Published for SISSA by Springer

Received: April 7, 2011 Accepted: May 12, 2011 Published: May 24, 2011

Dynamic SU(2) structure from seven-branes

Ben Heidenreich," Liam McAllister" and Gonzalo Torroba

aDepartment of Physics, Cornell University, Ithaca, NY 14853, U.S.A.

bSLAC and Department of Physics, Stanford University, [^0

Stanford, CA 94309, U.S.A.

E-mail: bjh77@cornell.edu, mcallister@cornell.edu, torrobag@slac.stanford.edu

Abstract: We obtain a family of supersymmetric solutions of type IIB supergravity with dynamic SU(2) structure, which describe the local geometry near a stack of four D7-branes and one O7-plane wrapping a rigid four-cycle. The deformation to a generalized complex geometry is interpreted as a consequence of nonperturbative effects in the seven-brane gauge theory. We formulate the problem for seven-branes wrapping the base of an appropriate del Pezzo cone, and in the near-stack limit in which the four-cycle is flat, we obtain an exact solution in closed form. Our solutions serve to characterize the local geometry of nonperturbatively-stabilized flux compactifications.

Keywords: Flux compactifications, Superstring Vacua, D-branes

ArXiv ePrint: 1011.3510v2

Open Access

doi:10.1007/JHEP05(2011)110

Contents

Introduction 1

1.1 Gaugino condensation in string compactifications 1

1.2 Nonperturbative effects from seven-branes and generalized complex geometry 2

1.3 Overview 4

2 Seven-branes and orientifolds

2.1 Orientifolds of resolved del Pezzo cones

2.2 The Calabi-Yau geometry of the P2 cone

3 Ansatz for the supergravity solution

3.1 The near-stack region

3.2 Seven-branes in the P2 cone

3.2.1 Metric ansatz for the P2 cone

3.2.2 Kahlerity and the Calabi-Yau metric

3.2.3 G3 ansatz for the P2 cone

3.2.4 D3-brane charge

3.3 The near-stack limit of the P2 cone

4 Supersymmetry conditions

4.1 Review of SU(2) structure and generalized complex geometry

4.2 SL(2, R) covariant supergravity

4.3 SL(2, R) covariant supersymmetry conditions for AdS4 compactifications

5 Supergravity solution in the near-stack region

5.1 Supersymmetry conditions in the near-stack region

5.2 Solutions in the near-stack limit

5.3 Beyond the near-stack limit

6 Geometry of the near-stack solution

6.1 Singularity structure

6.2 Constant dilaton solutions

6.3 Domain of validity of the near-stack solutions

6.3.1 Corrections to the near-stack limit

6.3.2 Curvature corrections

6.3.3 Regions of parametric control

7 Towards the physics of the solutions

7.1 Gaugino condensation in supergravity

7.1.1 Gaugino condensation and IASD flux

7.1.2 R-symmetry breaking and domain walls

7.2 Applications

8 10 11 12 13

15 19 21

28 29 31

Œ M •n o Cn

7.2.1 Ten-dimensional consistency of KKLT vacua

7.2.2 On del Pezzo transitions with seven-branes

8 Conclusions 39

A Equations of motion for the near-stack ansatz 41

1 Introduction C |

Two fundamental goals in string theory are characterizing the vacua of the theory and understanding strongly-coupled four-dimensional gauge theories from a ten-dimensional view- M point. These problems intersect when the dynamics of a strongly-coupled four-dimensional gauge theory determines the potential for compactification moduli, as in flux compactifi- ^^ cations of type IIB string theory, where gaugino condensation on seven-branes provides an important contribution to the potential for the Kahler moduli.

In this work we present a family of explicit local solutions that describe the region near a stack of seven-branes wrapping a rigid four-cycle. We argue that a subclass of our solutions encode seven-brane nonperturbative effects in ten-dimensional supergravity. We begin in section 1.1 by motivating the study of seven-brane gaugino condensation, then explain in section 1.2 why the corresponding solution will be a generalized complex geometry.

1.1 Gaugino condensation in string compactifications

In a compactification of type IIB string theory on a Calabi-Yau threefold, classical vacua I

involving nonvanishing fluxes and localized D-brane and orientifold plane sources provide |—*

a rich array of four-dimensional theories with N = 1 or N = 0 supersymmetry. The Kahler moduli are typically unfixed in the classical vacuum and mediate gravitational-strength interactions that preclude the construction of realistic models of particle physics and cosmology. Perturbative and nonperturbative effects may be expected to give mass to the Kahler moduli, and in certain special cases the dominant effects can be computed.

The proposals for Kahler moduli stabilization of [1-3] incorporate nonperturbative effects arising from branes wrapping four-cycles in the compact space [4]: these can be either Euclidean D3-branes, or a stack of (p, q) seven-branes giving rise to a four-dimensional gauge theory that is strongly coupled in the infrared and generates a nonperturbative superpotential. Considerable efforts have been directed at understanding the four-dimensional effective theory incorporating the classical flux superpotential and the nonperturbative superpotential arising from wrapped seven-branes, but the corresponding ten-dimensional configuration that encodes the effects of nonperturbative dynamics on the seven-branes remains mysterious.

Why understand the imprint of four-dimensional nonperturbative physics in ten dimensions, given that the four-dimensional theory itself is well-understood? One powerful motivation comes from the utility of higher-dimensional locality in model building. For

1.2 Nonperturbative effects from seven-branes and generalized complex geometry

The above configuration of seven-branes will preserve four supercharges that are embedded in the type IIB ten-dimensional Majorana-Weyl supersymmetry generators el as

ei = Z+ ® n+ + Z- ® n- , i = 1, 2 , (1-1)

where the conventions are those of [7]: Z+ is a positive-chirality four-dimensional spinor that generates four-dimensional N = 1 supersymmetry transformations, and the n+ are fixed positive-chirality six-dimensional spinors, with rf_ ,Z- the Majorana conjugates of n+ ,Z+, respectively.

xIn fact, negative D3-brane charge and tension induced on the seven-branes will contribute to the formation of a singularity near the four-cycle.

2 See section 7.1 for important comments on how the global anomaly of [5, 6] may constrain the resulting gauge group. For simplicity of presentation we will speak of gaugino condensation in pure super Yang-Mills throughout, while keeping in mind that the full gauge theory may be more complicated.

example, one can construct a supersymmetric visible sector on D-branes in one region of a compactification, and later incorporate soft terms induced by supersymmetry breaking in a distant region. A prerequisite for such an analysis is locality in the compact space, which manifestly requires a ten-dimensional solution. Thus, understanding locality in nonperturbatively-stabilized vacua requires a ten-dimensional solution encoding the effects that stabilize the Kahler moduli. A second motivation is that local geometries describing strong gauge dynamics on seven-branes could be glued into compact geometries as 'modules' effecting the stabilization of Kahler moduli. A third important motivation is that gravity solutions can shed new light on the dynamics of supersymmetric gauge theories on seven-branes via gauge-gravity duality, as we explain in section 1.2.

We are therefore led to search for explicit local solutions describing strong gauge dynamics on seven-branes wrapping rigid four-cycles. A natural class of local Calabi-Yau geometries containing rigid four-cycles are complex cones over del Pezzo surfaces. The possibilities for wrapping seven-branes on the del Pezzo base of such a cone are quite con- •n

strained: the total seven-brane charge must vanish (see section 2.1). A convenient choice ( )

obeying this constraint is a stack of four D7-branes that coincide with an O7-plane: the total seven-brane charge and tension vanish, but lower-dimensional brane charges may be induced on the stack, e.g. by a! corrections.1 The resulting four-dimensional gauge theory is pure2 super Yang-Mills, which exhibits gaugino condensation and chiral symmetry breaking at low energies.

We conclude that a promising setting for studying the backreaction of four-dimensional

nonperturbative effects is a Calabi-Yau cone whose base, a del Pezzo surface, is the fixed- |_\

point locus of an orientifold action and in addition is wrapped by four D7-branes. In this work we formulate the problem in this general setting but provide a detailed solution for a simpler special case, in which one zooms in on the region near the seven-branes. Results I *

for del Pezzo cones will be provided elsewhere.

3An earlier example that displays gaugino condensation and (1, 2) three-form flux is the Polchinski-Strassler solution [11]; the ten-dimensional analysis of [12, 13] revealed the presence of an SU(2) structure.

4See [15] for a ten-dimensional description of gaugino condensation in the heterotic string.

5 Anomaly-free gauge theories from seven-branes on del Pezzo cones were constructed recently in [16].

Taken as strictly classical sources, four D7-branes wrapping the del Pezzo fixed-point locus of an orientifold will preserve 'type B' supersymmetry [7], with = ±in+. However, a supergravity solution encoding gaugino condensation requires a different supersymmetry, as we now argue. In a type B background (as characterized e.g. in [8]) a D3-brane experiences vanishing potential, while gaugino condensation on seven-branes is known to lift the D3-brane moduli space. Further evidence comes from [9], where it was established that in a type B solution, gaugino condensation from D7-branes on a rigid cycle sources imaginary anti-self-dual flux, of Hodge type (1,2), which is incompatible with the supersymmetry of the background. In fact, a supergravity solution encoding seven-brane gaugino condensation requires that the internal spinors and be distinct, as argued in section 4, and a direct analysis of the dilatino and gravitino variations becomes more involved. Mathematically, the two internal spinors define a local or dynamic SU(2) structure that can be dealt with most efficiently in terms of generalized complex geometry [7], as originally proposed for D7-brane gaugino condensation in [10].3 '

Our goal in this work is to investigate supergravity solutions with dynamic SU(2) structure that describe the gauge theory dynamics of compact seven-branes. The approach taken here is purely ten-dimensional, and we do not assume the existence of a four-dimensional nonperturbative superpotential. Rather, in the spirit of [11, 14], the supergravity solution describing seven-branes wrapped on an appropriate rigid cycle should already encode the effects of gaugino condensation.4 (Z^)

Let us comment on the relation between our approach and well-understood AdS/CFT |—^

descriptions of gaugino condensation in other systems. The gravity dual of pure super Yang-Mills is not known, and is expected to correspond to a regime of large curvature and strong corrections to classical supergravity. However, pure super Yang-Mills can be embedded into a branch of a large N quiver theory, or into a higher-dimensional gauge theory, leading to well-defined supergravity solutions, albeit with extra fields that are absent from the pure glue theory.

A celebrated example is the Klebanov-Strassler solution [14], in which an SU(N) x SU(N + M) quiver theory, which confines and breaks chiral symmetry in the infrared, is dual to a conformally-Calabi-Yau solution, the warped deformed conifold. In this case the boundary supersymmetry is of type B, facilitating embedding in well-understood flux compactifications [8]. In contrast, there is no known supergravity solution dual to a gauge theory with seven-branes wrapped on a rigid four-cycle. One obstacle is that taking a four-dimensional limit by shrinking the four-cycle generically gives chiral matter and an anomalous gauge theory.5 The anomaly has to be canceled by adding extra ingredients, such as orientifolds, but these do not decouple from the low energy theory and complicate both the gauge theory analysis and the supergravity solution. Furthermore, the number of seven-branes cannot be taken to be large, so that it is difficult to obtain parametric control of the curvatures appearing on the gravity side.

One might expect that our noncompact supergravity solutions should capture the backreaction associated to the gauge theory dynamics. However, there is at present no fully-realized gauge-gravity duality for our system: our solutions do not include a large number of D3-branes, and the asymptotic geometry is very different from AdS5. We will suggest that our solutions describe the behavior of the gauge theory in the deep infrared, after most of the degrees of freedom have renormalized away. Introducing a large number of color branes might yield a solution in which a more precise gauge/gravity dictionary can be constructed, but we leave this question for the future.

1.3 Overview

Our main result is an explicit supersymmetric solution with dynamic (and in general, type-changing) SU(2) structure, which describes the generalized complex geometry near a stack of four D7-branes and one O7-plane. This solution arises as a limiting case of configurations in which the seven-branes wrap a compact four-cycle: in the example we •n

provide, this cycle is the base of the Calabi-Yau cone over P2, O(-3)P2. We construct our solution in closed form after solving the supersymmetry conditions for an AdS4 vacuum in the limit of vanishing cosmological constant. The form of the supersymmetry conditions used here is SL(2, R) covariant, which is very useful in classifying the resulting solutions.

The organization of this paper is as follows. In section 2, we present some essential geometric background for our analysis. In section 3.1, we describe a simple ansatz that o

will be our primary focus. In section 3.2, we extend our considerations to the P2 cone, and |—

show that the ansatz of section 3.1 arises in a scaling limit that zooms in on the P2. In |_

section 4, we briefly review dynamic SU(2) structure, then present the full supersymmetry conditions for compactification to AdS4 in an SL(2, R) covariant form. In section 5 we solve these conditions, as well as the equations of motion, to obtain the most general 'AdS-like' supersymmetric solution to our ansatz. In section 6 we describe the regime of I—*

validity of supergravity and discuss a few key geometric properties of our solutions. In section 7 we present a preliminary analysis of the relation between the above solutions and nonperturbative effects on seven-branes, and indicate a few interesting applications and directions for future research. We conclude in section 8. The equations of motion for our ansatz are assembled in appendix A.

