Toroidal Backgrounds Research Articles

Homotopy Transfer and Effective Field Theory II: Strings and Double Field Theory

AbstractWe continue our study of effective field theory via homotopy transfer of ‐algebras, and apply it to tree‐level non‐Wilsonian effective actions of the kind discussed by Sen in which the modes integrated out are comparable in mass to the modes that are kept. We focus on the construction of effective actions for string states at fixed levels and in particular on the construction of weakly constrained double field theory. With these examples in mind, we discuss closed string theory on toroidal backgrounds and resolve some subtle issues involving vertex operators, including the proper form of cocycle factors and of the reflector state. This resolves outstanding issues concerning the construction of covariant closed string field theory on toroidal backgrounds. The weakly constrained double field theory is formally obtained from closed string field theory on a toroidal background by integrating out all but the ‘doubly massless’ states and homotopy transfer then gives a prescription for determining the theory's vertices and symmetries. We also discuss consistent truncation in the context of homotopy transfer.

Noncommutativity and Nonassociativity of Closed Bosonic String on T‐dual Toroidal Backgrounds

AbstractIn this article we consider closed bosonic string in the presence of constant metric and Kalb‐Ramond field with one non‐zero component, , where field strength H is infinitesimal. Using Buscher T‐duality procedure we dualize along x and y directions and using generalized T‐duality procedure along z direction imposing trivial winding conditions. After first two T‐dualizations we obtain Q flux theory which is just locally well defined, while after all three T‐dualizations we obtain nonlocal R flux theory. Origin of non‐locality is variable defined as line integral, which appears as an argument of the background fields. Rewriting T‐dual transformation laws in the canonical form and using standard Poisson algebra, we obtained that Q flux theory is commutative one and the R flux theory is noncommutative and nonassociative one. Consequently, there is a correlation between non‐locality and closed string noncommutativity and nonassociativity.

Probing the string winding sector

We probe a slice of the massive winding sector of bosonic string theory from toroidal compactifications of Double Field Theory (DFT). This string subsector corresponds to states containing one left and one right moving oscillators. We perform a generalized Kaluza Klein compactification of DFT on generic 2n-dimensional toroidal constant backgrounds and show that, up to third order in fluctuations, the theory coincides with the corresponding effective theory of the bosonic string compactified on n-dimensional toroidal constant backgrounds, obtained from three-point amplitudes. The comparison between both theories is facilitated by noticing that generalized diffeomorphisms in DFT allow to fix generalized harmonic gauge conditions that help in identifying the physical degrees of freedom. These conditions manifest as conformal anomaly cancellation requirements on the string theory side. The explicit expression for the gauge invariant effective action containing the physical massless sector (gravity+antisymmetric+gauge+ scalar fields) coupled to towers of generalized Kaluza Klein massive states (corresponding to compact momentum and winding modes) is found. The action acquires a very compact form when written in terms of fields carrying O(n, n) indices, and is explicitly T-duality invariant. The global algebra associated to the generalized Kaluza Klein compactification is discussed.

Revisiting the two formulations of Bianchi identities and their implications on moduli stabilization

In the context of non-geometric type II orientifold compactifications, there have been two formulations for representing the various NS-NS Bianchi-identities. In the first formulation, the standard three-form flux ($H_3$), the geometric flux ($\omega$) and the non-geometric fluxes ($Q$ and $R$) are expressed by using the real six-dimensional indices (e.g. $H_{ijk}, \omega_{ij}{}^k, Q_i{}^{jk}$ and $R^{ijk}$), and this formulation has been heavily utilized for simplifying the scalar potentials in toroidal-orientifolds. On the other hand, relevant for the studies beyond toroidal backgrounds, a second formulation is utilized in which all flux components are written in terms of various involutively even/odd $(2,1)$- and $(1,1)$-cohomologies of the complex threefold. In the lights of recent model building interests and some observations made in arXiv:0705.3410 and arXiv:0709.2186, in this article, we revisit two most commonly studied toroidal examples in detail to illustrate that the present forms of these two formulations are not completely equivalent. To demonstrate the same, we translate all the identities of the first formulation into cohomology ingredients, and after a tedious reshuffling of the subsequent constraints, interestingly we find that all the identities of the second formulation are embedded into the first formulation which has some additional constraints. In addition, we look for the possible solutions of these Bianchi identities in a detailed analysis, and we find that some solutions can reduce the size of scalar potential very significantly, and in some cases are too strong to break the no-scale structure completely. Finally, we also comment on the influence of imposing some of the solutions of Bianchi identities in studying moduli stabilization.

Elliptic inflation: interpolating from natural inflation to R 2-inflation

We propose an extension of natural inflation, where the inflaton potential is a general periodic function. Specifically, we study elliptic inflation where the inflaton potential is given by Jacobi elliptic functions, Jacobi theta functions or the Dedekind eta function, which appear in gauge and Yukawa couplings in the string theories compactified on toroidal backgrounds. We show that in the first two cases the predicted values of the spectral index and the tensor-to-scalar ratio interpolate from natural inflation to exponential inflation such as R 2- and Higgs inflation and brane inflation, where the spectral index asymptotes to n s = 1 − 2/N ≃ 0.967 for the e-folding number N = 60. We also show that a model with the Dedekind eta function gives a sizable running of the spectral index due to modulations in the inflaton potential. Such elliptic inflation can be thought of as a specific realization of multi-natural inflation, where the inflaton potential consists of multiple sinusoidal functions. We also discuss examples in string theory where Jacobi theta functions and the Dedekind eta function appear in the inflaton potential.

