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Diego Marques

Publications and source records attributed to Diego Marques.

At least 55 records · Page 3Linked to original sources
arXiv2016

Second order higher-derivative corrections in Double Field Theory

HSZ Double Field Theory is a higher-derivative theory of gravity with exact and manifest T-duality symmetry. The first order corrections in the massless sector were shown to be governed solely by Chern-Simons deformations of the three-form field strength. We compute the full action with up to six derivatives ${\cal O} (α'{}^2)$ for the universal sector containing the metric, two-form and dilaton fields. The Green-Schwarz transformation of the two-form field remains uncorrected to second order. In addition to the expected Chern-Simons-squared and Riemann-cubed terms the theory contains a cubic Gauss-Bonnet interaction, plus other six-derivative unambiguous terms involving the three-form field strength whose presence indicates that the theory must contain further higher-derivative corrections.

hep-th↗
arXiv2016

On a problem posed by Mahler

E. Maillet proved that the set of Liouville numbers is preserved under rational functions with rational coefficients. Based on this result, a problem posed by Kurt Mahler is to investigate whether there exist entire transcendental functions with this property or not. For large parametrized classes of Liouville numbers, we construct such functions and moreover we show that it can be constructed such that all their derivatives share this property. We use a completely different approach than in a recent paper, where functions with a different invariant subclass of Liouville numbers were constructed (though with no information on derivatives). More generally, we study the image of Liouville numbers under analytic functions, with particular attention to $f(z)=z^{q}$ where $q$ is a rational number.

math.NT↗
arXiv2015

T-duality and $α'$-corrections

We construct an $O(d,d)$ invariant universal formulation of the first-order $α'$-corrections of the string effective actions involving the dilaton, metric and two-form fields. Two free parameters interpolate between four-derivative terms that are even and odd with respect to a $Z_2$-parity transformation that changes the sign of the two-form field. The $Z_2$-symmetric model reproduces the closed bosonic string, and the heterotic string effective action is obtained through a $Z_2$-parity-breaking choice of parameters. The theory is an extension of the generalized frame formulation of Double Field Theory, in which the gauge transformations are deformed by a first-order generalized Green-Schwarz transformation. This deformation defines a duality covariant gauge principle that requires and fixes the four-derivative terms. We discuss the $O(d,d)$ structure of the theory and the (non-)covariance of the required field redefinitions.

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arXiv2015

Perturbative Double Field Theory on General Backgrounds

We develop the perturbation theory of double field theory around arbitrary solutions of its field equations. The exact gauge transformations are written in a manifestly background covariant way and contain at most quadratic terms in the field fluctuations. We expand the generalized curvature scalar to cubic order in fluctuations and thereby determine the cubic action in a manifestly background covariant form. As a first application we specialize this theory to group manifold backgrounds, such as $SU(2) \simeq S^3$ with $H$-flux. In the full string theory this corresponds to a WZW background CFT. Starting from closed string field theory, the cubic action around such backgrounds has been computed before by Blumenhagen, Hassler and Lüst. We establish precise agreement with the cubic action derived from double field theory. This result confirms that double field theory is applicable to arbitrary curved background solutions, disproving assertions in the literature to the contrary.

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arXiv2015

A positive answer for a question proposed by K. Mahler

In 1902, P. Stäckel proved the existence of a transcendental function $f(z)$, analytic in a neighbourhood of the origin, and with the property that both $f(z)$ and its inverse function assume, in this neighbourhood, algebraic values at all algebraic points. Based on this result, in 1976, K. Mahler raised the question of the existence of such functions which are analytic in $\mathbb{C}$. In this work, we provide a positive answer for this question by showing the existence of uncountable many of these functions.

math.NT↗
arXiv2014

Heterotic $α$'-corrections in Double Field Theory

We extend the generalized flux formulation of Double Field Theory to include all the first order bosonic contributions to the $α'$ expansion of the heterotic string low energy effective theory. The generalized tangent space and duality group are enhanced by $α'$ corrections, and the gauge symmetries are generated by the usual (gauged) generalized Lie derivative in the extended space. The generalized frame receives derivative corrections through the spin connection with torsion, which is incorporated as a new degree of freedom in the extended bein. We compute the generalized fluxes and find the Riemann curvature tensor with torsion as one of their components. All the four-derivative terms of the action, Bianchi identities and equations of motion are reproduced. Using this formalism, we obtain the first order $α'$ corrections to the heterotic Buscher rules. The relation of our results to alternative formulations in the literature is discussed and future research directions are outlined.

hep-th↗
arXiv2014

Powers of two as sum of two generalized Fibonacci numbers

For $k\geq 2$, the $k$-generalized Fibonacci sequence $(F_n^{(k)})_{n}$ is defined by the initial values $0,0,\ldots,0,1$ ($k$ terms) and such that each term afterwards is the sum of the $k$ preceding terms. In this paper, we search for powers of two of the form $F_n^{(k)}+F_m^{(k)}$. This work is related to a recent result by Bravo and Luca concerning the case $k=2$.

math.NT↗
arXiv2013

The Schanuel Subset Conjecture implies Gelfond's Power Tower Conjecture

As an alternative to the famous Schanuel's Conjecture (SC), we introduce the Schanuel Subset Conjecture (SSC): Given $α_1,...,α_n\in \mathbb{C}$ linearly independent over $\mathbb{Q}$, if $\{α_1,...,α_n, e^{α_1},...,e^{α_n}\}$ is $\overline{\mathbb{Q}}$-dependent on a subset $\{β_1,...,β_n\}$, then $β_1,...,β_n$ are algebraically independent}. (A set $X\subset \mathbb{C}$ is called $\overline{\mathbb{Q}}$-dependent on $Y\subset \mathbb{C}$ if $\overline{\mathbb{Q}}(X) \subset \overline{\mathbb{Q}}(Y)$.) We discuss whether SC is equivalent to the a priori weaker SSC. Assuming SSC, we give conditional proofs of Gelfond's Power Tower Conjecture and of two other results.

