AdS5 compactifications with punctures in massive IIA supergravity

TL;DR

This paper constructs explicit AdS5 solutions in massive IIA supergravity that are dual to six-dimensional N=(1,0) theories compactified on Riemann surfaces with punctures, revealing new classes with diverse brane configurations.

Contribution

It simplifies previous analyses and finds three explicit classes of solutions with brane setups, including smeared D4-branes and novel dual field theory interpretations.

Findings

Three classes of explicit AdS5 solutions with different brane configurations.

Identification of a class dual to E-string theories coupled to quiver gauge theories.

A class with central charge scaling as n^{5/2}, indicating a 5D origin.

Abstract

We find AdS5 solutions holographically dual to compactifications of six-dimensional N=(1,0) supersymmetric field theories on Riemann surfaces with punctures. We simplify a previous analysis of supersymmetric AdS5 IIA solutions, and with a suitable Ansatz we find explicit solutions organized in three classes, where an O8--D8 stack, D6- and D4-branes are simultaneously present, localized and partially localized. The D4-branes are smeared over the Riemann surface and this is interpreted as the presence of a uniform distribution of punctures. For the first class we identify the corresponding six-dimensional theory as an E-string theory coupled to a quiver gauge theory. The second class of solutions lacks D6-branes and its central charge scales as $n^{5/2}$, suggesting a five-dimensional origin for the dual field theory. The last class has elements of the previous two.

Figures

Equations

$$a=\frac{27}{32}\left(\frac{1}{5}(g-1)N^3{M}^2+\frac{1}{3}n{N}^{2}M\right),$$

$$d{s}_{10}^2$$

$$d{s}_3^2$$

$${\eta }_{\psi }=d\psi -\frac{1}{2}{\star }_2{d}_2{D}_s$$

$$e^{-6\lambda }=\frac{1}{8s}\frac{\det (h)}{\det (g)},e^{4W}=-\frac{{\partial }_s{D}_s}{3\det (h)},e^{2A}=\frac{1}{24}\det (h)e^{D_s},$$

$$\begin{array}{rl}\det (g)& ={\partial }_u{D}_u{\partial }_s{\tilde{D}}_s-{\left({\partial }_u{D}_s\right)}^2,\\ \det (h)& ={\partial }_u{D}_u{\partial }_s{D}_s-{\left({\partial }_u{D}_s\right)}^2.\end{array}$$

$$e^{2\varphi }=e^{6W}{e}^{-6\lambda }.$$

$$H$$

$$V_2$$

$$F_0$$

$$F_2$$

$$-F_0\frac{1}{18\sqrt{2s}}du\wedge {\eta }_{\psi }-F_0\frac{1}{12\sqrt{2}{s}^{3/2}\det (g)}\left({\partial }_u{D}_{u}du+{\partial }_u{D}_{s}ds\right)\wedge {\eta }_{\psi },$$

$$V_1$$

$$F_4$$

$$\displaystyle\times\Bigl{\{}\Bigr{.}\left[\partial_{u}D_{u}\partial_{s}V_{0}-\partial_{u}D_{s}\left(\partial_{u}V_{0}+\tfrac{4}{3}\right)\right]du+\left[\partial_{u}D_{s}\partial_{s}V_{0}-\partial_{s}\widetilde{D}_{s}\left(\partial_{u}V_{0}+\tfrac{4}{3}\right)\right]ds\Bigl{.}\Bigr{\}}$$

$$\wedge d\psi \wedge d{x}_1\wedge d{x}_2$$

$$+\frac{1}{24\sqrt{2}}\frac{(\det (h){)}^2}{\det (g)}\frac{1}{s^{3/2}{\partial }_s{D}_s}\star d_2{V}_0\wedge du\wedge ds\wedge D\psi ,$$

$$V_0$$

$${\Delta }_2{D}_s={\partial }_s\left(s\det (g)e^{D_s}\right)+\frac{1}{24\sqrt{2s}}{F}_0{\partial }_s{e}^{D_s}.$$

$${\Delta }_2\left({\partial }_u{D}_u\right)={\partial }_u^2\left(s\det (g)e^{D_s}\right)+\frac{1}{36\sqrt{2s}}{F}_{0}s{\partial }_s\left(\det (h)e^{D_s}\right).$$

$$e^{2A}=f(s,u)e^{2A_0(x_1,x_2)}$$

$$D_s=F_s(s,u)+2A_0(x_1,x_2),D_u=F_u(s,u)+2v{\tilde{A}}_0(x_1,x_2),$$

$$\left({\partial }_{x_1}^2+{\partial }_{x_2}^2\right)A_0+\kappa e^{2A_0}=0.$$

$$e^{A_0}=\frac{2}{1+\kappa \left(x_1^2+x_2^2\right)}.$$

$${\tilde{A}}_0=\{\begin{array}{ll}-\left(x_1^2+x_2^2\right)& \text{mbox}for\kappa =0,\\ \kappa A_0& \text{mbox}for\kappa \ne 0.\end{array}$$

$$V=-\frac{1}{2}{\alpha }_{\kappa }{\star }_2{d}_2{\tilde{A}}_0={\alpha }_{\kappa }\frac{x_{1}d{x}_2-x_{2}d{x}_1}{1+\kappa \left(x_1^2+x_2^2\right)},{\alpha }_{\kappa }=\{\begin{array}{ll}\frac{1}{4\pi }& \text{mbox}for\kappa =0;\text{vspace}.4cm\\ \frac{\kappa }{1-g}& \text{mbox}for\kappa \ne 0.\end{array}$$

$${\Delta }_2{\tilde{A}}_0=-e^{2A_0},dV=\frac{{\alpha }_{\kappa }}{2}{e}^{2A_0}d{x}_1\wedge d{x}_2,\int dV=2\pi .$$

$$d{s}^2\left({\Sigma }_g\right)=e^{2A_0(x_1,x_2)}\left(d{x}_1^2+d{x}_2^2\right).$$

$${\eta }_{\psi }=d\psi -\frac{1}{2}{\star }_2{d}_2{D}_s=d\psi -2\left(g-1\right)V.$$

$$F_s(s,u)$$

$$F_u(s,u)$$

$$t_1=u+b_1(s),t_2={\ell }_{0}u+b_2(s),$$

$${\tau }_i\equiv {\partial }_s{b}_i(s),{\partial }_s\frac{1}{{\tau }_i^2}=\frac{F_0}{18\sqrt{2s}}.$$

$$f_2=c_2+t_2,f_1=c_1+2\kappa t_1+f_0{\left(c_0+t_1\right)}^{1/3}$$

$${\tau }_1=-{\tau }_2={\left(\frac{18}{F_0\sqrt{2s}}\right)}^{1/2}.$$

$$c_0+t_1=2F_0{z}^3,c_2+t_2=2F_0{z}^3(1-k^3).$$

Full text

1. 1 Introduction

2. 2 Supersymmetric AdS5 solutions: general system

3. 3 New solutions

4. 3.1 Ansatz

5. 3.2 Solutions

6. 3.3 Regularity and brane sources

7. 3.3.1 O8-plane–D8-branes

8. 3.3.2 D6-branes

9. 3.3.3 D4-branes

10. 3.4 The range of the coordinates

11. 3.5 NS–NS Flux quantization

12. 3.6 Central charge

13. 4 Three classes of solutions

14. 4.1 O8, D6, D4

15. 4.1.1 Genus greater than one

16. 4.1.2 Genus one

17. 4.1.3 Genus zero

18. 4.2 O8, D4 distribution

19. 4.3 O8, D6, D4 distribution

20. 5 The six-dimensional theory and its gravity dual

21. 5.1 The ${\rm AdS}_{7}$ solution

22. 5.2 The field theory interpretation

23. A Comparison with [1]

24. B D4-branes inside D8-branes

25. C The limiting case $z_{1}=z_{0}$

26. D $a$ anomaly

1 Introduction

An interesting way of producing a conformal field theory is by reducing a higher-dimensional one. Among four-dimensional superconformal field theories, there is a vast number of examples that come from compactifying six-dimensional theories. This has been demonstrated for the ${\cal N}=(2,0)$ theory living on a stack of M5-branes: upon compactification on Riemann surfaces, it produces interesting ${\cal N}=2$ theories in four dimensions, known as “class $\mathcal{S}$” theories. This can be understood field-theoretically [2, 3, 4] and from a holographic perspective [5, 6]. It is also possible to compactify the ${\cal N}=(2,0)$ theory in a more complicated way, so as to produce ${\cal N}=1$ theories [5, 7, 8, 9, 10].

In six dimensions, string theory also suggests the existence of a much larger class of superconformal field theories with ${\cal N}=(1,0)$ supersymmetry; see for example [11, 12, 13, 14, 15, 16, 17]. Recently, certain classifications have been proposed, based on anomalies and supersymmetry [18] and on F-theory [19]. Moreover, a holographic description has been found [20, 21, 22, 23] for a large class of such theories [15, 16]. These theories have an effective description in terms of a chain of unitary gauge groups, coupled to tensor multiplets and hypermultiplets. They are engineered by an NS5–D6–D8 brane system. Their holographic duals are AdS${}_{7}\times M_{3}$ solutions with D8/D6-brane sources, where $M_{3}$ has the topology of the three-sphere $S^{3}$.

