CERN-TH.6736/92 FTUAM-38/92 Topics in String Theory and Quantum Gravity 1 L. Alvarez-Gaum´e Theory Division CERN CH-1211 Geneva 23 Switzerland and M.A. V´azquez-Mozo Departamento de F´ısica Te´orica C-XI Universidad Aut´onoma de Madrid E-28049 Madrid, Spain CERN-TH.6736/92 November 1992 1Based on the lectures given by L. Alvarez-Gaum´e at Les Houches Summer School on Gravitation and Quantization, July 6 31, 1992. − Contents 1 Field-theoretical approach to Quantum Gravity 5 1.1Linearizedgravity................................... 5 1.2Supergravity...................................... 6 1.3Kaluza-Kleintheories................................. 8 1.4Quantumfieldtheoryandclassicalgravity..................... 10 1.5EuclideanapproachtoQuantumGravity...................... 17 1.6Canonicalquantizationofgravity.......................... 21 1.7Gravitationalinstantons............................... 23 2 Consistency Conditions: Anomalies 24 2.1Generalitiesaboutanomalies............................. 24 2.2 Spinors in 2n dimensions............................... 27 2.3Whencanweexpecttofindanomalies?....................... 30 2.4 The Atiyah-Singer Index Theorem and the computation of anomalies ...... 34 2.5 Examples: Green-Schwarz cancellation mechanism and Witten’s SU(2) global anomaly........................................ 47 3 String Theory I. Bosonic String 50 3.1BosonicString..................................... 51 3.2ConformalFieldTheory............................... 59 3.3QuantizationoftheBosonicString......................... 66 3.4InteractioninStringTheoryandthecharacterizationofthemodulispace.... 68 3.5 Bosonic strings with background fields. “Stringy” corrections to the Einstein equations....................................... 73 3.6 Toroidal compactifications. R-duality........................ 74 3.7Operatorformalism.................................. 77 4 String Theory II. Fermionic Strings 97 4.1FermionicString................................... 97 4.2HeteroticString....................................111 4.3Stringsatfinitetemperature.............................114 4.4Isstringtheoryfinite?................................119 5 Other Developments and Conclusions 121 5.1StringPhenomenology................................121 5.2BlackHolesandRelatedSubjects..........................122 General Introduction The goal of this Les Houches school is to bring together the practitioners of different research lines concerned with the quantization of gravity ([1] and references therein). In this collection, 2 the present lectures represent in part the point of view of string theorist (for details and ref- erences see for example [2, 3, 4, 5]). The most ambitious approach to the problem is certainly embodied by Superstring Theories where one obtains together with the gravitational field many other interactions with enough richness to account for many features of the Standard Model (SM). This is not an accident. One can consider String Theory as the culmination of several decades of effort dedicated to incorporate within the same framework gravity together with the other known interactions. After nearly a decade of renewed interest in String Theory it seems reasonable to recapitulate how this theory has come to be one of the leading candidates of the unification of all known interactions. We still do not know the basic physical or geometrical principles underlying String Theory. There has been some progress in the formulation of a String Field Theory (see the lecture by B. Zwiebach, [6] and references therein) but we are still far from having a complete theory of string fields. Progress has been made by following a number of consistency requirements which worked quite well in ordinary field theories, and in particular for the SM. These requirements when applied to the unification of gravity with the SM lead to very strong constraints which for the time being seem to be fulfilled only by String Theory. One of the basic properties of the SM which supports many of its successes is its renormal- izability, or equivalently, the fact that after a finite number of parameters are given, we have a rather successful machinery to calculate and explain a large number of high and low energy phenomena. This machinery is renormalizable Quantum Field Theory (QFT). This renormal- izability, or rather, predictability of the SM is one of the desired features we would like to have in a quantum theory of gravity. There are also several features of the SM that should be explained in any theory trying to go beyond. Namely, the chiral nature of the families of quarks and leptons with respect to the SM gauge group SU(3) SU(2) U(1), the origin of the gauge interaction, the origin of the symmetry breaking, the× vanishing× of the cosmological constant etc. One of the more