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Closed Bosonic Lightcone Spectrum

Physical scope

The flat closed tensile bosonic string on a periodic cylinder with p^+ nonzero, equal left/right target momentum, D=26, and intercepts a=tilde-a=1.

Statement

Use the target $M = R^{1, 25}$ with $η = diag (- 1, + 1, \dots, + 1)$ , define $X^{\pm} = (X^{0} \pm X^{25}) / \sqrt{2}$ , $p^{\pm} = (p^{0} \pm p^{25}) / \sqrt{2}$ , and oscillator components $α_{n}^{\pm} = (α_{n}^{0} \pm α_{n}^{25}) / \sqrt{2}$ , ${\tilde{α}}_{n}^{\pm} = ({\tilde{α}}_{n}^{0} \pm {\tilde{α}}_{n}^{25}) / \sqrt{2}$ , and let $i = 1, \dots, 24$ denote transverse indices. The closed worldsheet is the dimensionless periodic cylinder $τ \in R$ , $σ \in R / (2 π Z)$ . The target light-cone chart is $p^{+} \neq 0$ with gauge condition $X^{+} (τ, σ) = x^{+} + α^{'} p^{+} τ$ . In this chart the nonzero-mode left and right Virasoro constraints determine $α_{n}^{-}$ and ${\tilde{α}}_{n}^{-}$ from the transverse modes, while the gauge sets $α_{n}^{+} = {\tilde{α}}_{n}^{+} = 0$ for $n \neq 0$ . Define the weighted transverse levels $N = \sum_{n > 0} \sum_{i = 1}^{24} α_{- n}^{i} α_{n}^{i}$ and $\tilde{N} = \sum_{n > 0} \sum_{i = 1}^{24} {\tilde{α}}_{- n}^{i} {\tilde{α}}_{n}^{i}$ ; a monomial with occupations $r_{n, i}$ , ${\tilde{r}}_{n, i}$ has levels $\sum_{n, i} n r_{n, i}$ , $\sum_{n, i} n {\tilde{r}}_{n, i}$ . The intercept convention is $L_{0}^{m} - a = 0$ and ${\tilde{L}}_{0}^{m} - \tilde{a} = 0$ . With $M^{2} = - p_{μ} p^{μ}$ , at $D = 26$ and $a = \tilde{a} = 1$ the physical Fock space has 24 left and 24 right transverse oscillator families, obeys level matching $N = \tilde{N}$ , and satisfies $α^{'} M^{2} / 4 = N - 1 = \tilde{N} - 1$ .

Statement references

David Tong, Lectures on String Theory, arXiv:0908.0333v3, full 218-page lecture notes, §§2.2–2.3.3 (PDF file pp.39–52; printed pp.32–45); printed pp.32–45); printed pp.32–45); printed pp.32–45); printed pp.32–45); printed pp.32–45); printed pp.32–45); left/right modes, level matching, and closed mass formula.

Facts & Assumptions

[F1]

Bosonic Matter Oscillator Modes fixes $p \in R^{1, D - 1}$ with dimension $L^{- 1}$ , dimensionless independent left/right modes, $α_{0} = {\tilde{α}}_{0} = \sqrt{α^{'} / 2} p$ , and the weighted transverse levels $N = \sum_{n > 0, i} α_{- n}^{i} α_{n}^{i}$ , $\tilde{N} = \sum_{n > 0, i} {\tilde{α}}_{- n}^{i} {\tilde{α}}_{n}^{i}$ .

[F2]

Critical Dimension Flat Bosonic String gives $D = 26$ and $a = 1$ in the standard flat tensile model.

[F3]

The draft physics theorem Closed Classical Mode Expansion locally proves the $C^{2}$ periodic-cylinder left/right mode expansions, common target zero mode, and chiral classical constraints used in the gauge reduction.

[F4]

Conformal Gauge and Residual Symmetry defines the local world-sheet conformal gauge, $σ^{\pm}$ coordinates, and residual world-sheet conformal maps. It does not define the target light-cone coordinates or target gauge; those are stated in this item's Statement.

