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Duality Invariant Cosmology to All Orders in Α′ PHYSICAL REVIEW D 100, 126011 (2019) Editors' Suggestion Duality invariant cosmology to all orders in α0 Olaf Hohm1 and Barton Zwiebach2 1Institute for Physics, Humboldt University Berlin, Zum Großen Windkanal 6, D-12489 Berlin, Germany 2Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA (Received 14 June 2019; published 10 December 2019) While the classification of α0 corrections of string inspired effective theories remains an unsolved problem, we show how to classify duality invariant α0 corrections for purely time-dependent (cosmological) backgrounds. We determine the most general duality invariant theory to all orders in α0 for the metric, b-field, and dilaton. The resulting Friedmann equations are studied when the spatial metric is a time- dependent scale factor times the Euclidean metric and the b-field vanishes. These equations can be integrated perturbatively to any order in α0. We construct nonperturbative solutions and display duality invariant theories featuring string-frame de Sitter vacua. DOI: 10.1103/PhysRevD.100.126011 I. INTRODUCTION geometry where the diffeomorphism invariance of general relativity is replaced by a suitably generalized notion of String theory is arguably the most promising candidate diffeomorphisms, which in turn partly determines the α0 for a theory of quantum gravity. As a theory of gravity, the “ ” prospect for a confrontation between string theory and corrections. These ideas play a key role in the chiral observation seems to be particularly promising in the realm string theory of [15], which is the only known gravitational of cosmology, where the effects of fundamental physics at field theory that is exactly duality invariant and has α0 very small scales may be amplified to very large scales. infinitely many corrections. This program, however, Since the early days of string theory there have been has not yet been developed to the point that it can deal with α0 intriguing ideas of how its unique characteristics could play the set of all corrections for bosonic or heterotic string a role in cosmological scenarios [1–3], ideas that have been theory. (See, however, the recent proposal for heterotic revisited and extended recently [4–7]. Two such features of string theory [22].) classical string theory will be central for the present paper: This state of affairs is unsatisfactory since the inclusion α0 (i) the existence of dualities that for cosmological back- of a finite number of higher-derivative corrections is grounds send the scale factor aðtÞ of the universe to 1=aðtÞ, generally insufficient. Gravitational theories with a finite in sharp contrast to Einstein gravity, and (ii) the presence of number of higher derivatives typically display various infinitely many higher-derivative α0 corrections. pathologies that are an artifact of the truncation and not It is a natural idea that the higher-order corrections of present in the full string theory [23]. In this paper we will string theory play a role, for instance, in resolving the big- bypass these difficulties by classifying the higher-derivative α0 bang singularity. The first step is the inclusion of the α0 corrections relevant for cosmology to all orders in . α0 corrections of classical string theory. Such corrections have Rather than finding the complete corrections in general been computed to the first few orders in the 1980s [8–10], dimensions and then assuming a purely time-dependent but a complete computation or classification of these cosmological ansatz, we immediately consider the theory corrections is presently out of reach. In recent years reduced to one dimension (cosmic time) and determine duality-covariant formulations of the string spacetime the complete higher-derivative corrections compatible theories (double field theory [11–14]) have been used to with duality. String theories, being duality invariant, must make progress in describing α0 corrections; see [15–21]. correspond to some particular points in this theory space. These developments hint at a completely novel kind of While we do not know which points those are, the full space of duality invariant theories is interesting in its own right, and may exhibit phenomena that are rather general and apply to string theory. Published by the American Physical Society under the terms of Our analysis is based on the result by Veneziano and the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to Meissner [24], extended by Sen [25], concerning the the author(s) and the published article’s title, journal citation, classical field theory of the metric, the b-field, and the and DOI. Funded by SCOAP3. dilaton arising from D ¼ d þ 1 dimensional string theory. 