Two questions about gravity
Shiraz Minwalla
Department of Theoretical Physics
Tata Institute of Fundamental Research, Mumbai.

Strings 2018, 50 Years of String Theory

Shiraz Minwalla

Two Questions

In this talk I ask two questions about the theory of gravity. In myopinion these questions
Feel interesting and important and appear to touch on thefundamental issues.
Nonetheless may prove possible to answer in the not too
distant future.

Shiraz Minwalla

Classical Gravity
Challenge 1: Enumerate all minimal classical consistenttheories of gravity ‘in flat space’.
Clarrifications
Classical gravity= Diff inv system of metric + optionallyother fields.
In flat space= Has a stable D dimensional Minkowski‘vacuum’ soln. Excitations about this solution include D
dim gravitons.
Minimal: Admits no further consistent trunctations.
Consistent= Obeys a set of physically motivatedrequirements (e.g. causality, positive energy . . .).
Formulating the reasonable list of conditions is part of thechallenge.

Shiraz Minwalla

Classical Gravity: Warm up
Special case: The fields of the dynamical system consist of theD dimensional metric plus a finite number of additional Ddimensional matter fields. Equations of motion have a finite
number of derivatives. Conjecture: Only solution to warm up
problem is given by the vacuum Einstein equations in D
dimensional space.
Comments
All familiar two derivative gravity-matter theories admit
consistent trucation to vacuum Einstein equations for
Einstein frame metric.
Higher derivative theories are not necessarily counterexamples as they may prove to be inconsistent. E.g.
Einstein Gauss Bonnet violates causality.
Warm up challenge: Either prove conjecture or provide a
counter example. Suggestion: a study of the second law forblack holes might prove useful.
Shiraz Minwalla

Gravity an infnite number of other fields.
Let us now allow gravity to interact with an infinite number ofother fields. There are atleast two new solutions: (the universalsector of) the type II string and the Heterotic String.
Conjecture: There are no other consistent solutions.
Comments
Type II theory on R 4 × Y admits a consistent truncation (tomodes whose vertex operators are constructed only out of
the R 4 CFT). This minimal ‘universal sector’ is independentof Y .
Each solution of our challenge generates a tree level Smatrix for gravitions. Our conjecture implies that there areexactly 3 such consistent S matrices (with poles but nocuts). These are the Einstein, Type II (Virasoro Shapiro)and Heterotic S matrices. Striking that we have found noother examples in almost 50 years of work on string theory.
Perhaps this is evidence for our conjecture?
Challenge: Either prove conjecture or find a counterexample.
Shiraz Minwalla

Cosmological Constant in Quantum Gravity
Consider a model of quantum gravity in D ≥ 4 dimensions
that is non supersymmetric at low energies. Let thecurvature mass scale of the vacuum geometry be given by
Mcurv . Let the cut off scale for gravitational contributions tothe vacuum energy be given by Mcut . (Mcut could, forinstance, be the SUSY restoration scale or a string scale).
Let the Planck scale be given by MPl . Let

Mcurv
Mcut2



McutMPlD−2

where the inequality holds parametrically.
The challenge is to show that no such model exists or findan example of such a model that is explicit enough to allow
you to do actual calculations (e.g. 1 loop graviton‘scattering amplitudes’)Shiraz Minwalla

Cosmological Constant: Comments
Comments
2
The contribution of g loop gravitational graphs to Mcurv
is#"gMcut (D−2)
22.
δMcurv ∼ McutMPl
A model in which

Mcurv
Mcut2


McutMPlD−2

2is thus one in which the physical value of Mcurv
is smaller
than the g loop contribution upto parametrically large g.
Real world: MMcurv≤ 10−45 while MMcut≥ 10−15 . This ourcut
Plchallenge is to find any model of quantum gravity whichshares this gross feature of the real world.
Emphasize: A model that does not allow practical
calculations does not solve our challenge
Shiraz Minwalla

Possible Strategy
Use the AdS/CFT correspondence.
For the purposes of this slide I’m going to assume that thewarm up conjecture above is correct. Now consider, e.g., asequence of conformal large N matter Chern Simons
theories with level k fermions in some representation of thegauge group. labelled by the level k . Large k implies weakcoupling. Theory simple and boring. Make theory moreinteresting by reducing k down to values of order unity.
Let us suppose it is possible to keep k small while
maintaining conformal invariance. If we can somehow
analyze the theory (e.g. bootstrap??) and demonstratethat all operators other than the stress tensor andmultitraces become parametrically heavy in this limit, wewould have found a holographic solution to our challenge.
Can one find such a sequence of field theories or showthat no such exists?
Shiraz Minwalla