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Non-geometric Fluxes, Asymmetric Strings and Nonassociative Geometry

Ralph Blumenhagen, Andreas Deser, Dieter Lust, Erik Plauschinn, Felix Rennecke

arXiv:1106.0316v1hep-thmath-ph

TL;DR

The paper asks how closed strings behave in constant H-flux and T-dual non-geometric backgrounds, where exact conformal descriptions and target-space interpretations are limited. It constructs a linear-order flux conformal field theory and computes tachyon amplitudes, finding crossing-symmetric tri-product phases, new tachyonic modes, and evidence that R-flux flows toward an asymmetric conformal field theory.

  • Problem

    Closed-string backgrounds with H-flux and its T-duals challenge conventional conformal and geometric descriptions, especially for R-flux backgrounds associated with nonassociativity.

  • Method

    The paper constructs a conformal field theory capturing linear effects in H-flux, analyzes corrected correlators and operator products, and computes tachyon scattering amplitudes and their poles.

  • Results

    The analysis yields crossing-symmetric fluxed four-tachyon amplitudes, relative R-flux phases encoded by a nonassociative tri-product, and two new types of tachyonic modes.

  • Takeaways & Limitations

    The results support a relation between non-geometric R-flux, nonassociative target-space structures, and left-right asymmetric string backgrounds.

  • Takeaways & Limitations

    The conformal field theory is controlled only to linear order in H because flux back-reaction on the metric appears at second order.

Abstract

from arXiv · show

We study closed bosonic strings propagating both in a flat background with constant H-flux and in its T-dual configurations. We define a conformal field theory capturing linear effects in the flux and compute scattering amplitudes of tachyons, where the Rogers dilogarithm plays a prominent role. For the scattering of four tachyons, a fluxed version of the Virasoro-Shapiro amplitude is derived and its pole structure is analyzed. In the case of an R-flux background obtained after three T-dualities, we find indications for a nonassociative target-space structure which can be described in terms of a deformed tri-product. Remarkably, this product is compatible with crossing symmetry of conformal correlation functions. We finally argue that the R-flux background flows to an asymmetric CFT.

1 Introduction

The paper investigates how closed strings probe non-geometric and potentially nonassociative backgrounds, building a linear-in-flux conformal field theory and studying tachyon scattering. Its results connect R-flux, deformed tri-products, crossing symmetry, new tachyonic modes, and asymmetric conformal field theories.

  • Motivation: Strings can propagate in left-right asymmetric conformal field theories, including backgrounds without a clear target-space interpretation.T-duality itself acts asymmetrically on left- and right-moving sectors.
  • Flux backgrounds: Successive T-dualities map constant H-flux to geometric, non-geometric, and finally R-flux backgrounds associated with proposed nonassociative geometry.The R-flux background arises after a formal third T-duality that is not along an isometry direction.
  • Approach: The paper constructs a conformal field theory for H-flux at linear order and computes corrected coordinate correlators, operator products, vertex operators, and T-dual observables.The construction uses field redefinitions and conformal perturbation theory, with logarithmic terms appearing in operator products and correlators.
  • Scattering amplitudes: Tachyon amplitudes remain crossing symmetric after momentum conservation, while R-flux introduces phases encoded by a nonassociative tri-product.The four-tachyon calculation yields a conformally invariant, crossing-symmetric fluxed Virasoro-Shapiro amplitude.
  • Consequences: The pole structure reveals two new types of tachyonic modes, interpreted as instabilities associated with higher-order flux corrections.The paper also proposes that R-flux flows toward a left-right asymmetric WZW-type conformal field theory.

2 Open and closed strings in flux backgrounds

The paper adapts conformal-field-theory methods from open strings to closed strings in flux backgrounds. It uses flux-corrected correlators and T-duality relations to organize momentum-winding states and investigate noncommutative structures in closed-string scattering.