2 Seven-branes and orientifolds

We will study a stack of four D7-branes atop an O7-plane, wrapping a rigid four-cycle in a local Calabi-Yau geometry and preserving four supercharges. The specific local geometries that we will consider are resolved Calabi-Yau cones over del Pezzo surfaces. In each case, the del Pezzo base is a rigid shrinking divisor within the Calabi-Yau. These properties make Calabi-Yau cones over del Pezzos the natural choice for the purpose of obtaining local gauge theories that undergo gaugino condensation. In this work we will focus primarily on the simplest del Pezzo surface, P2, but we begin with topological considerations that are valid for any del Pezzo cone, and then proceed to describe the geometry of the resolved cone over P2, O(-3)P2.

2.1 Orientifolds of resolved del Pezzo cones

•n o

Wrapping seven-branes over the del Pezzo surface yields a supersymmetric field theory in four dimensions. In order to obtain a pure glue theory in which gaugino condensation can occur, we do not add seven-branes wrapping non-compact divisors. Then, owing to the rigidity of the del Pezzo within the Calabi-Yau, the four-dimensional field theory has no light matter, and develops a gaugino condensate in the infrared. One can imagine altering this situation in various ways to remove the gaugino condensate, for instance by adding D3-branes close to the seven-branes, giving light 3-7 strings. Thus, it makes sense to consider the supergravity background both with and without the effects of gaugino condensation. We first consider the situation without a condensate, in which case the background contains only D3-branes and seven-branes, and can be studied using F-theory.

The only compact divisor in these cones is the del Pezzo itself. This places a strong constraint on the allowed brane content consistent with tadpole cancellation, as we now show. Since stacks of seven-branes carry an SL(2, Z) monodromy depending on their total charge, we consider the SL(2, Z) monodromy structure of the solution. For a manifold M without branes, the allowed monodromies are given by homomorphisms A : n1(M) ^ SL(2,Z). O"!

Seven-branes may be thought of as topological defects in the type IIB vacuum; thus, the —^

seven-brane charges and monodromy structures associated with a given configuration of branes are classified by homomorphisms A : n1(M') ^ SL(2, Z), where M' is given by M minus the worldvolumes of the seven-branes.

A del Pezzo cone may be viewed as a real cone M over a Sasaki-Einstein manifold which we refer to figuratively as the 'horizon.' Upon excising the tip, the resulting M' is homotopically equivalent to its horizon. Thus, the allowed seven-brane charges are constrained by the fundamental group of the horizon. It is known that these horizons are always simply connected up to torsion, and the only torsion groups that appear are Z3 (for the P2 cone) and Z2 (for the P1 x P1 cone) [17]. Except in these special cases, we must cancel the D7-brane tadpole locally.6 A well-understood way to do this is to wrap eight D7-branes on the four-cycle and then orientifold by a Z2 involution whose fixed point locus is the cycle itself.

It is important to distinguish between the 'upstairs' and 'downstairs' geometries of the resulting orientifold. From the perspective of perturbative string theory, it is natural to do computations in the upstairs geometry, in which we have eight D7-branes coinciding with an O7-plane wrapping the base of a resolved del Pezzo cone. At energies below the Kaluza-Klein scale the resulting gauge theory is pure glue N = 1 super Yang-Mills, where all open string fields arise from 7-7 strings and live in the adjoint representation. Classically, the composite object carries zero seven-brane charge and tension, and generates no deficit angle.

From the perspective of F-theory, on the other hand, it is more natural to work in the downstairs geometry, where we identify under the involution to obtain a Z2 orbifold of the del Pezzo cone which carries a Z2 monodromy coming from the —1 £ SL(2, Z). The eight

6This can be seen from the field theory perspective as well: the gauge theories corresponding to disallowed configurations of seven-branes will be rendered inconsistent by anomalies [16].

D7-branes of the upstairs geometry are reduced to four by this identification. In addition, the orientifold plane appears in F-theory as a combination of two coincident (p, q) seven-branes that separate at strong coupling [18]. Thus, in the downstairs geometry, we obtain a stack of six (p,q) seven-branes. Each carries a deficit angle of n/6, for a total deficit angle of n, as required to match the deficit angle of (O(-3)P2)/Z2.

Although seven-branes are often conveniently treated in F-theory, the more general supersymmetric backgrounds that will be relevant in our analysis are not well-studied in F-theory, and we opt to work in ten-dimensional supergravity. (In section 6.3 we will explicitly demonstrate parametric control of the supergravity approximation.) We will generally work in the upstairs geometry, removing modes forbidden by the involution.

2.2 The Calabi-Yau geometry of the P2 cone

We now review the well-known geometry of the Calabi-Yau cone over P2, i.e. the resolution of the orbifold C3/Z3, where the Z3 acts by

z* e2ni/V

EC M •n o

on the C3 coordinates z*. The Calabi-Yau metric has a U(3) = SU(3) x U(1)^ isometry that acts naturally on the z*, where the SU(3) subgroup acts on the P2 base in the natural way, and we normalize the U(1)^ such that the z* carry charge +1/3. Let us define the U(3) invariant radial coordinate

P2 = E |z*|2 . (2.2)

Each surface of constant p is diffeomorphic to the horizon, S5/Z3. It is useful to think of this space as a Hopf fibration over P2, where U(1)^ rotates the fibers. Locally, we can define a circle coordinate — with periodicity 2n, such that for some arbitrarily

chosen i, and U(1)^ rotations are equivalent to shifts in 0.

The Z3 orbifold singularity may be resolved into a P2. Viewed from the C3/Z3 — {0} region, the size of the resolution is visible as a normalizable perturbation to the conical metric, as we now review. The conical Calabi-Yau metric for this space, either singular or resolved, can be obtained by taking the Kahler potential K to depend only on p, so that (cf. e.g. [19])

gmn(y)dymdyn = zjK'(p2)) dz*dzj = K'(p2)^ dzidz* + K"(p2) E ^*z-jdzidzj . (2.3)

Here primes denote derivatives with respect to p2 and z* = 5*jzj. The Ricci-flatness condition det(didjK) = 1 admits the solution

p2K'(p2) = (p6 + p6)1/3 . (2.4)

Plugging (2.4) into (2.3) gives the metric on the resolved cone, where the origin p = 0 has been blown up into a finite P2 with size controlled by p0.

We wrap eight D7-branes on the P2 base and orientifold under the involution

a : zi ^—z1, (2.5)

combined with —1 £ SL(2, Z). This involution is holomorphic and reverses the holomorphic three-form

Q « eijk dz' A dzj A dzk , (2.6)

as required to preserve N = 1 supersymmetry. The fixed point locus is the P2 itself. Since the stack carries no D7-brane charge or tension in the upstairs geometry, the Calabi-Yau metric we have just derived remains valid. However, there is a net D3-brane charge and C_|

tension induced by a' corrections to the D7-brane action [20]. For the del Pezzo orientifold that we consider, the results of [21, 22] imply7 that there is an induced negative D3-brane charge proportional to the Euler characteristic of the base. There is also a corresponding negative tension. These charges will backreact on the warp factor and C4 in a manner which we derive in section 5. o

Since the P2 is rigid, we expect gaugino condensation to ensue at low energies. The condensate will source imaginary anti-self-dual fluxes [9], and therefore the background will no longer be conformally Calabi-Yau. Our approach to this problem is to search for new supersymmetric supergravity solutions as candidates for the backreaction from nonpertur-bative effects, in some region away from the branes where the supergravity approximation is valid. Let us reiterate that we do not directly incorporate gaugino condensation as a I

localized source (cf. [9, 10]), but instead obtain supergravity solutions that are consistent |—^

with the possibility of such backreaction.

The holomorphic three-form (2.6) is charged under U(1)^, which will therefore be an |_

R-symmetry in the gauge theory. The R-charge of Q is the same as that of the four-dimensional superpotential, R(Q) = +2, so e^ £ U(1)R corresponds to e2^ £ U(1)^. As in the gauge theory, we expect that U(1)r is anomalous, breaking to a discrete subgroup. o

Nonzero G3 flux will further break the R-symmetry since G3 must be odd under the subgroup generated by the spatial involution, z' ^ — z\ We will see in section 7.1 that the R-symmetry breaks to Z2 for our SU(2) structure solutions. We anticipate that, in regions of parameter space where our solution provides a gravity dual to gaugino condensation, this will be the gravity analogue of the expected spontaneous R-symmetry breaking in the gauge theory due to the expectation value of the gaugino bilinear. We assume that the backreacted solution does not break the remaining SU(3) symmetry.

3 Ansatz for the supergravity solution

To simplify the problem, we will first focus on a small R6 patch near the stack. The supergravity ansatz for this limit is presented in section 3.1. In section 3.2 we present the supergravity ansatz for the full Calabi-Yau cone over P2, and in section 3.3 we show that the two ansatze are related by a particular near-stack limit.

7There are subtleties here involving cancellation of the global anomaly found in [5, 6]; cf. section 7.1.

3.1 The near-stack region

We consider the R6 neighborhood of a small piece of the seven-brane stack, and approximate the stack as flat within this region. Imposing the SU(3) symmetry group, the full geometry can be recovered from its local form in the p ^ p0 limit (where p0 determines the size of P2 as in (2.4)).

The first step is to define the correct 'near-stack limit'. We focus on the decomposition of the U(3) = SU(3) x U(1)^ isometry group in this limit, leaving a more detailed mapping of the supergravity fields to section 3.3. Consider the region z3 = 0. We define coordinates

ua = za/z3 , a = 1,2 , z = V)3 , (3.1)

which are invariant under the Z3 orbifold action, and carry charges 0 and +1, respectively, under the U(1)^. The SU(3) decomposes into SU(2) xU(1)T, where the SU(2) acts naturally on the ua, and the U(1)t takes the form

•n o

ua ^ e*dua , z ^ e-2Wz. (3.2) (j~\

In addition, there are four generators of SU(3) that mix the ua and z. These take the infinitesimal form

za ^ za + eaz3 + o(e2) , z3 ^ z3 — ebzb + o(e2) , (3.3)

for complex ea, where eb = ¿baea. Thus,

ua (1 + ube^j + ea + o(e2) , z ^ z (1 — 3ub^) + o(e2). (3.4)

This is a nonlinear transformation on the coordinates, even at first order in ea. However, if we additionally approximate that ru = a |ua |2 ^ 1, then we obtain

ua ^ ua + ea + o(rue,e2) , z ^ z + o{rue,e2), (3.5)

corresponding to C2 translations on the ua. To accomplish this formally, we rescale

ua ^ eua , z ^ ez , (3.6)

and then truncate to leading order in e.

We take (3.6) as the initial definition of the near-stack limit. (In section 6.3 we obtain a more precise definition (6.17) in terms of the effective codimension of the stack: in the near-stack limit, the seven-branes are real codimension two sources, while at longer distances they appear as real codimension six sources.) Geometrically, this limit corresponds to zooming in on a small neighborhood of a specified patch of the D7/O7 stack. As we have shown, 'small' SU(3) transformations — those which map the small neighborhood to itself — decompose locally into C2 xi (SU(2) x U(1)T) transformations; different local patches are related by 'large' SU(3) transformations.

The stack of seven-branes is located in the plane z = 0, and the involution takes z ^ —z. We choose a circle coordinate ^ locally such that z = rze^ where rz is real. In the near-stack limit, we find

rz p3 = r, (3.7)

where r is an alternate radial coordinate on the P2 cone that will be useful below. Thus, r and rz match in the near-stack limit, and except where the distinction is important, we will denote both by r.

In our notation, the low energy effective action8 for type IIB string theory written in Einstein frame is

d1(W -g(10)

R - 1 (W)2 + е-ф|Яз|2 + е2ф|Fi|2 + еф | F312 + 21|2

C4 Л H3 Л F3

where |FP|2 = pFMl^F^1...Mp, H3 = dB2, Fp = dCp-i, and

F3 = F3 — C0H3 , F5 = F5 — 2C2 A H3 + 2B2 A F3 . We adopt a warped ansatz, with ten-dimensional metric

ds20 = e2A(y) (x) dxMdxv + e-2A(y) gmra(y) dymdyn , and five-form field strength9

F5 = (1 + *io) Q4 A d [a(y)] ,

(3.10)

(3.11)

where a is a scalar related to C4, (x) is a maximally symmetric metric on R3'1 with cosmological constant Л = R(4)/4 and volume form Q4, and gmn(y) (times the conformal factor e-2A) gives the internal space metric. We do not assume that gmn(y) is Calabi-Yau, or indeed even complex. The axiodilaton т(y) = C0 + ге-ф = ti + ir2 varies over the compact space, and the three-form flux

Œ И Tl

Сз =

(F3 - тНз)

(3.12)

points along the internal directions only.

Backreaction from the seven-branes changes the metric and sources various supergravity fields. We consider the most general ansatz compatible with the assumed symmetry group C2 x (SU(2) x U(1)t). The internal metric gmn must take the form

gmn(y)dymdyn = grr(r)dr2 + 2g^(r)drd^ + дфф(r)d^2 + e2C(r) ^ du°du° . (3.13)

8The equations of motion must be supplemented by the self-duality constraint F5 = *i0F5.

9With our sign conventions, the Hodge star associated with a D-dimensional metric g with volume form fi(9) is defined by *(dxmi A ... A dxmp)

(D-py. °r9)'"mp mp+1...mDl A ... A dxmD ).

