Recent Progress in String Theory
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Recent Progress in String Theory Shiraz Minwalla Department of Theoretical Physics Tata Institute of Fundamental Research, Mumbai. DAE Symposium, IITG, Dec 2014 Shiraz Minwalla Plan for this talk Very broadly speaking, the string theory research effort may be characterized as the study of consistent theories of gravity plus those non gravitational theories that can consistently be coupled to gravity This is clearly a very broad mandate, and research in string theory proceeds in diverse directions. This makes it very hard comprehensively survey progress in the field. I will not try. To convey a sense for recent progress in string theory, in this talk I simply pick out and describe five specific results obtained in the field. Each of the results I describe was obtained in the last three years. My choice of topics is ideosynchratic: I have chosen five papers I find happen to be particularly familiar with and find particularly interesting but I could well have chosen others. Shiraz Minwalla Consistent modifications of classical gravity Einstein gravity is defined by the action 1 Z p = S d−2 gR lp where lp is the Planck length. The Planck length lp defines the length scale below which quantum fluctuations of gravity are important. Consider a theory of gravity that has a second length scale, ls, s.t. ls lp Dimensional analysis permits the Einstein Lagrangian to be modified length scale ls. For instance wecould have 1 Z p = + 2 2 S d−2 g R ls R lp Shiraz Minwalla Consistent modifications of classical gravity As ls lp the modified theory is effectively classical. Question: are there any constraints imposed by the general principles of classical field theory on the form of this modified Langrangian? Answer (Edelstein, Maldacena, Zhibedeov a few months ago): yes. We must demand causality. A gravity wave that propagates through (for instance) a shock wave in this theory must emerge at infinity later than a partner wave that propagates in flat space. Any modification of the Einstein Lagrangian that modifies the Einstein form of three graviton scattering may be shown to violate this condition, unless there are also tree level graviton graviton particle couplings to to particles of arbitrarily high spin . Shiraz Minwalla Consistent modifications of classical gravity Let us summarize. Einstein gravity is - as far as we know - a consistent classical theory of gravity. If it proves possible to remove the technical restriction above about three point couplings, we will have demonstrated that every consistent classical deformation of Einstein gravity involves an infinite number of higher spin particles. The only known consistent classical deformations of gravity are classical string theories, which always have particles of arbitrarily high spin, coming from the higher harmonics of the string. We now see that something like this had to be true on grounds of general consistency Shiraz Minwalla Semiclassical corrections to black hole entropy Over forty years ago Beckenstein and Hawking demonstrated that general low energy consideration force us to assign an entropy A S = 4hG¯ to any stationary black hole in general relativity (A here is the area of the event horizon of the black hole). About twenty years ago Wald discovered the modifications of the Hawking Beckenstein induced by classical corrections to Einstein’s equations. The corrections obtained by Wald scale like l n Ld−2 × s L Shiraz Minwalla Semiclassical corrections to black hole entropy For large black holes, in other words, the new Wald contributions to the entropy are supressed compared to Hawking Beckenstein by inverse powers of the black hole radius. The details of these corrections depends on the details of the modified classical Lagrangian. Two years ago Ashoke Sen discovered that the leading quatnum correction to black hole entropy is, in contrast, universal. This result follows from a simple analysis of the one loop determinant of the massless fields in the theory around the Euclidean black hole background. Shiraz Minwalla Semiclassical corrections to black hole entropy In particular Sen demonstrated that A S = + N ln A 4hG¯ where N is a pure number determined by the massless field content of the theory. The correct accounting for the Hawking Beckenstein formula in terms of state counting has been used as a consistency check on proposed quantum theories of gravity. Sen’s new universal result adds fine structure to this consistency check. All known examples of the ennumeration of the spectrum of black holes in string theory pass this consistency check: some other proposals are in tension with this result. Shiraz Minwalla A c function for 3 and 4 d theories Unlike dynamical equations of physics, the renormalization group flow equations of quantum field