popular “Beyond the SM”avenues is the study of theories incorporating supersymmetry [7]. Supersymmetry is a central property of Superstring Theory. Unfortunately there is no evidence in present accelerators that supersymmetry is realized in Nature. However, if present or future accelerators would find evidence of supersymmetry partners of the known elementary particles, the temptation to extrapolate and to believe that Superstring Theory should play a central rˆole in the unification of the known interactions would be nearly irresistible. When one tries to look for a predictive theory containing Quantum Gravity together with the chiral structure of the low energy degrees of freedom, one seems to be left only with String Theory. These two requirements alone pose very stringent constraints on candidate Theories. As we will see in the next chapter various proposals including combinations of Supergravity [8] and Kaluza-Klein Theories [9] do not satisfy any of these requirements. If one consider only the quantization of the gravitational field independently of other interactions, there has been some substantial progress in the recent past which is reviewed in the lectures by A. Ashtekar. The outline of these lectures is as follows. In Chapter 1 we present many of the approaches used to quantize gravity and to unify it with other interactions in the framework of Quantum Field Theory. We briefly analyze the difficulties one encounters when one tries to apply standard field theory techniques to perturbative computations including graviton loops. We also sumarize the approaches including Supergravity and Kaluza-Klein Theories. These theories are not only 3 afflicted with uncontrollable infinities, but also by the fact that low energy chiral fermions are hard to obtain. We also present a rather brief study of the conceptual problems encountered in the quantization of fields in the presence of external gravitational backgrounds. We conclude the chapter with a collection of general remarks concerning the Euclidean and Hamiltonian approaches to the quantization of gravity. If taking seriously, the na¨ıve approach of summing over all possible topologies and geometries meets with formidable mathematical difficulties. Furthermore, the use of semiclassical methods based on instantons, which could in principle provide some insights into quantum-gravitationally induced processes is also presented (and criticized). This chapter gives a rather negative portrait of the field theoretic attempts to understand Quantum Gravity, and needless to say, this view was not shared by many of the others lecturers. In Chapter two we come to study the constraints imposed by the requirement of having consistent gauge and gravitational interactions between chiral fermions. This brings us to the analysis of Anomalies, both local and global. We present the general conceptual features of the computation of local anomalies in diverse dimensions; we present some efficient computa- tional techniques, and apply them to few interesting examples, in particular the Green-Schwarz anomaly cancelation mechanism [96] which opened the way to the formulation of the Heterotic String [10]. We also present briefly the analysis of global anomalies, and in particular Wit- ten’s SU(2) anomaly [11, 12]. In this chapter we want to exhibit how difficult it is to have anomaly-free chiral theories with gravitational and gauge interactions. In the Green-Schwarz mechanism for example we are left only with the possible gauge groups SO(32), E8 E8 and 248 × E8 U(1) . Thus, the spectrum of low energy excitations (massless states) is severely re- stricted× by the requirement of absence of anomalies. Global anomalies (anomalies with respect to diffeomorphisms or gauge transformations not in the identity component) play a central rˆole in determining the spectrum of possible Superstring Theories. We will analyze in detail in chap- ter four how the absence of global world-sheet diffeomorphism anomalies (modular invariance), a purely two-dimensional condition, restricts the space-time spectrum in String Theory. In Chapter three we begin our study of String Theory. In this chapter we restrict for simplicity to the bosonic string. We present briefly the quantization of the bosonic string, the difference between critical and non-critical strings, and the appearance of a tachyon and a graviton in the spectrum in the critical dimension (d = 26). These two states are ubiquitous in critical bosonic string theories. We then present a general analysis of String Perturbation Theory. We use the operator formalism presented in [13], (other approaches can be found in [14, 15]). This approach emphasizes the geometrical nature of first quantized String Theory. String scattering amplitudes and physical state conditions are