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From closed to open strings: the tensionless route in Kalb-Ramond background and noncommutativity

Sarthak Duary, Sourav Maji

TL;DR

The paper studies tensionless bosonic strings in a constant Kalb–Ramond background and shows a universal closed→open transition on the worldsheet as the tension vanishes, realized through a Bogoliubov transformation and a background-induced gluing matrix that yields a squeezed boundary state. A toroidal extension demonstrates that worldsheet Bose–Einstein condensation persists in the presence of the $B$-field, while a unified symplectic treatment derives the boundary noncommutativity parameter in both tensile and tensionless regimes, with the tensionless limit yielding $oldsymbol{ ext{Θ}}=(oldsymbol B)^{-1}$ intrinsically from the Carrollian worldsheet. The results knit a coherent picture in which open-string behavior, boundary noncommutativity, and generalized vacua emerge from null-string geometry in a $B$-field background, and they outline rich directions for flux backgrounds, supersymmetry, slowly varying backgrounds, and covariant formulations. The analysis connects the Seiberg–Witten noncommutativity in tensile strings to an intrinsic tensionless analogue, highlighting a deep geometric unity between worldsheet Carrollian dynamics and open-string boundary physics.

Abstract

We study tensionless bosonic strings propagating in the presence of a constant Kalb-Ramond background and show how closed strings undergo a transition into open strings in this regime. The tensionless (Carrollian) limit induces a universal gluing between the worldsheet oscillators, which we generalize to include the effects of a constant $B$-field. We derive the modified mixed boundary conditions, construct the corresponding gluing matrix and obtain the generalized induced vacuum as a squeezed boundary state. This vacuum continuously emerges from the closed string vacuum through a Bogoliubov transformation, providing a precise realization of the closed-to-open string transition. We further extend the analysis to toroidal compactifications, and show that the mechanism of worldsheet Bose-Einstein condensation persists unaltered. In the second part of the paper, we give a unified symplectic derivation of the open string noncommutative parameter. We show that the boundary symplectic form alone determines the noncommutative geometry, and that in the tensionless limit the $B$-field term becomes the unique surviving contribution. Its inverse produces a noncommutative parameter similar to the Seiberg-Witten result, but arising intrinsically from the Carrollian worldsheet dynamics. Our results establish a coherent picture in which the emergence of open strings, boundary noncommutativity, and generalized vacuum structure all arise naturally from the worldsheet geometry of null strings in a $B$-field background.

From closed to open strings: the tensionless route in Kalb-Ramond background and noncommutativity

TL;DR

The paper studies tensionless bosonic strings in a constant Kalb–Ramond background and shows a universal closed→open transition on the worldsheet as the tension vanishes, realized through a Bogoliubov transformation and a background-induced gluing matrix that yields a squeezed boundary state. A toroidal extension demonstrates that worldsheet Bose–Einstein condensation persists in the presence of the -field, while a unified symplectic treatment derives the boundary noncommutativity parameter in both tensile and tensionless regimes, with the tensionless limit yielding intrinsically from the Carrollian worldsheet. The results knit a coherent picture in which open-string behavior, boundary noncommutativity, and generalized vacua emerge from null-string geometry in a -field background, and they outline rich directions for flux backgrounds, supersymmetry, slowly varying backgrounds, and covariant formulations. The analysis connects the Seiberg–Witten noncommutativity in tensile strings to an intrinsic tensionless analogue, highlighting a deep geometric unity between worldsheet Carrollian dynamics and open-string boundary physics.

Abstract

We study tensionless bosonic strings propagating in the presence of a constant Kalb-Ramond background and show how closed strings undergo a transition into open strings in this regime. The tensionless (Carrollian) limit induces a universal gluing between the worldsheet oscillators, which we generalize to include the effects of a constant -field. We derive the modified mixed boundary conditions, construct the corresponding gluing matrix and obtain the generalized induced vacuum as a squeezed boundary state. This vacuum continuously emerges from the closed string vacuum through a Bogoliubov transformation, providing a precise realization of the closed-to-open string transition. We further extend the analysis to toroidal compactifications, and show that the mechanism of worldsheet Bose-Einstein condensation persists unaltered. In the second part of the paper, we give a unified symplectic derivation of the open string noncommutative parameter. We show that the boundary symplectic form alone determines the noncommutative geometry, and that in the tensionless limit the -field term becomes the unique surviving contribution. Its inverse produces a noncommutative parameter similar to the Seiberg-Witten result, but arising intrinsically from the Carrollian worldsheet dynamics. Our results establish a coherent picture in which the emergence of open strings, boundary noncommutativity, and generalized vacuum structure all arise naturally from the worldsheet geometry of null strings in a -field background.

Paper Structure

This paper contains 52 sections, 288 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Tensionless strings in the presence of a uniform Kalb--Ramond field
  3. Generalized induced vacuum condition in the presence of a uniform Kalb--Ramond field
  4. Kinetic term.
  5. $\mathfrak B$-term.
  6. Total variation.
  7. Gluing matrix in the presence of a uniform Kalb--Ramond field
  8. Squeezed-state boundary state and gluing condition
  9. Bogoliubov generator
  10. Physical interpretation of the gluing condition
  11. Tensionless strings with a background Kalb--Ramond field compactified on a torus $T^{d}$
  12. Boundary conditions
  13. Gluing matrix
  14. Bogoliubov generator
  15. Bose--Einstein condensation on the worldsheet
  16. ...and 37 more sections

Figures (1)

  • Figure 1: Logical flow of the paper: The first part (\ref{['reviewbfield', 'sec:boundary_conditions', 'torustd']}) develops the physics of open string from a closed string theory in the tensionless regime in a Kalb--Ramond background, while the second part (\ref{['sec:symplectic_tensile', 'sec:symplectic_tensionless']}) presents the symplectic analysis and noncommutative structure. Appendices provide calculations for the second part \ref{['sec:symplectic_tensile', 'sec:symplectic_tensionless']}.