theory are not invariant under t ! −t (here t = ln Λ). A much stronger statement has been known for over 30 years for RG flows in 2 dimensional field theories. In the 1980s Zamalodchikov was able to construct a c function for arbitrary unitary two dimensional field theories, and demonstrate that this function always decreases under renormalization group flow. At fixed points of the renormalization group the Zamalodchikov c function reduces to the central charge of the CFT. Shiraz Minwalla A c function for 3 and 4 d theories The Zamalodchikov result establishes in particular that the UV central charge is always greater than the IR central charge in the case of an RG flow that goes from one fixed point to another. This result establishes a clear irreversablity for RG flows (rougly speaking they are gradient flows with the central charge function playing the role of the potential). Important general property of quantum field theories. However established only in 2 dimensions. Shiraz Minwalla A c function for 3 and 4 d theories In the last three years a c function has been constructed, and demonstrated to decrease under RG flows, in 3 and 4 dimensions. The four dimensional result was first conjectured by Cardy, and proved by Komargodski and Schwimmer. In general structure the four dimensional c function is very similar to Zamalodchikov’s construction. As in two dimensions it reduces, at fixed points, to a particular anomalous coefficient in the general formula for the trace of the stress tensor. ‘Weyl anomaly’. Shiraz Minwalla c functions in 3 and four dimensions The three dimensional c function, on the other hand, is necessarily very different from its two and four dimensional counterparts, as the Weyl anomaly vanishes in all odd dimensions. Consider the entanglement entropy S(r) of a disk of radius R with the rest of the two plane. Define 0 c0(r) = S(r) − rS (r): Two years ago Casini and Huerta used strong subadditivity of the entanglement entropy to demonstrate that c0(r) ≥ 0. At conformal points S(r) = ar + c and c(r) = c. It follows that c increases in any RG flow from a UV to an IR fixed point. The constant term c may be demonstrated to equal the partition function of the conformal field theory on S3. Shiraz Minwalla Nonsupersymmetric strong weak coupling dualities Over the last twenty years we have got used to the fact that supersymmetric quantum field theories often exhibit strong weak coupling dualities whose underlying mechanisms remain quite mystereous. E.g. S duality of N = 4 Yang Mills theory. Completely un obvious from the Lagrangian. Is thsi proeerty an oddity of the dynamics of supersymmetric field theories, or does it a more general property of quantum field theories in general? No known examples outside 2d Shiraz Minwalla S duality in nonsupersymmetric There is now very good evidence that (atleast in the large N limit) a theory of 3 dimensional fundamental boson minimally coupled to a U(NB) Chern Simons gauge theory at level kB is dual to a theory of fundamental fermions minimally coupled to a fundamental fermion minimally coupled to a U(kB) Chern Simons gauge theory at level NB. N The t Hooft coupling of a Chern Simons theory is k , so this 1 duality takes λ to λ and is a strong weak coupling duality. ‘Bosonization’ in 2+1 dimensions. Shiraz Minwalla S duality in nonsupersymmetric theories This result, developed over the last 3 years by groups in TIFR, Weizmann, Princeton and Harvard suggests that dualities are ubiquitious in field theories : the reason we have not yet discovered them is simply that we lack the tools to study strongly coupled dynamics in non supersymmetric field theories above two dimensions. In the case at hand it turned out that large N techniques were powerful enough to obtain exact results even at strong coupling, enabling the discovery of this duality. Shiraz Minwalla Hydrodynamics About seven years ago it was demonstrated that the set of regular long wavelength solutions of negative cosmological constant Einstein Maxwell Chern Simons gravity are in one to one correspondence with the solutions to the equations of conformal relativistic hydrodynamics, with gravitationally determined transport coeffficients. This result can be regarded as a specialization of the AdS/CFT correspondence to long wavelength thermal solutions. It has motivated an ongoing wave of investigations into the theory of hydrodynamics regarded as a long wavelength effective field theory, with interesting results, some of which I briefly review. Shiraz Minwalla Borel summability of hydrodynamics The equation of hydrodynamics are the dynamical equations for locally thermally equilibriated stuff. These equations of motion are presented in a power series expansion in derivatives. While the most familiar terms in these equations (like the pressure force and viscocity) occur at