Copyright by Stephen Christopher Young 2012

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1 Copyright by Stephen Christopher Young 01

2 The Dissertation Committee for Stephen Christopher Young Certifies that this is the approved version of the following dissertation: Non-Supersymmetric Holographic Engineering and U-duality Committee: Willy Fischler, Supervisor Elena Caceres, Co-Supervisor Sonia Paban Duane Dicus Dan Freed

3 Non-Supersymmetric Holographic Engineering and U-duality by Stephen Christopher Young, B.S.; M.A. DISSERTATION Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY THE UNIVERSITY OF TEXAS AT AUSTIN August 01

4 Dedicated to my Parents

5 Acknowledgments Thanks foremost to my parents for their unwavering financial and moral support. Also to my advisor, Willy Fischler, for his refreshing enthusiasm and informality, and for helping me to push through in times when motivation was waning. Special thanks to Elena Cáceres for collaboration and suggestion of a project that made me learn quite a bit in a very short time period. Thanks to Vadim Kaplunovsky for incredibly comprehensive courses and problem sets in quantum field theory and supersymmetry. Thanks to Barry S., who always told me that math was cool. Thanks to Dan A., whose accomplishments and artistic genius have always been a prime source of motivation. Thanks to Mike S. for inspiring me to remain true to the hacker ethic which he embodies and to not neglect my friends or let my social skills atrophy whilst holed up in the Ivory Tower. 1 Thanks to Matthias I. for long term friendship, collaboration, and for not letting me die in that blizzard on Mt. Rainier. Thanks to Anindya D. for friendship and collaboration. Thanks to Tim H. for being a co-collaborator-in-suffering in the early years of grad school. Thanks to Rob M. for longterm friendship and moral support. Thanks to Tom. M. for explaining crazy math to me despite my reluctance to learn any. 1 This completely happened anyway. v

6 Non-Supersymmetric Holographic Engineering and U-duality Stephen Christopher Young, Ph.D. The University of Texas at Austin, 01 Supervisors: Willy Fischler Elena Cáceres In this Ph.D. thesis, we construct and study a number of new type IIB supergravity backgrounds that realize various flavored, finite temperature, and non-supersymmetric deformations of the resolved and deformed conifold geometries. We make heavy use of a U-duality solution generating procedure that allows us to begin with a modification of a family of solutions describing the backreaction of D5 branes wrapped on the S of the resolved conifold, and generate new backgrounds related to the Klebanov-Strassler background. We first construct finite temperature backgrounds which describe a configuration of N c D5 branes wrapped on the S of the resolved conifold, in the presence of N f flavor brane sources and their backreaction i.e. N f /N c 1. In these solutions the dilaton does not blow up at infinity but stabilizes to a finite value. The U-duality procedure is then applied to these solutions to generate new ones with D5 and D3 charge. The resulting backgrounds are a non-extremal deformation of the resolved deformed conifold with D3 and D5 sources. It is tempting to interpret these solutions as gravity duals of finite temperature field theories exhibiting phenomena such as Seiberg dualities, Higgsing and confinement. However, a first necessary step in this direction is to investigate their stability. We study the specific heat of these new flavored backgrounds and find that they are thermodynamically vi

7 unstable. Our results on the stability also apply to other non-extremal backgrounds with Klebanov-Strassler asymptotics found in the literature. In the second half of this thesis, we apply the U-duality procedure to generate another class of solutions which are zero temperature, non-supersymmetric deformations of the baryonic branch of Klebanov-Strassler. We interpret these in the dual field theory by the addition of a small gaugino mass. Using a combination of numerical and analytical methods, we construct the backgrounds explicitly, and calculate various observables of the field theory. vii

8 Table of Contents Acknowledgments Abstract List of Figures v vi xi Chapter 1. Introduction The one-parameter family of wrapped D5 brane solutions The Maldacena-Nunez limit The U-duality procedure and the limit of the Klebanov-Strassler baryonic branch Chapter. The Stability of Non-Extremal Conifold Backgrounds with Sources 16 Chapter 3. The Flavored Wrapped D5 Backgrounds, T = The smearing process Background Ansätz and Langrangian Master equation and UV asymptotics Chapter 4. Non-extremal flavored backgrounds Non-extremal Ansätz, GTV reduction, and equations of motion New non-extremal flavored solutions with stabilized dilaton UV expansions IR asymptotics Numerics Temperature of the solutions viii

9 Chapter 5. Non-Extremal Flavored Backgrounds with Flavored Resolved Deformed Conifold Asymptotics Equations of motion with D3 and D5 charges and smeared flavor branes Rotating flavored non-extremal solutions Asymptotics after the rotation Chapter 6. Thermodynamic Properties Energy and specific heat Chapter 7. The Non-SUSY Baryonic Branch 68 Chapter 8. The SUSY system Aspects of the SUSY solutions Chapter 9. The SUSY-breaking deformation Asymptotic expansions Numerical analysis The ADM Energy and Free Energy Free Energy Chapter 10. Field Theory Aspects Form of the B field Interesting Asymptotic Behaviors Energy Charges Gauge couplings and beta functions K-strings The Non-SUSY Seiberg-like duality Domain Walls Central Charge Force on a probe D3-brane Field Theory comments Some words on (meta)-stability ix