This metric is not in general Hermitian with respect to the complex structure defined by (.z,u%), but can always be made Hermitian by a suitable coordinate redefinition that alters the complex structure,

z ^ \(r)z. (3.14)

Using this, the metric may be brought to the form

ds2 = e-4B(r) (dr2 + r2d^2) + e2C(r) ^ duadua . (3.15)

The metric is now manifestly Hermitian, with Kahler form

e-4B(r)dz A dz + e2C(r) ^ dua A dua . (3.16)

The metric is Kahler if and only if e2C(r) is a constant.

We now consider the general form of G3. Since there are no invariant one-forms pointing along the base (the dua and dua directions), invariant three-forms must have two legs along the base and one along the fiber, and descend from invariant two-forms along the base. These are:

= -i E dua A dua , w2;0 = z du1 A du2 . (3.17)

Invariant three forms are constructed by wedging 1 dz and its conjugate into w1;1, w2>0, and w0>2 = ^2)0. Three-forms built out of w1;1 are even under z ^ —z, whereas those built from w2>0 and w0>2 are odd. G3 then takes the general form:

G3 = g3,0dz A du1 A du2 + g2,1 e2i^dz A du1 A du2 + g1)2e-2^dz A du1 A du2 + g0,3dz A du1 A du2 ,

(3.18)

where the gp,q are complex-valued functions of r only. Comparing with (3.16), we see that J A G3 = 0, so that G3 is automatically primitive.

The scalars a, A, and t can depend only on r. As previously remarked, the U(1)^ symmetry is broken for solutions with non-vanishing G3. In section 7.1 this breaking will be identified with the spontaneous breaking of the exact R-symmetry on the gauge theory side.

Although one can obtain an exact supersymmetric solution to the above system by writing down the equations of motion10 and solving them directly, it is far easier to use the conditions for unbroken supersymmetry, which we will present in section 4 and solve in section 5. First, however, we generalize the preceding ansatz to a Calabi-Yau cone.

3.2 Seven-branes in the P2 cone

Having proposed an ansatz for the relatively simple geometry near a seven-brane stack, we now extend our analysis to seven-branes wrapping the P2 base of the resolved orb-ifold C3/Z3. We will verify that the complete supergravity ansatz proposed in section 3.1 emerges upon taking the near-stack limit of our result for the P2 cone. This connection

0For reference, we summarize the equations of motion in appendix A.

provides valuable insight into the near-stack solution, in particular because knowledge of the solution for a compact four-cycle provides a regulator for divergences associated with the noncompact nature of the four-cycle in the near-stack ansatz.11

We develop the ansatz using the C3/Z3 coordinates, and later shift to a different chart appropriate for studying a small neighborhood of the resolved P2. As before, we wrap eight D7-branes on the resolved P2 and orientifold using the involution zi ^ — zi combined with — 1 £ SL(2,Z). The Calabi-Yau metric has an isometry group U(3) which acts naturally on the z\ However, z% ^ — z% lies within the U(1) factor, so nonzero G3 will spontaneously break the U(1), leaving an SU(3) symmetry group. We now study the ansatz that arises upon imposing this symmetry group. Since the orbifold action Z3 c SU(3), SU(3) singlets are never projected out by the orbifold, and we need not consider Z3 invariance separately.

3.2.1 Metric ansatz for the P2 cone

The backreacted ten-dimensional metric will be of the form (3.10), where the internal '

metric gmn must be invariant under the SU(3) symmetry. Since the symmetry group acts transitively on the horizon, we are free to choose a particular point or region on the horizon S5/Z3, and we select the z1 = z2 = 0 plane. The symmetry group within this plane decomposes to SU(2) x U(1). The metric evaluated in this plane must take the SU(2) x U(1) invariant form:

gmndymdyn = f1(|z3|) (z3/z3) dz3dz3 + c.c. + f2(|z3|)dz3dz3 + f3(|z3|)(dz1dz1 + dz2dz2).

(3.19)

Since f1 is complex, the metric depends on four real functions. We can construct the global I *

form of the metric by combining the invariant one-forms dp2, 9p2, and i dzidzi:

gmndymdyn = ^f1(p)(dp2)(dp2) + c.c. + p2 (f2(p) — fa(p^ (dp2)(<9p2) + fs(p) e dzMzf.

i (3.20)

This reduces to the local form given above at z1 = z2 = 0, as dp2 = ^i zidzi. Since p2 is SU(3) invariant, as is the complex structure, the ansatz (3.20) is manifestly invariant.

The metric (3.20) can always be made Hermitian by a suitable redefinition of the complex structure that preserves the SU(3) symmetry,

zi ^ A(p)zi, (3.21)

where A £ C2. One can show that f1 can always be set to zero by an appropriate choice of A. We define

f2(p) = p4e-4B(p) , f3(p) = "~"2e2C(p), (3.22)

to make the positivity of the metric explicit. Thus,

gmn(y)dymdyn = i'p2e-4B(p) — -1 e2C(pA e zizjdzidzj + e2C(p) ^^ e dzidzi. (3.23)

11For example, a' corrections induce D3-brane charge on certain compact four-cycles, but the topological information determining this charge is lost in taking the near-stack limit.

The choice of notation is not accidental; we will see shortly that in the near-stack limit this form reduces to (3.15).

The corresponding Kahler form,

J = -L (p4e-4B(p - 1e2CW A <9p2 + -Le2C(p d9p2 , (3.24)

2p2 V p2 / 2p2

can be rewritten as J = e-4B%i,i + e2Cwi,i, where

wi,i = fd9p2 - -1dp2 A 9p2) , xi,i = ^dp2 A <9p2 . (3.25)

2p2 V P2 7 ' ' 2

Note that w1;1 points along the base, and %1;1 points along the fiber. In components, the metric is

P4 1 1 -

9ij =ye-4BPj + 2p2 e2C (Sij — Pj , where Pj = ^ *z_ = %5_jP_, z_ = %zj, (3.26)

and P™ is a real rank one projector satisfying P^z- = z\ The determinant of the metric is

8G = 8 det gi_ = 3eijk£1m_£i_£jm£kn = Td^t^ = e-4B+4C . (3.27)

3.2.2 Kahlerity and the Calabi-Yau metric

Let us rewrite the metric in terms of the alternate radial coordinate r = |p3. We find

W1,1 = ^ — 1 dr2 A dr2^ , X1,1 = ^dr2 A 9r2 . (3.28)

Note that dw1;1 = 0 and d%1)1 = —3dr2 A w1;1. The Kahler condition now takes a particularly simple form:

-3 e-4B + 1 (e2C

Thus, Kählerity requires e 4B = 6r (e2C)', where f = gT /. For a Kahler metric, (3.27) implies

8G = - (e2C)' e4C = — (e6C)' . (3.30)

6r v ; 18r v !

As a check, we consider the special case in which the metric is Calabi-Yau. The Ricci

form is

R = —idd log G =--—

dr2 Л <9r2 — d<9r2 = —

2rG r dr

rG' 2G

(3.31)

Xl,1--•

Thus, G' = 0. We find the solution e6C = 9 g0 [r2 + r§], so that

e2C = flo [9 (r2 + r02)]1/3 , e-4B = flo [9 (r2 + r02)]-2/3 . (3.32)

For r0 > 0 this is the Calabi-Yau metric for the resolved P2 cone, and for r0 = 0 it describes the singular P2 cone, for which the metric reduces to the canonical one on C3 with overall scale g0.

cn КЗ

dr2 Л W1,1 =0 • (3.29)

3.2.3 G3 ansatz for the P2 cone

We now enumerate the SU(3)-invariant p-forms of various ranks. The only invariant one-forms are dp2 and its conjugate, which point along the fiber. In addition to the invariant two-forms %i,i and wi,i discussed above, pointing along the fiber and the base, respectively, there exists a complex invariant (2,0) form pointing along the base:

W2,o = 1 £ijkzMzj A dzk , (3.33)

Including the conjugate w0,2 = , this exhausts the list of invariant two-forms. All invariant p-forms for p > 3 can be written as wedge products of invariant one-forms and two-forms. Note that w2,0 A w1,1 = 0, as both point along the base. These results can be checked by considering the z1 = z2 = 0 plane as in section 3.2.1.

We now consider the most general form of G3 that preserves the SU(3) and is odd under z* ^ — z\ Since w1,1 and %1,1 are even under the involution, whereas w2,0 and w0,2 •n

are odd, G3 takes the general form

G3 = g3,0(r) W3,0 + g2,1 (r) W2,1 + g1,2(r) W1,2 + g0,3(r) W0,3 , (3.34)

We immediately find w1,1 A G3 = %1,1 A G3 = 0; therefore G3 is automatically primitive. In components, Gj = £jk£3,0 and Gj = jiz*£ jkizlg2,1.

3.2.4 D3-brane charge

The seven-brane stack wrapping the P2 can carry additional charges besides its seven-brane charge, which is fixed by the —1 e SL(2, Z) monodromy. However, since the horizon S5/Z3 has vanishing third Betti number, F3 and H3 must be exact, and the stack cannot carry five-brane charge. We now show how to compute the D3-brane charge of the stack.

We define the D3-brane charge enclosed in a region R via the generalized Gauss's law:

QD3(R) = — / F5 = (2n)V2Nd3 , (3.37)

where D3-branes carry positive charge 2k20 p3 = (2n)4a'2, and there is a bulk contribution from the three-form fluxes, QD3 = f F3 A H3 + Qloc. Using (3.11), we obtain:

F5 = Q4 A da — e-8A da. (3.38)

Integrating over the S5/Z3 at constant radius and accounting for the Z2 involution, we find

QD3(r) = / re4C-8A da f1 w2 1 A ^ = — re4C-8A ^ , (3.39)

w3 0 = dz1 A dz2 A dz3 = 1 dr2 A w2 0 = dw2 0 = Wn 3 , (3.35)

r2 . . ,

W2 ,1 = r^<9r2 A W2 , 0 = £kimjzkdzJ' A dzl A dzm = 2 . (3.36)

dr 2 1,1 2 dr

where w = I(d — <9)r, dr = 2 J2 A dr, and we use the periods

1 2 _ 1 „2 1 1 , ,2 A ,, _ „3

2 w2,i = 2 n2 , / - w2,i A w = n3 . (3.40)

2 2 ./«5/7.,, -

ZF2 2 2 .7S5/Z

Since the integrands are closed, these integrals can be computed over any surface in the specified homology class. The periods (3.40) can also be used to compute the volume of the resolved P2:

vol(P2) = 1 n2 e4C(r)

(3.41)

The D3-brane defined by (3.37) is sourced in the bulk, and therefore not quantized. C_|

We define the Page charge [23] via the flux integral12 |~[~|

QDa3ge(R) = — $ ( + ^ C A H3 — ^ B2 A F3 ) . (3.42)

2 C2 A H3 — 2 B2 A F3J . (3.42)

The integrand is closed in the absence of sources, so the Page charge is not sourced in the bulk. In the absence of local sources coincident with dR, the Page charge is invariant Cn

under small gauge transformations of B2 and C2. It is not invariant under large gauge x-—^

transformations unless F3 and H3 are exact when pulled back to dR. It has been argued [24] that the Page charge is quantized.

One can solve the Bianchi identity for G3 to obtain

A2 = —U (C2 — tB2) = (g3,0 — g2,l) W2,0 + (g0,3 — gl,2) Wo,2 . (3.43)

It is then straightforward to compute the Page charge of the stack by the same method as above:

~ _4C-8A du , o „.2i„ „ 12 o^U ~ 12

12As there is more than one way to solve the F5 Bianchi identity, there are inevitable ambiguities in defining the Page charge [24]. This difficulty does not arise in our setup due to the vanishing of the third Betti number for the horizon dR = S5 /Z3, so that all definitions are related by integration by parts.

QDa3ge = Y (re4C-8A da +2 r2 |g3,0 — g2,i|2 — 2 r2 |gi,2 — g0,a!^ , (3.44) Q

where the right-hand side is independent of r as a consequence of the Bianchi identities. Since F3 and H3 are exact, the Page charge is gauge invariant. Henceforward, when we refer to the D3-brane charge of the solution, we mean the Page charge given by (3.44).

3.3 The near-stack limit of the P2 cone

We now apply the near-stack limit (3.6) to the above ansatz to recover the near-stack ansatz described in section 3.1 and to fix the relationship between the fields. As before, we apply the coordinate transformation (3.1), rescale as in (3.6), and truncate to leading order in e. Finally, since e is a formal expansion parameter, we set e = 1. We find

dr2 ^ dr2 , %1j1 ^ i dz A dz , w1,1 ^ |i ^ du" A du"j . (3.45)

Thus, in particular

J ^ (e-4Bdz A dz + e2C ^ A du") . (3.46)

We also find

G3 ^ $3,odzAdu1 Adu2 dzAdu1 Adu2 dzAdu1 Adu2 +g0,3dzAdu1 Adu2 . (3.47)

Therefore the quantities r, B, C, and gp,q defined in this section are appropriate generalizations far from the stack of the quantities r, B, C, and gp,q defined in section 3.1, and the two systems correspond in the near-stack limit. As an example, applying the near-stack limit to the Calabi-Yau metric for the resolved cone (3.32) gives a flat metric with B and C_|

C constant. The parameters r0 and g0 of the P2 ansatz can be recovered from |~[~|

1 ^2Bns +Cns - = e4(Cns-Bns)/3 to

r0 = - e ns+Cns , g0 = e4(°ns-Bns)/3 , (3.48)

where Bns and Cns are the constant values of B and C in the near-stack ansatz. 4 Supersymmetry conditions

13While type-changing SU(3) structure loci are possible, there is still a local SU(2) structure away from these loci. Global, i.e. "strict," SU(3) structure is impossible in AdS4 compactifications [26].

