Recent Progress in Theory

Shiraz Minwalla

Department of Theoretical Physics Tata Institute of Fundamental Research, .

DAE Symposium, IITG, Dec 2014

Shiraz Minwalla Plan for this talk

Very broadly speaking, the 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 √ = 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 √   = + 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 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 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 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 ’.

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 first and second order in the derivative expansion, the expansion continues to arbitrary order. Focussing attention on terms that are activated in flows that preserve a particular symmetry (boost invariance) Janik and collaborators were able to use the fluid gravity map to compute the coefficients of all such terms in the hydrodynamical expansion of N = 4 Yang Mills upto 250th order in the derivative expansion!

Shiraz Minwalla Borel summability of hydrodynamcis

They were able to use these results to argue that the hydrodynamical expansion is Borel summable, to compute the radius of convergence of the Borel summed expansion, and to demonstrate that the Borel expansion fails precicely at the location of the first non hydrodynamical mode of the theory. In other words gravity has yielded we an appreciation of the formal structure of the hydrodynamical expansion of strongly coupled field theories that would be impossible to obtain from any other method.

Shiraz Minwalla The gravitational anomaly in hydrodynamics

The standard model has gravitational anomalies: triangle anomalies involving a chiral global U(1) current and two These anomalies are one loop exact and so exactly computatble. From a practical point of view, however, their effect on scattering amplitudes etc is so tiny that they are utterly negligible. Loganayagam, Jensen and Yarom have recently demonstrated implies the existence of a new term (missed by Landau and Lifshitz) in the equations of hydrodynamics of hot QCD. This term could well have experimentally measurable consequences for fluid flows at RHIC/LHC. Thus a careful analysis of fluid flows at RHIC could lead to the observation of the gravitational anomaly!

Shiraz Minwalla Equilibrium and the second law

Ordinary non dissipative equations of motion are tightly constrained by the requirement that they follow from the variation of an action. The equations of motion of dissipative systems do not follow from an action. The general set of constraints on thermal equations of motion, like the equations of hydrodynamics are still not completely understood. One constraint, proposed on physical grounds by Landau and Lifshitz, is that the equations of motion be consistent with the existence of a point wise positive divergence entropy current. This requirement is a local form of the second law of thermodynamics, and appears to be rather deep.

Shiraz Minwalla Equilibrium and the second law

There is another apparently more elementary set of constraints on the same equations. Namely that the equations of hydrodynamics admit an equilibrium solution in every time independent background metric (and external gauge field). Moreover that the equilibrium solution of hydrodynamics is always stable against small fluctuations. It was demonstrated by S. Bhattacharyya - following up on work at TIFR - that these two conditions are actually equivalent. The existence and stability of equilibrium actually implies a local form of the second law. This observation seems conceptually important. It is also practically useful, as the constraints from equilibrium are effectively summarized in a partition function as a function of background metrics and gauge fields.

Shiraz Minwalla Summary

In this talk I have described five results, in the study of gravity, quantum field theory and non equilibrium statistical physics obtained in studies performed over the last three years. The studies I have described are not outstandingly exceptional. As mentioned above they are five of a pool of several fascinating results I could have described to you. I hope I have succeeded in illustrating my conviction that the string research effort is an exciting investigation into the basic strucures of theoretical physics. This effort often addresses structural questions that are of broad general interest in theoretical physics. Progress occurs in diverse directions and is quite rapid and often very exciting

Shiraz Minwalla