10 Chapter 11. Conclusions The Stability of Non-Extremal Conifold Backgrounds with Sources The non-susy Baryonic Branch Appendices 116 Appendix A. U-Duality for the wrapped D5 branes black hole 117 Appendix B. The Equations of Motion before rotation in the GTV formalism 11 B.1 Flavored, finite-temperature EOMs B. Non-SUSY EOMs Appendix C. Exact UV asymptotics for the finite-temperature flavored case 16 C.1 The UV asymptotic expansion Appendix D. Form of the B field 131 D.1 B for the rotated, flavored, finite temperature backgrounds D. B for the rotated, non-susy backgrounds Bibliography 137 Vita 147 x

11 List of Figures 1.1 (a) and (b) show the standard cartoons for the deformed and resolved conifolds. (c) represents the D5 branes wrapping the S of the resolved conifold. (d) represents the resulting backreacted geometry where the D5s have been replaced by geometry and flux. This final geometry has the topology of the deformed conifold with flux on the S 3. The near brane region is the (Chamseddine-Volkov-)Maldacena-Nunez solution [4, 5, 6]. (Image credit: Maldacena, Martelli - arxiv: ) (a) The moduli space of the conifold without flux or branes has two branches, denoted here by the vertical and horizontal axes. One represents deformation, and the other is resolution, which has two sides differing by a flop transition. After we add flux, we have a one-parameter family that interpolates continuously between a deformed conifold with flux in region D and a resolved conifold with branes in region R. (b) In region D of (a), the solution looks like the deformed conifold with flux. (c) In region R of (a) the solution looks like the resolved conifold with some branes, where the branes have been replaced by their near brane geometry. (Image credit: Maldacena, Martelli - arxiv: ) Horizon-shot and UV-shot solutions for metric functions at s = 1, c + = 3, C = e k - orange, e g -blue,e h -purple,e 8x -pink Horizon-shot and UV-shot solutions for metric functions at s = 1, c + = 3, C = Zoomed into region near horizon at ρ h Horizon-shot and UV-shot solutions for metric functions at s = 1, c + = 50, C = Horizon-shot and UV-shot solutions for metric functions at s = 1, c + = 50, C = Zoomed into region near horizon at ρ h Temperature at the horizon versus N f /N c. c + = 50,C = Temperature at the horizon versus N f /N c. c + =3,C = Temperature at the horizon versus C. c + = 50,s= Temperature at the horizon versus C. c + =3,s= g ρρ metric element for solutions after rotation, c + = 50, C = From bottom to top: s =0, 1, 19/10, 51/10, 91/ g ρρ metric element for solutions after rotation, c + =3,C = From bottom to top: s =0, 1, 19/10, 51/10, 91/ xi

12 5.3 g xx (red) and g tt (blue) metric elements for the unflavored solution after rotation: c + = 50, C = 5000,s= g xx (red) and g tt (blue) metric elements for the unflavored solution after rotation: c + =3, C = ,s= g xx (red) and g tt (blue) metric elements for flavored solutions after rotation: c + = 50, C = From top to bottom: s =0, 1, 19/10, 51/10, 91/ g xx (red) and g tt (blue) metric elements for flavored solutions after rotation: c + =3, C = From top to bottom: s =0, 1, 19/10, 51/10, 91/ g ψψ (blue), g θθ (red), and g θ θ (yellow) metric elements for unflavored solutions after rotation, c + = 50, C = g ψψ (blue), g θθ (red), and g θ θ (yellow) metric elements for unflavored solutions after rotation, c + =3, C = g ψψ (blue), g θθ (red), and g θ θ (yellow) metric elements for flavored solution after rotation with s =1/5, c + = 50, C = g ψψ (blue), g θθ (red), and g θ θ (yellow) metric elements for flavored solution after rotation with s =1/5, c + =3, C = g ψψ (blue), g θθ (red), and g θ θ (yellow) metric elements for flavored solution after rotation with s = 91/10, c + = 50, C = g ψψ (blue), g θθ (red), and g θ θ (yellow) metric elements for flavored solution after rotation with s = 91/10, c + =3, C = ADM energy density versus horizon temperature after rotation. s = 1, c + = ADM energy density versus horizon temperature after rotation. s = 1, c + = ADM energy density versus entropy density after rotation. s =1,c + = ADM energy density versus entropy density after rotation. s =1,c + = Plots of the functions g, h, k, Φ, log a and log b, obtained numerically (solid blue), together with the IR (dotted red) and UV (dashed orange) expansions, with small deviations from the SUSY values of the parameters. The SUSY solution (grey) is included for comparison xii

13 Chapter 1 Introduction In the past decade and a half, the gauge/gravity duality [1,, 3] has seen myriad applications in the study of strongly coupled gauge theories. The conjecture was originally formulated in the context of describing strongly coupled N = 4super Yang-Mills theory in terms of type IIB string theory propagating on an AdS 5 S 5 spacetime, but it has since been generalized to more phenomenologically relevant gauge theories with minimal or even completely broken supersymmetry, as well as systems at finite temperature and finite density. One long-standing goal has been to understand aspects of quantum chromodynamics, at zero and finite temperature, through the use of a suitable string dual. An important milestone in this endeavor was the discovery of two models whose dual field theory has behavior similar to N = 1 super Yang-Mills in the IR, and whose geometry encodes both chiral symmetry breaking and confinement: the wrapped branes model of Chamseddine, Volkov, Maldacena and Nunez [4, 5, 6], and the deformed conifold background of Klebanov and Strassler [7]. The Maldacena- Nunez model starts with an N = 1 SUSY-preserving compactification of D5 branes on the non-vanishing S of the resolved conifold. The near-brane geometry of the resulting system is dual in the IR to N = 1 SYM, coupled to a Kaluza-Klein tower of massive chiral and vector multiplets. A complication of this model is that the KK modes enter the spectrum at an energy scale proportional to the inverse size of the S, and this is comparable to the scale at which one wants to study nonperturbative phenomena such as confinement or chiral symmetry breaking. The 1