•n o Cn

Our approach to learning about the gauge dynamics of compact seven-branes is to study the allowed supersymmetric backgrounds surrounding the seven-brane stack subject to some symmetry group, in the same spirit that early constructions of supersymmetric extremal black holes in supergravity (cf. e.g. [25]) presaged the appearance of D-branes as localized sources in supergravity.

In general, one expects AdS4 vacua from compactifications that are stabilized by gaug-ino condensation on seven-branes [1]. This is because the superpotential generically devel- |—

ops a vev. If the compactification has finite warped volume, and therefore a finite four- |_\

dimensional Newton constant, the superpotential vev generates a negative cosmological constant, leading to an AdS4 compactification. Noncompact solutions may be Minkowski, but we expect such solutions to arise in an appropriate decompactification limit of an AdS4 solution.

We first consider general properties of supersymmetric AdS4 compactifications, and return to the question of noncompactness below. As we will see, supersymmetric compactifications of type IIB supergravity to AdS4 always have SU(2) structure.13 In our case, the SU(2) structure is dynamic. As dynamic SU(2) structure is less familiar than the more commonly studied strict SU(3) structure, we now review dynamic SU(2) structure in AdS4 compactifications of type IIB supergravity, using tools from the more general field of generalized complex geometry.

4.1 Review of SU(2) structure and generalized complex geometry

Having argued in section 1.2 that the supergravity solution should admit two spinors (n+, n+) that define a dynamic SU(2) structure, the next step is to analyze the gravitino

and dilatino variations for the supergravity ansatz introduced above. As shown in [7], the supersymmetry analysis simplifies considerably in terms of bispinors. Here we review the basic results from G-structures and generalized complex geometry needed for the rest of the work. For a recent review with further explanations and references, see e.g. [27]. We work in ten-dimensional Einstein frame, with metric ansatz (3.10),

ds?0 = e2A(y) V(x)dx^dxv + e-2A(y Wy)dymdyn . (4.1)

The warp and conformal factors are already made explicit and all the geometric quantities are constructed in terms of the internal metric gmn. The four-dimensional metric is that of AdS, with cosmological constant

A = — 3M2. (4.2) ¡U

The supersymmetry conditions were obtained in [7, 28], in string frame. The conversion M

from string frame to Einstein frame is done by modifying their warp factor A(s) and pure ' ^J

spinors *(S) as follows:

A(S) = A + ^ , = e^4-A)p*, (4.3)

where the definition of the operator p is given by

p Cp = pCp (4.4)

for a p-form Cp. The rescaling in the pure spinor takes into account that the Mukai pairing I

(*, *) is normalized by the volume of the internal metric d6y v7det . |—*

As in (1.1), we decompose the ten-dimensional Majorana-Weyl supersymmetry generators e* in Einstein frame,14

e* = Z+ ® + Z- ® n- . (4.5)

The internal spinors n* must have equal norms for an AdS4 compactification [28]. Preservation of four-dimensional N =1 supersymmetry then imposes the normalization

ln+12 = !n+12 = eA , (4.6)

up to an arbitrary overall rescaling.

The two internal spinors may be combined into even and odd bispinors,

= —8ie-A n+ ® n+ , = —8ie-A n+ ® n-, (4.7)

where the extra warp factor dependence has been included for normalization purposes. Using the Clifford map, these bispinors are sums of forms of different degrees (polyforms). The supersymmetry conditions of [7] then become

dH (e(^/4-A)pe4A Re*+) =—3e(^/4-A)p e3A-<^4 Re(p*-)+e(2A-^/2)(3-p)e4A+^ *6 A(F),

dH (e(^/4-A)p e2A-^/2 Im*+) = 0,

dH (e(^/4-A)p e3A-^/4 *-) = —e(^/4-A)p e2A-^/2 Im *+ . (4.8)

14Using the conventions of [29], n- = Cn* , where C is the charge conjugation matrix.

Here dH = d — H A, and all fluxes are internal, with

F = Fi + F3 + F5(int) , A(Ap) = (—1)p(p-1)/2Ap . (4.9)

When supplemented by the p-form Bianchi identities, the above supersymmetry conditions imply all of the supergravity equations of motion (which we will verify explicitly in our examples).15

Now, let us consider the geometric properties of manifolds with SU(2) structure [31]. It is convenient to introduce two globally defined orthonormal spinors n+ and x+. They are related by a vector Om,

X+ = 20mYmn- , (4.10) C_|

where |O|2 = 2. An SU(2) structure is then characterized by the following invariant forms:

n-7mX+ , (J2)mn = -n+ - X+7mnX+) , (^2)mn = ¿X+Ymn^ . (4-H) These satisfy

J2 A Q2 = ^2 A Q2 =0 , J2 A J2 = 2^2 A Q2 , «©^2 = «©^2 = 0 . (4.12)

Algebraically, the tangent bundle has a product structure, where Q2 and J2 may be thought of as the holomorphic two-form and Kahler form for a complex-dimension two subspace of the tangent bundle, and 0 and J1 = 20 A 0 as the holomorphic one-form and Kahler form for the complex-dimension one complement. However, this product structure is typically not integrable, and the manifold need not be a direct product. Instead, we will think of the I—

manifold as a line bundle with J2 and Q2 pointing along the base and 0 and 0 pointing "

along the fiber. This structure will turn out to be integrable in our examples, but this is |_\

not guaranteed in general.

The SU(2) structure can be viewed locally as the intersection of two SU(3) structures, each associated to one of the spinors. In particular, the SU(3) structure from n+ is defined o

Jmn = ¿n+7mnn+ , "mnr = ¿n_7mnr n+ . (4.13)

The forms (4.11) and (4.13) are related by

J = J2 + 0 A 0 , Q = 0 A Q2 . (4.14)

There are different ways of writing the spinors in terms of (n+, X+). In geometries with orientifold planes it is most convenient to average n+ rc (n+ + ), where ¿e^ is the relative phase between the two spinors [32, 33]. Thus we take

n+ = i e^/2 eA/2 (cos f n+ + sin | x+) , n+ = e_^/2 eA/2 (cos | n+ - sin f x+) . (4.15)

The warp factor is fixed by the normalization (4.6), and § and f parameterize the angle between the spinors,

e_A n+fn+ = ¿e^ cos f. (4.16)

5In the presence of sources, one must also impose calibration conditions on the sources; cf. [30].

Using this, the bispinors may be expressed in terms of the SU(2) forms (4.11), yielding

*+ = e^e 2eAe

cos ^ ( 1 — 1 J| ) + sin Im Q2 — iJ2

sin ^ ( 1 — 1 J| ) — cos Im Q2 + i Re Q2

(4.17)

16Here the 'type' of a polyform ^ refers to the rank of the lowest-rank non-zero component of as in [34]. In what follows, type (m,n) refers to of type m (n).

It is straightforward to check by substituting this result into (4.8) that we must take e^ = ±1 for AdS4 solutions, where the extra sign can be absorbed by redefinitions. Thus, we take $ = 0 without loss of generality.

The supersymmetry structure is characterized by the angle which in turn determines (~ |

the types of *+ and *-.16 For static SU(2) structure (type (2,1)), ^ = n/2 and the two internal spinors are everywhere orthogonal. For strict SU(3) structure (type (0,3)), ^ = 0 and the spinors are everywhere parallel; the polyforms simplify to *+ = e^e-tJ and = i Q3. For intermediate SU(2) structure (type (0,1)), 0 < ^ < n/2, and the spinors are neither parallel nor orthogonal.

If varies along the internal manifold, the SU(2) structure is said to be dynamic. Dynamic SU(2) structure can be 'type-changing' if ^ = 0 or ^ = n/2 on some locus. Our solution will turn out to be dynamic; for certain values of the parameters, it is also type-changing with a ^ = n/2 locus.

We now impose the orientifold projection O = Qp(—1)Flct, where a is the involution, o FL is the number of left-moving fermions, and Qp is the worldsheet parity. The involution |— on the pure spinors (*+, *-) should reduce to the known result |_\

ct(j) = J , ct(Q) = —Q , (4.18)

for an O3/O7, so that we have [28], I_\

= A(*+) , ) = A(*-), (4.19)

with A defined in (4.9). Applying (4.19) to (4.17) gives

ct(j2) = J2 , ct(Q2) = —Q2 , ct(0) = 0 , ff(p)= p. (4.20)

In other words, in the basis (4.15) the orientifold action is realized as an explicit n rotation in the (ReQ2, ImQ2) 'plane' of the space (J2, ReQ2, ImQ2) [33].

For strict SU(3) structure compactifications, we must take the holomorphic three-form Q to carry R-charge +2 [35]. Generalizing this, we see that should carry R charge +2. Thus, 0 and p carry R-charge +2, while Q2 and J2 are invariant.

Substituting (4.17) into (4.8), we obtain the supersymmetry conditions for compactify-ing type IIB supergravity to AdS4. As remarked above, supersymmetric compactification to AdS4 requires n+^n+ = n+^n+

and Re n+fn+ = 0. We refer to Minkowski (p = 0) solutions that satisfy these conditions as 'AdS-like.' The set of 'AdS-like' solutions may

be thought of as the closure of the set of AdS solutions, since a solution that arises upon taking a limit in parameter space of an AdS solution must still satisfy nV"n+ = n+^n+ and Re= 0, provided that n1 and n2 vary continuously in this limit. Thus, we expect that noncompact solutions with gaugino condensation will be AdS-like. AdS-like solutions with strict SU(3) structure are the well known type B supersymmetric solutions which arise from F-theory compactifications.

The supersymmetry conditions for AdS and AdS-like compactifications can be rewritten in an SL(2, R) covariant form. In section 4.2, we briefly review SL(2, R) covariant supergravity. In section 4.3, we present the covariant supersymmetry conditions, deferring a detailed derivation to [36], and discuss their implications. ç-

4.2 SL(2, R) covariant supergravity \ I |

It is well known that the action (3.8) is invariant under SL(2, R) transformations, where

F3 \ aM F3 H J ^ I c d J I H3

Q ^ ( I ' J Q, (4.24)

where Q is the 'S-charge' of Q. In this language, G3 and A2 carry charge +1/2 and -1 dr carries charge +1.

Since the t-dependent phase is in general nonconstant, the derivative of an S-covariant quantity is not covariant. We introduce a covariant derivative dM ^ DM = dM + ¿QKM where KM is a one-form connection. One can check that KM = i dMt1 transforms appropriately, where t1 = Re t. Thus, we define the covariant exterior derivative:

DQ = d Q + iQ — dri A Q . (4.25)

t = C0 + ie ^ transforms as t ^ aT+d, F3 and H3 mix as:

(4.21)

with ad — bc = 1, and the metric and C4 are invariant. Accounting for brane sources as required by string theory, this SL(2, R) breaks to the discrete subgroup SL(2, Z), but this subgroup is gauged: monodromies are allowed, and indeed charged seven-branes carry SL(2, Z) monodromies. I—

Since we are interested in studying nonperturbative effects on seven-branes, an ap- |—

proach that makes SL(2, Z) (and indeed SL(2, R)) invariance manifest is indispensable (cf. e.g. [37]). It will be convenient to work with -1 dT and the complex field strength and potential

G3 = -^(Fa — TH3) , A2 = -^(C2 — TB2) . (4.22)

VT2 VT2 o

These transform by a t-dependent phase under SL(2, R):

|ct + d| ^ 1 . /|ct + d|\ 2 1

G3 -—f G3 , — dT ^M-—^ — dT. (4.23)

ct + d t2 \ ct + d / t2

These are both examples of a more general transformation law,

•|ct + d|

It is easy to check that D is not nilpotent: D2 = ^ dri A dr2. We define the modified

covariant exterior derivative for Q of charge +1/2:

D± Q = D Q ±-dr A Q* .

± 2 T2

(4.26)

Similarly, for Q of charge -1/2, we define D± Q = D Q ± dr A Q* = (DT Q*)*. One can check that D+ and D- are both nilpotent.17 The G3 Bianchi identity now takes the simple form D- G3 = 0, with the local solution G3 = D- A2.

Expressed in terms of t and G3, the type IIB supergravity action (3.8) becomes

J d10.^

R - |dr |2 + |G3|2 + 1 |fF5|2

- A G3 A G* , (4.27)

where, in terms of G3 and A2, F5 takes the local form

F5 = dC4 + ( 4 A A G3 + c.c. ) ,

(4.28)

with the Bianchi identity d F5 = 2 G3 A G3 . One can then rewrite the equations of motion

D- *10 G3 = iG3 A F5 , D *io( ^dr)=|G3 A *WG3 ,

Rmn = -pä (VmT Vwr + c.c.) + 1

4 To 2

|G3|Mw - 1 |G3|2 ¿MN

(4.29)

+ 4 |F5|MN , (4.30)

and in addition one must impose the self-duality constraint F5 = *10F5. The action and equations of motion are now manifestly covariant under SL(2, R) transformations.19

Consider the additional Z2 symmetry of type IIB supergravity under which all of the RR fields are reversed (i.e. Cp ^ —Cp and Fp ^ — Fp). This is an exact (gauged) symmetry of string theory which we denote z2RR), corresponding for instance to the involution associated with the O9 plane of the type I theory composed with the —1 e SL(2, Z). In the S-covariant language developed above, z2RR) takes the form:

G3 ^ -G3

T — T

F5 ^ -F5

(4.31)

If Q carries S-charge Q and transforms as Q ^ —Q3 under z2RR), then DQ ^ —(DQ)3. For Q = ±1/2, D± Q ^ —(D± Q)3. Viewing z2RR) as diag(—1,1) e SL±(2, R), it combines with the SL(2, R) invariance discussed above to generate the classical symmetry group SL±(2, R), of which the subgroup SL±(2, Z) is an exact (gauged) symmetry of string theory.

