Ads/CFT Correspondence and Type 0 String Theory∗

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Quantum aspects of gauge theories, supersymmetry and uniﬁcation.

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AdS/CFT Correspondence and Type 0 String Theory∗

Dario Martelli SISSA, Via Beirut 2-4 Trieste 34014 INFN Sezione di Trieste, Italy [email protected]

Abstract: We review some applications of Type 0 string theory in the context of the AdS/CFT correspondence.

Most of the success of the AdS/CFT corre- mf ∼ TH at tree level. This leads also spin 0 spondence is so far devoted to those situations bosons to acquire nonzero masses at one loop and where at least some fraction of supersymmetry spoils conformal symmetry because cancellations is preserved. The case of Type IIB supergravity in the β-function do not occur any more. At large 5 1 5× comp ∼ in AdS S is a typical example [1], where a de- distances (L R TH ) the infrared effec- tailed mapping between quantities living in the tive theory is then expected to be pure YM in two sides of the correspondence can be exploited. one lower dimension. However, an important issue to address is how to This approach captures some of the expected embed the physically relevant nonsupersymmet- qualitative features of the quantum field theory, ric gauge theories in the correspondence, eventu- as a confining behavior of the Wilson loop (ba- ally recovering asymptotic freedom and confine- sically due to the presence of a horizon in the ment. metric) and a mass-gap in the glue-ball spec- There are different proposals for giving a holo- trum. Moreover it provides a Lorentz invariant graphic description of nonsupersymmetric gauge regularization scheme, as opposed for instance theories, which basically deal with possible mech- to the lattice regularization. Nonetheless it has anisms for breaking supersymmetry. After re- some drawbacks: There is still coupling with the calling some of them, we will discuss some conse- physics in higher dimension, as both the masses quences of the approach proposed by Polyakov in of glue-balls and of fermions are of the same or- [2]1 that consists in considering a nonsupersym- der of the Hawking temperature. In addition, the metric string theory with non-chiral GSO projec- p-dimensional ’t Hooft coupling obeys tion, known as Type 0 [4, 5]. ∼ Before going to the main subject, we will λp λp+1TH briefly recall some of the alternative approaches where TH acts as UV cutoff. As the cutoff is presented in the literature. removed (TH →∞), λp+1 should approach zero, It was pointed out by Witten [6, 7] that the but this is opposite to the regime in which the AdS/CFT correspondence can be formulated at supergravity approximation applies2. finite temperature. In doing so one identifies the Among various means to break supersymme- Hawking temperature TH of the supergravity so- try there are the deformations of supersymmetric lution with that of the field theory in a ther- solutions. One searches for (domain wall) solu- mal bath. The antiperiodic boundary conditions tions interpolating between some AdS vacuum for fermions along the compactified (time) direc- with N supersymmetries and a different vacuum tion break supersymmetry and give them masses 2In this respect the situation is similar to the lattice ∗Preprint SISSA 17/2000/EP approach, where computations are possible at strong cou- 1See [3] for recent developments. pling. Quantum aspects of gauge theories, supersymmetry and unification. Dario Martelli that can be AdS or not, and has N 0 < N .3 In all tree level correlators of vertex operators of the field theory side this is interpreted as turning (NS+,NS+) and (R+,R+) fields are the same as on some relevant operators that drive the initial Type II. Using these and other properties it is theory to an effective IR theory via RG flow of possible to derive an expression for the effective the couplings. Later we will mention an applica- gravity action [9] that, split in the NSNS and RR tion of this construction in the context of Type contributions, reads: 0 theory. Z 10 √ −2Φ Another way to break supersymmetry is that SNSNS = d x −ge R of considering orbifolds theories, or branes at sin- 1 2 2 1 2 gularities. These two methods have been pursued − |dB| +4|dΦ| − |dT | − V (T ) (1.1) 12 2 extensively in the literature and here we will not Z comment them further on. 10 √ 2 SRR = d x −g f(T )|Fp+2| + ··· (1.2) 1. Type 0 Strings and Gravity The main novelties are coming from the tachyon In this section we summarize some basic features couplings. In particular, there is a potential V (T ) of Type 0 string theories [4, 5] and their gravity which is an even function of the tachyon field, effective actions [9]. as well as functions f(T ) multiplying RR terms, Type 0 string theories are purely bosonic strings that can be worked out perturbatively. with modular invariant partition function and four sectors whose low laying fields are here sum- 2. A couple of examples marized4 (NS+,NS+) (NS−,NS−) Here we recall the basic ideas behind applica- 0A/0B Φ,Bµν ,gµν T tions of the Type 0 construction, with the help of two of the earlier examples provided by Kle- (R+,R∓)(R−,R) banov and Tseytlin in the papers [10, 11]. As (1) (1) (2) (2) in the usual Type II case, the idea is to consider 0A Aµ Aµνρ Aµ Aµνρ (1) (1) (+) (2) (2) (−) a setup of D-branes of the theory, work out the 0B A Aµν Aµνρσ A Aµν Aµνρσ gauge theory living on their world-volume, and There is a corresponding doubled set of D-branes eventually find the corresponding solution of the coupling to RR-fields. gravity theory in order to make comparisons and, As pointed out in [2] there are no open string hopefully, predictions. tachyons on the world-volume of these branes, while there is a perturbative closed string tachyon 2.1 YM theory in 4 dimensions that renders the Minkowski vacuum unstable. Nev- Consider the stack of N D3 branes of the same ertheless one should regard this as an indication type in Type 0B string theory. The configuration that ten dimensional flat background is not sta- is stable and the low energy spectrum on their ble, while there should exist other vacua in which world-volume consists of SU(N) gauge bosons the theory makes sense. AdS space seems in this plus 6 adjoint scalars in 3 + 1 dimensions. respect a good candidate in that it allows for In [10] the authors find two approximate so- tachyonic modes. lutions of the gravity equations of motions that N =(1,1) supersymmetry on the world- involve nontrivial tachyon, dilaton, together with sheet makes these theories similar in some re- RR 4-form, and whose asymptotics are given in spect to supersymmetric Type II. For example the UV and IR regions5. 3 5 See for instance the talks related to [8] in this confer- The metric approaches in both cases AdS5×S ence. (with different radii) and the behavior of the other 4The two entries in parenthesis refer respectively to left and right movers and the signs correspond to the choice 5The identifications of these regimes follows from the of GSO projection. The upper (lower) signs in the RR usual identification of the radial coordinate transversal to sectors define Type 0A (0B) theory. the stack, with energy scale of the gauge theory.