14 second model, due to Klebanov and Strassler, describes the near-brane geometry of N D3 branes and M fractional D3 branes living at the tip of the deformed conifold. It is dual to a cascading N =1SU(N + M) SU(N) quiver gauge theory with two bifundamental matter fields A, B transforming as SU() SU() doublets and interacting via a quartic superpotential. In the IR, the theory displays confinement and chiral-symmetry breaking, and becomes essentially an N = 1SU(M) SYM theory. The presence of some remaining supersymmetry played an important role in the successes of these two models, both in simplifying the search for solutions and in guaranteeing their stability. Despite this, considerable progress has been made towards finding dual descriptions which have no supersymmetry. One natural way to do this is to consider the dual field theory at finite temperature, which corresponds on the gravity side to finding solutions in which a black hole is present [8]. Such solutions finite temperature deformations of field theories ranging from N = 4 SYM [9] to the baryonic branch of Klebanov-Strassler [10, 11, 1, 13, 14] have recently been used as toy models for studying behavior of the quark-gluon plasma produced at the RHIC and LHC. Alternatively, one can consider field theories in which supersymmetry is softly broken by the insertion of relevant operators into the Lagrangian. By starting from a theory where the duality is well understood, it is possible to find gravity duals which are deformations of the SUSY case, as achieved in e.g. [15, 16, 17, 18, 19, 0, 1,, 3]. The deformed backgrounds will match the original ones asymptotically in the UV. They will also share many features of the SUSY backgrounds, such as symmetries, considerably simplifying the problem of finding solutions. Both the Maldacena-Nunez and Klebanov-Strassler models describe field theories whose IR behavior includes only adjoint degrees of freedom. A remaining

15 issue in attempting to make contact with QCD is including fundamental degrees of freedom to describe the quarks. Initial attempts [4] studied this problem in the quenched approximation (when the number of flavors is negligible compared to the number of colors) and used a probe brane approximation on the gravity side, in which the backreaction of the probes on the geometry is neglected. An important development that allowed one to go beyond this and study unquenched, dynamical flavor degrees of freedom was the model of Casero, Nunez, and Paredes [5]. In this model, a large number N f N c of flavor D5 branes is added to the Maldacena-Nunez solution by smearing them over cycles of the internal geometry simplifying the problem of finding the resulting gravity backgrounds and taking their backreaction into account. Backreacted flavor has also been added to the Klebanov-Strassler model, in which case the flavor branes are D7s. In these models N f /M is non-zero, but in order to compute the effects of the unquenched quarks, we must take N f M, 1 so we cannot truly have the number of flavors of the same order as the number of colors in the dual field theory. The model of Casero, Nunez, and Paredes avoids this issue by using D5s as the flavor branes, which allows for a decoupling limit where N f N c a desirable property if we are trying to model a theory like QCD. In this thesis, we will construct a number of new type IIB supergravity backgrounds which are modifications of the two N = 1 supersymmetric backgrounds mentioned above: the Klebanov-Strassler deformed conifold model and the Maldacena-Nunez wrapped branes model. In [6] it was shown that the Klebanov- Strassler background is a particular point in a one-parameter family of IIB supergravity solutions. On the field theory side, this family corresponds to different 1 To compute the effects of unquenched quarks in this model, we have to take an expansion in a parameter λn f /M, which must be kept small. Here M is the number of fractional D3 branes, and λ is the t Hooft coupling. Since λ is large at strong coupling, this corresponds to N f M. 3

16 expectation values of a baryonic operator. By varying the string coupling constant, one can smoothly interpolate to another one-parameter family describing the backreaction of D5 branes wrapping the S of the resolved conifold in which the Maldacena-Nunez background is a particular point. In [7], the connection between these two families was codified in terms of a supergravity U-duality transformation. Our interest in this U-duality procedure stems from its more general use as a solution generating mechanism: starting with any type IIB supergravity background with only the dilaton and either H 3 or F 3 flux turned on, we can generate another solution which contains F 5 flux as well as both H 3 and F 3 flux. We will apply this procedure to various modifications of the family connected to Maldacena-Nunez, and generate new solutions related to the Klebanov-Strassler baryonic branch. The results in this thesis are based on the papers [8] and [9]. In chapters ( 6), based on [8], we first construct new backgrounds which are finite temperature generalizations of a flavored version of the family of SUGRA solutions connected to the Maldacena-Nunez background. These describe the backreacted geometry of N c D5 branes wrapped on the S of the resolved conifold, modified by the presence of N f backreacting (i.e. N f /N c 1) D5 flavor brane sources. The flavor branes are wrapped on non-compact two-cycles of the same manifold, and smeared over the remaining transverse angular coordinates. In our solutions, the dilaton does not blow up at infinity but stabilizes to a finite value. This feature allows us to apply the U-duality procedure and generate a new set of solutions which contain D3 as well as D5 brane charge. We study various aspects of these solutions, including how their temperature depends on both the degree of non-extremality and the amount of flavor s = N f /N c added. We then calculate their ADM energy, using one of the family of extremal (supersymmetric) solutions as a reference background. From this, we determine their specific heat, 4