It is natural then to ask whether these six-dimensional ${\cal N}=(1,0)$ theories also produce superconformal field theories (SCFTs) upon compactification on Riemann surfaces $\Sigma$. In [1, 24] it was shown holographically that by compactifying the class in [20] on $\Sigma$ of genus $g>1$ one indeed obtains SCFTs with ${\cal N}=1$ supersymmetry in four dimensions.111The particular case $g=1$ was also considered for more general theories in [25, 26, 27, 28]. A next step would be to understand these theories more directly. Although they are not expected to admit a Lagrangian description, they might be amenable to a decomposition similar to the one obtained for the class $\mathcal{S}$ theories in [3, 4]. There it was shown how to associate a “generalized quiver” description to a pair-of-pants decomposition of a Riemann surface $\Sigma$, namely to a choice of representation of $\Sigma$ as a union of three-punctured spheres and tubes. To each three-punctured sphere one associates a class of theories that correspond to compactifying the ${\cal N}=(2,0)$ theory in presence of codimension-2 defects of various types. The theories associated to $\Sigma$ can be constructed by gluing three-punctured spheres [3, 4]. In [6] the holographic dual for this construction was described.

Obtaining a similar picture for ${\cal N}=(1,0)$ theories compactified on $\Sigma$ would greatly enlarge our knowledge of ${\cal N}=1$ SCFTs in four dimensions. So far, this has been attempted [29, 30, 31, 32, 28] for one particular ${\cal N}=(1,0)$ theory, namely the one describing a stack of M5-branes on top of a $\mathbb{Z}_{k}$ singularity. (This theory is also part of the class obtained in IIA in [20]: it is the only one with vanishing Romans mass.)

In this paper, we find AdS5 solutions which are dual to compactifications on punctured Riemann surfaces of any genus, of certain ${\cal N}=(1,0)$ theories belonging to the family of [20]. These ${\cal N}=(1,0)$ theories are obtained by coupling a six-dimensional quiver tail to the E-string theory of rank one (see figure 6), a construction following [15, 16]. The E-string theory has a tensor multiplet but no gauge group, and an $E_{8}$ flavor symmetry. It first appeared as a description of an M5-brane near an M-theory $E_{8}$ wall [33, 34, 11], but since then, has often played a role in other contexts, for example in the various theories engineered in F-theory [19, 35]. In our case, it appears because the six-dimensional theories we are compactifying have an AdS7 dual which contains, in addition to more “customary” D6-branes, an O8–D8 stack.222This possibility was already implicitly present in [22, 1], as a particular limit of the parameters, but was not noted at that time. The corresponding brane diagram (see figure 5(a)) describes $N$ NS5-branes near the O8–D8 stack, in the presence of Romans mass; because of the latter, D6-branes are also created. The resulting theory can be thought of as an analogue with Romans mass, of the rank-$N$ E-string theory describing $N$ M5-branes near an $E_{8}$ wall. For this reason, we will refer to our six-dimensional models as “massive E-string theories”. The holographic identification between the AdS7 solution and these theories is also bolstered by an anomaly computation similar to [23], which yields a perfect match.

The AdS5 solutions we find have in addition to the aforementioned D6-branes and O8–D8 stack, D4-branes, which represent the punctures on the Riemann surface. The D4-branes are extended along the AdS5 and smeared over the Riemann surface $\Sigma$. The latter means that there are many punctures distributed over $\Sigma$. In the rest of the internal space which is topologically an $S^{3}$ (or rather, half-$S^{3}$, because of the presence of the O8-plane), the D4-branes are completely localized on a point on top of the O8-plane. Our solutions thus contain sources of three different kinds (O8–D8, D6, D4), almost all completely localized; this is an exceptionally complex set of localized ingredients. Identifying the various sources requires comparing the field behavior near them to the one known in flat space. This is nontrivial especially for the D4-branes inside the O8-plane, and is performed here with a variant of the analysis in [36].

We have checked the field theory interpretation of our solutions by computing the central charge:

[TABLE]

where $N$ is the number of NS5-branes, $M$ is the number of D6-branes, and $n$ is the number of D4-branes. The ratio $27/32$ is typical of ${\cal N}=1$ theories obtained by mass deforming $\mathcal{N}=2$ theories [38]. In the parenthesis, the first term comes from the compactification of the massive E-string theory on a genus $g$ surface $\Sigma$. The $N^{3}M^{2}$ behavior is typical of ${\cal N}=(1,0)$ theories engineered from $N$ NS5-branes and $M$ D6-branes [21, 23]. The fact that it is proportional to $(g-1)$ is also standard [6, 7]. The second term in (1.1) is, quite sensibly, proportional to the number of punctures $n$; the contribution of each puncture scales as $N^{2}M$, suggesting that these are the analogue of the “simple” punctures of class $\mathcal{S}$ theories. Notice that for the case $\Sigma=S^{2}$ ($g=0$) the first term is negative; however, the second term is always positive enough to keep $a>0$, due to a lower bound $n\geq NM$ on the number of punctures in this case. (No such bound exists for $g>0$.) This again is in agreement with the intuition from class $\mathcal{S}$ theories, where a sphere cannot have too low a number of simple punctures.

In order to find the solutions, we started from the classification of supersymmetric AdS5 solutions of massive type IIA supergravity[1], but in a simpler reformulation which reduces the number of partial differential equations (PDEs) that characterize the classification. The latter acquire a form which is reminiscent of the Toda–Monge–Ampère system in [10], itself a generalization of the Toda equation in [37]. This observation can be useful for broader aims than the ones in this paper. We then used a separation of variables Ansatz inspired again in part by [10], in order to solve the PDEs.

In the process of looking for our punctured compactifications, we have also found some superficially similar solutions, which however appear to represent rather different physics. In the $\Sigma=S^{2}$ case, if one varies a parameter beyond a certain value, one finds a solution without any D6-branes, and with zero NS–NS flux integral, indicating also an absence of NS5-branes before the near-horizon limit. Moreover, some of the D4-branes have now moved off the O8-plane. In this case $a$ scales as $n^{5/2}$, which is the same scaling as of the action of the AdS6 solution of [39], arising as the near-horizon geometry of a O8–D8–D4 system. This might suggest a relation between the two solutions, and hence a five-dimensional origin of the four-dimensional SCFTs dual to the AdS5 solutions. For $g>1$, we also find a solution where the NS5-branes and D6-branes are still present, but all the D4-branes have moved off the O8-plane. In this case $a$ exhibits a mix of the behavior in (1.1) and of $a\sim n^{5/2}$. All these alternative solutions are intriguing, but we are not giving them an interpretation here.

The rest of this paper is organized as follows. In section 2 we present a reformulation of the classification of supersymmetric AdS5 IIA solutions of [1]. In section 3 we obtain our new analytic solutions and reduce their study to three classes, which we analyze in detail in section 4. The solutions of section 4.1 are the ones which we interpret as punctured compactifications of six-dimensional ${\cal N}=(1,0)$ SCFTs. In section 5 we review these SCFTs and their holographically dual AdS7 solutions.

2 Supersymmetric AdS5 solutions: general system

We begin by presenting a refinement of the classification of supersymmetric AdS5 solutions of massive type IIA supergravity worked out in [1]. In particular, by introducing a new set of functions characterizing the solutions, we are able to reduce the number of partial differential equations that control the classification, as well as simplify their form. This simpler system of equations was inspired by the one obtained for M-theory AdS5 solutions in [10], and its reduction along a flavor isometry to ten dimensions. The connection between the formalism presented here and the one in [1] is summarized in appendix A.

The metric for a general supersymmetric AdS5 solution is

[TABLE]

where

[TABLE]

and $\widetilde{D}_{s}=D_{s}-\frac{3}{2}\ln s$333The labeling of the potential $D_{s}$ is related to the one in [10] as $(D_{s},\widetilde{D}_{s})\to(\widetilde{D}_{s},D_{s})$.. The Hodge star operator $\star_{2}$444The convention for its action is $\star_{2}dx_{1}=dx_{2}$ and $\star_{2}dx_{2}=-dx_{1}$. This convention is opposite of the one used in [10]. and the exterior derivative $d_{2}$ are taken over the $(x_{1},x_{2})$ plane. The functions $(D_{u},D_{s})$, which determine the solution, depend on $(x_{i},s,u)$. The rest of the functions appearing in the metric are given in terms of these as follows:

[TABLE]

and

[TABLE]

The geometry has a U$(1)$ isometry whose orbits are parameterized by $\psi$. It is in fact a symmetry of the full solution and corresponds to the R-symmetry of the dual superconformal field theory.

The dilaton can be expressed as

[TABLE]

The NS–NS field strength $H$ is given by

[TABLE]

The R–R field strengths read

[TABLE]

In the above $\Delta_{2}$ is the Laplace operator $\Delta_{2}=\partial^{2}_{x_{1}}+\partial^{2}_{x_{2}}$.