Closed string partition functions in toroidal compactifications of doubled geometries

We revisit partition functions of closed strings on toroidal backgrounds, including their $\mathbb{Z}_N$ shift orbifolds in the formalism where the dimension of the target space is doubled to make T-duality manifest. In such a T-duality covariant formalism, the constraint equation imposes a form of chiral factorization. Our computation furnishes a non-trivial consistency check for the quantum worldsheet theory of the doubled sigma model, when strings are placed on general toroidal backgrounds. The topological term that mixes the physical space and its T-dual is crucial in demonstrating that chiral factorization works, and that we obtain the correct partition function after imposing the constraints. Finally, we discuss how our results extend to $\mathcal{N}=1$ worldsheet supersymmetry and string worldsheets of higher genus.

(Non-)commutative closed string on T-dual toroidal backgrounds

Abstract In this paper we investigate the connection between (non-)geometry and (non-)commutativity of the closed string. To this end, we solve the classical string on three T-dual toroidal backgrounds: a torus with H-flux, a twisted torus and a non-geometric background with Q-flux. In all three situations we work under the assumption of a dilute flux and consider quantities to linear order in the flux density. Furthermore, we perform the first steps of a canonical quantization for the twisted torus, to derive commutators of the string expansion modes. We use them as well as T-duality to determine, in the non-geometric background, a commutator of two string coordinates, which turns out to be non-vanishing. We relate this non-commutativity to the closed string boundary conditions, and the non-geometric Q-flux.

Fluxbrane inflation

As a first step towards inflation in genuinely F -theoretic setups, we propose a scenario where the inflaton is the relative position of two 7-branes on holomorphic 4-cycles. Non-supersymmetric gauge flux induces an attractive inter-brane potential. The latter is sufficiently flat in the supergravity regime of large volume moduli. Thus, in contrast to brane–antibrane inflation, fluxbrane inflation does not require warping. We calculate the inflaton potential both in the supergravity approximation and via an open-string one-loop computation on toroidal backgrounds. This leads us to propose a generalisation to genuine Calabi–Yau manifolds. We also comment on competing F-term effects. The end of inflation is marked by the condensation of tachyonic recombination fields between the 7-branes, triggering the formation of a bound state described as a stable extension along the 7-brane divisor. Hence our model fits in the framework of hybrid D-term inflation. We work out the main phenomenological properties of our D-term inflaton potential. In particular, our scenario of D7/D7 inflation avoids the familiar observational constraints associated with cosmic strings.

Thresholds for intersecting D-branes revisited

Gauge threshold corrections for intersecting D6-brane string models on toroidal orbifold backgrounds are reconsidered. Both by dimensionally regularising the appearing open string one-loop diagrams in tree-channel as well as by zeta-function regularisation of the corresponding loop-channel one-loop diagrams, we arrive at a result which takes into account the infrared divergence from the contribution of the massless states in the running of the gauge coupling constant as well as the contribution of states, which become light in certain regions of the moduli space.

Gauge symmetries of the N = 2 string

We study the underlying gauge symmetry algebra of the N = 2 string, which is broken down to a subalgebra in any spacetime background. For given toroidal backgrounds, the unbroken gauge symmetries (corresponding to holomorphic and antiholomorphic worldsheet currents) generate area-preserving diffeomorphism algebras of null 2-tori. A minimal Lie algebraic closure, containing all the gauge symmetries that arise in this way, is the background-independent volume-preserving diffeomorphism algebra of the target Narain torus T 4,4. The underlying symmetries act on the ground ring of functions on T 4,4 as derivations, much as in the case of the d = 2 string. A background-independent spacetime action valid for noncompact metrics is presented, whose symmetries are volume-preserving diffeomorphisms. Possible extensions to N = 2 and N = 1 heterotic strings are briefly discussed.

Target Space Duality as a Symmetry of String Field Theory

Toroidal backgrounds for bosonic strings are used to understand target space duality as a symmetry of string field theory and to study explicitly issues in background independence. Our starting point is the notion that the string field coordinates X(σ) and the momenta P(σ) are background independent objects whose field algebra is always the same; backgrounds correspond to inequivalent representations of this algebra. We propose classical string field solutions relating any two toroidal backgrounds and discuss the space where these solutons are defined. String field theories formulated around dual backgrounds are shown to be related by a homogeneous field redefinition, and are therefore equivalent, if and only if their string field coupling constants are identical. Using this discrete equivalence of backgrounds and the classical solutions we find discrete symmetry transformations of the string field leaving the string action invariant. These symmetries, which are spontaneously broken for generic backgrounds, are shown to generate the full group of duality symmetries, and in general are seen to arise from the string field gauge group.

Target Space Duality as a Symmetry of String Field Theory

Toroidal backgrounds for bosonic strings are used to understand target space duality as a symmetry of string field theory and to study explicitly issues in background independence. Our starting point is the notion that the string field coordinates $X(\sigma)$ and the momenta $P(\sigma)$ are background independent objects whose field algebra is always the same; backgrounds correspond to inequivalent representations of this algebra. We propose classical string field solutions relating any two toroidal backgrounds and discuss the space where these solutions are defined. String field theories formulated around dual backgrounds are shown to be related by a homogeneous field redefinition, and are therefore equivalent, if and only if their string field coupling constants are identical. Using this discrete equivalence of backgrounds and the classical solutions we find discrete symmetry transformations of the string field leaving the string action invariant. These symmetries, which are spontaneously broken for generic backgrounds, are shown to generate the full group of duality symmetries, and in general are seen to arise from the string field gauge group.