sum to zero, due to charge conservationin the seed solution without monodromy. But the imaginary parts in general do not. Thiswas clearly exhibited in the example solutions discussed in sec. 4.1 and 4.2, where the sumover the imaginary parts of the residues was non-vanishing. For general [ p, q ] 7-branes thecorresponding SL (2 , R ) rotated statements hold, and we likewise have one real charge con-servation constraint on the complex residues. For [0 ,
1] 7-branes this simply corresponds toswitched roles for the real and imaginary parts of the residues, as exhibited in the example insec. 4.3. We therefore find that the solutions correspond, in general, to 5-brane intersectionswith only one linear combination of the ( p, q ) 5-brane charges conserved.We now come to the punctures themselves. The parabolic SL (2 , R ) monodromies givenin (3.1) have the expected form for a [ p, q ] 7-brane [22], and for multiple coincident braneswe expect precisely a monodromy of the form given in (3.15). As discussed in sec. 3.7 thepunctures can indeed be identified with [ p, q ] 7-branes, and as reviewed in the introductionthe addition of 7-branes into 5-brane webs is well motivated. The way they appear in oursolutions indeed matches well with their role in the 5-brane webs. To recall, if we take the34-branes in the string theory construction to extend along the directions 0 − − − − SL (2 , R ) orbits of [ p, q ] 7-branes in sec. 3.7,instead of anti D7-branes, also has a natural interpretation from the brane web perspective.While for a 7-brane alone both choices are possible and supersymmetric, the differencebecomes crucial in the presence of the 5-branes. To preserve supersymmetry, the 7-branesadded to a 5-brane web have to be compatible with precisely the supersymmetries preservedby the 5-branes, hence explaining the restriction to D7-branes and their SL (2 , R ) orbits.The presence of 7-branes also provides a natural brane web explanation for the fact that theresidues Y (cid:96) ± , corresponding to the charges of the external 5-branes, do not necessarily sum tozero, as discussed in the previous paragraph. 5-branes may cross the branch cuts introducedby the 7-branes, where their charges undergo the corresponding SL (2 , R ) transformationand thus potentially change. Moreover, 5-branes can terminate on the 7-branes, such thattheir charges do not contribute to the total charge of the external 5-branes at all. The totalcharges of the external 5-branes therefore do not necessarily sum to zero in the presence of7-branes, precisely as realized in the supergravity solutions. We thus find a coherent generalpicture where the supergravity solutions constructed in sec. 3 correspond to the conformallimit of 5-brane webs with additional 7-branes.Establishing a precise map between specific brane webs and our supergravity solutionsis beyond the scope of this work, but we will close this section with a speculative generaldiscussion of a possible relation. Since the supergravity solutions correspond to the conformallimit of 5-brane webs and the 7-branes are accessible in the supergravity description, a naturalpossibility would be that the solutions correspond to 5-brane webs with 7-branes inside thefaces of the web. This interpretation aligns well with the fact that we find 7-branes in a non-trivial background, as discussed in sec. 3.7: Taking the conformal limit of a 5-brane web witha 7-brane kept inside a face means the 7-brane ends up precisely on the 5-brane intersection.The geometry created by the 5-branes at that point is AdS × S warped over Σ, and wethus find the 7-brane wrapping AdS × S . There is no limit of moving along the 7-brane35n the 5 , , NN NN
Figure 6: Brane web and intersection with a large- N limit. On the right hand side theconformal limit for generic N , on the left hand side for N = 2 a deformation correspondingto finite gauge coupling and a state on the Coulomb branch.context what the modulus corresponding to the position of the 7-brane in Σ, as exhibited inthe parameter count in sec. 3 and in the examples in sec. 4, would correspond to in the braneweb picture when the 7-brane is trapped at the intersection point. An explanation can begiven by the fact that we are considering solutions corresponding to brane webs in a “large-N”limit. Such brane webs can have a complex internal structure, as illustrated for an examplein fig. 6. The web for N = 2 has four distinct faces, and in the limit where the chargesof the external branes are large this becomes a dense grid of faces in which we can have a7-brane. The discrete choice of which face the 7-brane is in remains in the conformal limitwhere the web collapses to an intersection, and in the large- N limit it becomes effectivelycontinuous. In our supergravity solutions we expect the internal structure of the web to beencoded in Σ, and the choice of position of the branch point could then naturally correspondto the choice of face in which the 7-brane is located. A similar argument can explain thechoice for the orientation of the branch cuts, determined by the continuous parameters γ i .The trajectories of the branch cuts in Σ could have a natural interpretation as correspondingto their trajectory through the dense grid of faces in the corresponding brane webs in thelarge- N limit. This choice again remains meaningful in the conformal limit, giving a possibleinterpretation for all additional parameters associated with the punctures.36 Discussion