7Note that D± are not C-linear (iD+ = D-i) and do not always obey the product rule.

8Here | Fp

p | MN

2(p-1)! (Fp)

Mi...Mp_i

p ) M1...MV

+ c.c., so that gMN |Fp|Mw = P |Fp|2.

The full supersymmetric action is also invariant [38].

4.3 SL(2, R) covariant supersymmetry conditions for compactifications

Henceforward, we work with the almost complex structure defined by the holomorphic three-form Q = Q2 A 0 with associated Kahler form J = Ji + J2, where Ji = (¿/2)0 A 6. Consistent with (4.20), we assign Q2 an S-charge of -1/2 and take J2, 0, and ^ to be invariant. G3 can be decomposed into pieces with zero, one, or two legs along the 'fiber' (0, 6) directions. The supersymmetry conditions directly imply G3 A OA J2 = G3 A6 A J2 = 0. Thus, a general decomposition takes the form [31]:

G3 = Ji AGi + J2 A©1 + g3,0 Q2 A0 + g2,i Q2 A8+gi,2 Q2 A0+ go,3 Q2 A8 + Gi,i A0 + ©i,i A8 ,

(4.32)

where Gi and ©i are complex one-forms pointing along the base, the are complex

d/ = (d©/) 0 + (d©/) 8 + dn/. (4.33) o

scalars, and Gi,i and ©1,1 are complex primitive (1,1) forms pointing along the base. We also decompose the gradient into fiber and base directions:

Applying these decompositions to the supersymmetry conditions and simplifying, we obtain:

d [e4Ac^] = -3 e2A s^ Re (p8) + da, d [e2As^8] = 2ip (c J + J2) , d [c^ J1 + J2] = 0,

(4.34)

D+ [e2As^] = c^e4AG3 - ie4A * G + 3^(1 + A 8 - 3B(1 - c^Q A B , (4.35)

fl3,0 = -^ e-2A^d©r , g2,i = ^ e-2A^d©r, (4.36)

2s^ T2 2s^ T2

gi,2 = pe +—2-e 0©$_ , go,3 = pe--2-e d©$+ (4.37)

Gi,o A J2 = -s-1e-2A ^^ dn ^ A Q2 , ©1,0 = -c^Gi,o , (4.38)

Go,i A J2 = -s-1e-6A (dn [e4A] - c^dna) A Q2 , (4.39)

©0,1 A J2 = s-1e-6A (c^dn [e4A] - dna) A Q2 , (4.40)

where c^ and s^ are shorthand for cos ^ and sin and

$± = e4A ± a. (4.41)

By referring to the charge assignments of table 1, one can verify that the above equations are manifestly invariant under SL(2,R), and also under a U(1)r symmetry.

In fact, the supersymmetry conditions (4.34)-(4.40) are covariant under the full SL±(2, R) classical symmetry group of type IIB supergravity, which is generated by SL(2, R) transformations and by diag(-1,1) e SL±(2, Z). The latter transformation, (4.31), takes G3 ^ —G3, r ^ —r*, and a ^ —a, along with ^ —so that

Q2 ^ —Q* , ^ ^ + n , J2 ^ — J2 , (4.42)

S-charge R-charge S-charge R-charge

G3, Q 2 +1/2 0 fl1,2 0 -2

5t/t2 +1 0 fl0,3 0 +2

P, 0, <9© 0 +2 G1, ©1 +1/2 0

fl3,0 +1 -2 £1,1 +1/2 -2

fl2,1 +1 +2 ©1,1 +1/2 +2

Table 1. The S-charge and R-charge of various fields; Jt represents any differential of t.

with the appropriate transformations on the components of G3. In addition, the super- C_|

symmetry conditions possess the Z2 symmetry |~[~|

0 ^ —0 , ^ ^ —^ , ^ -Q2, (4.43)

under which the polyforms and are invariant.

The supersymmetry conditions for type B solutions are well known, and are easily derived from (4.8): t must be holomorphic, J must be closed, $_ must vanish, G3 must be primitive and of Hodge type (2,1), and d(e^/2 Q) = 0, which can be restated covariantly as DQ = 0. Unlike type B solutions, AdS and AdS-like SU(2) structure solutions need not be Kahler or even complex. Taking the (1,2) component of (4.35) and applying (4.36), (4.38) we obtain

[d Q2]i,2 = 1 e2A J2 A (Gi,o)* - e2A 0 A (Gi,i)* . (4.44)

2 V |_\

Applying (4.34), we find [dQ]22 = [dQ2]12 A 0. Thus, the almost complex structure associated with Q = Q2 A 0 is integrable if and only if £1)0 and G1,1 both vanish. Similarly, the Kahler form J is not in general closed. Applying (4.34), we find I *

d J = -2 p e_2A J2 A Re 0 - e_4A () d $+ A J1. (4.45)

\1 + c^ J

Adding (4.39) and (4.40), we see that dn vanishes if and only if G0)1 + ©0)1 = 0. Thus, J is closed if and only if the solution is Minkowski with G0)1 = -®0)1. Using (4.38), the above conditions on G0)1 and on the pair G0)1 and ©0>1 can be efficiently restated as the requirement that G2>1 and G1>2 be primitive, respectively.

In [10] it is argued that the appropriate generalization of the Gukov-Vafa-Witten on-shell flux superpotential [39, 40] to generalized complex geometry solutions is the on-shell superpotential

W = ^ J (e3A(s)_^ A A(F) , (4.46)

where A(s) and ) are related to the Einstein frame quantities A and by (4.3), A is defined in (4.9), and the integral is over the compact manifold. Supersymmetric AdS solutions correspond to a nonvanishing superpotential vev. Using the supersymmetry conditions (4.34)-(4.40), we can evaluate (4.46) explicitly. We find

p I n-, „_4A _ p

„2 10

W = -4 /d6y Vge_4A = -4 , (4.47)

K10 J K4

in exact agreement with the four-dimensional supergravity result A = —3 k4 |W|2, where is the four-dimensional Newton constant. The limit of rigid supersymmetry, ^ 0, is more subtle, as the integrand of (4.46) vanishes on-shell, but the integral is taken over an infinite volume.

5 Supergravity solution in the near-stack region

In this section we search for supersymmetric solutions to the ansatz described in section 3.1. First, as a warmup we describe type B solutions to this ansatz. The axiodilaton must be holomorphic, but can only depend on the real coordinate r, so it is a constant, t = C0 + g-. Thus, the solution is conformally Calabi-Yau, with B and C constant, consistent with the near-stack limit of the metric (3.32). The only nonvanishing component of G3 is G2,i, which must be closed by the G3 Bianchi identity; thus, g2,i = e k2,i/r2. Finally, we take M

a = e4A = $+/2, and solve the F5 Bianchi identity, (A.9), which gives:

5.1 Supersymmetry conditions in the near-stack region

We now apply the supersymmetry conditions for AdS and AdS-like vacua (section 4.3) to the near-stack ansatz of section 3.1. We assume that the internal spinors n1 and n2 are singlets under the SU(3) symmetry group, so that Q2, J2, 0 and ^ are also singlets under the SU(3).

The only invariant one-forms are 1 dz and its conjugate, up to a radially-dependent factor. In order to satisfy the orthonormality conditions g-1(0, 0) = 0 and g-1(0, 0) = 2, we must take

0 = e-2^ r dz, (5.3)

for some real d = 0(r). Q2 must be a complex two-form that is odd under the involution z ^ —z, and that satisfies i© Q2 = 0, Q2 A Q2 = 0, with the normalization vol6 = 4 J1 A Q2 A Q2. Therefore, we must have

Q2 = e2C+i« - du1 A du2 , (5.4)

for some phase factor £ = £(r), and the complex structure defined by Q = Q2 A 0 is integrable by inspection. Then, (5.3) and (5.4) uniquely determine J1 and J2:

J1 = |e-4Bdz A dz , J2 = | e2C (du1 A du1 +du2 A du2) . (5.5)

1 dr(r >0 = —2e-4C |g2,i|2, (5.1)

where primes denote derivatives with respect to r. We integrate to obtain = k0 + ki logr — |k2,i|2. Comparing with (3.44), we see that ki is related to the D3-brane charge:

QD3 = —n3 r e4C ($-i); + n3 r2 |g2,i |2 = —n3 e4C ki, (5.2)

where (n2/2)e4C is the volume of the resolved P2 and the near-stack approximation is valid |—* for r ^ r0, where r0 = | e2B+C, as in (3.48).

Applying (4.34), we see that J2 is closed, since ^ = ^>(r) is a singlet scalar, and therefore c^ J1 is closed by (5.5). Thus, C is constant and the metric is Kahler. By (4.45), the solution must be Minkowski (p = 0).20

Applying p = 0 to the first equation of (4.34) and integrating, we find

a = c^ e4A , (5.6)

where we fix the a shift symmetry. The middle equation of (4.34) gives

e2A s^ 0 = ci dz, (5.7)

where ci is a complex constant. Thus,

rs^ e2(A-B)+^ = ci, (5.8)

and 9 must be constant. Combining (5.6), (5.8) to eliminate we find

$+= ic2- e4(A+B). (5.9) ^

Henceforth, we assume that c1 = 0, since the special case c1 = 0 is just a type B solution, as discussed above. For c1 =0, ^ = 0,n, and so (5.6) implies that |a| < e , and therefore

$± > 0 . (5.10)

Now consider the decomposition of G3, (4.32). SU(3) invariance constrains G1, G1, |—^

G1,1, and ©1,1 to vanish. Thus, we decompose:

G3 = g3,0 C A 0 + g2,1 C A 0 + g1,2 C2 A 0 + go,3 C2 A 0 , (5.11)

where the gp,q are related to the gp,q of section 3.1 by

<3,o = e2(C-B)+*«+%3,o , <2,1 = e2(C-B)+i(^-0)02,1, <71,2 = e2(C-B)-iV,2 , go,3 = e2(C-B)-i«+%o,3 . (5.12)

The conditions (4.38)-(4.40) are now trivially satisfied. We use the above decomposition to write out the remaining conditions. Applying (5.6), (4.35) becomes

D+ [e2A s^ Q2] = $+ (G3,o + G12)* - $- (G2,1 + Go,3)* , (5.13)

since G3 is primitive. Next, (4.36), (4.37) become:

—4A • / —4A • /

re ir' re ir' _

g3,o =---$ - — , fl2,1 = , _ $+— , (5.14)

4 c1 t2 4 c1 t2

r e r e

fl1,2 = —:-, flo,3 =--7— , (5.15)

4 c1 - 4 cl1 - +

20This is an artifact of the near-stack approximation: (4.34) requires ^ = 0 for p = 0 solutions on the P2 cone. The AdS-like solutions obtained in this section may be viewed as 'approximately AdS,' in that they approximate solutions on the P2 cone with small cosmological constant.

where we have

„ , r

(5.16)

df (r) = f '(r) dr = f '(r) e2A s^ Re

so that

— в ci

do/ = e2A s, —/'(r). (5.17)

It is convenient to fix an SL(2, R) frame. To do so, we choose some radius r and perform an SL(2, R) transformation to make t'/t2 imaginary. Thus, at radius r, g3,0 g0,3 and g2,i gi,2 are both real. However, in this case the axion equation of motion (A.11) reads C0' = 0. Thus, in this frame the axion is constant, so that ¿t'/t2 = 0'. More general solutions can be recovered from this special case by an SL(2, R) transformation. j~yj

We evaluate the left-hand side of (5.13), taking dC0 = 0:

D+ [e2A s, Q2] =d [e2A s, Q2] — - e2A s, d0 A Q2 . (5.18)

Using (5.16), it is straightforward to check that the second term cancels against the (1, 2) + Cn

(0,3) component of the right-hand side of (5.13).2i We are left with ^

d [e2A s, Q2] = $+ Gi,2 — G0,3 . (5.19)

We evaluate the exterior derivative using (5.4) and (5.16): I *

d [e2As, Q2] = s,^(r^s,) 8AQ2 + 2- e2A-^s,dL(ie2A+^s,) 8AQ2 .

i i (5.20)

Comparing with (5.15), we see that £ must be a constant. After some manipulation, (5.19) |_\

becomes

1 A (V2e-4A Ф+Ф-) = e-8A Ф-Ф+ , r2 d e-4A Ф+Ф-) = e-8A Ф+Ф- . (5.21) Defining S± = Ф-1, both halves of (5.21) reduce to

2(H+ + E-) = E- - E+ . (5.22)

Now (5.9) can be rewritten as

B = - 4 log

|ci|2 p + * )

(5.23)

The supersymmetry conditions for the near-stack ansatz of section 3.1 are therefore equivalent to (5.10), (5.14), (5.15), (5.22), (5.23), where, in a general SL(2, R) frame, D = 0.22

21This could have been anticipated from the integrability of the complex structure defined by 02 A 0.