The Bianchi identity of the Romans mass $F_{0}$ sets it to a constant. The one of the NS–NS field strength, $dH=0$, yields an equation for $D_{s}$:

[TABLE]

Given the above, the Bianchi identity of $F_{2}$, $dF_{2}-F_{0}H=0$, becomes

[TABLE]

A potential extra integrability condition, obtained by acting on (2.10) with $\partial_{u}^{2}$ and on (2.11) with $\partial_{s}$, is automatically satisfied upon using (2.7). The Bianchi identity of $F_{4}$ is also automatically satisfied.

Summarizing, the solution is determined by $(D_{u},D_{s})$ which are subject to equations (2.7), (2.10) and (2.11).

3 New solutions

We will now look for new AdS5 solutions, by using the system obtained in the previous section and introducing a suitable Ansatz. It involves the presence of a Riemann surface $\Sigma$, coordinatized by $x_{1}$, $x_{2}$. The remaining coordinates describe a three-manifold $M_{3}$, topologically fibred over $\Sigma$.

3.1 Ansatz

In this paper, we study solutions whose metric on the $(x_{1},x_{2})$ plane has constant sectional curvature. This requires the warp factor $A$ to be separable:

[TABLE]

for some function $f(s,u)$. In order to satisfy the separability condition (3.1) we make an appropriate Ansatz for the potentials $(D_{u},D_{s})$. The most general one is

[TABLE]

where $v$ is an arbitrary parameter.

Riemann Surface

In the Ansatz for the $D$ potentials, $A_{0}(x_{1},x_{2})$ is a solution of the equation

[TABLE]

The parameter $\kappa$ is restricted to the values $\{-1,0,1\}$, without loss of generality. The three choices correspond to the hyperbolic space $H_{2}$, the torus $T^{2}$ and the two-sphere $S^{2}$ respectively. The hyperbolic space $H_{2}$ can be replaced by the quotient $H_{2}/\Gamma$ to obtain a constant curvature Riemann surface of genus $g$. $\Gamma$ is a Fuchsian subgroup of the PSL$(2,\mathbb{R})$ isometry group of $H_{2}$. A representative solution to (3.3) is

[TABLE]

The function $\widetilde{A}_{0}$ is given as

[TABLE]

It is convenient to introduce the connection one-form, $V$, which is defined as

[TABLE]

The normalizations are such that

[TABLE]

The local metric on the Riemann surface is

[TABLE]

The one-form dual to the $\partial_{\psi}$ Killing vector can be written as

[TABLE]

Ansatz for $(F_{u},F_{s})$

When we plug the Ansatz for the $D$ potentials into the system of equations of section 2, we obtain Monge–Ampère equations for $(F_{u},F_{s})$. By studying how the known AdS5 solutions in type IIA and 11D supergravity solve these, we refine our Ansatz as

[TABLE]

where $\nu$ is a constant and

[TABLE]

with the parameter $\ell_{0}\in\{0,1\}$. Furthermore,

[TABLE]

The aim is to find all solutions for $f_{i}$ in terms of the variables $t_{i}$. The solutions of [7] are obtained by taking $f_{i}$ linear and by fixing $\ell_{0}=1$ and $F_{0}=0$. The solutions of [1] are also obtained by taking $f_{i}$ linear and $\ell_{0}=1$ but with non-zero $F_{0}$. The solutions of [40] can be obtained by fixing $\ell_{0}=0$ and by taking $f_{1}$ linear.

The class of solutions we study in this paper are

[TABLE]

where the set of parameters $(c_{2},c_{1},c_{0},f_{0})$ are integration constants. In this class of solutions we have

[TABLE]

The $\tau$’s can be integrated for the $b$’s. In order to study the solutions, we make a further coordinate transformation from $(t_{1},t_{2})$ to $(z,k)$:

[TABLE]

We now write and study the solutions.

3.2 Solutions

The metric that follows from the solution (3.14) in the $(z,k)$ coordinates is

[TABLE]

where

[TABLE]

and a prime denotes differentiation with respect to $z$. Explicitly,

[TABLE]

The parameters $z_{0}$, $z_{1}$ and $\ell$ are real, and $\ell\in\{-1,1\}$555These parameters are related to the ones in (3.14) as

$f_{0}=-6\left(2F_{0}\right)^{2/3}\ell z_{1}^{2}\,,\qquad c_{1}-2\kappa c_{0}=4z_{0}F_{0}\left(3\ell z_{1}^{2}-\kappa z_{0}^{2}\right)\,.$

(3.20)

. Without loss of generality $z_{1}$ is chosen to be non-negative. The parameter $\kappa$ is the curvature of the Riemann surface $\Sigma_{g}$ of genus $g$, with local metric given in (3.8). The one-form $\eta_{\psi}$ is also given above in (3.9).

The warp factor is given by the expression

[TABLE]

The metric on AdS5 is taken to be of unit radius. A radius $L$ can be reinstated by rescaling

[TABLE]

Positivity of the metric requires

[TABLE]

The metric (and indeed the complete solution) is invariant under the simultaneous reflection $z\to-z$ and $z_{0}\to-z_{0}$, and so we will restrict our study to $z\geq 0$.

The dilaton reads

[TABLE]

Finally, we present the expressions of the fluxes in terms of potentials $B$ for the NS–NS flux and $C_{1}$, $C_{3}$ for the R–R fluxes:

[TABLE]

subject to the gauge transformations

[TABLE]

Their expressions are

[TABLE]

This gauge choice is particularly convenient for the purpose of presentation. As we will see, the above potentials are actually singular at certain loci. They will however be sufficient for computing the charges of the various sources in our solutions. We postpone a more careful treatment to section 3.5.

3.3 Regularity and brane sources

In this section we study the geometry in special regions where the $S^{1}$ parameterized by $\psi$ shrinks or where the metric is singular. The latter regions are (i) $k=0$, (ii) $z=0$ and (iii) $p^{\prime}=0$. We will demonstrate that these singularities correspond to brane sources. This will be one of the main results in this paper.

The regions where the $S^{1}$ shrinks are (i) $p=0$ and (ii) $k=1$. We will begin by examining these.

First we consider a region where $p=0$ and $p^{\prime}\neq 0$. Let $z_{r}$ be a single root of $p$. We expand $p$ around $z_{r}$, $p(z)=p^{\prime}(z_{r})(z-z_{r})+O\left((z-z_{r})^{2}\right)$, and introduce a new coordinate, $\varrho$, as $\varrho^{2}=\frac{4}{3}(1-z/z_{r})$. The metric in the region $z=z_{r}$ takes the form

[TABLE]

and hence is regular provided that the period of $\psi$ is fixed to be $2\pi$. The dilaton reads

[TABLE]

Similarly, near $k=1$ we introduce the coordinate $\varrho^{2}=\frac{4}{3}(1-k)$ and take the $\varrho\to 0$ limit. The metric takes the form

[TABLE]

where the warp factor is $e^{4W}=-3zp/p^{\prime}$. Fixing the period of $\psi$ to be $2\pi$, the $S^{1}$ shrinks in a regular way. The dilaton reads

[TABLE]

Although in the two separate limits above the $S^{1}$ can shrink regularly, when the double limit is considered a singularity appears. As we will see later this is due the presence of D6-branes.

3.3.1 O8-plane–D8-branes

In this section we study the region near $k=0$, away from $z=0$ or $p^{\prime}=0$. We first make the coordinate transformation $k=f(z)r$ with

[TABLE]

which maps the region near $k=0$ to the one near $r=0$. The differential $dk$ becomes $dk=f(z)\left(dr+rf^{\prime}(z)/f(z)dz\right)$. Following the coordinate transformation, the metric near $r=0$ takes the form

[TABLE]

The dilaton reduces to

[TABLE]

The solution in this region describes a stack of $2n_{8}$ D8-branes, with $n_{8}<8$, stuck on an O8-plane. The metric and dilaton for the latter system are given by

[TABLE]

where $ds^{2}_{||}$ is the line element of the flat spacetime parallel to the wordlvolume of the D8-branes, and $x_{9}$ is the transverse coordinate. $g_{s}$ is the string coupling constant. The function $H_{8}$ reads

[TABLE]

with $F_{0}$ being the Romans mass, $\ell_{s}$ the string length and $c$ an arbitrary constant.

The O8–D8 metric is matched to the solution near $k=0$ by the identification666The $(g_{s}F_{0})^{-3/4}$ factor in $ds^{2}_{||}$ can be removed by rescaling the coordinate $z$ in $ds^{2}_{9}$.

[TABLE]

While $c$ can be of any value in general, for our solution $c=0$. Since for us $F_{0}>0$, we have $n_{8}<8$. Looking at (3.38), we see that this implies that the dilaton diverges on the O8-plane. The same behavior occurs for example in the AdS6 solution of massive type IIA supergravity [39], which arises as the near-horizon geometry of an O8–D8–D4 system.

In the region near $k=0$ the fluxes are non-zero, but there are no cycles that yield quantization conditions.