17 and find that in all cases both before and after the U-duality procedure the specific heat is negative and the solutions are thus unsuited for description of a dual field theory at finite temperature. Our analysis also includes unflavored solutions with s = 0: these are non-extremal generalizations of the one-parameter family of SUGRA solutions that smoothly connects to the Maldacena-Nunez background [7], this family in turn being related to the baryonic branch of the Klebanov-Strassler background [6] by the U-duality procedure. Our thermodynamic analysis indicates that these backgrounds both before and after the U-duality are unstable at finite temperature as well. In chapters (7 10), based on [9], we again begin by modifying the oneparameter family of SUGRA solutions connected to Maldacena-Nunez. This time our modification does not involve flavor branes or finite temperature; instead we break supersymmetry by changing a parameter in the UV asymptotics. We concentrate on the case where the SUSY breaking parameter is small compared to the other parameters in the SUSY backgrounds. This results in backgrounds dual to the strongly coupled dynamics of well-understood SUSY field theories, where SUSY has been softly and controllably broken. We next apply the U-duality procedure and generate new backgrounds with Klebanov-Strassler asymptotics; these then describe a non-susy deformation of the baryonic branch of the Klebanov-Strassler background. After calculating the ADM energy of our backgrounds, we present a detailed study of various field theory quantities, whose strong coupling results will point us towards an interpretation of the dual field theory as being deformed by the insertion of relevant operators, like gaugino masses that break SUSY and may also influence VEVs. In chapter 11, we present conclusions from both parts of the thesis, as well as possible interesting directions for further research. 5

18 In the following sections of this chapter, we give a brief introduction to the one-parameter family of wrapped D5 brane solutions that contains the Maldacena- Nunez background, the related family which represents the baryonic branch of Klebanov-Strassler, and the U-duality procedure which connects them. 1.1 The one-parameter family of wrapped D5 brane solutions In this section we describe the one-parameter family of supergravity solutions that is associated with N c wrapped D5 branes on the S of the resolved conifold. These solutions take into account the backreaction of the branes on the geometry, and are smooth with F 3 flux on the S 3. After backreaction the solutions have the topology of the deformed conifold, with a vanishing S and finite sized S 3 at the tip see Fig.(1.1 (d)). We will use the form of the solution in [7] which is given in terms of wrapped NS5 branes. The form corresponding to wrapped D5 branes can be obtained by performing an S-duality. The solution is ds str = dx α N c 4 ds 6 ds 6 = c (dt +( 3 + A 3 ) )+ c tanh t ( e 1 + e c )+ sinh t ( 1e 1 + e ) t + tanh t 1 ( 1 + e 1 e ) e φ = e φ f 1/ c 0 sinh t, H 3 = α N c 4 w 3, w 3 = ( 3 + A 3 ) ( 1 + e 1 e )+ t sinh t ( 1 e + e 1 ) + (t coth t 1) dt ( 1 e 1 + e ) (1.1) sinh t The D5 form of the solution was also discussed in [5]. 6

19 where e 1 = dθ 1, e = sin θ 1 dφ 1, A 3 = cos θ 1 dφ 1, 1 + i = e iψ (dθ + i sin θ dφ ), 3 = dψ + cos θ dφ. (1.) The SU() left-invariant one-forms i obey d 1 = 3 and cyclic permutations. The functions c(t) and f(t) appearing in equations (1.1) obey f = 4sinh tc (1.3) c = 1 f [c sinh t (t cosh t sinh t) ] (1.4) where the primes denote derivatives with respect to t. The range of t is between zero and infinity. We will be interested in solutions to these equations with the following boundary conditions for small and large t c = γ t +, f = t 4 γ +, for t 0 c = 1 6 e (t t ) 3 +, f = 1 16 et e 8(t t ) 3 +, for t U 1e t 3 (1.5) where the dots indicate higher order terms. γ and t are IR and UV parameters which are related by the equations of motion; varying one or the other takes us along the one-parameter family of solutions. We have also related the parameter t to a parameter U, which will represent the VEV of a baryonic operator in the baryonic branch of the Klebanov-Strassler theory after we perform the U-duality procedure. Solutions of equations (1.3) and (1.4) can in general only be found numerically, by interpolating between the UV and IR behaviors. These solutions vary between the conifold at t and the t = 0 region where there is an S 3 which does not shrink and has flux N c for the H 3 field, see Fig.(1.1(d)). The dilaton is at a maximum at t = 0, decreases as we go to large values of t, and achieves a constant value φ asymptotically as t. In further calculations, we will set e φ = 1. 7

20 deformed S 3 resolved S 3 S S (a) (b) S 3 Flux S 3 Branes (c) S near brane region (d) S Figure 1.1: (a) and (b) show the standard cartoons for the deformed and resolved conifolds. (c) represents the D5 branes wrapping the S of the resolved conifold. (d) represents the resulting backreacted geometry where the D5s have been replaced by geometry and flux. This final geometry has the topology of the deformed conifold with flux on the S 3. The near brane region is the (Chamseddine-Volkov-)Maldacena-Nunez solution [4, 5, 6]. (Image credit: Maldacena, Martelli - arxiv: ) Let us examine the behavior of the solutions as the parameter γ or t is varied. The range of t is from to, which corresponds to U having a range from to 0. As t decreases from to, γ increases from 1 to a very large number. The S 3 at the origin has radius squared equal to rs = α N 3 c γ, and so attains a minimum of α N c for large positive t, and becomes very large for large negative t. In the regime that t is very large and positive one can show that the solution has a region where it looks very close to the resolved conifold with some branes wrapping the S. See Fig.(1.(c)). In this case γ is very close to one. As we get close to these branes the solution takes backreaction into account and the 8