22Like fi2, e^ carries S-charge -1/2.

5.2 Solutions in the near-stack limit

We define

A(r) = 1(s- - S+) , f (r) = S- + S+ = rA':

(5.24)

where the final equality follows from (5.22). To further constrain the solution, we must impose the equations of motion. In fact, we will only need two: the 0 equation of motion and the B equation of motion.

First, from the dilaton equation of motion (A.11) we find

1dr [гФ] = 8 e4A-4C GW'3 + g2'V'2) = -S^ ф ,

in the constant axion frame, so that

r (S+ + S-) '

where c2 is another constant.

Now consider the B equation of motion (A.17), which can be written

(5.25)

(5.26)

Œ И TI

1 Я S S' 1

^ [rB/] =-S+S- 2 + 1 (ф/)2

r dr L J 2 (S+ + S-)2 8

Inserting (5.23), we find

(5.27)

1А [r (S+ +S' )] = (S+)2 + (S-)2

r dr L ^ + (S+ + S-)

- 1(S+ + S-) (Ф')2

(5.28)

It is straightforward to check that the E± equations of motion (A.9), (A.10) are satisfied provided that (5.28) and the supersymmetry constraints are obeyed. Substituting (5.26) and (5.24), (5.28) takes the form

1- [f = 1 r dr f

;(f о2(f2 - c2)

(5.29)

A general solution is of the form

f (r) = C3r + C4r + C5 .

(5.3О)

Substituting this result into (5.29), one obtains

С22 = С52 - 4 Сз С4

(5.31)

We integrate once more to find

A(r) = 2 Сз r2 - 2 C4 r 2 + C5 log r + C6 .

(5.32)

23To see this, first differentiate (5.29) with respect to r, obtaining a third-order equation whose nonlinear terms can be eliminated using (5.29). The result gives [rfl'j = 725 where g(r) = rf'(r). The solution is then easy to guess.

s+ = (C5/2 — Co) + C4 r 2 — C5 log r , = (C5/2 + Co) + C3 r2 + C5 log r, (5.33) and (5.26) becomes:

0' = "7-C2 , . (5.34)

r (c3 r2 + c4 r_2 + c5)

This can be integrated to give

" (C5 + C2) r2 +2 C4

0(r) = 00 + log

(5.35)

5.3 Beyond the near-stack limit

We now briefly consider the extension of our methods to the full P2 cone. We must have

8 = 1 e_2dr2 , (5.36)

which agrees with (5.3), since r2 ^ |z|2 in the near-stack limit. We compute

d [e2As,8] = d [rs,e2(A_B)+^] xi,i + 6irs,e2(A_B)+^

= 2 i p (c, e_4B xi,i + e2C wM) , (5.37)

where we used (3.28), (4.34) and the decomposition Ji = 2 8 A0 = e_4Bxm and therefore J2 = J — Ji = e2C wi,i. Comparing the terms proportional to wi,i in (5.37), we find

3rs,e2(A_B)+^ = pe2C . (5.38)

Thus, for s, = 0, we must take p = 0, and the solution is always AdS. Using methods similar to those in section 5.1, one can check that supersymmetric AdS solutions with global SU(3) symmetry of this type do exist. The resulting ODEs are nonlinear and difficult to solve except in certain special cases, and we defer further consideration of this problem to a later work.

Comparing (5.38) with (5.8), we see that ci of the near-stack ansatz is related to p via

ci = Pe2 Cns, (5.39)

where Cns is the constant near-stack value of C.

(C5 — C2) r2 + 2 C4 where we use (5.31).

The full solution is given by (5.33), (5.35), (5.14), (5.15), (5.23), where S± = $±i as above. It is straightforward to check that the remaining equations of motion (A.12)-(A.15), (A.18) are automatically satisfied. An interesting special case is where c2 = 0, or c5 = 4 c3 c4, so that 0' = 0. This implies that g3,0 = g2,:L = 0, but g^2 and g0,3 are nonvanishing. These are explicit examples of constant t solutions that are not type B solutions.

6 Geometry of the near-stack solution

6.1 Singularity structure

The solution found in section 5 depends on several parameters: the a (i = 1... 6), = , C, £, and Co. The a must obey the relation (5.31), and in particular we must have

Cj; ^ 4C3C4 .

The explicit solution is given by

+ (r) = (c5/2 - C6 ) + C4 r2 - C5 log r, H_(r) = (c5/2 + Co) + C3r2 + C5 log r, (6.2)

T = C + — f (C5 - C2)r2 + 2C4 o 7s V (C5 + C2)r2 + 2C4

B = 4 log

|C112 (C4 + C5r2 + C3r4)

with fluxes specified in (5.14), (5.15). The spinor angle ^ varies with radius,

-4A -C4 + 2 co r2 + 2 C5 r2 log r + C3 r4 cos = a e =-5--,

c4 + c5 r2 + c3 r4

so the solution has dynamic SU(2) structure.

For the = 0, n (i.e. non-type B) case that we are considering, the c must all be finite or vanishing, and the solution is regular and supersymmetric whenever the E± are both positive. We now show that a singularity (i.e. a zero in either or ) always occurs at finite radius. Consider the sum (cf. (5.24)):

r2f (r) = r2(H+ + S_ ) = c4 + r2 c5 + r4 c3

This must be positive as a necessary but insufficient condition for regularity and super-symmetry.

Assume that there exists a solution that is smooth at all finite radii. To maintain regularity in the large and small r regions, we must have c4 ^ 0 and c3 ^ 0, and moreover, since the discriminant of the quadratic polynomial r2f (r) is nonnegative by (6.1), c5 must be nonnegative, or else + H_ acquires a root at some finite radius. However, under these conditions will become negative at small r unless c5 = 0. If this is the case, then either c3 or c4 must vanish by (6.1), and c6 must be respectively either positive or negative to obtain a regular solution anywhere. In either case, one of or H_ is constant, whereas the other crosses zero at some finite radius. Thus, a singularity will always occur at some radius. The constraint (6.1) plays a crucial role in this argument.

A singularity at finite radius is always of the eA ^ to type. It is straightforward to check that curvature invariants diverge and the singularity is physical. Moreover, horizons, characterized by eA ^ 0, can only occur for r ^ 0 and/or for r ^ to, so the singularity is naked.

We now classify the available regions of parameter space for which a regular solution exists at some radius. Clearly we must have f = + S_ > 0. However, this is also sufficient at any given point for some choice of c6, since we can always set = H_ at any point of interest by adjusting c6. If either c3 or c4 is positive, then f is positive at large or

Œ M •n o Cn

Figure 1. (c3, c4, c5) parameter space. The region inside the cones is excluded. The surface of the negative cone is also excluded, whereas the surface of the positive cone consists of constant dilaton solutions. Hyperbolae of constant c3 c4 are related by radial rescaling.

small r, respectively, and c6 can be chosen such that a regular region exists. For solutions with neither c3 nor c4 positive, one can check that a region of positive f exists so long as c5 is positive and the inequality c5 ^ 4c3c4 is not saturated.

The space of available (c3,c4,c5) can be imagined as R3 minus two cones, a 'positive' cone in the region c3,c4 > 0 given by c5 < 4c3c4, and a 'negative' cone in the region c3, c4 < 0 given by c5 ^ V4c3c4. The surface of the positive cone is available, and consists of the constant dilaton solutions (c2 = 0), whereas the surface of the negative cone is unavailable. Not all points in this space are physically distinct, as radial rescalings r ^ Ar trace out hyperbolae c3 c4 = const. The (c3, c4, c5) parameter space is depicted in figure 1.

Œ M •n o Cn

6.2 Constant dilaton solutions

The space of solutions is large and complicated, and without an explicit mechanism for resolving the singularity and lacking an asymptotic AdS region for comparisons with a boundary field theory, it is hard to anticipate which of these solutions will be realized physically. We will focus on one of the simplest classes, in which the dilaton is constant (c2 = 0).24 These are the solutions that lie on the positive cone c2 = 4c3c4 with c3, c4 ^ 0, as noted above. We will not consider the special cases where either c3 or c4 vanishes, so

24One motivation for considering this class of solutions is that string loop corrections can be controlled parametrically by taking gs C 1. Although gs is still available as a control parameter, the situation is more complicated when the dilaton runs.

c5 = 0 in general. It is convenient to reparameterize:

Co = 1 5 |C51 - C5 log r*

where c5 is related to the D3-brane charge. Using (3.44), we compute

Q = n3 e4CC5 , where Q is the Page charge. We find that

IQI e_4C

ï_(r) =

IQI e_4C

-§ + sgn(Q) (1 - 2 log r/r*) - 5

-, + sgn(Q) (1 + 2 log r/r*) + 5

(6.8) (6.9)

where sgn(Q) is the sign of Q.

We consider the case of positive and negative Q separately. For positive Q, H_ is a strictly increasing function of r, whereas S+ is a strictly decreasing function of r. Then, f = + S_ rc ( r'+T,2c; is positive everywhere, so the solution can be made regular in any region by an appropriate choice of 5. However, ^ —to as r ^ to and H_ ^ —to as r ^ 0, so the solution is only valid between two radii n and r2 > n where S_ and cross zero, respectively. One can easily show that

r2 = r*

Wo(eô_1 )

r1 = r*

Wo(e_ô_1)

(6.10)

where Wo is the main branch of the Lambert W-function. In particular, the ratio of the two scales is

(6.11)

(r2/n)2 = Wo(eô-1)Wo(e-д-1)

This ratio is minimized at 5 = 0, where it takes the value r2/ri = [Wo(1/e)] 1 ~ 3.59. For > 0, the ratio increases, and asymptotically for large |5|, we find

(r2/r1)2 ^ |5|-1e1+|01

(6.12)

so the region of regularity can be made very large for modest values of 5.

For the case of negative Q, S+ and S_ have a single minimum at r*, and f « ( t,2t,2c) = 0 at r = r*, so the solution can be made regular anywhere but at r*. For any choice of 5, the solution is regular for r > r2 and r < r1, where for 5 > 0 both singularities are due to H+ crossing zero, and for 5 < 0 both are due to S_ crossing zero. The radii of the singularities for 5 > 0 are

r1 = r*

-W_1(-e-1-0 )

r2 = r*

-Wo(-e-1-0 )

(6.13)

where W-1 is the lower branch of the Lambert W-function. For 5 < 0, we have instead

-Wo(-e0-1)

-W_1(-e0-1)

(6.14)

Œ M •n o Cn

Figure 2. The singularity structure of the constant dilaton solutions. The horizontal axis is r/r* and the vertical axis is S. The nonsingular region is unshaded and S+ (S_) is negative in the blue (red) region. The first (second) plot corresponds to positive (negative) Q.

For the special case S = 0, ri = r2 = r*, and the solution is regular everywhere else. This special case has interesting properties. For instance, the spinor angle is finite everywhere:

r4 — r* — 4 r2 log(r/r*)

(r2 _ r2)2

(6.15)

We see that ^ interpolates between ^ = 0 as r ^ to and ^ = n as r ^ 0, passing through a type-changing locus (^ = n/2) coincident with the singularity at r = r*.

The singularity structure of constant dilaton solutions is illustrated in figure 2 for both positive and negative Q. The special case discussed above corresponds to S = 0 in the second plot of the figure.

6.3 Domain of validity of the near-stack solutions

While we have obtained a family of closed-form solutions to the supergravity equations, our result is subject to corrections from a number of sources. Though string loop corrections can be controlled parametrically for constant dilaton solutions, the solutions we have obtained are always singular at some finite radius, and so a' corrections will invariably be large in some regions. Furthermore, the near-stack approximation employed in section 3.1 also has a finite region of validity. Generically we expect an interplay between these two sources of corrections, as curvatures should fall off at large distances whereas the near-stack approximation is valid at short distances. We now estimate the size of corrections from various sources, and show that in some regions of parameter space we can parametrically suppress all corrections in some finite interval rnear < r < rfar.

6.3.1 Corrections to the near-stack limit

Corrections to the near-stack limit of section 3.1 come in two forms. At large distances from the stack wrapping the resolved P2, the resolution appears as a small perturbation

EC M •n o Cn

to the geometry, and the seven-branes appear to be codimension six sources, rather than codimension two sources as they do in the near-stack limit. This effect manifests itself in the Einstein equations for the P2 cone as extra terms of order (r/ro)2 and (r/ro)4, where ro is the radial scale of the blow-up, as in (3.48); that is, one can show that the Einstein equations (A.16), (A.17), (A.18), written in the form

r2 B'' = ... , r2 C'' = ... , 2 rC' = ..., (6.16)

receive corrections of the form r2 e-4B-2C and r4 e-8B-4C. (In the remainder of this section, we will omit numerical factors in expressions of the form a « b.)