2 Quantum aspects of gauge theories, supersymmetry and uniﬁcation. Dario Martelli

fields is as follows. The first one displays a van- theory (or gravity) side. In fact the stability of ishing ’t Hooft coupling and a tachyon conden- the CFT at weak coupling implies that Type 0B 5 sate with ∼−1, while the second repre- theory on AdS5×S should be stable for suffi- sents an IR conformal point at infinite coupling ciently small radius. In [13] it is suggested that — in fact the dilaton blows up while the tachyon the instability of the background at large radius condensate is ∼0. translates in a phase transition occurring in the Interestingly, the asymptotic freedom is re- large N CFT at strong coupling, with anomalous produced. It should be noted however that in dimension of the operator dual to the tachyon this situation α0 corrections become important, field developing a singularity at a critical value while in the IR one has to worry about string λc. loop corrections. Moreover, connecting the two asymptotic solutions is still an open question. 3. Non-critical Type 0 2.2 Nonsupersymmetric CFT In this section we expose some results obtained Another interesting problem that has been stud- in [14, 15] concerning the extension of Type 0 ied is the construction of gauge theories that are string theory in non-critical dimensions, together conformal though nonsupersymmetric and their with a proposal for the effective action and the dual gravitational description. Let us illustrate discussion of some solutions. the construction in [11]. In Type 0B theory one The original proposal in [2] was to consider starts with a stack of N D3 branes of one type a string theory in dimension d<10. Even if (say, electric) and N of the other (say, magnetic). a microscopic description of string theory out of This configuration is argued to be stable in the criticality is still far, there are at least indications large N limit6. The theory living on the world- for a possible extension of Type 0 theories in non- volume of this stack is a SU(N)×SU(N) gauge critical dimensions: theory in 3 + 1 dimensions, whose field content comprises, in addition to gauge bosons, 6 ad- • a diagonal partition function, whose mod- joint scalars for each SU(N) factor, 4 bifunda- ular invariance doesn’t rely on d =10(as ¯ mental fermions in the (N, N)andtheirconju- opposed to Type II theories) gates. The dual gravity solution found in [11] is 5 again AdS5×S with constant dilaton and van- • the tachyon should condense, providing an ishing tachyon. effective central charge ceff ∼ V (hT i). It String loop suppression requires large N as doesn’t seem unnatural to shift ceff by the usual, while tachyon stability translates in a con- central charge deficit (10 − d) dition on the ’t Hooft coupling, namely 2 ceff provides a tree level cosmological constant λ = gYMN ∼<100. Then the dual theory should be a CFT in the large N limit, for not very large in the low energy theory, and this agrees with coupling. the expectation that the inconsistencies possibly The gravity solution resembles very much the arise only in flat background. Type IIB familiar case. In fact, it was pointed We work at the level of the effective gravity out in [12] that the gauge theory belongs to the action. In d<10 one should guess the field con- class of orbifold theories of N = 4 SYM, where tent of the theory and write down the relative ac- the Z2 projection belongs to the center of the R- tion. In [15] we assume the NSNS sector (gravity symmetry group. So it is exactly conformal in + tachyon) is always present and the RR sector the large N limit. This information can be used is worked out on group theory grounds, consid- to reverse the argument, that is, via AdS/CFT ering tensor products of SO(d − 2) spinors. In the gauge theory may be predictive on the string five dimensions for instance, 2 × 2 = 1 + 3,and one is led to include a scalar potential A and a 6At finite N branes of different types repel each other because of fermionic degrees of freedom in their world- vector Aµ. We further assume the existence of a volume. 4-form potential to accommodate the would be