21 geometry in this near-brane region is the Maldacena-Nunez solution. The metric is very close to the metric of the resolved conifold for a large range of distances when t 1. In the limit that t, the UV behavior of the dilaton qualitatively changes and no longer asymptotes to a constant, rather inheriting a nearly linearly decreasing behavior. 3 This corresponds to going to the field theory or near-brane limit, and the solution becomes the Maldacena-Nunez solution for all values of t after performing an S-duality to obtain the wrapped D5 solution. On the other hand, when t is very large and negative, the solution looks like the deformed conifold with a very large S 3, see Fig.(1. (b)). The size of the S 3 is determined by the IR parameter γ, which becomes very large as t. For very large γ it is possible to find an approximate solution of equations (1.3) and (1.4) by ignoring the second term in equation (1.4). This gives a standard solution for the deformed conifold. Thus we see that this family of solutions displays a geometric transition completely within the supergravity description. For large positive t we can view the solutions as branes wrapping the S of the resolved conifold, and for large negative t we have a deformed conifold with flux. When the flux is zero, the deformed and resolved conifold represent distinct branches of the moduli space. With non-zero flux these branches are smoothly connected, see Fig.(1. (a)). More background and details of geometric transitions in the context of the conifold can be found in [30]; a review is [31]. 3 In the S-dualized description in terms of wrapped D5 branes, the asymptotic behavior of the dilaton is nearly linearly increasing, see the next section for details. 9

22 deformation D Z R flopped resolution (a) resolution Flux S 3 3 Flux S S S (b) near brane region (c) Figure 1.: (a) The moduli space of the conifold without flux or branes has two branches, denoted here by the vertical and horizontal axes. One represents deformation, and the other is resolution, which has two sides differing by a flop transition. After we add flux, we have a one-parameter family that interpolates continuously between a deformed conifold with flux in region D and a resolved conifold with branes in region R. (b) In region D of (a), the solution looks like the deformed conifold with flux. (c) In region R of (a) the solution looks like the resolved conifold with some branes, where the branes have been replaced by their near brane geometry. (Image credit: Maldacena, Martelli - arxiv: ) 1. The Maldacena-Nunez limit As discussed in the last section, the near-brane (or decoupling) limit corresponds to taking the parameter t, which yields the exact Maldacena-Nunez solution of [6], dual in the IR to a four-dimensional N =1SU(N c ) super Yang-Mills theory coupled to a Kaluza-Klein tower of massive chiral and massive vector multiplets. Here, we will briefly describe the construction of this solution. For a more detailed discussion, see [6] 10

23 The model starts with N c D5 branes in flat space. The field theory that lives on them is 6d super Yang-Mills with 16 supercharges. Suppose that we wrap two directions of the D5-branes on a curved two-manifold that can be chosen to be a sphere. In order to preserve some fraction of SUSY, we must implement a twisting procedure. We will be interested in a twisting that preserves four supercharges and hence N = 1 SUSY. In this case the two-manifold must live inside a Calabi-Yau 3- fold. Then a dimensional reduction on the two-sphere will give a 4d theory, at least for energies below the inverse volume of the S. Thus the corresponding supergravity solution can be argued to be dual to 4d SYM only for low energies (small values of the radial coordinate). As the energy scale of the 4d theory becomes comparable to the inverse volume of the S, the Kaluza-Klein tower of massive chiral and vector multiplets begins to enter the spectrum. The infinite number of Kaluza-Klein modes reflects the fact that the UV completion is not given by a quantum field theory (in fact, it is given by a Little String Theory). As the energy is increased further, the theory first becomes six-dimensional, and then the blowing up of the dilaton 4 forces one to S-dualize, and we regain the NS5 description. More details about the twisting and the spectrum are contained in [3, 33]. 1.3 The U-duality procedure and the limit of the Klebanov-Strassler baryonic branch. In this section we describe the U-duality procedure used in [7, 34] and how its application to the one-parameter family of wrapped D5 solutions gives a new family of solutions dual to the baryonic branch of the Klebanov-Strassler theory. For our purposes, the key function of this U-duality (henceforth referred to as a rotation as in the literature) is to map a solution of type IIB supergravity 4 The dilaton for this solution behaves as e φ er r for large values of the radial coordinate r. 11

24 with only dilaton and either NS or RR three-form flux into another solution which includes non-trivial RR five-form flux in addition to both NS and RR three-form fluxes. If we start with the family in equations (1.1), describing wrapped NS5s, our initial solution will include NS three-form flux, and the first step of the rotation procedure will involve the S-duality referred to in the previous section. If we instead start with an ansätz which describes wrapped D5s (as will be the case in subsequent chapters), our initial solution will contain RR three-form flux, and we will omit this first S-duality. Here are the steps in the rotation procedure. It goes without saying that these must be performed with the utmost concentration, and practiced many times in a sheltered environment before attempting an invocation in the wild. A spurious sign error, 5 committed in a moment of laxity, could summon a foul demon of the bulk, a tachyonic instability or rouge singularity, or even a bubble of nothing placing your life and the lives of those around you in peril. 1. (only if starting with NS5 ansätz) S-dualize, yielding a type IIB solution describing the backreaction of wrapped D5s, with F 3 flux.. Compactify the three spatial Minkowski directions spanned by the D5s on a T 3 and T-dualize along these directions, obtaining a type IIA configuration with Ds wrapped on the S, and F 4 (dual of F 6 )flux. 3. Lift this to M-theory, and perform a boost in the t x 11 plane: t cosh βt sinh βx 11 x 11 sinh βt + cosh βx 11 (1.6) We now have M branes, and the boost has generated KK momentum charges with A 1 potential. So we have G 4 and A 1 in 11d. 5 I have made plenty. *shudders* 1