Assuming that dimensionless terms of the form rB', etc., are of order unity, the extra terms are suppressed if

r2 e-4B-2C « 1. (6.17)

By (3.48), this is equivalent to the requirement (r/r0)2 « 1. ' (J

The other source of corrections to the near-stack limit is the finite cosmological constant A = —3 |^|2. This sources new terms in the Einstein equations and the warp factor equation of the form r2 e-4(A+B) A, as in (A.10), (A.16), (A.17), (A.18). Applying the relation (5.39), we obtain the requirement

r2 e-4(A+B+C) |Ci|2 « 1. (6.18)

Using (5.9), (5.23) the conditions (6.17), (6.18) can be rewritten in the form:

|ci|2 e-2C (S+ + S-) « 1 , |ci|4 e-4C H+ S_ « 1. (6.19)

However, for S± > 0,

|E+ H_|< 4(S+ + S-)2 . (6.20)

Therefore, for supersymmetric solutions, the second bound in (6.19) is implied by the first. For solutions with nonvanishing Q, it is convenient to extract an overall scale from S±:

H± = e-4 C F± , (6.21)

where for constant dilaton solutions, the F± are given by the bracketed terms in (6.8), (6.9). The bound (6.19) becomes

Z |Q| (F+ + F-) « 1, (6.22)

where Z = |c1|2 e-6C. The constants c1 and eC naturally appear in physical quantities in this combination, since both can be rescaled by redefining the four-dimensional coordinates ^ and absorbing the rescaling into the warp factor and the six-dimensional metric, leaving the ten-dimensional metric invariant; Z is invariant under this rescaling.

6.3.2 Curvature corrections

To estimate the size of a' corrections, we compute the ten-dimensional Riemann tensor components in Einstein frame. We assume that corrections can be suppressed if these components (expressed in an orthonormal basis) can be made parametrically small in

units of a'. The size of a' corrections in Einstein frame will actually depend on since the F-string tension depends on in this frame. However, if we restrict our attention to constant dilaton solutions, and fix some small value of to suppress string loop corrections, this will only modify the curvature scale at which corrections set in by a fixed factor, and parametric control of the Einstein-frame curvature is still sufficient to suppress corrections. We will not consider corrections involving the other supergravity fields.25

The ten-dimensional Riemann tensor for the metric (3.10) can be computed using standard methods. Expressing everything in an orthonormal basis, one finds terms of the form e-2A [R(4)] e2A (VmA)(V„A), and e2A VmV„A, where [R(4)]

are the components of the four-dimensional Riemann tensor computed from , [R(6)]mnpq are the components of the (unwarped) six-dimensional Riemann tensor computed from and V is the connection computed from . We assume that the near-stack approximation holds in the region of interest, and use the ansatz of section 3.1 to compute

e2A[%)]r^ =2 e2A+4B (B'' + , e2A(VrA)(VrA) = e2A+4B (A')2 , (6.23)

e2A VfVfA = e2A+4B(A'' + 2A'B'), e2A V^A = e2A+4B (1 A' - 2A'B^ ,

where all other terms vanish. If we repeat this computation for the full P2 geometry, we ^^

obtain extra terms which are suppressed in the near-stack limit, but which scale similarly o

to those above. However, we also obtain a contribution from the cosmological constant |— which scales distinctly:

-2A[R(4)] V = -H2 e-2A (^ nv. - nv^ . (6.24)

Taking rA' and other dimensionless derivatives of A and B to be of order unity, all the components in (6.23) are of the same order, as determined by the prefactor, whose square is:

1 e4A+8B ( |C114 „ „ (w + „ )\-1 ( |Q| Z—2 2

r4 e = ^ + V Z F+ F— (F+ + F—) , (6.25)

where we have used (6.21). By contrast, the square of the curvature induced by the cosmological constant, (6.24), is:

e—4A = ^ ff+ft ■ (-26)

using (5.39).

6.3.3 Regions of parametric control

To suppress a' corrections, we should parametrically suppress (6.25), (6.26) while also satisfying (6.22). Rewriting Z|Q| = e1 and |Q|—3Z—2 = e2, we see that near-stack corrections can

25One might hope that these corrections are also suppressed when the Riemann components are small, but not all of these corrections are known, even at leading order, so more detailed estimates may be misguided.

be controlled by taking e1 ^ 1, while curvature corrections can be controlled by taking e2 to be well below some fiducial curvature scale. The curvature (6.26) is then suppressed by e^ e2. However, in this limit |Q| = 1/(eie2), so the D3-brane charge must be taken to be large.

In a physical solution, we expect that the D3-brane charge is determined by the world-volume dynamics on the seven-brane stack, and is not a free parameter. If we add D3-branes near the tip, the 3-7 strings become light and introduce light matter into the worldvolume theory, precluding gaugino condensation. Thus, |Q| is not a free parameter, and we must look elsewhere for parametric control of the curvature. Comparing (6.22) and (6.25), we see that control may be possible for F± » 1 with Z scaled appropriately (e.g. Z rc F-4/3). For constant dilaton solutions, (6.8) gives (~ j

r 2 r 2

F+ + F- = -2 + ^ +2sgn(Q). (6.27)

Thus, F± » 1 requires r » r* or r ^ r*. ^^

Solutions in the region r ^ r* are unlikely to be physical for the following reason. Referring to figure 2, we see that such a solution can only be regular for r < rs where rs < r*. The large-distance singularity at r = rs is not obviously due to localized sources, and should be removed by compactification, rather than by curvature corrections. However, corrections to the near-stack limit (the first step towards compactification) decrease with o

increasing r in the region r < r*, since S+ + has a single minimum at r = r*.

Thus, we restrict our attention to the r » r* case. For simplicity, we consider the case |_

where and H_ cross at some large radius req » r*. Thus, at req, F+ = F_ w r2q/(2 r*). In this limit, one can show that the curvature terms are indeed suppressed. At leading order in r* ^ req, r, we find

6 r2 r2 r4 r2 3 r2 r2 r2A"(r) - n6 Ir2eq 9re,q2 , rA'(r) 2req 2, , r2B "(r) -- ^ , rB'(r) - % ,

2(2 r2 - r2)2 ' v ' 2(2 r2 - r2q) ' w r2 ' v ' r2

(6.28)

so the derivatives are of order one or smaller for r > req, and the size of the Riemann components is determined by the prefactor (6.25). Thus, our previous arguments apply, and we conclude that both curvature and near-stack corrections can be suppressed at r ~ req.26 Depending on the parameters, near-stack corrections will become important at some rfar > req and curvature corrections at some rnear < req. Near r = req both types of corrections are small, but ^ ~ n/2 and the solution cannot be described by perturbations about a type B background.

As illustrated by this example, the solutions we have obtained describe physics inaccessible to previous approaches. We stress that the above is not a complete classification of regions of parametric control. We leave further exploration of the large parameter space of solutions, including solutions where the dilaton runs, to a future work.

26For req ^ r* the singularity occurs at req/v^, but the ten-dimensional distance between req and the singularity is proportional to re-2B-A, and will be large in string units when the curvatures are small.

7 Towards the physics of the solutions

Our choice to study four D7-branes atop an O7-plane on a rigid four-cycle was strongly motivated by the fact that the corresponding seven-brane gauge theory confines in the infrared, but our analysis so far has exclusively involved ten-dimensional supergravity, without any input of gauge theory physics. We have obtained a family of exact noncompact solutions parameterized by a number of integration constants, but we expect that some of these constants are actually fixed by a proper inclusion of nonperturbative source terms localized on the seven-branes (cf. [9, 10]), or by matching to the seven-brane gauge theory.

In this section we present a preliminary analysis of the relation between the solu- C_|

tions obtained in section 5 and the dynamics of the four-dimensional gauge theory on j~yj

seven-branes in the P2 cone. Potential applications of our results to the ten-dimensional description of KKLT vacua and geometric transitions for seven-branes are also discussed.

7.1 Gaugino condensation in supergravity

M •n o cn

Let us start by briefly discussing the gauge theory supported on the seven-brane stack x-—>v

wrapping the rigid holomorphic P2. An important subtlety in obtaining the worldvolume gauge theory is the following. Since P2 is not spin, wrapping D7-branes on it gives rise to global anomaly [5, 6], whose cancellation requires a nontrivial gauge bundle that will break SO(8) down to (at most) U(4). While it is not clear to us how the anomaly constraint is modified in the presence of the O7-plane, if we assume that the anomaly does persist, the resulting gauge group need not be asymptotically free.

In this work we will assume that the seven-brane gauge group generates a gaugino condensate in the infrared, postponing a proper treatment of the Freed-Witten anomaly to future work. Fortunately, the methods we have developed apply equally well to the Calabi-Yau cone over P1 x P1. The gauge theory analysis there is simpler, because P1 x P1 is spin, but the supergravity analysis becomes slightly more involved than in the P2 cone.

Let us now turn to gaugino condensation. Denoting the dynamical scale by A and the dual Coxeter number of the nonabelian group by c2 (not to be confused with the integration constant c2), we have

(AA) - ac2A3 , (7.1)

where ac2 is a c2-th root of unity. The nonperturbative superpotential is also proportional to the gaugino bilinear (AA). Recall that the U(1)R symmetry that acts on the gauginos is anomalous at the quantum level:

A ^ eieA , tym ^ tYM + - d. (7.2)

The exact symmetry is thus reduced to a discrete Z2c2. Moreover, this is spontaneously broken to Z2 by (7.1), leading to c2 inequivalent vacua.

7.1.1 Gaugino condensation and IASD flux

To connect the gauge theory dynamics to our supergravity solution,27 we use the results of [9], which showed that gaugino condensation on D7-branes sources imaginary anti-self-dual (IASD) flux in the space surrounding the branes. Using the classical DBI coupling between D7-brane gauginos A and bulk fluxes [41],

; J VgG3 ■ fi AA + c.c., (7.3)

the flux equation of motion, expanded around a background containing exclusively imagi nary self-dual fluxes, was found to be [9]

e4A (*G - ¿G3) =4iaK|0 d AA fi ¿(P2) . (7.4)

gs a'2

27A precise matching between supergravity and gauge theory requires having a smooth solution, which we lack at present. Our discussion here will be qualitative, and limited to showing that the family of supersymmetric solutions found above has the required ingredients to encode gaugino condensation.

28Consideration of more general D-brane probes [42, 43] of our solutions is likely to lead to substantial physical insight, but is beyond the scope of the present work.

Here * is the six-dimensional Hodge star, ¿(P2) is a delta-function localizing on P2, and a C D

is a dimensionless constant.

The nonzero expectation value (7.1) provides a localized source for flux in ten dimensions via (7.4), and the resulting flux is IASD with Hodge type (1,2). Compelling evidence for this proposal comes from the fact that for D7-branes wrapping a given four-cycle in a local geometry, the G1,2 flux background induced by the coupling (7.4) precisely en- o

codes the superpotential of the four-dimensional gauge theory: a D3-brane probing the |—*

flux background sourced by (7.4) experiences the superpotential derived upon reduction to four dimensions [9].

The analysis of [9] was performed in an expansion around type B backgrounds [8], but focused on the theory of probe D3-branes28 and consistently neglected perturbations to the metric and dilaton: such perturbations are clearly present as a consequence of the IASD flux sourced by the gaugino condensate, but do not contribute to the D3-brane scalar potential until third order. However, it was natural to expect that the full solution of all the equations of motion would unite the proposals of [10], in which the background is a generalized complex geometry, and of [9], in which G1,2 flux plays the central role. In this work we have exhibited a solution with dynamic SU(2) structure that crucially involves G1,2 flux, thereby illuminating the relationship between [10] and [9].

7.1.2 R-symmetry breaking and domain walls

Having explained the relation between three-form fluxes and gaugino condensation, let us discuss how the pattern of R-symmetry breaking described above may be encoded in the supergravity solution. Referring to table 1, we see that R(6) = +2 and R(fi2) = 0. Comparing (5.7), (5.4) with the geometric action of U(1)^ (appropriately normalized as

in section 2.2), we see that c1 and e^ must carry R-charges +2 and —2 respectively. This suggests that, in a four-dimensional off-shell formulation (where the equations of motion of the effective action reproduce the ten-dimensional equations), some combination of c1 and e^ becomes a fluctuating space-time field with nonvanishing R-charge, whose expectation value spontaneously breaks the exact R-symmetry to Z2.

Gaugino condensation has a similar off-shell description in terms of the glueball su-perfield S = — 32^tr WaWa and the Veneziano-Yankielowicz superpotential [44]. On-shell this field acquires a nonzero expectation value proportional to (7.1) and reproduces the nonperturbative superpotential. It is natural to conjecture that the combination of c1 and e^ mentioned above is dual to S; then the on-shell superpotential (4.46), (4.47) would have to agree with the gaugino-condensate superpotential.29

We expect the appearance of domain walls due to the spontaneous breaking of the exact R-symmetry Z2c2 ^ Z2; these should correspond to wrapped branes in the gravity solution. We now argue that D3-branes wrapping a loop inside the S5/Z3 have the right ' U

properties to be domain walls in our solution.

The S5/Z3 has fundamental group Z3 generated by a loop where ^ runs from 0 ^ 2n. However, due to the involution, a D3-brane wrapping from 0 ^ n is permitted, where the two ends are identified under the involution ^ ^ ^ + n. This corresponds to the generator of the Z6 fundamental group of the horizon in the downstairs geometry. We compute the tension of the domain wall by evaluating the DBI action for the brane in the probe C D approximation: |_\

T(waii) = j^y ds(6) e2A = j nre2(A-B) = j n |c11 p"_)2 ^ ^3 n |C11, (7.5)

where the bound is saturated if and only if = at the location of the domain wall. Thus, the tension is minimized at a ^ = n/2 locus, in which case the wall is a half-BPS state [43]. The analysis of section 6.3 describes one example where the supergravity approximation is valid at such a locus.

