# Symmetries and Cosmology

## Research Statement

Fundamental and high-energy physics has matured in the last 40 to 50 years to such a remarkable extent that nowadays virtually all observational phenomena are well described by either quantum field theory in the form of the standard model of particle physics or Einstein's theory of general relativity describing astrophysics and cosmology. Despite this striking success of fundamental physics, it is almost certain that these theoretical frameworks are incomplete. Most importantly, a complete theory of quantum gravity is needed that would unify general relativity with quantum field theory. In fact, the standard model and general relativity, while tremendously successful at currently accessible energy or length scales, break down in extreme situations like close to the big bang or in the center of a black hole. Moreover, there are concrete theoretical puzzles that need to be resolved. Examples are the nature of the 'dark energy' driving the accelerated expansion of the universe (and the related cosmological constant problem) or the precise dynamics governing the evolution of the very early universe such as inflation or similar mechanisms.

String theory has emerged as arguably the most promising candidate for a complete theory of quantum gravity. Indeed, string theory is unique among proposals for a theory of quantum gravity in that it is UV finite and leads to Einstein's theory of general relativity at low energies. Moreover, string theory is characterized by a wealth of novel features and phenomena such as: supersymmetry, dualities or novel symmetries, higher-derivative $$\alpha'$$ corrections to the Einstein equations, and an infinite tower of massive higher string modes. While attempts to find empirical evidence in the realm of particle physics, as through the discovery of supersymmetric particles, so far have been unsuccessful, there are strong reasons to believe that, even without direct evidence in particle physics, string theory could become testable, namely in (quantum) gravitational physics and cosmology. Indeed, the data releases in recent years of the cosmic microwave background radiation such as by the PLANCK collaboration have reminded us that even the (quantum) dynamics of the very early universe is accessible to the methods of science.

Two of the most striking features of string theory, which distinguish it sharply from Einstein gravity, are the presence of dualities or new symmetries and an infinite number of higher-derivative corrections, governed by the (inverse) string tension $$\alpha'$$. On toroidal backgrounds there is a new symmetry or duality (T-duality) sending the metric to its inverse, $$g\rightarrow g^{-1}$$, thereby inverting all radii. Two configurations that are distinct from the point of view of conventional field theory and geometry can be equivalent in string theory. Moreover, new states emerge on such backgrounds, the so-called winding modes that accompany the conventional Kaluza-Klein modes of general relativity and which transform into each other under the T-duality group $$O(d,d,\mathbb{Z})$$.

There have been numerous proposals of how to employ these unique characteristics in cosmology and thereby to possibly test string theory. For instance, Brandenberger and Vafa proposed in the late 1980s that the winding modes and T-duality property of string theory could play a crucial role in the cosmological evolution of the early universe. Under certain assumptions, this so-called 'string gas cosmology' yields cosmological signatures that would allow it to be discriminated from, say, inflationary models. Another string cosmology proposal was put forward by Gasperini, Veneziano and others, which is a 'pre-big-bang' scenario in which T-duality connects a contracting phase with metric $$g^{-1}$$ to an expanding phase with metric $$g$$, assuming that the big-bang singularity is resolved by some 'stringy' mechanism. It has also been suggested that this may happen already classically, thanks to the higher-derivative $$\alpha'$$ corrections that are present in classical string theory.

The problem with such ideas and proposals, and the reason why (as of now) they have not led to falsifiable empirical predictions, is that they are based on certain theoretical assumptions that cannot be confirmed with present-day string theory techniques. We cannot verify theoretically that the early universe evolves as assumed by the string gas cosmology picture, because we simply do not have the exact 'stringy' version of the Friedmann equations governing the evolution of cosmological backgrounds. While in regimes in which $$\alpha'$$ corrections are suppressed the Einstein equations provide a reliable approximation, close to the big-bang it is to be expected that all $$\alpha'$$ corrections become important. Moreover, the dynamics of the winding modes is only understood approximately, and it remains to describe their complete dynamics, which is almost certainly non-local.