25 4. Reduce to type IIA, giving Ds and D0s. We have F 4 = da 3 + A 1 H 3 and H 3 from the dimensional reduction of G 4, as well as F = da 1 5. T-dualize back along the three Minkowski spatial directions, yielding a final type IIB solution with D3-brane and D5-brane charge. We now have F 3 (dual of F 7 ), H 3, and F 5 fluxes. The solution after the rotation is ˆφ = φ here = φ previous = φ d s str = 1 h 1/ dx e ˆφ Ñ c α h 1/ ˆφds 4 cosh β e 6 h = 1 + cosh β(e ˆφ 1) F 3 = α M 4 w 3 H 3 = tanh β Ñc α 4 e ˆφ 6 w 3 F 5 = tanh βe ˆφ ( )Vol 4 dh 1, (1.7) where we have defined M = 1 4π α F 3 = e ˆφ M cosh β N. (1.8) S 3 Thus we see that the original number of wrapped NS5 branes is not quantized in the transformed solution. Note also that the sign on the dilaton has flipped as a result of the initial S-duality, and we have introduced a new parameter ˆφ to account for the freedom to shift the dilaton, which determines the asymptotic value of the string coupling. The boost parameter β can be thought of as an interpolating parameter. The warp factor h 1/ is an increasing functions of t which goes to one at infinity. From the small t expansions of the functions given in section 1.1, we see that h 1/ becomes constant at t = 0, thus the IR geometry is essentially the same for all values of the parameter β. The fact that the warp factor asymptotes to a 13

26 constant is a sign that we are coupling the field theory modes to the string theory modes on the ordinary conifold. In order to recover a field theory interpretation, we take the limit that β, which eliminates the constant factor in the expression for h in equation (1.7). In this limit, the warp factor grows without bound as we go to large t, and its leading UV behavior is the Klebanov-Strassler warp factor. To obtain a finite limit, we rescale the world volume coordinates x e ˆφ Ñ c α U cosh βλ 0 x (1.9) The factor of U ensures that the UV asymptotic form of the metric is independent of U, 6 and the factor Λ 0 introduces a scale, which is the scale of the last step of the cascade. This then gives a final solution ˆφ = φ d s str = e ˆφ 1 Ñ c α ĥ 1/ UΛ 0dx ĥ1/ e ˆφds 6 ĥ = e ˆφ 1 F 3 = α Ñ c 4 w 3 H 3 = α Ñ c e ˆφ e ˆφ 6 w 3 4 F 5 = Λ 4 0(Ñcα ) U e ˆφ ( )Vol 4 dĥ 1 (1.10) This solution now describes a one-parameter family of solutions dual to the baryonic branch of the Klebanov-Strassler theory [6]. The parameter t differentiates solutions in the one-parameter family, and is related to the baryonic branch VEV U by U = 1e t /3. The discrete parameter Ñc gives the number of fractional D3 branes. After the β limit is taken, in the t or U 0limit,themetric becomes exactly the Klebanov-Strassler background. In particular, the unwarped 6 This follows from the fact that ĥ U te 4t 3 and c U 1 e t 3 for large t, see[7]. 14

27 internal metric is the deformed conifold metric. On the other hand, in the t or U limit, we have very large VEVs on the baryonic branch. It was argued in [7] that in this limit one approaches the wrapped fivebrane theory, but with a B field on the two-sphere, induced by the rotation procedure. In this case the theory is well described by fivebranes wrapped on a fuzzy two-sphere. 15

28 Chapter The Stability of Non-Extremal Conifold Backgrounds with Sources Two particularly important developments in AdS/CFT have been the generalization of Maldacena s original conjecture to explore the dynamics of strongly coupled gauge theories at finite temperature, and the expansion of the matter content of the gauge theories to include fields which transform in the fundamental representation. These are both fundamental issues in their own right, but can also be seen practically as steps towards understanding aspects of QCD at zero and finite temperature, through the use of a suitable string dual. A first step in constructing such models is to account for the gauge degrees of freedom. We do this by considering two supergravity backgrounds whose field theory dual reduces to four-dimensional N = 1 Super Yang-Mills theory in the IR: the deformed conifold model of Klebanov and Strassler [7], and the wrapped branes model of Maldacena and Nunez [6]. These theories have both been thoroughly studied, and the connections between them have been well understood. As described in chapter 1, each of these theories is known to be a point in a one-parameter family of type IIB supergravity solutions, and the two families can be smoothly connected through the use of a supergravity U-duality transformation [6, 7, 34]. The family on the Klebanov-Strassler side is known to provide a dual description of the baryonic branch of the Klebanov-Strassler field theory, where the parameter corresponds to the expectation value of a baryonic operator. On the Maldacena-Nunez side, the 16

29 parameter describes the size of a non-vanishing S 3, threaded by the F 3 flux of the backreacting D5 branes. In order to add dynamical fundamental flavor degrees of freedom to these models, we will add a large number N f of flavor D5 branes to the Maldacena-Nunez family of solutions, and include their backreaction on the geometry. This will be done in the Veneziano scaling limit: as we take the standard AdS/CFT scaling we also scale N c,λ= g YMN c =fixed, (.1) N f,s N f N c =fixed, (.) which will allow us to take s 1. In order to make the resulting equations of motion more tractable, the flavor branes will be smeared using a technique that has been used extensively in the literature [35, 36, 37, 38, 39, 40]. This results in various types of flavored solutions which were studied in [5, 41, 4, 43] and are dual to modified versions of N = 1 SQCD. We will be interested in a particular class of solution in which the dilaton asymptotes to a constant value in the UV; this will allow us to apply the rotation procedure and generate a flavored generalization of the Klebanov- Strassler baryonic branch. This construction was studied at zero temperature by Gaillard et al. in [43], where the class of rotated backgrounds was christened the flavored warped resolved deformed conifold, and the proposed dual field theory was found to exhibit a combination of Seiberg dualities and Higgsing. The scenario was developed further in [44, 45, 46]. Other models which incorporate the backreaction of flavor branes (in this case D7s) in the Klebanov-Strassler background include [47, 48]. In the first half of this thesis, we will study finite temperature (or nonextremal) deformations of both the flavored Maldacena-Nunez wrapped D5 branes 17