  • Open strings: Open-string correlators on D-branes with two-form flux produce a noncommutativity parameter through a discontinuous sign function.The symmetric matrix G is associated with the effective metric, while the antisymmetric matrix θ is proportional to the two-form flux.
  • Closed strings: Closed-string correlation functions are used to investigate whether the product structures familiar from open strings also appear in the closed-string sector.The relevant amplitudes include tachyon vertex operators and their momentum-winding dependence.
  • Open strings: The Moyal-Weyl product is extracted from open-string correlation functions because its phases modify scattering amplitudes and the low-energy gauge-theory description.The construction motivates applying analogous correlation-function methods to the closed-string sector.
  • Closed strings: Closed strings in constant H-flux require a flux-corrected treatment because the flux back-reacts on the metric and obstructs an exact flat-background conformal field theory.The paper treats the theory as conformal up to linear order in H, denoting it CFTH.
  • T-duality: T-duality exchanges flux configurations together with momentum and winding states in the compact three-dimensional background.The mapping for Q- and R-flux is obtained by generalizing the ω-flux result.

3 Conformal field theory with H-flux

The paper develops a closed-string conformal field theory for flat space with constant H-flux at linear order, where three-coordinate correlators probe nonassociativity. It then analyzes vertex operators, consistency conditions, flux-induced instabilities, and T-dual backgrounds, finding especially distinctive effects for R-flux.

  • 3 Conformal field theory with H-flux: Conformal perturbation theory yields a three-coordinate correlator proportional to H, providing the basic world-sheet signal of nonassociative geometry.The construction includes linear redefinitions of coordinates and currents and flux corrections to correlators and OPEs.
  • 3 Conformal field theory with H-flux: The flat constant-H background is a bona fide conformal field theory only at linear order in H, because flux back-reaction obstructs conformal invariance beyond that order.The three-bracket is also linear in H, so this order captures potential nonassociative effects.
  • 3 Conformal field theory with H-flux: At linear order, corrected currents have purely holomorphic or anti-holomorphic three-field correlators and generate two current algebras with opposite relative H-signs.The corresponding currents retain the free-theory central charge, while the vertex operators are tested for primary conformal dimensions.
  • 3 Conformal field theory with H-flux: The deformed tachyon vertex is physical when it is primary of conformal dimension (1, 1), subject to a flux-dependent momentum constraint also obtained from the classical equations of motion.The same constraint appears in the operator-product analysis and in the persistence of the H-corrected classical tachyon solution.
  • 3 Conformal field theory with H-flux: Logarithmic terms can signal a logarithmic CFT, whose spectrum includes massive and tachyonic longitudinal fluctuations, although level matching removes some states.The paper notes that fully determining the LCFT mass spectrum remains open.
  • 3 Conformal field theory with H-flux: T-duality distinguishes R-flux from geometric and Q-flux cases: pure-momentum R-flux scattering is reliably computable, while some other amplitudes may reflect only coordinate redefinitions.For R-flux, the relevant amplitudes are T-dual to pure-winding scattering in the H-flux background.

4 Tachyon scattering amplitudes

The tachyon amplitudes remain crossing symmetric after momentum conservation, while R-flux introduces phase factors captured by deformed products. The fluxed Virasoro-Shapiro amplitude reveals new poles, mass shifts, and tachyonic modes.

  • Factorization constraints: Three-tachyon amplitudes receive no linear corrections in θ, so factorization requires the corresponding three-tachyon amplitude to vanish at that order.This agrees with direct computation for external tachyons on shell.
  • Crossing symmetry and deformed products: R-flux phase factors are encoded in deformed N-products, yet momentum conservation makes all N-tachyon correlators crossing symmetric.The same mechanism applies to four-tachyon amplitudes through a deformed four-product.
  • Fluxed Virasoro-Shapiro amplitude: The conformally invariant fluxed Virasoro-Shapiro amplitude is explicitly crossing symmetric and includes corrections through first order in θ.Its pole structure is used to extract information about the spectrum and couplings.
  • Pole structure: At linear order in flux, higher Regge poles acquire double poles and new poles, while mass renormalization explains the higher-order pole structure.The first new simple pole has the momentum dependence of a new tachyonic mode.
  • Flux-dependent exchanges: In the factorization limit, R-flux corrects graviton and dilaton vertices and yields a graviton pole, whereas H-flux has no linear correction to the graviton pole.The roles of graviton and Kalb–Ramond fluctuations appear exchanged under T-duality.
  • New tachyonic modes: R-flux produces a type-I tachyon with half the negative mass squared of the bosonic ground-state tachyon, alongside type-II modes associated with continuous mass shifts.The two new tachyonic mode types are expected to correspond to zero-order graviton/dilaton modes in R-flux and B-field modes in H-flux.

