Feynman Path Integral Formulation

1.11 Supersymmetric Strings 53∫I matt = d d X √ {}1G − ¯ψ Dψ −4g 2 (φ) F2 μνσ + ... , (1.240)with g −2 ≡ e −φ /12, with the vacuum expectation value of the dilaton field, relatedto the original string coupling constant g st by g 2 st = 〈e 2φ 〉. Therefore the couplingconstant g st is a dynamical variable in string theory, unlike the case of quantumfield theory where it is usually considered a true constant, up to renormalizationgroup effects. As long as supersymmetry is unbroken, scalar fields like φ can takearbitrary values (they are referred to as moduli). However, supersymmetry breakingcould create a potential for these scalar fields, whose minima could then in principlebe calculable. But since the non-perturbative potential for the dilaton field is notknown, there is no theory for the dilaton mass either.Nevertheless there are two main, in principle observable, effects of the dilatonin low energy phenomena (Damour and Polyakov, 1994; Damour, Piazza andVeneziano, 2002). The first one is that the dilaton field enters the gravitational actionand modifies it, as shown in Eq. (1.239). It can therefore lead to violations ofthe Equivalence Principle, which could be large if the dilaton mass is small. Thereforehigh precision tests of the Equivalence Principle such as the universality of freefall, could be viewed as possible windows on string-scale physics. Similarly a smalldilaton mass could affect high precision tests of the inverse square law for gravityon sub-millimeter scales, although, by the nature of its couplings, a light dilaton isnot likely to play an important role in cosmological evolution.The second main effect of the dilaton is its influence on the gauge coupling constant,through the gluon field strength in Eq. (1.240) (Taylor and Veneziano, 1988;Kaplan and Wise, 2000). Again specific investigation of the effects associated withthe dilaton require some educated guess on its mass, which in some scenarios, basedon specific mechanisms for supersymmetry breaking, is assumed to be of the orderm ∼ Λ 2 susy/μ p ,givingforΛ susy ∼ 1TeV a dilaton Compton wavelength of a fewmillimeters. From these numbers on can then make appropriate estimates on themodifications of matter couplings.Finally one interesting aspect of string theories is how they relate to the physicsof black holes. In the field theory (supergravity) description of strings the D-branesthat appear in the perturbative string picture re-emerge in the supergravity frameworkas so-called black-branes. In the supergravity framework it is in fact morenatural to look at charged black holes in anti-DeSitter space, since these spaces aresupergravity solutions with maximal supersymmetry. It is believed that in string theoryblack hole evaporation then arise through the emission of closed strings fromexcited D-branes.In conclusion superstring theory provides a fascinating alternative to the traditionalfield theoretic approach to quantum gravity. Yet it still has to confront somevery basic issues: there is no known non-perturbative formulation of strings thatwould allow the investigation and selection of superstring vacua in ten dimensions.Furthermore, the dynamical mechanism for compactification is not understood, insteadthe usual avenue for compactification is the selection of a class of manifoldswhich appear to have desirable properties in four dimensions. Indeed there is to this