As observed earlier in (3.23), $k\in[0,1]$. The presence of an O8-plane at $k=0$ suggests that our solution is, in fact, half of a bigger solution for which $k\in[-1,1]$. We can then continue our solution to $k\in[-1,0)$ by imposing the standard O8-plane conditions: $g_{\mu\nu}(-k)=g_{\mu\nu}(k)$ for $\mu,\nu\neq k$, $g_{\mu k}(-k)=-g_{\mu k}(k)$, while for the fluxes $B(-k)=-B(k)$, $F_{4i}(-k)=-F_{4i}(k)$, $F_{4i+2}(-k)=F_{4i+2}(k)$.

3.3.2 D6-branes

In this section we study the singularity in the region near $k=1$ and $p=0$. Let $z_{r}$ be a (single) zero of $p$ and introduce the coordinates $(r,\theta)$ via

[TABLE]

The region of interest maps to $r=0$ in these coordinates.

The metric takes the form

[TABLE]

and the dilaton

[TABLE]

The neighborhood of $\{z=z_{r},\,k=1\}$ can be identified with the region near a stack of $M$ D6-branes with ${\rm AdS_{5}}\times\Sigma_{g}$ world-volume. The number of D6-branes $M$ is related to the parameters of the solution, as we will see shortly.

In the string frame, the type IIA supergravity solution, that describes the geometry of a stack of $M$ D6-branes is

[TABLE]

with

[TABLE]

The parameters $g_{s}$ and $\ell_{s}$ are respectively the string coupling and string length. The coordinate $r$ is the overall radial coordinate of the transverse space with metric $ds^{2}_{\perp}$. The metric on the world-volume of the D6-branes correspond to $ds^{2}_{||}$. For the solution of interest in (3.42), we fix the spaces as

[TABLE]

where $L$ is the radius of the AdS5. We can identify $H_{6}$ from the warp factor of the transverse space and from the expression of the dilaton. These, respectively, yield the following relations for $L_{6}$

[TABLE]

These relations taken with the expression of $L_{6}$ in terms of $M$ yield

[TABLE]

The AdS5 radius is fixed, by matching the warp factor of the world-volume space, to $L=3z_{r}$.

The identification in (3.49) is made with the assumption that $r$ in (3.42) is the radius of the space transverse to the D6-branes. Since it was done by matching the warp factor with $H_{6}$ in the near-brane region, there is an overall scaling between $r$ and the radius of the transverse space that is not fixed. However, we will compute the number of D6-branes more accurately by using the flux.

In the $\{z\to z_{r},\,k\to 1\}$ limit, at leading order, the R–R fluxes read

[TABLE]

The explicit expressions for the functions $\alpha_{1}$ and $\alpha_{3}$ are unnecessary for the present discussion: in the region near $r=0$ the terms with a $dr$ component do not yield a quantization condition.

The field strength $dC_{1}$ can be integrated over the two-sphere $S^{2}_{z_{r}}$ with coordinates $(\theta,\psi)$, and $dC_{3}$ can be integrated over the four-cycle $S^{2}_{z_{r}}\times\Sigma_{g}$. The quantization conditions

[TABLE]

for $dC_{1}$ and $dC_{3}$ respectively give

[TABLE]

The first parameter $M$ counts the number of D6-branes and its expression is consistent with (3.49). The parameter $m$ counts the number of D4-branes in this region, which are dissolved in the D6-branes. The volume $V_{g}$ of the Riemann surface $\Sigma_{g}$ is fixed by the curvature so that

[TABLE]

There are no cycles that yield a quantization condition for the NS–NS flux.

3.3.3 D4-branes

In this section we analyze the singularities at $z=0$ and $p^{\prime}=0$. As we will see, these regions describe smeared D4-branes. In the $z=0$ case the smearing is on the Riemann surface $\Sigma_{g}$; in the $p^{\prime}=0$ case, it is on $\Sigma_{g}$ and two more internal directions.

$z\to 0$ limit

In the region near $z=0$ we make the coordinate transformation $\{z^{3}=(\frac{3}{2}r)^{2}$, $k^{3}=\cos^{2}(\theta)\}$, and write the metric as

[TABLE]

where $Q\equiv\left(\frac{3}{2}\right)^{-4/3}\ell z^{2}_{1}/p(0)$. The dilaton reads:

[TABLE]

This metric describes a stack of D4-branes with AdS5 world-volume, which are smeared on the Riemann surface, $\Sigma_{g}$, while being inside a stack of D8-branes. We describe this system in more detail in appendix B. To identify the number of the D4-branes, we look at the $dC_{3}$ field strength in this region:

[TABLE]

The quantization condition for $dC_{3}$ then yields

[TABLE]

$n$ is an integer: its absolute value the number of D4-branes in this region.

The R–R field strength $dC_{1}$ vanishes in this region, and there are no cycles that can provide a quantization condition for the NS–NS flux. However, we point out that the latter is singular near $z=0$:

[TABLE]

The singularity of $H$ in this region is a phenomenon which we do not fully understand; an analogous singularity in the region near the D6-branes is discussed in section 3.5.

$p^{\prime}\to 0$ limit

We now turn to the $p^{\prime}=0$ region. The function $p^{\prime}$ can vanish at $z=z_{1}$ for $\kappa=\ell$. In this region, we introduce the coordinate $r=\frac{3}{2}\kappa(1-z/z_{1})$. In the limit $r\to 0$ and upon making the coordinate transformation $k=\cos^{2/3}(\theta)$, the metric takes the form:

[TABLE]

where $\tilde{Q}\equiv\frac{4}{3}z_{1}/p(z_{1})$. The dilaton reads:

[TABLE]

In this case the local behavior of the metric can be recognized as an example of the “harmonic function rule” for delocalized branes [41, 42, 43]. It describes a stack of D4-branes with AdS5 world-volume, smeared on the Riemann surface $\Sigma_{g}$ and on the two-sphere with coordinates $(\theta,\psi)$. The overall warp factor blows up at $\theta=\pi/2$, and this due to the fact that there is a O8–D8 system localized there.

The $dC_{3}$ field strength in this region reduces to

[TABLE]

and yields the following quantization condition:

[TABLE]

This coincides with (3.57), the integer $n$ counting the number of D4-branes. There are no cycles that can provide a quantization condition for the R–R field strength $dC_{1}$ and the NS–NS flux in this region.

3.4 The range of the coordinates

In this section, we determine the possible ranges for the coordinates $\psi$, $k$ and $z$. Each choice of ranges will correspond to different classes of solutions.

The regularity analysis in the previous sections restricts the circle coordinate $\psi$ and the coordinate $k$ as

[TABLE]

There are more options for $z$ since there are three “special” regions: [math], $z_{r}$ and $z_{1}$ where there are brane sources. The possible intervals depend on the ordering of these points along $z$ and on the positivity conditions (3.23) for a given choice of $\kappa$ and $\ell$. In what follows we use these conditions to identify the possible choices; we will summarize the result at the end.

First consider the cases when $\kappa=0$ or when $\kappa=-\ell$. The only real root of $p$ is $z_{0}$, whereas $p^{\prime}$ has no real roots. The positivity condition on $p^{\prime}$ implies that $\kappa=-\ell=-1$. For these cases $z\in[0,z_{0}]$, and obviously $z_{0}>0$.

When $\kappa=\ell$, $p^{\prime}$ has roots at $z=\pm z_{1}$ and $p$ can have three real roots. The system is analyzed by writing $p$ as

[TABLE]

where the roots satisfy

[TABLE]

Without loss of generality we can take $z_{0}$ to be always real, whereas

[TABLE]

When $z_{\pm}$ are imaginary, the positivity conditions on $p$ and $p^{\prime}$ yield

[TABLE]

When the roots are all real, there is a permutation symmetry among them; moreover at least one is negative and at least one is positive. Without loss of generality $z_{-}$ can be chosen to be the smallest negative root and $z_{0}$ the smallest positive root. Depending on whether $z_{+}$ is negative or positive there are two orderings of the roots of $p$ and $p^{\prime}$: (a) $(z_{-},-z_{1},z_{+},0,z_{1},z_{0})$ and (b) $(z_{-},-z_{1},0,z_{0},z_{1},z_{+})$. The positivity conditions yield

[TABLE]

The possible choices of intervals for $z$ consistent with the positivity conditions in (3.23) can be summarized as follows:777Recall that we are taking $z\geq 0$ and $z_{1}\geq 0$ without loss of generality.

$z\in[0,z_{0}]$

This interval is realised for $\ell=1$ and all values of $\kappa$, with the restriction $z_{0}\leq z_{1}$ for $\kappa=1$ and $z_{1}>0$ for $\kappa=0$. 2. 2.

$z\in[0,z_{1}]$

This interval is realised for $\ell=\kappa=1$, with the restriction $z_{0}<-2z_{1}$ and $z_{1}>0$. 3. 3.

$z\in[z_{1},z_{0}]$

This interval is realised for $\ell=\kappa=-1$, with the restriction $z_{1}<z_{0}$.

Each of these cases correspond to a different branch of the space of solutions, and describes a different class of theories. We will explore these three cases in detail in section 4.