2470-0010=2019=100(12)=126011(21) 126011-1 Published by the American Physical Society OLAF HOHM and BARTON ZWIEBACH PHYS. REV. D 100, 126011 (2019) This field theory displays an Oðd; d; RÞ symmetry to all The pattern is clear: the general term at order α0k involves 0 orders in α provided the fields do not depend on the d traces with 2k factors of S_ . Each trace must have an even spatial coordinates. This symmetry, henceforth referred to number of S_ factors, where the even number cannot be “ ” → −1 as duality, contains the scale-factor duality a a . two. The c’s are a priori undetermined coefficients, duality The work of Meissner [26] implies that in terms of standard holding for any value they may take. Except for a few of α0 fields the duality transformations receive corrections, but them, their values for the various string theories are α0 it was shown that to first order in there are new field unknown. In establishing the above result, we made variables in terms of which dualities take the standard form. repeated use of field redefinitions iteratively in increasing We will assume that there are field variables so that duality orders of α0. The result is striking in that only first α0 transformations remain unchanged to all orders in (this derivatives of the fields appear in the action. All duality certainly happens in conventional string field theory invariant terms with more than one time derivative on S can variables [27]). With this assumption we are able to classify be redefined away. All terms with one or more derivatives α0 completely the duality invariant corrections. This work of the dilaton can also be redefined away. The resulting amounts to an extension and elaboration of the results higher derivative actions are in fact actions with high obtained by us in [28]. We use the freedom to perform numbers of fields acted on by one derivative each. This is a duality-covariant field redefinitions to show that only first- major, somehow unexpected, simplification. order time derivatives need to be included and that the In the second part of this paper we investigate this dilaton does not appear nontrivially, thereby arriving at a general α0-complete theory for the simplest cosmological minimal set of duality invariant higher-derivative terms ansatz, a Friedmann-Lemaître-Robertson-Walker (FLRW) α0k to all orders. We prove that at order the number of background whose spatial metric is given by a time- independent invariants, and thus the number of free dependent scale factor times the Euclidean metric and a R parameters not determined by Oðd; d; Þ,isgivenby vanishing b-field. The resulting Friedmann equations are 1 − pðk þ Þ pðkÞ, with pðkÞ the number of partitions of the determined to all orders in α0. While there is of course still integer k. an infinite number of undetermined ck;l parameters in (1.3), Let us briefly summarize the core technical results of the we can write the equations efficiently in terms of a single first part of the paper. The two-derivative spacetime theory function FðHÞ of the Hubble parameter whose Taylor ϕ for the metric g, the b-field b, and the dilaton , restricted to expansion is determined by these coefficients: depend only on time, is described by the one-dimensional action [24] X∞ ≔ 4 −α0 k−122k−1 2k Z FðHÞ d ð Þ ckH ; ð1:4Þ 1 −Φ −Φ_ 2 − S_ 2 k¼1 I0 ¼ dte 8 trð Þ ; ð1:1Þ where, in terms of (1.3), ck ≔ ck;0 þ 2dck;1 þÁÁÁ. The where Φ is the O d; d; R invariant dilaton, defined by Friedmann equations then take the concise form pffiffiffiffiffiffiffiffiffi ð Þ e−Φ ¼ det ge−2ϕ, and we introduced the Oðd; d; RÞ d valued matrix ðe−ΦfðHÞÞ ¼ 0; dt −1 −1 bg g − bg b ̈ 1 S ≡ ; ð1:2Þ Φ þ HfðHÞ¼0; g−1 −g−1b 2 2 Φ_ þ gðHÞ¼0; ð1:5Þ in terms of the spatial components g and b of the metric and b-field. Our classification implies that the most general where the functions fðHÞ and gðHÞ are determined in terms 0 duality invariant α corrections take the form of FðHÞ as Z −Φ 0 _ 4 02 _ 6 f H ≔ F0 H ;gH ≔ HF0 H − F H ; 1:6 I ≡ I0 þ dte α c2;0trðS Þþα c3;0trðS Þ ð Þ ð Þ ð Þ ð Þ ð Þ ð Þ 03 _ 8 _ 4 _ 4 where 0 denotes differentiation with respect to H. þ α ½c4;0trðS Þþc4;1trðS ÞtrðS Þ We show that solutions of the lowest-order equations α04 S_ 10 S_ 6 S_ 4 þ ½c5;0trð Þþc5;1trð Þtrð Þ given by Mueller in [29] can be extended, perturbatively, to 0 05 _ 12 _ 8 _ 4 _ 6 2 arbitrary order in α . We then turn to the arguably most þ α ½c6 0trðS Þþc6 1trðS ÞtrðS Þþc6 2ðtrðS ÞÞ ; ; ; intriguing implication of our results: the potential existence _ 4 3 þ c6;3ðtrðS ÞÞ þÁÁÁ; ð1:3Þ of interesting cosmological solutions that are nonperturba- tive in α0.
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