Quantum aspects of gauge theories, supersymmetry and unification. PROCEEDINGS 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 <T >∼−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 <T >∼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.