3.5 NS–NS Flux quantization

In the discussion of the various sources in the main solution (3.17), the possible quantized fluxes from the R–R potentials were computed and identified with the charges of various branes. There is an additional quantization condition that comes from integrating the NS–NS flux $H$ over the whole internal space $M_{3}$: it measures the number of NS5-branes wrapping the Riemann surface, whose near-horizon limit yields the AdS5 solution. In this section we discuss this quantization condition, carefully considering some of the regularity issues regarding the NS–NS potential $B$ given in (3.27).

The NS–NS flux is $H=dB$ on the domain $D$ defined by the range of the coordinates $k$ and $z$. While $k\in[0,1]$, there are three possible ranges for $z$, as discussed in the previous section. Let us denote a given range of $z$ as $[z_{\rm L},z_{\rm R}]$ and write the domain as $D=[0,1]\times[z_{\rm L},z_{\rm R}]$. By Stokes’ theorem, the flux integral is $\int_{D\times S^{1}}H=\int_{\partial D\times S^{1}}B$ where the $S^{1}$ is the $\psi$-circle. From the component under consideration in (3.27) i.e. the $dk\wedge\eta_{\psi}$ term, we obtain

[TABLE]

The relevant values of the function in brackets are

[TABLE]

From the summary in section 3.4, we see that there is nonzero flux only for cases 1 and 3, given as

[TABLE]

Flux quantization requires

[TABLE]

In the quantization of the NS–NS flux, there are no subtleties regarding the regularity of $B$ at the boundary of $D$. This is not the case in the quantization of the R–R flux. For example, the gauge choice for $B$ could influence the result for the number of D6-branes computed in section 3.3.2. So we need to worry about regularity near that region. First, let us analyze the $k\to 1$ and $z\to z_{r}$ limits separately.

The standard way to recognize a regular form is to transform to local Cartesian coordinates. If $r$ is the local radial coordinate where a circle shrinks, so that the metric contains a $dr^{2}+r^{2}d\psi^{2}$ “piece”, we can transform to local Cartesian coordinates $x=r\cos\psi$, $y=r\sin\psi$. Then we see for example that $r^{2}d\psi=xdy-ydx$ and $rdr\wedge d\psi=dx\wedge dy$ are regular forms, while $d\psi=\frac{xdy-ydx}{x^{2}+y^{2}}$ is not.

Around $k=1$, the radial coordinate is $r=\sqrt{1-k}$ and $B\sim rdr\wedge d\psi+(1-r^{2}){\rm vol}_{\Sigma_{g}}$, which is regular (the dependence on $z$ is suppressed). On the other hand, around $z=z_{r}$ the radial coordinate is $r=\sqrt{z_{r}-z}$ and $B$ has a $dk\wedge d\psi$ component which is not multiplied by a vanishing function of $r$; this is not regular.

Thus we have to find another gauge for $B$. A simple choice (but by no means the only one) is to perform the gauge transformation

[TABLE]

where $C_{1}$ was given in (3.28). This choice does not spoil regularity of $B$ at $k=1$, while at $z=z_{r}$ it cures the regularity problem we just saw: the coefficient of $dk\wedge d\psi$ now has a double zero in $r$.

We now have to take care of the simultaneous limit $\{z\to z_{r},\,k\to 1\}$ which, as we saw in section 3.3.2, yields a region where D6-branes are located. At that locus we cannot expect $B$ to be regular and in fact this is not the case even for $H$, as near that region it has a $rdr\wedge{\rm vol}_{S^{2}}\sim r^{-1}{\rm vol}_{{\mathbb{R}}^{3}}$ component. This is the same behavior that the NS–NS flux of the AdS7 solutions of [20] exhibits near a similar locus. In particular, this is the case for the solution one obtains by reducing the AdS${}_{7}\times S^{4}/{\mathbb{Z}}_{k}$ solution of M-theory to IIA. As can be seen from [20, Eq. (5.7)], the radial coordinate there being $r=\alpha^{2}$, $H\sim rdr\wedge{\rm vol}_{S^{2}}\sim r^{-1}{\rm vol}_{{\mathbb{R}}^{3}}$. While this might look puzzling at first, it comes about as one reduces a regular 4-form flux in M-theory along a shrinking $S^{1}$.

What we want to impose on $B$ then, is that its exterior derivative does not contain any delta functions.888This criterion was also imposed for the aforementioned AdS7 solutions. This can be checked by performing an integral of $B$ over a path $\gamma$ that goes from $\{k=1\}$ to $\{z=z_{r}\}$, and taking the limit of the result $\int_{\gamma}B$ as $\gamma$ shrinks to the point $\{z=z_{r},\,k=1\}$. If $\int_{\gamma}B\to 0$, then $dB$ does not contain any delta functions. For the gauge (3.73), this is indeed the case.

Although R–R flux quantization, which we carried out in section 3.3.2 to count the number and charges of D6-branes, would be more strictly performed in the gauge (3.73), the result is eventually what we obtained in that section, namely (3.52). This is because the gauge transformation $\Lambda_{1}$ in (3.73) goes to zero at $\{z=z_{r},\,k=1\}$. In any case, the result for the number of D6-branes could also be obtained more simply from the Bianchi identity $dF_{2}-HF_{0}=\delta$. By integrating this over the entire $M_{3}$ and using (3.51) and (3.72), we obtain

[TABLE]

The above equation, combined with (3.71) indeed gives (3.49).

3.6 Central charge

A handle on a CFT4 is provided by the central charges which given a dual AdS5 solution can be computed holographically, following [44]. In particular, the central charge $a$ is related to the five-dimensional Newton constant $G_{5}$ via

[TABLE]

$G_{5}$ can be obtained by compactifying the ten-dimensional action. In our case, $R_{{\rm AdS}_{5}}$ in fact depends on the internal coordinates, since we are dealing with a warped compactification. A reasonable procedure (as in [21, 23], for example) is to average the warping function coming from $R_{{\rm AdS}_{5}}^{3}$ over $M_{5}$. This leads to

[TABLE]

where ${\rm vol}_{5}$ is the volume form of the internal manifold, defined by $(-p^{\prime}/9z^{2})ds^{2}_{5}$ in (3.17). The factor $e^{-2\phi}$ appears as we switch from the string to the Einstein frame. We find

[TABLE]

The central charge is then

[TABLE]

In the next section we will evaluate $a$ for the three different classes identified at the end of section 3.4.

4 Three classes of solutions

At the end of section 3.4, we concluded that there are three possibilities for the range of the coordinate $z$. These actually correspond to different physics. We will now study them in turn.

4.1 O8, D6, D4

We start from the case $z\in[0,z_{0}]$ and $\ell=1$. Following the analysis of sections 3.3.1, 3.3.2 and 3.3.3, there are

• •

an O8–D8 stack at $k=0$;

• •

D6-branes at $\{z=z_{0},\,k=1\}$;

• •

D4-branes at $z=0$, smeared on the Riemann surface $\Sigma_{g}$.

The number $M$ of D6-branes and $n$ of D4-branes are related to the parameters of the solution by (3.52) and (3.57), which we repeat here for the reader’s convenience:

[TABLE]

The flux quantum $n_{0}=2\pi\ell_{s}F_{0}$ is related by $n_{0}=8-n_{8}$ to the number $n_{8}$ of D8-brane pairs on top of the O8-plane. From (3.52) and (3.53) we also see that the D6-branes have a D4-brane charge $m=n+(1-g)M^{2}/n_{0}$, or

[TABLE]

in terms of the NS–NS flux quantum $N$; see (3.74). This is manifestly an integer. The situation is summarized in figure 1.

Because of the rescaling (3.22), the space of solutions can be parameterized by $z_{1}/z_{0}$. From (4.1) and(3.74) it follows

[TABLE]

Hence, the space of solutions is discretized by flux quantization.

By specializing (3.78) to the $[0,z_{0}]$ interval, we find

[TABLE]

as anticipated in (1.1). The $N^{3}M^{2}$ term points to a compactification of a six-dimensional field theory. We will soon make this more precise, and identify the theory. $n$ is the number of D4-branes, and it is natural to interpret them as punctures, similar to the interpretation of M5-branes wrapping the R-symmetry circle as punctures, in [6, 45]. Thus, the $nN^{2}M$ term is the contribution of the punctures to the central charge. Since the D4-branes are uniformally distributed on the Riemann surface, we interpret these as simple punctures.

Let us now have an additional look at the various cases for the Riemann surface $\Sigma_{g}$.

4.1.1 Genus greater than one

For $\kappa=-1$ there are no constraints on $z_{1}$ and $z_{0}$, other than $z_{0}$ being positive ($z_{1}$ was fixed to be non-negative, without loss of generality, in section 3.) As mentioned before, the space of solutions can be parameterized by the ratio $z_{1}/z_{0}$. In this case it is a half-line.

It is worth investigating what happens at the extremum of this half-line, namely for $z_{1}=0$. Given (4.3), it corresponds to $n=0$, a solution without punctures. The change of coordinates

[TABLE]

transforms the metric of the solution to

[TABLE]

The warp factor and dilaton read

[TABLE]

This solution is a member of the family of solutions obtained in [1],999It corresponds to the value $18$ of the parameter $b_{2}$ that parameterizes the solutions in [1]. The coordinate $w$ is related via $w^{2}=y/y_{0}$ to the coordinate $y$ appearing there, where $2y_{0}=F_{0}z_{0}^{2}$. which, as argued there, are dual to compactifications of six-dimensional field theories on Riemann surfaces $\Sigma_{g>1}$ of negative curvature, without punctures.101010The central charge for similar theories was computed in [22] and agrees with (4.4), although the factor $27/32$ was mistakenly omitted there. This argument was later strengthened in [24], where the holographic RG flow connecting the AdS7 duals of the six-dimensional field theories and the AdS${}_{5}\times\Sigma_{g>1}$ solutions was obtained.