We have constructed physically regular
AdS solutions to Type IIB supergravity with 16supercharges, that realize the unique five-dimensional superconformal algebra F (4) geomet-rically. Similarly to the solutions in [6, 7], the geometry takes the form AdS × S warpedover a two-dimensional Riemann surface Σ. Moreover, there are once again mild isolated sin-gularities on the boundary of Σ that correspond to semi-infinite 5-branes. The new featurecompared to the existing solutions is that Σ has punctures around which the supergravityfields undergo non-trivial SL (2 , R ) monodromy. The solutions may in that sense also be re-garded as solutions to F-theory [28]. We have identified the punctures with [ p, q ] 7-branes,and the fact that we can identify both, 5-branes and 7-branes, suggests a direct identifi-cation of the solutions with the conformal limit of 5-brane webs with additional 7-branes,as introduced in [9]. The solutions therefore provide compelling candidates for holographicduals of the UV fixed points of five-dimensional gauge theories that are described by branewebs with additional 7-branes. This offers a clear path for quantitative analyses of the UVfixed points, e.g. of their spectra, entanglement entropies and free energies. We will closewith a discussion of open questions and of some directions for future research.We have collected a number of arguments for the identification of the punctures with7-branes already, and found a coherent general picture for the interpretation of the solutionswe have constructed. To further specify and substantiate the relation to 5-brane webs withadditional 7-branes, a natural next step is to compare supergravity computations, e.g. of thefree energy, to the corresponding field theory or string theory calculations. Moreover, for theidentification of the punctures with the addition of 7-branes additional consistency checkscan already be performed directly in the supergravity description. Namely, via the relationof 5-brane webs with 7-branes to 5-brane webs without 7-branes by the Hanany-Wittenbrane creation effect [9]. It suggests that certain supergravity solutions with punctures,as constructed here, should yield equivalent results in holographic computations as certainsolutions without 7-branes, as constructed previously in [6, 7]. Identifying precisely whichsolutions are equivalent in that sense would provide interesting information about the internalstructure of the webs and further support the identification of the supergravity solutions withbrane webs. A more technical question in that context concerns the role of the puncturesfor holographic computations: As shown in [8], the isolated singularities on the real line donot interfere with supergravity computations at least of the free energy and entanglemententropy. We expect the same to be true for the punctures since the singularities are of asimilarly mild type, but leave an explicit verification for the future.Concerning the solutions themselves, a natural next question is for an extension of theconstructions presented here to include punctures with non-commuting monodromies. We In the context of AdS /CFT , solutions with non-trivial monodromy were recently constructed in [29]. Acknowledgements
We are happy to thank Oren Bergman, Andreas Karch and Diego Rodriguez-Gomez formany interesting discussions on five-brane webs. We also acknowledge the Aspen Centerfor Physics, which is supported by National Science Foundation grant PHY-1066293, forhospitality during the workshop “Superconformal Field Theories in d ≥
4” and thank theorganizers and participants for the enjoyable and inspiring conference. ED is grateful tothe Kavli Institute for Theoretical Physics in Santa Barbara for their hospitality during thecompletion of this work. The work of all three authors is supported in part by the NationalScience Foundation under grant PHY-16-19926. The work of ED is also supported in partby the National Science Foundation under grant NSF PHY-1125915.38