5 Asymmetric backgrounds and nonassociative geometry

The paper relates non-geometric R-flux to asymmetric conformal field theories and proposes a nonassociative tri-product for closed-string amplitudes. The tri-product requires N-ary definitions, yet its deformation becomes invisible under integration, supporting consistent closed strings on nonassociative backgrounds.

  • Tachyon condensation: For the H-flux background, type I and type II tachyons are associated speculatively with flows toward the su(2)_k WZW model and the trivial vanishing-flux background, respectively.The flat constant-H background is conformal only to linear order, motivating flows driven by relevant tachyonic modes.
  • Asymmetric backgrounds: The R-flux background is conjectured to flow to an asymmetric su(2)_k conformal field theory, whose target-space interpretation differs from the geometric model despite identical CFT data.The two models have the same representations, characters, and modular invariant partition functions, while one is geometric and the other non-geometric.
  • A tri-product: In R-flux scattering, relative phase factors between vertex-ordering channels motivate a generalized Moyal-Weyl product called a tri-product.The construction is intended to make the proposed relation between non-geometric R-flux and nonassociative geometry precise.
  • A tri-product: Inside integrals, the tri-product differs from the ordinary product only by a total derivative, causing its effect to vanish in the relevant scattering amplitudes.The authors therefore argue that closed on-shell strings are blind to this deformation and can be consistently defined on nonassociative backgrounds.
  • A tri-product: The N-products cannot generally be generated by repeated three-products, so a separate deformed product must be specified for every number of functions.For N = 5, the five-fold product differs from an iterated three-product expression.
  • Scope and limitations: The paper limits its mathematical claims because a full analysis of the nonassociative spaces lies beyond its scope, while its flux-dependent methods are reliable only to linear order.The proposed T-duality and tachyon-condensation outcomes are illustrated for a T3 compactification with constant flux.

6 Conclusions

The paper studies closed strings in three-form flux backgrounds and finds evidence linking R-flux, nonassociative geometry, and asymmetric conformal field theories. It derives crossing-symmetric tachyon amplitudes while identifying several open questions and extensions.

  • The analysis treats H- and R-flux configurations to linear order, where they are expected to satisfy the string equations of motion.
  • The R-flux three-point coordinate function involves the Rogers dilogarithm and agrees with a limiting result from the SU(2) WZW model.
  • R-flux tachyon amplitudes acquire permutation-dependent phases that vanish after momentum conservation and can be encoded by a nonassociative tri-product.The tri-product generalizes the Moyal-Weyl product and is compatible on-shell with two-dimensional conformal field theory.
  • A conformally invariant, crossing-symmetric four-tachyon amplitude was derived, with two new tachyon types interpreted through apparent system instabilities.
  • The R-flux model was conjectured to flow toward an asymmetric WZW model, connecting non-geometric flux with asymmetric strings and nonassociative target-space geometry.
  • Future work includes constructing the boundary CFTH, extending the analysis to superstrings, and analyzing logarithmic-CFT-like features in graviton OPEs.The boundary construction could make the Freed-Witten anomaly directly visible because it is linear in H.
  • The paper leaves open whether antisymmetric polarizations represent B-field modes or R-flux fluctuations that promote θabc to a field Θabc(X).Conformal symmetry is expected to yield on-shell equations of motion for Θabc(X).

A.1 T-dual flux backgrounds

This appendix reviews backgrounds with H-, geometric, non-geometric, and R-flux.

  • The appendix surveys H-, geometric, non-geometric, and R-flux backgrounds.