This limiting case supports our earlier claim that the solutions in this class should be interpreted as dual to compactifications of a six-dimensional field theory. As we have just seen, they generalize the AdS5 solutions of [1] by including D4-branes smeared over $\Sigma_{g}$. In [1], where no punctures were present, only the case of genus $g>1$ was realised. With the inclusion of punctures, we are able to obtain also $g=0$ and $g=1$. In section 5 we will discuss the aforementioned six-dimensional field theory and its AdS7 gravity dual.

4.1.2 Genus one

For $\Sigma_{g=1}=T^{2}$ ($\kappa=0$), the only constraint is $z_{1}>0$. This matches with the fact that in this case there should be no solutions without punctures ($n=0$ or $z_{1}=0$), as found in [1]. Since $p(z)$ in (3.18) becomes linear, the solution is particularly simple. Moreover, the directions of $T^{2}$ are isometries.

One can T-dualize to type IIB supergravity along one of these isometries. The dilaton is not constant on the IIB side; in particular the T-dual solution is not a Sasaki–Einstein compactification.

From (4.4) we see that the $N^{3}M^{2}$ term drops out of the central charge. This is similar to what happens for the class $\mathcal{S}$ theories, where the compactification on a torus results in circular quiver of ordinary gauge theories, with $a\sim nN^{2}$ growth.

4.1.3 Genus zero

For $\Sigma_{g=0}=S^{2}$ ($\kappa=1$), we saw in 3.4 that $z_{1}\geq z_{0}$. Via (4.3) and (3.53) this constraint translates to the bound

[TABLE]

In other words, there is a minimum amount of punctures one can have on the sphere. This again has a counterpart in the class $\mathcal{S}$ theories, where for example one cannot have three simple punctures on a sphere.

It is natural to investigate what happens when the bound is saturated. We analyze this in appendix C. Unfortunately, the physical interpretation of this limiting case is challenging, and it is not clear what it represents.

4.2 O8, D4 distribution

The second case we found in section 3.4 is $z\in[0,z_{1}]$. As explained there, this is only possible for $\ell=\kappa=1$. We can again parameterize the solutions by the ratio $z_{1}/z_{0}$, which has to lie in the range $(-\frac{1}{2},0)$.

In this case, $z$ does not reach the $z=z_{0}$ locus; as a result, there are no D6-branes in the solution. On the other hand, at $z=z_{1}$ there is a uniform distribution of D4-branes smeared not only on $\Sigma_{g=0}=S^{2}$ (as was the case for the solutions of section 4.1) but also on the ($k,\psi$) directions of the internal $M_{3}$. The total number of the D4-branes is equal to

[TABLE]

which is the same total number as that of the D4-branes at $z=0$. The situation is depicted by figure 2.

The absence of D6-branes ($M=0$) is also reflected in the vanishing of the NS–NS flux quantum, $N=0$. This can be seen from (3.74), or directly from (3.69). Using (3.78) we evaluate the central charge:

[TABLE]

This $n^{5/2}$ scaling is the case also for the AdS6 solution of [39], which similar to the present setup involves $n$ D4-branes on top of an O8–D8 system. (This scaling was also reproduced in field theory [46].) This seems to suggest that the CFT4 dual to the AdS5 solution under consideration, has a five-dimensional origin.

The limiting case $z_{1}/z_{0}=-1/2$ is the same as the one discussed in appendix C.

4.3 O8, D6, D4 distribution

Finally we describe the case $z\in[z_{1},z_{0}]$. As we saw in section 3.4, this is only possible for $\ell=\kappa=-1$. The ratio $z_{1}/z_{0}$ has to lie in the range $[0,1)$.

As in the class of section 4.1, there are D6-branes at $\{z=z_{0},\,k=1\}$. However, at $z=z_{1}$ there is a uniform distribution of D4-branes similar to the one of section 4.2. Again, the number of D6-branes and D4-branes are given by (3.52) and (3.57):

[TABLE]

The central charge is given this time by

[TABLE]

which seems to signal a mix of the six-dimensional origin of section 4.1 and of the conjectural five-dimensional origin of section 4.2.

The limiting case $z_{1}=0$ turns out to be again the punctureless solution with $\kappa=-1$ discussed in section 4.1.1, a member of the family of solutions of [1].

All the various cases we discussed are summarized in figure 4 in terms of the parameter $z_{1}/z_{0}$.

5 The six-dimensional theory and its gravity dual

In this section we will describe the six-dimensional origin of the solutions of section 4.1. We will first describe the AdS7 solutions (which are a particular case of [1], but which were not appropriately discussed there), and then their CFT6 duals.

5.1 The ${\rm AdS}_{7}$ solution

Let us first review the family of AdS7 solutions of type IIA supergravity [20, 22]. We will present the solutions as in [23]111111We will call here $\zeta$ the coordinate that in [23] was called $z$.. The metric, fluxes and dilaton read121212$B$ is to be understood up to a gauge transformation.

[TABLE]

Here $\ddot{\alpha}=\ddot{\alpha}(\zeta)$ is a piecewise linear function on a closed interval $I$ parameterized by $\zeta$. Flux quantization restricts its derivative to be integer-valued (with the integer related to the Romans mass $F_{0}$), and its points of discontinuity to be located at integer values of $\zeta$. $\alpha$ is a double integral of $\ddot{\alpha}$, as the notation implies. The two integration constants can be fixed in various ways. In [23] they were fixed so that $\alpha$ vanishes at the endpoints of $I$. This corresponds to the $S^{2}$ shrinking smoothly there. Various boundary conditions for $\alpha$ correspond to various physical situations. The cases that occur are:

• •

at an endpoint of $I$ where $\alpha$ and $\ddot{\alpha}$ have a single zero, and $\dot{\alpha}^{2}\neq 2\alpha\ddot{\alpha}$, the metric is regular,

• •

at an endpoint of $I$ where $\alpha$ has a single zero, $\ddot{\alpha}\neq 0$ and $\dot{\alpha}^{2}\neq 2\alpha\ddot{\alpha}$, there are D6-brane sources,

• •

at an endpoint of $I$ where $\ddot{\alpha}$ has a single zero, $\alpha\neq 0$ and $\dot{\alpha}^{2}\neq 2\alpha\ddot{\alpha}$, there are O6-plane–D6-brane sources,

• •

at an endpoint of $I$ where $\dot{\alpha}^{2}-2\alpha\ddot{\alpha}$ and $\ddot{\alpha}$ have a single zero, there are O8-plane–D8-brane sources.

The first three situations were already identified in [22]. The last one we present here and is the AdS7 solution of interest for this paper. It occurs for $b_{2}=18$ in the formulation of [22]. This O8–D8 system is of divergent dilaton type, in the sense discussed after (3.40). As noticed there, this is the same that occurs for example in the AdS6 solution of type IIA [39] supergravity, describing an O8–D8–D4 system. In a sense the solutions of this section are an AdS7 analogue of [39]. AdS7 solutions with an O8–D8 system and non-divergent dilaton should exist, but are not relevant for the present paper.

For the solution of interest

[TABLE]

According to the analysis above, at $\zeta=0$ there is an O8–D8 source, while at $\zeta=\zeta_{0}$ a stack of D6-branes. The presence of the O8-plane means that the solution should be thought of as the “right half” of a solution where $\zeta\in[-\zeta_{0},\zeta_{0}]$. The internal manifold for this “full” solution would have the topology of a three-sphere $S^{3}$. However we only consider the right half for which $\zeta\in[0,\zeta_{0}]$, and correspondingly the internal manifold has the topology of a half-$S^{3}$. From (5.1) we obtain

[TABLE]

Applying the map [1, Eq. (5.19)] (or [22, Eq. (5)]) to the metric in (5.3) we find (4.6), with $w=\zeta/\zeta_{0}$ and $z_{0}=3\pi\zeta_{0}$. As we anticipated in section 4.1.1, this provides the link between the AdS5 solutions found in this paper and the AdS7 solutions discussed in this section.

The gauge of $B$ is fixed by demanding regularity at $\zeta=\zeta_{0}$. Flux quantization (3.51) fixes

[TABLE]

where

[TABLE]

$M$ can be interpreted as the number of D6-branes at $\zeta=\zeta_{0}$. It is also easy to compute the integral of the NS–NS flux $H$: by integration by parts, it reduces to the integral of $B$ over $S^{2}$ near $\zeta=\zeta_{0}$ (where it vanishes) and near $\zeta=0$. This gives

[TABLE]

This result can be obtained more easily by integrating the Bianchi identity $dF_{2}-F_{0}H=\delta$, since $F_{0}$ is constant.