30 backgrounds, and the rotated flavored warped resolved deformed conifold backgrounds of [43]. Our formalism will also apply to non-extremal deformations of the unflavored Maldacena-Nunez family, and the rotated family describing the nonextremal baryonic branch of Klebanov-Strassler. The study of non-extremal deformations of the unflavored case both before and after rotation was initiated by Cáceres, Núñez, and Pando Zayas in [34]. However, they only considered a specific solution, whereas our treatment generalizes their results to the entire baryonic branch. Our results therefore include theirs as a special case (N f =0,a(r) = 0). As a prototype of a non-conformal theory, a finite temperature deformation of the KS baryonic branch could serve as a more realistic dual of finite temperature QCD. Examples of this type are notoriously difficult to find [10, 11, 1], but it is undoubtedly a fascinating subject with rich new physics [1, 13, 14]. After constructing our new backgrounds numerically, we perform a thermodynamic analysis to determine their specific heat. We find that the specific heat is negative, and thus the backgrounds are generally unstable and unsuited for dual descriptions of the field theories of [43] at finite temperature. Our solutions include as a special case the unflavored ones of [34], and we expect our results to hold for those backgrounds as well. The remaining chapters (3-6) in the first half of this thesis are organized as follows. In chapter 3, we review the smearing procedure and the construction of the zero temperature flavored wrapped D5 backgrounds of [5, 41, 4, 43]. The UV asymptotics of these solutions will be of use to us when deriving a set of conditions for the UV asymptotics of our non-extremal solutions. In chapter 4, we begin by introducing an ansätz that allows for non-extremal solutions. We then obtain the form of the Einstein equations of motion, and use them to derive a set of UV asymptotic relations that our non-extremal solutions must satisfy. Then, for a simple 18

31 choice of UV boundary parameters, we numerically find solutions with a stablized dilaton that contain event horizons. We conclude the chapter with an analysis of how the temperature of these new solutions depends on the non-extremality parameter and on the amount of flavor s N f /N c added to the background. In chapter 5, we discuss some details of the application of the U-duality procedure to our non-extremal flavored backgrounds, and explicitly show that the EOMs after the rotation procedure is applied are satisfied given they are satisfied beforehand. This holds independently of the choice of boundary parameters, for all values of s, and we take it as a check of the consistency of the rotated solutions. We then apply the U-duality and produce the new solutions which have event horizons as well as F 5 and H 3 flux possible finite temperature generalizations of the field theories dual to both the flavored resolved deformed conifold backgrounds of [43] and the unflavored Klebanov-Strassler baryonic branch of [6]. In chapter 6, we study the thermodynamics of these new flavored finite temperature solutions, both before and after the rotation. We find that they have negative specific heat, and hence are thermodynamically unstable. The same analysis is applied to the unflavored case, with similar conclusions. This indicates that the solutions we have found so far inherit the instability of the solutions before the U-duality in particular, they are not useful as dual descriptions of finite temperature field theories at strong coupling. Conclusions and directions for further research are given at the end of the thesis in chapter

32 Chapter 3 The Flavored Wrapped D5 Backgrounds, T =0 In [5, 41, 4], the authors studied a setup that added N f flavor D5 branes to the background generated by N c D5 branes wrapped on the S of the resolved conifold. The flavor branes share the same Minkowski directions as the color branes and also wrap a holomorphic, non-compact two-cycle in the CY3-fold. The branes are then smeared along the remaining four compact directions, and their backreaction is taken into account. The resulting supergravity backgrounds were conjectured to be dual to versions of zero temperature N = 1 SQCD at strong coupling. We will be interested in constructing finite temperature deformations of a particular class of these backgrounds with a dilaton displaying asymptotically constant UV behavior. The UV asymptotics of the zero temperature solutions will therefore be important to us as a zero temperature limiting behavior, so in this chapter we review the setup of [5, 41, 4]. We begin by summarizing the smearing process. Next, we give the ansätz for the metric and background gauge field that accommodates this setup, as well as the form of the Lagrangian that incorporates the smeared flavor sources. Lastly, we present a choice of variables which allows for a full description of the solutions in terms of a single variable which obeys a single second order differential equation, referred to in the literature as the master equation for these backgrounds. We also give the explicit form of the UV asymptotics for the metric, dilaton, and gauge fields, as these will be important for constructing our finite temperature flavored solutions. 0