Our research group tries to make progress on the following problems:

• What is the underlying symmetry or geometry of $$\alpha'$$ corrections?
• How can we encode massive string modes in a field theory generalizing Einstein gravity?
• Are there possible cosmological signatures of such 'stringy' gravity theories?
Specifically, we pursue these research questions in the framework of double field theory, which was originally proposed by Hull and Zwiebach in order to describe the Kaluza-Klein and winding modes in one theory, by having fields that depend on doubled coordinates. The subsequent work on double field theory was mainly concerned with a sub-sector of the full theory that essentially can be viewed as a reformulation of the usual spacetime theories, however, in a way that makes the T-duality symmetries manifest by working on a generalized, extended spacetime. Conceptually, the advancement of this theory (referred to as strongly constrained double field theory), is analogous to the geometrical reformulation of the special theory of relativity due to Minkowski in 1908: By working with a four-dimensional spacetime rather than with the more familiar three-dimensional space, the invariance under Lorentz transformations becomes manifest. More importantly, this reformulation provides a calculus that allows one to construct new Lorentz invariants or to verify invariance by pure inspection. Similarly, the strongly constrained double field theory provides a calculus for writing theories that are automatically compatible with T-duality, which in turn led to powerful applications.

Particularly in the realm of higher-derivative $$\alpha'$$ corrections such a calculus would be extremely useful, for there is no other practical way to determine these corrections beyond a few orders in $$\alpha'$$. Recently it was shown that the $$\alpha'$$ corrections in double field theory can be described via a generalization of the Green-Schwarz transformations needed for anomaly cancellation, showing that the higher-derivative corrections are at least partially determined by a gauge symmetry principle. While we have thus already seen the first enticing glimpses of a completely new geometry, we are still far away from a complete understanding.

In relation to the second research question, we hope to construct a true (or weakly constrained) double field theory in which the fields depend genuinely on all coordinates of the doubled space. Until fairly recently, it was arguably not even clear whether such a theory has to exist, because it would contain massive string modes but truncate other string modes of a similar mass scale. Such a theory would thus not be a low-energy effective theory in the usual Wilsonian sense. It was proved by Sen, however, that in the full string theory (in the form of closed string field theory) there is a generalized Wilsonian procedure of integrating out massive string modes, except for the massive Kaluza-Klein and winding modes on toroidal backgrounds. Concretely, there must be a theory for the usual massless fields of string theory (generally containing metric, two-form, and a scalar dilaton), but depending on doubled coordinates. Moreover, Sen's results imply that the resulting theory, which must be UV-finite and a consistent truncation of the full string theory, is governed by the same algebraic structures that underly the gauge symmetries and dynamics of string field theory. These are the strongly homotopy Lie algebras or $$L_{\infty}$$ algebras that generalize familiar Lie algebras. Thus, there is a concrete strategy for finding such theories: constructing the corresponding $$L_{\infty}$$ algebra.

Once the theoretical frameworks envisioned in the first two parts are developed, there will be various potential applications in Cosmology. First, one may compute the dynamics of the universe close to the big-bang using the full $$\alpha'$$-corrected Einstein equations. Second, one may test proposals like string gas cosmology and verify whether the detailed dynamics of the winding modes really does have the anticipated effects. Third, we will be able to develop the cosmological perturbation theory including all $$\alpha'$$ corrections and certain massive string modes. This, in turn, will be instrumental in order to compute higher-point functions for cosmological fluctuations, which may become measurable in the future and thus test string theory.

If successful, this program will open up and make accessible new areas of research in string theory. So far string theory was arguably confined to two extremes: the low-energy effective description given by two-derivative supergravity on the one hand, and the complete string theory, encoded in some world-sheet CFT or a target space string field theory, on the other. While the low-energy effective actions have been heavily used in various string-inspired scenarios, they capture truly stringy effects only in a very limited sense. The full string theory, on the other hand, is simply too involved, and its principles too poorly understood, to use it for applications (say in cosmology) directly. The research proposed here provides a bridge between these two extremes by aiming for a theory 'in between' supergravity and the full string theory that would capture truly stringy effects like $$\alpha'$$ corrections and higher string modes, yet only describe a (consistent and duality-invariant) sub-sector.