H-flux background

The H-flux background is formulated on a flat rectangular three-torus with constant flux and is consistent only to linear order in H. Viewing the torus as a T2 fibered over an S1 exposes its moduli and parabolic monodromy.

  • The starting geometry is a flat rectangular three-torus with coordinates x1, x2, x3.
  • The torus has radii R1, R2, R3 and carries a constant H-flux.
  • The sigma-model equations hold only at linear order in H, so the configuration is consistent only to that order.
  • Treating the three-torus as a T2 fibered over an S1 defines complex-structure and Kähler moduli for the fiber.
  • The modulus ρ varies with x1 and encodes the background as a parabolic monodromy, realized by an SL(2, Z) transformation around the base circle.

Geometric flux background

A T-duality along x3 exchanges the torus moduli and produces a twisted torus whose metric depends explicitly on the base coordinate. Its globally defined one-forms yield geometric flux and Lie-algebraic consistency conditions.

  • T-duality along the x3 direction exchanges the complex structure and Kähler moduli.
  • The dual metric depends explicitly on x1, requiring a restored periodic identification that defines a twisted torus.
  • A dual globally defined basis of one-forms is introduced to interpret the geometric role of the flux parameter N.
  • Cartan’s structure equation identifies the nonzero connection component with geometric flux.
  • The condition d^2ηa = 0 imposes a Jacobi identity, allowing the geometric fluxes to serve as Lie-algebra structure constants.
  • For compact spaces, the additional condition ωaab = 0 is satisfied by nilpotent algebras, producing nilmanifolds.

Non-geometric flux background

A second T-duality produces a T-fold: locally defined metric and B-field data require transition functions that mix them globally. Its non-geometric character is encoded by Q-flux N.

  • A second T-duality yields a parabolic SL(2, Z)ρ transformation.
  • The resulting metric and B-field are well-defined locally but not globally.
  • Transition functions mix the B-field with the metric, defining a T-fold.
  • The parameter N is identified with the non-geometric flux Qx1x2x3 = N.

R-flux background

A further formal T-duality along the base direction is unavailable to the Buscher rules because that direction lacks an isometry. The resulting locally non-geometric background is characterized by R-flux.

  • The base-direction T-duality is not captured by the Buscher rules because the direction has no isometry.
  • The formally T-dualized background is characterized by R-flux Rx1x2x3 = N.

A.2 The Rogers dilogarithm

The appendix reviews the complex Rogers dilogarithm and a generalized version used to rewrite the four-tachyon amplitude. This reformulation makes its dependence on the cross-ratio and SL(2, C) invariance explicit.

  • The generalized Rogers dilogarithm is used to rewrite the four-tachyon amplitude in terms of the cross-ratio.
  • The rewritten amplitude is manifestly invariant under SL(2, C) transformations.

Definition and fundamental properties

The appendix develops the Rogers dilogarithm, its complex and extended forms, and the five-term relation used to reorganize four-point tachyon correlators. Cross-ratio geometry and flattening conditions make the resulting correlator crossing symmetric and SL(2, C)-invariant.

  • Definition and fundamental properties: For real 0 < x < 1, the Rogers dilogarithm is defined using the Euler dilogarithm and logarithmic terms.
  • Definition and fundamental properties: Analytic continuation to C \ {0, 1} makes the complex Rogers dilogarithm multivalued, requiring the universal cover as its domain.
  • Definition and fundamental properties: The extended Rogers dilogarithm restores a five-term relation with integer parameters constrained by flattening conditions.
  • Definition and fundamental properties: Cross-ratios of five points correspond to the arguments in the five-term relation and admit an interpretation through ideal tetrahedra in hyperbolic three-space.
  • Four-tachyon correlator: Applying the extended dilogarithm formalism to the four-tachyon correlator rewrites its terms using cross-ratios and fixes integration constants through crossing symmetry.
  • Four-tachyon correlator: The four-point tachyon correlator is crossing symmetric and depends only on the SL(2, C)-invariant cross-ratio X = 1 −z.
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