The Z6 fundamental group in the downstairs geometry implies that the number of domain walls of this type is conserved modulo six. Thus, the gauge theory has six vacua, consistent with gaugino condensation in SO(8) pure super Yang-Mills. We anticipate that these vacua are related by exact R-symmetry transformations. The BPS domain wall tension (7.5) can then be used to infer the precise value of the superpotential vev. We postpone a more detailed study of these domain walls and other wrapped branes to a future work.

Finally, we recall that the extension of the gravity solution to the full P2 cone using the ansatz of section 3 reveals that the space-time becomes AdS, as explained in section 5.3. We have not obtained a satisfactory interpretation of this restriction from the viewpoint of the gauge theory.

7.2 Applications

Our solution has a range of interesting applications, two of which we now describe.

9For D5-branes or D6-branes in the conifold this matching was obtained in the large N duality of [45, 46].

7.2.1 Ten-dimensional consistency of KKLT vacua

One application would be to study the ten-dimensional consistency conditions for KKLT vacua. As vacua of the four-dimensional effective theory, these solutions are reasonably well understood, but are known to violate constraints that emerge from the ten-dimensional type IIB supergravity equations of motion with classical sources. Specifically, from the external Einstein equations and the five-form Bianchi identity, one obtains

v2$- = 4 e8A |*G - i G3I2 + R4 + e-4A |V$_|2 + Siocal, (7.6)

where R4 is the four-dimensional Ricci scalar, and [8]

Siocal = 2k20 e2A (e4ATm - t; - ^3) i i, (7.7)

4 4 local

where T™ and TV are the internal and external components of the ten-dimensional stress- ( )

energy tensor T^ (with indices raised by the unwarped metrics and , respectively), and p3 is the D3-brane charge density. An anti-D3-brane at position y0 provides a positive localized source,

SiDai = 4 k2o e8A ¿6(y - yo), (7.8)

whereas D3-branes, O3-planes, and O7-planes, like all other local sources allowed in the solutions of [8], provide a vanishing contribution to S|oca|. Then, noting that the integral of the left-hand side of (7.6) over a compact space vanishes, one learns that a de Sitter solution is possible only if a suitable localized negative contribution to the right-hand side is present. We will denote such a contribution as p-(y).

We emphasize that negative tension alone does not suffice to produce a contribution to p-, as is evident from the fact that O3-planes and O7-planes do not contribute. Localized classical sources that do contribute to p- include anti-O3-planes and O5-planes [8], but to our knowledge such objects have not played a role in the construction of consistent de Sitter vacua in the framework of [1].

A natural guess is that the four-dimensional nonperturbative effects that led to stability in the effective theory — i.e., gaugino condensation on D7-branes — will provide new sources in the ten-dimensional equations of motion. In fact, the stress tensor contribution given in (7.6) arises from varying the classical action for bosonic fields with respect to the metric. The stress tensor by definition involves the variation of the complete action, but fermionic expectation values vanish in classical vacua, and the corresponding contributions may then be omitted. However, in the solutions we have considered, the fermion bilinear AA plays an essential role, and contributions proportional to (AA) should be retained.

Specifically, to incorporate the effects of gaugino condensation one should also vary the coupling (7.3) with respect to the metric. The result is a new contribution to (7.7) that is negative and proportional to |(AA)|2. We speculate that this negative contribution could suffice to establish that KKLT vacua can be lifted to consistent ten-dimensional solutions, but we leave a thorough investigation of this point for the future.

7.2.2 On del Pezzo transitions with seven-branes

•n o

A more speculative application of our result is to a geometric transition for seven-branes. It would be interesting to understand when a divisor wrapped by seven-branes can be contracted in such a way that the resulting singularity can be deformed. The only divisors in a Calabi-Yau threefold that admit birational contraction followed by deformation to a new Calabi-Yau are the del Pezzo surfaces P1 x P1 and dPk, k > 2 [47, 48].30 (The del Pezzo surfaces dP0 = P2 and dP1 can be contracted, but the resulting singular varieties cannot be deformed to smooth Calabi-Yau threefolds.)

We would like to understand when del Pezzo transitions can occur for divisors wrapped by seven-branes, motivated by the rich physics of geometric transitions involving D5-branes. The role of gaugino condensation in the conventional D5-brane geometric transition [45, 46] j | j

is well understood, and strongly suggests that for seven-branes it is also important to characterize the effect of gaugino condensation in the geometry. Our result prepares the tools for such an analysis, but exploring a seven-brane geometric transition in detail is beyond the scope of this work. We also observe that in any case where a del Pezzo transition with seven-branes is possible, so that a smooth geometry is obtained after the deformation,

the resulting absence of a local source for p_ as described in section 7.2.1 presents an obstacle to obtaining consistent de Sitter vacua.

8 Conclusions |—\

We have obtained a family of exact, noncompact, supersymmetric solutions of type IIB supergravity with dynamic SU(2) structure. The core of each solution is a stack of four D7-branes atop an O7-plane on a flat non-compact four-cycle. We argued that this solution I *

describes the region near a small patch of a compact, rigid four-cycle, within which we have assumed rotational symmetry. We gave strong evidence for this claim by presenting an ansatz for a corresponding configuration of four D7-branes and an O7-plane wrapping the P2 base of the simplest del Pezzo cone, and showing that in the near-stack limit our flat ansatz is recovered. Decompactifying the four-cycle destroys information about induced charges and sends the seven-brane gauge coupling to zero, while drastically simplifying the equations of motion. By comparing the flat-space solution to the P2 cone configuration, we were able to interpret certain aspects of our solution as arising from seven-brane gauge dynamics or from induced D3-brane charge and tension.

For compact, rigid four-cycles, the seven-brane gauge theory undergoes gaugino condensation at low energies. In this work we have identified a class of exact solutions as candidates for the ten-dimensional backreaction of seven-brane gaugino condensation. Our approach was strictly ten-dimensional and did not incorporate nonperturbative source terms, in contrast to, but not in contradiction with, [10],31 which proposed that gaugino conden-

30Strictly speaking, del Pezzo surfaces are the only possible exceptional divisors for 'primitive' contractions, from which more general contractions can be constructed [48].

31See e.g. [11-13] for earlier connections between gaugino condensation and generalized complex geometry, and [9] for evidence that gaugino couplings to flux source ten-dimensional deformations.

sation provides a localized source term that induces a deformation to a generalized complex geometry. It would be valuable to understand the relationship between these approaches.

Although the seven-brane charge necessarily vanishes in our solutions, the three-brane charge and tension induced on seven-branes wrapping P2 can be negative, so that at short distances one expects singular behavior typical of orientifolds. We indeed find a singularity with divergent warp factor near the seven-branes. It would be very interesting to understand if this singularity is ultimately removed by strong gauge dynamics on the seven-branes.

Stacks of seven-branes wrapping rigid four-cycles are ubiquitous in type IIB compact-ifications, and understanding their effects in ten-dimensional supergravity is an important step toward characterizing the resulting four-dimensional effective theories. In particular, j~|~j

decoupling arguments analogous to [49] that invoke extradimensional locality require a ten-dimensional description, and the effective theory of D3-branes is most efficiently described by geometrizing seven-brane gauge dynamics [9], as we have done here. The configuration we have presented is arguably the simplest nontrivial example of the backreaction o of seven-brane nonperturbative effects, because the seven-brane charges vanish and the four-cycle is highly symmetric. Our approach can be extended to configurations with less symmetry, such as seven-branes wrapping the base of a suitable del Pezzo cone; we argued that analogous solutions exist for the P2 cone. It would be very interesting to construct additional examples and explore their implications, both as windows into the dynamics of the seven-brane gauge theory, and as descriptions of local regions of stabilized type IIB compactifications.

Acknowledgments

We are grateful to D. Baumann, M. Berg, P. Cámara, A. Collinucci, F. Denef, A. Dymarsky, S. Franco, T. Grimm, S. Kachru, L. Martucci, G. Moore, E. Silverstein, T. Weigand, and T. Wrase for helpful discussions. We thank D. Baumann, S. Kachru, J. Liu, L. Martucci, and P. Szepietowski for useful comments on the manuscript. L.M. specially thanks T. Weigand for valuable discussions and correspondence. G.T. is particularly grateful to S. Kachru for many enlightening conversations. The research of L.M. was supported by the Alfred P. Sloan Foundation and by the NSF under grant PHY-0757868. The research of B.H. was supported in part by a Cornell University Olin Fellowship and by NSF grant PHY-0757868. B.H. and L.M. gratefully acknowledge support for this work by the Swedish Foundation for International Cooperation in Research and Higher Education. G.T. is supported by the US DOE under contract number DE-AC02-76SF00515 at SLAC. We thank the organizers of String Phenomenology 2010 for providing a stimulating environment for a portion of this work. G.T. would like to thank the KITP, where part of this work was done, for hospitality.

Portions of this document were prepared with the help of TEXmaos, free software available at http://www.texmacs.org.

A Equations of motion for the near-stack ansatz

We now present the general equations of motion for the warped ansatz (3.10), (3.11), and then specialize to the near-stack limit. Rewriting the covariant action (4.27) in terms of a four-dimensional Lagrangian density,

S = j d4 x V-h L,

where L is given by an integral over the internal space, and applying the warped ansatz (3.10), (3.11),32 we find the vacuum Lagrangian:

1 a2 oCV7 A\2 , A A „-4A 1 l 1 IJ_|2 , |2

R + 2e"8A(V«)2 - 8(VA)2 + 4Ae-4A - 2 _|dr|2 + e4A|Gs|

2 2 \ To

a G3 A G3 ,

where A = R(4)/4 is the four-dimensional cosmological constant, t2 = Imt, contractions are made using the unwarped metric gmn, and G3 = D_ A2, so that D- G3 = 0. The corresponding equations of motion are:

d (e_8A da) = -i G3 A G3 , V2A =1 e4A|G3|2 + 1 e_8A(Va)2+Ae_4A , (A.3)

D- (e4A *6 G3) = -i da A G3 , D *6 (-1 dfj = |e4AG3 A ^3

Œ M •n o Cn

Rmn = 8 VmA VnA - 1 e-8A Vma V„a + [VmrV„f + c.c.] + 1 e4A Tmn - Ae-4Agmra ,

rpm _ _( fimjpq r^ i r^tmp^ \ _ ^pqr ^ cm

— 4 lG Grapq "T" G GrapqJ 12G GpqrSra .

(A.5) (A.6)

To work out the Einstein equations, we compute T in a complex basis. It is straightforward to check that TV = 0 if j and v are both holomorphic indices; this is a consequence of the primitivity of G3. Of the mixed, TV components, all except TZ = (Tf )* must vanish by symmetry, and we find

Tzz = 4 e_ Z^g2,1 + g1,2^3] . After a straightforward computation, we find the Ricci components

C'' + 4B'C' + (C')2 - 1C' r

B'' - C'' + -(B' - C') - (C')

R . , =__e

^Vuj 2e

C '' + - C ' + 4(C ')2

Using these formulae, one can write down the Einstein equations in terms of B and C. Applying the remaining equations of motion to the ansatz of section 3.1 in like fashion, we

2 As usual, there are some surmountable subtleties relating to the self-duality of F5.

find four real second order equations of motion for A, a, B, and C, along with one complex second order equation of motion for t, four real first order equations of motion for the , and one complex constraint coming from the r, ^ component of the Einstein equations. The a and A equations of motion are

iJ! (re4(C-2A)a') = 4 ^(-l)P|gP,3-P|2 , (A.9)

1 d (re4CA') = e4A >T |gp,3-p|2 + 1 e4(C-2A)(a')2 + Ae4(C-B-A), (A.10)

r r p=0 C_|

where primes denote derivatives with respect to r. The t equation of motion is j | |

^dr (re4Ct') + -Le4C(t')2 = -8ie4A (g3,0g0,3 + g2,y,2) . (A.ll) ^

The G3 equations of motion and Bianchi identities are

1 d- [r—2e4V'°] + ^e4A[g3'° + g1'2] = —g3'° + 1(e4A - a)'[g3'° - g2'1] , (A.12)

dr 2t2 r 2

^dr [r^V1] + ^e4A [g2'1 + g°'3] = -^g2'1 - 2(e4A + a)'[g3'°- ' (A.13) O

1 d-[r—T2e4Ag1'2] + e4A[g3'° + g1'2] = -g1'2 - 1(e4A - a)'[g°'3- g1'2], (A.14)

r^/T2 dr L J 2t2

1 d [™ „4A„°'3] , %T' 2'1 , „°'3] e4A „°'3 , 1

[r—T2e4Ag°'3] + f"e4A [g2'1 + g°'3] = —g°'3 + ^(e4A + a)' [g°'3- g1'2] . (A.15)

r dr 2t2 r 2

The B and C equations of motion are ^—^

C'' + 1C' + 4(C')2 = Ae-4(A+B) , (A.16)

B'' + 1B' + 3(C')2 = 2(A')2 - 1 e-8A(a')2 + '|2 + 1 e-4(A+B)A, (A.17) r 8 8t2 2

and the constraint takes the form C'

2 + 3C' - 4B'

Of /1/Ni2 ^A2 , 1 U'l2

= 2(A)- ^e (a) +8Tf|T|

+2e4(A-C) (g3'° g2'1 + g1'2 g°'3) + e-4(A+B)A. (A.18)

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