As we remarked earlier, the dilaton diverges near $\zeta=0$. However, this issue is localized in a small region around that locus. Moreover, as we will see, this singularity does not affect the computation of the $a$ anomaly.

5.2 The field theory interpretation

We will now give the field theory interpretation of the AdS7 solution (5.3).

Following the logic of [21], we can think of the solution as the result of a near-horizon limit of a brane configuration, which looks like the diagram in figure 5(a). The vertical line represents an O8-plane with $n_{8}=8-n_{0}$ D8-brane pairs, all extended along the directions $(012345789)$. The nodes represent NS5-branes, extended along the $(012345)$ directions. The horizontal lines represent D6-branes extended along the $(0123456)$ directions. Just like in [21], the idea is that in the near-horizon limit the direction 6 and the radius of the directions $(789)$ mix to produce the radial direction of AdS7 and the coordinate $\zeta$ – this was recently made more precise in [47, 48]. The D6-branes in the brane diagram become the D6-branes at the pole $\zeta=\zeta_{0}$ in (5.3). The $N$ NS5-branes in the brane diagram become the flux integer (5.6).

Reading off the field theory from the brane diagram is mostly a straightforward application of the standard string theory techniques [16, 15]. As usual in six dimensions, the field theory one reads off this way is an effective description of the tensor moduli space of a SCFT. The conformal point is really obtained at the origin of this tensor moduli space, which corresponds in the brane diagram to putting all the NS5-branes on top of each other and on top of the O8-plane.

Having said this, the effective theory is as follows. It consists of a chain of gauge groups ${\rm SU}(i\cdot n_{0})$, $i=1,2,\ldots,(N-1)$, coupled to hypermultiplets and tensor multiplets. At the end of the chain there is an ${\rm SU}(Nn_{0})$ flavor symmetry. (For details, see [16, 15] or the summary given in [23, Sec. 2.1].) The tensor multiplets couple to the gauge fields via a Green–Schwarz–Sagnotti–West mechanism, and via a term of the type $(\phi_{i}-\phi_{i+1}){\rm Tr}(F_{i}^{2})$, where $\phi_{i}$ is the real scalar in the $i$-th tensor multiplet and $F_{i}$ is the $i$-th gauge field strength.

All this is like in the theory corresponding to the NS5–D6 system, shown for example in [23, Fig. 6], whose gravity dual is the tear-drop shaped, “simple massive” solution of [22] and [20, Sec. 5.2]. There is, however, an additional subtlety here, due to the presence of the O8-plane. (This was discussed in section [15, Sec. 5.1], although the theories we need are a “piece” of the ones in that reference. The theory we are describing appeared recently in [49].) Via a chain of dualities, the O8–D8 system can be mapped to the $E_{8}$ wall in M-theory [50]. An unbroken $E_{8}$ corresponds to a situation where there are 7 D8-brane pairs on the O8-plane; notice that in this situation $n_{0}=2\pi\ell_{s}F_{0}$ is equal to 1. When there are fewer D8-brane pairs, $n_{8}=8-n_{0}$, the flavor group is broken to $E_{n_{8}+1}=E_{9-n_{0}}$. For $n_{0}=6,5$ this gives $E_{7}$ and $E_{6}$ respectively. For $n_{8}$ smaller still, the sequence was determined in [51] in the context of the five-dimensional CFTs obtained by putting D4-branes next to an O8–D8 system (whose gravity dual [39] was mentioned earlier):

[TABLE]

Under the same duality chain, the NS5-branes are mapped to M5-branes. An M5-brane near an $E_{8}$ M-theory wall is described by a six-dimensional theory with a single tensor multiplet and an $E_{8}$ flavor symmetry, known as “E-string theory”. It has a one-dimensional tensor moduli space; conformal symmetry is unbroken at its origin. It also has an F-theory realization, which consists of a single $-1$-curve touching an $E_{8}$ singularity. This is usually denoted by

[TABLE]

This is actually a rank-1 E-string theory. There is also a rank-$N$ E-string theory, which describes $N$ M5-branes near the $E_{8}$ wall. This has an $N$-dimensional tensor moduli space. At a generic point, there is an effective description with $N-1$ tensors coupled to (5.8). The F-theory realization consists of a sequence of $-2$-curves ending with (5.8):

[TABLE]

In our setup, the NS5-brane which is nearest to the O8–D8 gives rise to a rank-1 E-string theory (5.8). However, this E-string theory is also coupled to the chain of gauge groups we described earlier, whose first gauge group is ${\rm SU}(n_{0})$. The coupling is made by gauging a subgroup ${\rm SU}(n_{0})$ of the $E_{8}$ flavor symmetry of the E-string theory. The leftover flavor symmetry is the commutant of ${\rm SU}(n_{0})$ inside $E_{8}$.

The algebra of the flavor symmetry can be identified by looking for the maximal subalgebras of $\mathfrak{e}_{8}$ of the form $\mathfrak{g}\oplus\mathfrak{su}_{n_{0}}$. One way is to consider semisimple regular subalgebras, which are obtained by deleting a node from the affine $E_{8}$ Dynkin diagram. The list one obtains this way is

[TABLE]

We may also consider non-semisimple regular subalgebras, which are obtained by deleting a node from the ordinary (non-affine) $E_{8}$ Dynkin diagram; the removed dot becomes a $\mathfrak{u}_{1}$. We get $\mathfrak{su}_{2}\oplus\mathfrak{u}_{1}\oplus\mathfrak{su}_{7}$ this way. We can now obtain the commutant of $\mathfrak{su}_{n_{0}}$ inside $\mathfrak{e}_{8}$: for example, the commutant of $\mathfrak{su}_{2}$ is $\mathfrak{e}_{7}$, the commutant of $\mathfrak{su}_{3}$ is $\mathfrak{e}_{6}$, and so on. This reproduces the full list of “exceptional” flavor groups $E_{9-n_{0}}$ in (5.7), which was originally computed in [51] using heterotic strings.

The end result is the theory shown in figure 5(b) for the brane diagram in figure 5(a), and in figure 6 for the most general case.

For completeness we also write down the F-theory realization of these theories:

[TABLE]

The $n_{0}=1$ case is rather special, because the first gauge group is in fact empty:

[TABLE]

These theories are similar to the rank-$N$ E-string theory in (5.9), except for the presence of the linearly growing gauge groups on the $-2$-curves. They are the analogue of a rank-$N$ E-string theory in the presence of Romans mass $F_{0}$. For this reason, we will call (5.11), (5.12) rank-$N$ “massive E-string theories”.

As a cross-check of our field theory identification, we have computed using the methods in [52, 53], the leading order behavior of the $a$ anomaly, and compared it against a holographic computation from the gravity dual. For solutions with only D8-branes, which are dual to chains of ${\rm SU}(r)$ gauge groups, this comparison was carried out in [23], finding perfect agreement for all the infinitely many theories in that class. For the theories discussed in this paper, the computation is a bit different because of the presence of the rank-1 E-string. Recalling the caveat pointed out at the end of section 5.1 about the divergent dilaton, one might be skeptical of a holographic comparison. Indeed for example in [46], for the aforementioned AdS6 solution with an O8-plane [39], it was found that the on-shell action diverges, hindering comparisons with field theory quantities. In our case, however, the holographic $a$ anomaly is found to be finite, and to reproduce the $a$ anomaly:

[TABLE]

at the leading order in the holographic limit; for details, see appendix D. Notice that (5.13) is different, even at leading order, from the theory with linearly growing quivers but without the rank-1 E-string, which was found to have [22, 23] $a\sim\frac{16}{7}\frac{4}{15}N^{3}M^{2}$.

Acknowledgements

We would like to thank Ken Intriligator and Vasilis Stylianou for discussions. We thank the organizers of the Simons Center for Geometry and Physics 2015 summer workshop for hospitality at the initial stages of this project. We also thank the organizers of the workshop “String Theory in London 2016”, where part of this work was completed. The work of I.B. is supported in part by UC president’s post-doctoral fellowship and in part by DOE grant DE-SC0009919. The work of A.P. and A.T. is supported by INFN and the European Research Council under the European Union’s Seventh Framework Program (FP/2007-2013) – ERC Grant Agreement n. 307286 (XD-STRING). A.P. is also supported by the Knut and Alice Wallenberg Foundation under grant Dnr KAW 2015.0083. A.T. was also supported by the MIUR-FIRB grant RBFR10QS5J “String Theory and Fundamental Interactions”, and by the ERC-STG grant 637844- HBQFTNCER.

Appendix A Comparison with [1]

In this appendix we summarize the relation between the presentation of the results of [1] in that paper and the one in section 2 of the present work.

The metric of the AdS5 solutions was expressed in [1, Sec. 3] as

[TABLE]

The functions $(W,b,a_{1},a_{2},\varphi)$131313$W$ is denoted by $A$ in [1]. and the dilaton $\phi$ depend on the coordinates $(x,y,x_{1},x_{2})$, and obey the relations [1, Eqs. (3.6), (3.7)]

[TABLE]

and [1, Eqs. (3.17b), (3.17c), (3.22)]

[TABLE]

The one-form $\rho$ is determined by the above functions as [1, Eq. (3.17a), (3.23)]

[TABLE]

where the Hodge star operator $\star_{2}$ and the exterior derivative $d_{2}$ are taken over the $(x_{1},x_{2})$ plane.