[A1]

Work at fixed momentum with $p^{+} \neq 0$ and the standard left/right transverse Fock vacua; use the target chart and gauge conditions stated above.

[A2]

In this light-cone quantization, impose the normal-ordered left and right oscillator forms of the nonzero-mode Virasoro constraints from [F3], solve them for the two longitudinal oscillator families, and impose the zero-mode conditions $(L_{0}^{m} - a) ∣ ψ ⟩ = ({\tilde{L}}_{0}^{m} - \tilde{a}) ∣ ψ ⟩ = 0$ . This is the adopted gauge-fixed quantization prescription.

Given: Assume a flat closed tensile string on the periodic cylinder, with equal left/right target momentum and intercepts $a = \tilde{a} = 1$ .

Proof

1.1givenF1F3F4A1A2algebra

The gauge gives $α_{n}^{+} = {\tilde{α}}_{n}^{+} = 0$ for $n \neq 0$ and $α_{0}^{+} = {\tilde{α}}_{0}^{+} = \sqrt{α^{'} / 2} p^{+} \neq 0$ . With $η_{+ -} = - 1$ , [F3] and [A2] give $0 = L_{n}^{m} = - α_{0}^{+} α_{n}^{-} + L_{n}^{⊥}$ and $0 = {\tilde{L}}_{n}^{m} = - {\tilde{α}}_{0}^{+} {\tilde{α}}_{n}^{-} + {\tilde{L}}_{n}^{⊥}$ , where $L_{n}^{⊥} = \frac{1}{2} \sum_{r \in Z} \sum_{i} : α_{n - r}^{i} α_{r}^{i} :$ and likewise for tilded modes. Thus each longitudinal oscillator is solved by division by its nonzero plus zero mode. The separate transverse commutators are $[α_{m}^{i}, α_{n}^{j}] = m δ^{i j} δ_{m + n, 0}$ and the same for tilded modes, while mixed commutators vanish; hence the finite-particle left/right transverse Fock forms are positive.

2.1givenF1F2A1A2step 1.1algebra

Each chiral zero mode gives $L_{0}^{m} = \frac{1}{2} α_{0}^{2} + N = α^{'} p^{2} / 4 + N$ and ${\tilde{L}}_{0}^{m} = α^{'} p^{2} / 4 + \tilde{N}$ . The zero-mode conditions in [A2] and intercepts $a = \tilde{a} = 1$ from [F2] are $α^{'} p^{2} / 4 + N - 1 = 0$ and $α^{'} p^{2} / 4 + \tilde{N} - 1 = 0$ . Subtracting gives $N = \tilde{N}$ ; using $M^{2} = - p^{2}$ gives $α^{'} M^{2} / 4 = N - 1 = \tilde{N} - 1$ . The weighted levels have the nonnegative integer eigenvalues stated in [F1].

3.1A1A2step 1.1step 2.1algebra∎

Let $T_{N} (p)$ and ${\tilde{T}}_{\tilde{N}} (p)$ be the spans of transverse occupation monomials of weighted levels $N$ and $\tilde{N}$ in the two positive 24-family Fock spaces. At the fixed momentum in [A1], the physical space is precisely the algebraic sum of $T_{N} (p) \otimes {\tilde{T}}_{\tilde{N}} (p)$ over nonnegative integers satisfying both $α^{'} p^{2} / 4 + N - 1 = 0$ and $α^{'} p^{2} / 4 + \tilde{N} - 1 = 0$ . Necessity follows from step 2.1; conversely these scalar equations and the longitudinal solution of step 1.1 fulfill every condition of [A2]. Thus the space is zero if $1 - α^{'} p^{2} / 4$ is not a nonnegative integer, and otherwise it is the single level-matched summand with $N = \tilde{N} = 1 - α^{'} p^{2} / 4$ . Its nonzero vectors have positive oscillator norm and its masses are exactly the stated spectrum. No semi-relative or absolute BRST cohomology is identified here.

Depends on

Dependency tree · 2 levels

Sources

David Tong, Lectures on String Theory, arXiv:0908.0333v3, full 218-page lecture notes