3 Quantum aspects of gauge theories, supersymmetry and uniﬁcation. Dario Martelli

D3-brane: This really amounts to consider mas- the dilaton. It turns out that such solution is a d−p−2 sive gravity. AdSp+2×S space, with tachyon VEV, and With these assumptions, the equations of mo- fixed ’t Hooft coupling λ = eΦ0 N.8 tion following from the relative action have inter- Notice that string loop corrections are sup- esting solutions, whose interpretation may give pressed in the large N limit, while α0-corrections some insight into their field theory duals, and are important because the curvature is O(1). eventually provide hints in favor of the consis- By the AdS/CFT correspondence the field tency of either critical or even non-critical Type theory dual of this solution should be at a confor- 0 string theory. mal point. However it is difficult to make contact The relevant piece of the d-dimensional ac- with perturbative field theory because λ ∼ O(1). tion, for the ansatz that we consider is 7 Nevertheless one can still get additional informa- Z ( tion. It is in fact possible to count the number of √ S = ddx −g R degrees of freedom that should live in the theory dual to this background [14]. p 1 2 1 2 2 Φ Consider a thermal deformation of the solu- − (∂M Φ) − (∂M T ) − V (T )e d−2 2 2 tion. Identifying the Hawking temperature with p ) 2 f(T ) 1 2 (d−2p−4)Φ the finite temperature of the field theory one can − e2 d−2 FM ···M , 2(p +2)! 1 p+2 compute the entropy. By either computing the free energy from the Euclidean action, or com- d−2 2 puting the area of the horizon, it turns out that where V (T )=−10 + d − 8 T + ··· is the tachyon potential, including central charge deficit. it has the following behavior

One wants to find solutions of the equations 2 p S ∼ N VpT (3.3) of motion following from the action above and H give them a dual interpretation. Consider an for any value of p. This is an indication that the ansatz in which nonzero fields are the metric, dual field theory should have N 2 degrees of free- constant dilaton (Φ0)andtachyon(T0), and a dom, and could be YM theory in some non Gaus- RR (p + 2)-form field strength: sian limit. Notice the different scaling power in the analogous relation one gets from evaluating − 1 − Rµνρλ = 2 (gµρgνλ gµλgνρ) the entropies of black M2 and M5 branes (3/2 R0 and 3 respectively). 1 − Rijkl =+ 2 (gikgjl gilgjk) (3.1) Notice that this kind of approach is poten- L0 p tially predictive. Consider in fact the formula for Fµ1 ···µp+2 = F0 −g(p+2)µ1···µp+2 . scaling dimensions of dual operators9 Then the equations of motion become a set of p (d−1) + (d − 1) + 4m2R2 algebraic equations. The tachyon VEV is deter- ∆= 0 (3.4) 2 mined implicitly by the following equation s ! 0 2 2 2 τ 1 V (T0) 0 0 − 2 − f (T0) 1 V (T0) m R0 = d(d 1) 1+ τ +(2d 4) 2 = (d − 2p − 4) . (3.2) 2 2 V (T0) f(T0) 2 V (T0)