The transformation relating the coordinates $(s,u)$ of section 2 and $(y,x)$ is

[TABLE]

The functions $(D_{s},D_{u})$ in that section are defined by the ones appearing in [1] as

[TABLE]

Equations (A.3) and (A.4) are then solved by $(D_{s},D_{u})$ via the expressions

[TABLE]

See section 2 for the definition of $\widetilde{D}_{s}$, $\det(h)$ and $\det(g)$. Finally, $\rho=-\frac{1}{2}\star_{2}d_{2}D_{s}$, and the metric (A.2) becomes

[TABLE]

Appendix B D4-branes inside D8-branes

Solutions describing D$p$-branes inside the wordvolume of D$(p+4)$-branes were constructed in [54]. Specialising to the case $p=4$, let the D4-branes be extended along the directions $x_{0}$ to $x_{4}$ and the D8-branes along all directions except for $x_{9}$. The spacetime metric of the solution is given by

[TABLE]

where the functions $H_{4}$ and $H_{8}$ satisfy the equations

[TABLE]

These are solved by

[TABLE]

where $r^{2}=\sum_{i=5}^{8}(x_{i})^{2}$ is the radial coordinate in the $x_{5}$ to $x_{8}$ directions, and $Q_{4}$, $Q_{8}$ are constants.

We are interested in a solution where the D4-branes are localized only in the $x_{7}$, $x_{8}$ directions and smeared over the $x_{5}$, $x_{6}$ ones. The first of the equations (B.2) is thus modified as

[TABLE]

and the $H_{4}$ that solves it reads

[TABLE]

It is convenient to introduce the coordinate $\lambda$ via

[TABLE]

and rewrite the spacetime metric as:

[TABLE]

Near the core of the solution,

[TABLE]

and upon making the coordinate transformation,

[TABLE]

the metric takes the asymptotic form:

[TABLE]

Appendix C The limiting case $z_{1}=z_{0}$

In this appendix, we will analyze the limiting case $z_{1}=z_{0}$ in the space of solutions of section 4.1.3. For $z_{1}\geq z_{0}$, the solutions contain D6-branes, and an O8-plane with D4- and D8-branes on top. We argued in the main text that they represent gravity duals of a compactification on a two-sphere with punctures of the six-dimensional theories discussed in section 5. Here we will see that the limiting case $z_{1}=z_{0}$ is of unclear physical interpretation.

For $z_{1}=z_{0}$, $z_{0}$ becomes a double root of $p$, since $p^{\prime}(z_{1})=0$. Once this happens, the analysis of section 3.3 is not valid near $z=z_{0}$ and near $\{z=z_{0},\,k=1\}$. Indeed the geometry in these regions departs from the one where $z_{0}$ is a single root.

Near $z=z_{0}$, we introduce the coordinate $\varrho^{2}=\frac{8}{3}(1-z/z_{0})$. In the $z\to z_{0}$ limit the metric becomes

[TABLE]

Given that the period of $\psi$ is $2\pi$, $\frac{1}{4}\left(\eta_{\psi}^{2}+ds^{2}_{S^{2}}\right)$ represents the metric on a $\mathbb{Z}_{2}$ orbifold of a three-sphere of unit radius, with the $\mathbb{Z}_{2}$ acting on the R-symmetry circle. Including the direction parameterized by $\varrho$, we conclude that the internal space contains $\mathbb{R}^{4}/\mathbb{Z}_{2}$.

The locus $\{z=z_{0},\,k=1\}$ is of more challenging physical interpretation. In the neighborhood of $\{z=z_{0},\,k=1\}$, the local change of coordinates

[TABLE]

puts the metric in the form

[TABLE]

This is not reminiscent of a brane singularity familiar to us. Given that this solution arises as a limiting case of a solution with D6-branes present at $\{z=z_{0},\,k=1\}$, and the appearance of an $\mathbb{R}^{4}/\mathbb{Z}_{2}$ singularity near $z=z_{0}$, it is tempting to speculate that as $z_{r}=z_{0}$ in (3.42) becomes a double root, $p^{\prime}(z_{r})$ goes to zero and the base $\Sigma_{g}=S^{2}$ shrinks, thus turning the D6-branes into fractional D4-branes, whose near-horizon produces (C).

Appendix D $a$ anomaly

In this appendix we will explain how to compute the $a$ anomaly for the massive E-string theories (5.11) and (5.12), using both field-theoretic and holographic techniques. Part of this computation follows [23], to which we refer for further details.

The field theory computation can be performed using the methods of [52, 53]. We will focus directly on the holographic leading order contribution, rather than giving the full detailed computation as in [23]. As in that paper, we use [55] $a=\frac{16}{7}(\alpha-\beta+\gamma)+\frac{6}{7}\delta$, where $\alpha$, $\beta$, $\gamma$ and $\delta$ are the various coefficients in the ’t Hooft anomaly polynomial:

[TABLE]

where $c_{2}(R)$ is the second Chern class of the R-symmetry bundle and $p_{1}(T)$, $p_{2}(T)$ are the first and second Pontryagin classes of the tangent bundle respectively. The holographic limit consists in taking the number $N$ of gauge groups to be large. The limit can be taken in such a way as to leave the supergravity solution essentially unchanged; see [23, Sec. 2.2.4]. The leading order contribution comes from $\alpha$, the coefficient of $c_{2}(R)^{2}$. This in turn comes from a Green–Schwarz–Sagnotti–West mechanism. Let us first explain how this worked in [23], and then how it is modified for the case of this paper. The relevant terms of the anomaly polynomial read

[TABLE]

where $C_{ij}$ is the Cartan matrix of $A_{N-1}$. Gauge anomalies should be canceled: this leads us to postulate the presence of a Green–Schwarz–Sagnotti–West [56, 57] mechanism which gives a further contribution $\frac{1}{8}\sum_{i,j}C_{ij}I_{i}I_{j}$, $I_{i}={\rm tr}F_{i}^{2}+2C^{-1}_{ij}r_{i}c_{2}(R)$. This cancels the two terms appearing in (D.2), but introduces the term $\frac{1}{2}\sum_{i,j}C^{-1}_{ij}r_{i}r_{j}c_{2}(R)^{2}$. This is the leading contribution to $\alpha$ and hence to $a$:

[TABLE]

If we consider the theory with linearly growing ranks, $r_{i}=i\cdot n_{0}$ [23, Fig. 6], we can compute $\sum_{i,j}C^{-1}_{ij}r_{i}r_{j}=\frac{1}{180}N(1-N^{2})(1-4N^{2})\sim\frac{1}{45}N^{5}$, which gives

[TABLE]

where equation (5.6) is used.

In our case, the computation is modified by the presence of an E-string. The relevant part of its anomaly polynomial is

[TABLE]

Here $F$ is the $E_{8}$ field strength, and ${\rm Tr}\equiv\frac{1}{30}{\rm tr}_{\rm fund}$. There is also a ${\rm Tr}F^{2}p_{1}(T)$ term which would contribute to $\beta$ and $\gamma$, but their contributions to $a$ are subleading. When an ${\rm SU}(n_{0})$ subgroup of $E_{8}$ is gauged, (D.5) modifies (D.2) in two ways:

[TABLE]

Of these two, only $C\to\tilde{C}$ affects the leading order. Hence, we end up with

[TABLE]

In our case, again, $r_{i}=i\cdot n_{0}$. We evaluate $\sum_{i,j}\tilde{C}^{-1}_{ij}r_{i}r_{j}=\frac{1}{60}N(N+1)(N+2)(3N^{2}+6N+1)\sim\frac{1}{20}N^{5}$, which finally leads to

[TABLE]

We will now compare the above result with a holographic computation. As argued in [23, Sec. 4], the latter reduces to

[TABLE]

where $\alpha=\alpha(\zeta)$ is the function that characterizes the supergravity solution in (5.1). As explained in [23], this is a continuum version of $\sum_{i,j}r_{i}C^{-1}_{ij}r_{j}$, keeping in mind the fact that $C_{ij}$ is a “discrete double derivative”. For example, the gravity solution corresponding to the chain of gauge groups SU$(i\cdot n_{0})$, $i=1,\ldots,(N-1)$ is the so-called “simple massive” solution. In the present language, it is given by (5.1) with

[TABLE]

Performing the integral (D.9) one reproduces (D.4).

On the other hand, for the rank-$N$ massive E-string theories, we should use $\alpha$ as in (5.2). This reproduces (D.8).

Notice that the two solutions we just considered have the same $\ddot{\alpha}$, and only differ by its double indefinite integral $\alpha$. In [23], this integral was fixed by demanding it to vanish at the extrema. This, on the field theory side, corresponds in a way to the choice of $C_{ij}$ rather than $\tilde{C}_{ij}$ as a discrete version of the double derivative.

We expect that the match obtained in this section works for more general quivers with an E-string at their end. However, we will not demonstrate this here, since such theories are for the time being not relevant to generating AdS5 solutions similar to the ones studied in this paper.