[Paper Review] Noncommutative Open String: Neutral and Charged

Chong‐Sun Chu|ArXiv.org|Jan 23, 2000

Noncommutative and Quantum Gravity Theories13 references3 citations

TL;DR

This paper demonstrates that open strings ending on D-branes in a background NS-NS B-field exhibit noncommutative endpoints due to the modified boundary conditions, with the noncommutativity localized at the string ends and determined by the field strength. The same formalism is extended to charged open strings in background gauge fields, showing consistent noncommutativity, and suggests that RR backgrounds may underlie 'spacetime' noncommutativity distinct from worldvolume effects.

ABSTRACT

We review the quantization of open string in NS-NS background and demonstrate that its endpoint becomes noncommutative. The same approach allows us to determine the noncommutativity that arises for a charged open string in background gauge fields. While NS-NS background is relevant for ``worldvolume'' noncommutativity, a simple argument suggests that RR background is likely to be relevant for ``spacetime'' noncommutativity.

Motivation & Objective

• To derive the noncommutative structure of open string endpoints in a background NS-NS B-field using Hamiltonian quantization.
• To generalize the analysis to charged open strings in background U(1) gauge fields, showing consistent noncommutativity.
• To distinguish 'worldvolume' noncommutativity (from D-brane dynamics) from 'spacetime' noncommutativity (from quantum gravity effects).
• To explore the role of RR backgrounds in potentially generating spacetime noncommutativity via S-duality.
• To provide a unified framework for quantizing neutral and charged open strings in background fields, avoiding complications of worldsheet perturbation.

Proposed method

• Use of the Hamiltonian approach to quantize open strings in a constant NS-NS B-field, treating boundary conditions as constraints.
• Derivation of modified boundary conditions involving the field strength tensor $\mathcal{F}_{ij} = B_{ij} - F_{ij}$, which encode the noncommutative structure.
• Computation of equal-time commutators between string coordinates at the endpoints using mode expansions and constraints.
• Application of summation identities involving $\sum_{n} \frac{1}{n - \epsilon} \cos((n - \epsilon)\sigma + \gamma)$ to evaluate nontrivial commutators at $\sigma = 0, \pi$.
• Use of complex analysis and residue theorems (e.g., $\sum (-1)^n h(n)$) to evaluate divergent sums and show vanishing commutators away from endpoints.
• Consistency check via comparison with Dirac constrained quantization and earlier results, confirming endpoint-localized noncommutativity.

Experimental results

Research questions

• RQ1 How does the presence of a background NS-NS B-field lead to noncommutativity in the endpoints of an open string?
• RQ2 Can the same quantization framework be applied to charged open strings in background U(1) gauge fields to derive noncommutativity?
• RQ3 What is the physical distinction between 'worldvolume' noncommutativity (on D-branes) and 'spacetime' noncommutativity (from quantum gravity)?
• RQ4 Why might RR backgrounds be more relevant for spacetime noncommutativity than NS-NS backgrounds?
• RQ5 How does S-duality relate noncommutative D-branes to noncommutative worldsheet theories in RR backgrounds?

Key findings

• The string endpoints become noncommutative, with the commutator $[X_+(\tau,0), X_-(\tau,0)] = \frac{\pi\alpha}{1+\alpha^2}$, localized at $\sigma = 0$.
• At $\sigma = \pi$, the noncommutativity is $[X_+(\tau,\pi), X_-(\tau,\pi)] = \frac{\pi\beta}{1+\beta^2}$, showing field strength dependence at the other end.
• For $\sigma, \sigma'$ not both 0 or $\pi$, the commutator $[X_+(\tau,\sigma), X_-(\tau,\sigma')] = 0$, confirming noncommutativity is strictly endpoint-localized.
• The noncommutativity structure is independent of the probe string, confirming it is a property of the D-brane itself.
• The same formalism applies uniformly to both neutral and charged open strings, avoiding complications of worldsheet perturbation or delta-function actions.
• The results support the conjecture that RR backgrounds may be relevant for 'spacetime' noncommutativity, as suggested by S-duality mapping NS-NS flux to RR flux.