Now, without a precise knowledge of the func- 0 2 00 00 V (T0) − 2 f (T0) − V (T0) − tions f(T )andV(T) one cannot infer whether with τ = d 2 1 . V (T0) d f(T0) V (T0) it admits solutions. One should really assume it has, and extract some information. The re- First, note that the tachyon VEV T0 behaves maining equations ﬁx the value of the radii of as a “bare” quantity: it does not enter in deter- the two maximally symmetric spaces and that of mining physical quantities. It is very much as in 7 Uppercase indices run from 1 to d, while Greek and 8N is the number of branes, which has to be evaluated Latin indices run from 1 to p +2 andfrom p+3 to d in the string frame. respectively. 9These results are obtained in the case d = p +2.

4 Quantum aspects of gauge theories, supersymmetry and uniﬁcation. Dario Martelli

− the renormalization group. One can do a field − d 1 ˙ 2 γ˙ = − (φi) redefinition, this will shift T0, without affecting 2(d 2) λ, R0,∆. where γ =(d−1) d log A. Then, the scaling dimensions depend on a fi- dy nite number of parameters and with a good guess Now, suppose that (3.2) has more than one on the field theory side, one in principle should solution, say T1,T2 at least. This means that be able to predict an infinite tower of dimensions exist two AdS solutions, with fixed values for from KK analysis. scalars, and radii R2 >R1, say. Then, it can be shown (see [8] for instance) that there are interpolating solutions between the previous two. 4. Holographic RG Flows The explanation of this fact is roughly as follows. The extension of the AdS/CFT correspondence The set of equations (4.3) can be interpreted as to theories away from their conformal limit leads describing a point-like particle rolling in a po- −V naturally to the issue of giving a dual description tential , subject to an effective friction force of RG flows. This holographic description has encoded in γ. In fact, given that γ is monotoni- recently attracted much attention. Here we will cally decreasing and focus on a Type 0 application. d − 1 The construction of gravity solutions describ- γ → > 0 (4.4) R2 ing RG flows is quite general. Given 1) an effective potential for the scalars and 2) the existence as y →∞, it follows that it is strictly positive of AdS solution (at extrema of the potential), the along the trajectory. The solutions we are look- procedure is in principle straightforward. In fact ing for are then those starting at a minimum of the equations of motion for the coupled scalars V for y →−∞(IR) and ending at maximum for plus gravity system provide the RG equations for y →∞(UV). the couplings in the dual theory, once an appro- Some universal information can be read from priate ansatz for the metric is plugged in as a the local behavior of these solutions. Consider physical input. This setup can be achieved confor instance the UV fixed point. First it can sidering consistent truncations of Type IIB or be identified a coordinate which in the field the- 11d supergravities (for instance gauged super- ory can be consistently interpreted as the energy gravities) or other KK compactifications emerg- scale and parameterizes the interpolating solu- ing for example as compactifications on mani- tion. Such a coordinate can be chosen to be10 folds which are near horizon limit of branes at A2 conical singularities. Finally, such a framework U = . (4.5) ˙ can be provided by a simple gravity theory, such A as the Type 0 gravity. The linearized equation for the fluctuations eigen- In this case the effective action is: vectors δφ˜i(U) has the solutions Z d √ 1 2 S = d x −g R − (∂φi) −V(φi) . ˜ −(d−1−∆i) −∆i 2 δφi = AiU + BiU (4.6) (4.1) i where φi =(Φ,T). With the following ansatz where ∆ were given in the previous section. One can then extract the leading β-functions for the 2 2 2 2 ds =dy +A (y)d~x running couplings11:

φi = φi(y) (4.2) d βi(gi)=U gi =(∆i−(d−1))(gi − gi∗)+··· the equations of motions read dU (4.7)

φï + γφ˙i = ∂iV 10 ! Different definitions can be given, which reduce lo- 2 cally to the same one. A¨ A˙ 1 11 +(d−2) = V (4.3) Unless ∆i = d − 1, which means that there is a VEV A A d − 2 for some operator in the dual field theory [8].