33 3.1 The smearing process We can add a large number N f of flavor degrees of freedom to the dual field theory, without breaking supersymmetry, by adding N f D5 branes to the unflavored background generated by the original N c color D5 branes. The unflavored backgrounds have topology R 3,1 R S S 3 ; we use coordinates θ, ϕ on the S, and θ, ϕ, ψ on the S 3. The flavor D5s share the Minkowski directions spanned by the N c color D5s, and in addition wrap a holomorphic non-compact two-cycle Σ, which we take to span the ψ and r (radial) directions. They are thus extended along the six coordinates t, x 1,x,x 3,r,ψ, and localized in θ, ϕ, θ, ϕ. These are the cylinder solutions of [49]. The two-cycle must be holomorphic in order to preserve supersymmetry, and non-compact so that in the Veneziano limit, the gauge coupling of the effective 4d field theory on the flavor branes g4 = g 6 /V ol(σ ), vanishes. In this case, the only symmetry added to the theory will be the global U(N f ) coming from the open string modes between the flavor and color branes. Since the flavor branes extend down to r = 0 and intersect the color branes, these open string modes will be massless. The stack of N f flavor D5s will be heavy, so it will backreact and the original background will be modified. We can think of an action describing the dynamics of the backreacted system as S = S IIB + S sources. (3.1) S IIB describes the background of the backreacted N c wrapped color branes. On top of this we add a large number N f of flavor branes and find a new background that solves the Einstein equations which encode the presence of the flavor branes. The S sources part of the action corresponds to the Dirac-Born-Infield and Wess-Zumino 1

34 action of the flavor branes. In Einstein frame, it reads S sources = T 5 N f branes d 6 xe φ ĝ (6) + P [C 6 ] M 6 M 6, (3.) where the integrals are taken over the six-dimensional worldvolume of the flavor branes M 6 and ĝ (6) stands for the determinant of the pullback of the metric to this worldvolume. P [C 6 ] is the pullback of the RR six-form potential to the worldvolume. Again, in our case, this is for branes localized in the coordinates (θ, ϕ, θ, ϕ). S sources is six-dimensional, whereas S IIB is ten-dimensional, and the former will give delta functions in the Einstein equations depending on where the branes are localized. To avoid this and the associated (θ, ϕ, θ, ϕ) dependence in the action, we implement a process introduced in [37, 39] and smear the flavor branes over the two transverse S s parametrized by (θ, ϕ) and ( θ, ϕ), obtaining an action of the form S smeared = T 5N f (4π) d 10 x sin θ sin θe φ ĝ (6) + Vol(Y 4 ) C (6), (3.3) where Vol(Y 4 )=sinθsin θdθ dϕ d θ d ϕ and the new integrals span the entire space-time. The Wess-Zumino term depends on neither the metric nor the dilaton, so it does not affect the Einstein equations. It does however alter the equation of motion for the RR six-form C (6), which before was d F (7) df (3) = 0. Now we have a source term, so: df (3) = 1 4 N f Vol(Y 4 )= 1 4 N f sin θ sin θdθ dϕ d θ d ϕ, (3.4) which we can account for by adding a term to F 3. On the other hand, we still have df (7) = 0.

35 3. Background Ansätz and Langrangian Two types of backgrounds were proposed as possible solutions for this setup. In [41], these solutions were referred to as the type A and type N backgrounds. We will only concern ourselves with the type N backgrounds here. It was argued in [5, 41] that these backgrounds faithfully capture non-perturbative physics for any value of the number of flavors N f > 0. Also, it was proposed that they can describe vacua of the theory where the gaugino condensate is non-zero. In the next chapter we will construct non-extremal solutions which asymptote to the type N solutions with stabilized dilaton in the UV as their temperature goes to zero, so here we review the form of the type N background solutions and their UV behavior. The metric (in Einstein frame), representing a modification of the metric of the resolved conifold, has the following ansätz: ds 10 = α g s N c e φ(r)/ 1 α dx 1,3 + dr + e h(r) (dθ +sin θdϕ )+ g s N c + eg(r) 4 (ω1 + a(r)dθ) +(ω a(r)sinθdϕ) + ek(r) (ω 3 + cos θdϕ). 4 The RR 3-form field strength reads F 3 = α g s N c 4 [ (ω 1 + b(r)dθ) (ω b(r)sinθdϕ) (ω 3 + cos θdϕ) (3.5) +b dr ( dθ ω 1 +sinθdϕ ω )+(1 b(r) )sinθdθ dϕ ω 3, (3.6) and automatically satisfies the Bianchi identity df 3 = 0. The left-invariant oneforms ω a on the S 3 are ω 1 = cos ψd θ +sinψ sin θd ϕ, ω = sin ψd θ + cos ψ sin θd ϕ, ω 3 = dψ + cos θd ϕ. (3.7) 3

36 We also introduce the new radial coordinate ρ, and the standard notation dρ = e k(r) dr, (3.8) e 1 = dθ, e =sinθdϕ ω 1 = ω 1 + a(ρ)e 1, ω = ω a(ρ)e, ω 3 = ω 3 + cos θdϕ. (3.9) The metric then becomes ds 10 = e φ(ρ)/ dx 1,3 + e k(ρ) dρ + e h(ρ) (e 1 + e )+ + eg(ρ) 4 ω 1 + ω e k(ρ) + ( ω 3 ), (3.10) 4 where we have set α g s = 1 and N c has been absorbed into e h,e g, and e k. To be able to account for the backreaction of the flavor branes, the action must be augmented by the DBI and Wess-Zumino actions for the flavor D5-branes. complete action then reads The S = S grav + S sources, (3.11) where, in Einstein frame, we have S grav = 1 κ d 10 x g (10) R 1 ( µφ)( µ φ) 1 1 eφ F(3), (3.1) (10) and N f S sources = T 5 6 d 6 xe φ/ g (6) + 6 P [C (6) ]. (3.13) After smearing the N f flavor branes over (θ, ϕ, θ, ϕ) as described in the previous section, the functions f(ρ),h(ρ),g(ρ) and k(ρ) lose all but their radial dependence. After the smearing we have S sources = T 5N f (4π) d 10 x sin θ sin θe φ/ g (6) + Vol(Y 4 ) C (6), (3.14) 4