math.NT↗
arXiv2013

Double Field Theory: A Pedagogical Review

Double Field Theory (DFT) is a proposal to incorporate T-duality, a distinctive symmetry of string theory, as a symmetry of a field theory defined on a double configuration space. The aim of this review is to provide a pedagogical presentation of DFT and its applications. We first introduce some basic ideas on T-duality and supergravity in order to proceed to the construction of generalized diffeomorphisms and an invariant action on the double space. Steps towards the construction of a geometry on the double space are discussed. We then address generalized Scherk-Schwarz compactifications of DFT and their connection to gauged supergravity and flux compactifications. We also discuss U-duality extensions, and present a brief parcours on world-sheet approaches to DFT. Finally, we provide a summary of other developments and applications that are not discussed in detail in the review.

hep-th↗
arXiv2013

Exploring Double Field Theory

We present a flux formulation of Double Field Theory, in which geometric and non-geometric fluxes are dynamical and field-dependent. Gauge consistency imposes a set of quadratic constraints on the dynamical fluxes, which can be solved by truly double configurations. The constraints are related to generalized Bianchi Identities for (non-)geometric fluxes in the double space, sourced by (exotic) branes. Following previous constructions, we then obtain generalized connections, torsion and curvatures compatible with the consistency conditions. The strong constraint-violating terms needed to make contact with gauged supergravities containing duality orbits of non-geometric fluxes, systematically arise in this formulation.

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arXiv2012

The proof of a conjecture concerning the intersection of k-generalized Fibonacci sequences

For $k\geq 2$, the $k$-generalized Fibonacci sequence $(F_n^{(k)})_{n}$ is defined by the initial values $0,0,...,0,1$ ($k$ terms) and such that each term afterwards is the sum of the $k$ preceding terms. In 2005, Noe and Post conjectured that the only solutions of Diophantine equation $F_m^{(k)}=F_n^{(\ell)}$, with $\ell>k>1, n>\ell+1, m>k+1$ are [(m,n,\ell,k)=(7,6,3,2) and (12,11,7,3).] In this paper, we confirm this conjecture.

math.NT↗
arXiv2012

Gauged Double Field Theory

We find necessary and sufficient conditions for gauge invariance of the action of Double Field Theory (DFT) as well as closure of the algebra of gauge symmetries. The so-called weak and strong constraints are sufficient to satisfy them, but not necessary. We then analyze compactifications of DFT on twisted double tori satisfying the consistency conditions. The effective theory is a Gauged DFT where the gaugings come from the duality twists. The action, bracket, global symmetries, gauge symmetries and their closure are computed by twisting their analogs in the higher dimensional DFT. The non-Abelian heterotic string and lower dimensional gauged supergravities are particular examples of Gauged DFT.

hep-th↗
arXiv2011

Weighted Zero-Sum Problems Over $C_3^r$

Let $C_n$ be the cyclic group of order $n$ and set $s_{A}(C_n^r)$ as the smallest integer $\ell$ such that every sequence $\mathcal{S}$ in $C_n^r$ of length at least $\ell$ has an $A$-zero-sum subsequence of length equal to $\exp(C_n^r)$, for $A=\{-1,1\}$. In this paper, among other things, we give estimates for $s_A(C_3^r)$, and prove that $s_A(C_{3}^{3})=9$, $s_A(C_{3}^{4})=21$ and $41\leq s_A(C_{3}^{5})\leq45$.

math.NT↗
arXiv2011

The effective action of Double Field Theory

We perform a generalized Scherk-Schwarz dimensional reduction of Double Field Theory on a twisted double torus. The four dimensional effective action is shown to exactly reproduce the bosonic electric sector of gauged N = 4 supergravity. We present explicit expressions for the gaugings in terms of the twists, and analyze the associated string backgrounds. This framework provides a higher dimensional origin of the gaugings in terms of generalized fluxes.

hep-th↗
arXiv2011

Algebraic and transcendental solutions of some exponential equations

We study algebraic and transcendental powers of positive real numbers, including solutions of each of the equations $x^x=y$, $x^y=y^x$, $x^x=y^y$, $x^y=y$, and $x^{x^y}=y$. Applications to values of the iterated exponential functions are given. The main tools used are classical theorems of Hermite-Lindemann and Gelfond-Schneider, together with solutions of exponential Diophantine equations.

math.NT↗
arXiv2011

On Type IIB moduli stabilization and N = 4, 8 supergravities

We analyze D = 4 compactifications of Type IIB theory with generic, geometric and non-geometric, dual fluxes turned on. In particular, we study N = 1 toroidal orbifold compactifications that admit an embedding of the untwisted sector into gauged N = 4, 8 supergravities. Truncations, spontaneous breaking of supersymmetry and the inclusion of sources are discussed. The algebraic identities satisfied by the supergravity gaugings are used to implement the full set of consistency constraints on the background fluxes. This allows to perform a generic study of N = 1 vacua and identify large regions of the parameter space that do not admit complete moduli stabilization. Illustrative examples of AdS and Minkowski vacua are presented.

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