5 Quantum aspects of gauge theories, supersymmetry and uniﬁcation. Dario Martelli

5. R´esum´e and Perspectives References

We have illustrated how the Type 0 approach [1] J. Maldacena, “The large N limit of su- perconformal field theories and supergravity,” can be used to apply the AdS/CFT correspon- Adv. Theor. Math. Phys. 2 (1998) 231 [hep- dence to nonsupersymmetric cases. This method th/9711200]. offers some advantages compared to others ap- [2] A.M. Polyakov, “The wall of the cave,” Int. J. proaches, as the finite temperature one for in- Mod. Phys. A14 (1999) 645 [hep-th/9809057]. stance. Though, it has some limitations as well. [3] A. Polyakov and V. Rychkov, “Gauge fields: We conclude summarizing some problems Strings duality and the loop equation,” [hep- th/0002106]. •N=0→difficult to check12 [4] L.J. Dixon and J.A. Harvey, “String Theories • there is no proof of tachyon condensation In Ten-Dimensions Without Space-Time Super- symmetry,” Nucl. Phys. B274, 93 (1986). • α0 corrections are usually important [5] N. Seiberg and E. Witten, “Spin Structures In String Theory,” Nucl. Phys. B276, 272 (1986). • lack of a microscopic stringy description in [6] E. Witten, “Anti-de Sitter space and hologra- d<10 phy,” Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150]. and some interesting features [7] E. Witten, “Anti-de Sitter space, thermal phase transition, and confinement in gauge theories,” • cutoff independent results Adv. Theor. Math. Phys. 2 (1998) 505 [hep- th/9803131]. • confining and asymptotically free solutions [8] M. Petrini and A. Zaffaroni, “The holographic • no mixing with higher dimensional physics RG flow to conformal and non-conformal theory,” [hep-th/0002172]. • “conformal” solutions [9] I.R. Klebanov and A.A. Tseytlin, “D-branes and dual gauge theories in type 0 strings,” Nucl. In particular the last point is potentially inter- Phys. B546 (1999) 155 [hep-th/9811035]. esting. These solutions could be just an artifact [10] I. R. Klebanov and A. A. Tseytlin, “Asymp- of the approximation. However they can also in- totic freedom and infrared behavior in the type dicate some novel field theories or (strongly cou- 0 string approach to gauge theory,” Nucl. Phys. pled) conformal fixed points in YM theories as B547 (1999) 143 [hep-th/9812089]. suggested in [2]. [11] I. R. Klebanov and A. A. Tseytlin, “A non- We conclude noticing that, some control on supersymmetric large N CFT from type 0 the field theory may shed light on Type 0 string string theory,” JHEP 9903 (1999) 015 [hep- theory and/or on non-critical string theory via th/9901101]. AdS/CFT correspondence. In particular the lat- [12] N. Nekrasov and S. L. Shatashvili, “On Non- est developments in the context of holographic Supersymmetric CFT in Four Dimensions,” RG flows could provide an appropriate frame- [hep-th/9902110]. work for studying which constraints the dual field [13] I. R. Klebanov, “Tachyon stabilization in the B466 imposes on the gravity theory. AdS/CFT correspondence,” Phys. Lett. (1999) 166 [hep-th/9906220]. [14] G. Ferretti and D. Martelli, “On the construc- Acknowledgments tion of gauge theories from non critical type 0 strings,” Adv. Theor. Math. Phys. 3 (1999) 119 I wish to thank the organizers of the conference [hep-th/9811208]. for hospitality. This work was supported in part [15] G. Ferretti, J. Kalkkinen and D. Martelli, “Non- by European Union TMR program CT960045. critical type 0 string theories and their field theory duals,” Nucl. Phys. B555 (1999) 135 [hep- 12This is common to all nonsupersymmetric ap- th/9904013]. proaches.