The vanishing of G on ∂ Σ In this appendix we provide further technical details for the derivation of the regularityconditions to guarantee G = 0 in sec. 3.4.2. There are two auxiliary results for which weomitted the derivation in the main part and we will discuss the details in the following.The first result used in sec. 3.4.2 is that the ∆ k B contribution in (3.44) indeed reproducesthe first term, and to evaluate the result more explicitly to arrive at (3.45). Evaluating thefirst term in (3.44) explicitly, using (3.32), yields∆ k G = 2 πi (cid:16) ¯ A Y k + + A Y k − + (cid:88) (cid:96) (cid:54) = k Y [ (cid:96),k ] ln | p (cid:96) − p k | (cid:17) + iπY k (cid:16) I ( p k + (cid:15) ) − I ( p k + ε ) (cid:17) + iπf ( p k ) Y k (cid:16) η − A s + ( p k + ε ) − η + A s − ( p k + ε ) + c . c . (cid:17) + ∆ k B + ∆ k ¯ B (A.1)It remains to evaluate ∆ k B . Starting from (3.42) and using (3.17), (3.22) we find∆ k B = ∆ k B s − (cid:90) C k dz I ( η − ∂ z A s + − η + ∂ z A s − ) + (cid:90) C k dz F ( η − A s + − η + A s − ) (A.2)where B s denotes the part of B constructed from the single-valued differentials and functions,and C k is the half circle contour centered on p k . It is convenient to evaluate ∆ k B togetherwith its complex conjugate. For the last term we find (cid:90) C k dz F ( η − A s + − η + A s − ) + c . c . = iπf ( p k ) Y k (cid:16) η − A s + ( p k + ε ) − η + A s − ( p k + ε ) + c . c . (cid:17) (A.3)where (cid:15) > C k . For the second term in (A.2) wehave to evaluate the integral in I from ∞ to z ∈ C k . It is convenient to split it into the partfrom ∞ to the starting point of C k , p k + (cid:15) , and the remaining part along the half circle C k ,parametrized by p k + (cid:15)e iθ with θ ∈ (0 , π ). Namely, I ( p k + (cid:15)e iθ ) = I ( p k + (cid:15) ) + i(cid:15) (cid:90) θ dφ F ( p k + (cid:15)e iφ ) (A.4)The first term is constant along the integration contour in (A.2) and does not complicate theintegration there. The second term in (A.4) can be evaluated explicitly, since the contour islocalized around p k such that the integrand can be expanded. The contribution to ∆ k B + c . c . then becomes (cid:90) C k dz I ( η − ∂ z A s + − η + ∂ z A s − ) + c . c . = iπY k (cid:16) I ( p k + (cid:15) ) − I ( p k + (cid:15) ) (cid:17) (A.5)39valuating ∆ k B + ∆ k ¯ B using (A.2) with (A.3) and (A.5) shows that it exactly reproducesthe already existing terms in (A.1). We thus find for the shift of G given in (A.1)∆ k G πi = 2 ¯ A Y k + + 2 A Y k − + (cid:88) (cid:96) (cid:54) = k Y [ (cid:96),k ] ln | p (cid:96) − p k | + Y k (cid:16) I ( p k + (cid:15) ) − I ( p k + ε ) (cid:17) + f ( p k ) Y k (cid:16) η − A s + ( p k + ε ) − η + A s − ( p k + ε ) + c . c . (cid:17) (A.6)Using that f ( p k ) is imaginary, we can write the shift in G in the form given in (3.45),completing the derivation for that result.The second result for which we have not provided a detailed derivation in the mainpart concerns the integration by parts in (3.45), to arrive at (3.46). We repeat (3.45) forconvenience∆ k G πi = 2 ¯ A Y k + + 2 A Y k − + (cid:88) (cid:96) (cid:54) = k Y [ (cid:96),k ] ln | p (cid:96) − p k | + Y k (cid:16) f ( p k ) (cid:104) η − A s + ( p k + ε ) − η + A s − ( p k + ε ) (cid:105) − I ( p k + ε ) − c . c . (cid:17) (A.7)The individual terms in the second line are divergent as (cid:15) →
0, but their combination isfinite. We can use A (0)+ − A (0) − = η − A s + − η + A s − and integration by parts to rewrite I definedin (3.25) as I ( p k + (cid:15) ) = (cid:90) p k + (cid:15) ∞ dw f ( w ) (cid:0) η − ∂ w A s + − η + ∂ w A s − (cid:1) = f ( w ) (cid:0) η − A s + − η + A s − (cid:1) (cid:12)(cid:12)(cid:12) p k + (cid:15) ∞ − (cid:90) p k + (cid:15) ∞ dw (cid:0) η − A s + − η + A s − (cid:1) ∂ w f (A.8)The first term evaluated at p k + (cid:15) cancels the first term in the round brackets in the second lineof (A.7). The first term evaluated at ∞ becomes ( η − A − η + A − ) f (+ ∞ ), since (cid:80) (cid:96) Y (cid:96) ± = 0.The integrand of the last term in (A.8) is only logarithmically divergent as p k is approachedand in particular integrable, so we can now drop (cid:15) in the integration bound. The shift (A.7)therefore becomes∆ k G πi = 2 ¯ A Y k + + 2 A Y k − + (cid:88) (cid:96) (cid:54) = k Y [ (cid:96),k ] ln | p (cid:96) − p k | + Y k (cid:18) − ( η − A − η + A − ) f (+ ∞ ) + (cid:90) p k ∞ dw (cid:0) η − A s + − η + A s − (cid:1) ∂ w f − c . c . (cid:19) (A.9)The integral in the second line contains the integration constants A ± , which only multiply40 w f , and it will be convenient to extract them. This yields∆ k G πi = 2 ¯ A Y k + + 2 A Y k − + (cid:88) (cid:96) (cid:54) = k Y [ (cid:96),k ] ln | p (cid:96) − p k | + 2 f ( p k ) Y k ( η − A − η + A − ) + Y k (cid:32)(cid:90) p k ∞ dw L (cid:88) (cid:96) =1 Y (cid:96) ln( w − p (cid:96) ) ∂ w f − c . c . (cid:33) (A.10)This is the result quoted in (3.46), thus completing the derivation for that result as well.41 eferences [1] N. Seiberg, Five-dimensional SUSY field theories, nontrivial fixed points and stringdynamics , Phys. Lett.
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