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Free Closed String Graded Spectrum Under Radius Inversion

Physical scope

Exact algebraic duality of the flat free closed compact-boson graded oscillator spectrum, weights, and level matching only; it excludes full CFT/operator, cocycle, partition-function, sewing, curved-background, S-duality, and U-duality equivalence.

Statement

For the free compact-boson sector defined in the preceding item, set $R^{'} = α^{'} / R$ , $n^{'} = w$ , and $w^{'} = n$ . Then $p_{L}^{'} = p_{L}$ and $p_{R}^{'} = - p_{R}$ ; in the supplied $τ - σ$ / $τ + σ$ convention, reflect the right-moving compact-coordinate modes by $α_{m} \mapsto - α_{m}$ for $m \in Z$ and leave the left-moving modes ${\tilde{α}}_{m}$ fixed. This induces a bijection on occupation states and preserves $N, \overset{ˉ}{N}$ , so $h$ and $\overset{ˉ}{h}$ are unchanged. Also $n^{'} w^{'} = n w$ , hence level matching $n w + N - \overset{ˉ}{N} = 0$ is unchanged. Thus the perturbative free-circle graded oscillator state-space weights and level matching are preserved under radius inversion and momentum-winding exchange. This calculation does not establish vertex-operator locality or OPE equivalence, lattice cocycles, partition-function or sewing equivalence, or full worldsheet CFT/operator equivalence.

Statement references

Polchinski, Chaudhuri, and Johnson, Lecture II §2.1

Giveon, Porrati, and Rabinovici, §2.2

Facts & Assumptions

Given: $R > 0$ , $α^{'} > 0$ , integers $n, w$ , fixed left/right grades $N, \overset{ˉ}{N}$ , and the free-circle spectrum conventions.

[F1]

Compact Free String Circle Spectrum: $p_{L} = n / R + w R / α^{'}$ , $p_{R} = n / R - w R / α^{'}$ , $h = α^{'} p_{L}^{2} / 4 + N$ , $\overset{ˉ}{h} = α^{'} p_{R}^{2} / 4 + \overset{ˉ}{N}$ , and $h - \overset{ˉ}{h} = n w + N - \overset{ˉ}{N}$ . In its mode convention, $α_{m}$ are the right-moving $τ - σ$ modes and ${\tilde{α}}_{m}$ the left-moving $τ + σ$ modes; the occupation labels and their grades are defined there.

Proof

1.1givenF1algebra

Define $R^{'} = α^{'} / R$ , $n^{'} = w$ , and $w^{'} = n$ . Since $R, α^{'} > 0$ , also $R^{'} > 0$ . Substitution into [F1] gives $p_{L}^{'} = n^{'} / R^{'} + w^{'} R^{'} / α^{'} = w R / α^{'} + n / R = p_{L}$ and $p_{R}^{'} = n^{'} / R^{'} - w^{'} R^{'} / α^{'} = w R / α^{'} - n / R = - p_{R}$ .

2.1givenF1step 1.1constructalgebra

Let $H_{R}$ be [F1]'s Hilbert space at radius $R$ and define $H_{R^{'}}$ by the same construction with $R$ replaced by $R^{'}$ . On basis states define $T_{R} ∣ n, w; r_{L}, r_{R} ⟩_{R} : = (- 1)^{\sum_{k \geq 1} r_{R, k}} ∣ w, n; r_{L}, r_{R} ⟩_{R^{'}}$ . This signed permutation maps the orthonormal basis of $H_{R}$ onto that of $H_{R^{'}}$ . Its inverse is the same construction $T_{R^{'}}$ , so it is unitary and $T_{R^{'}} T_{R} = I$ . On the common occupation labels this gives $T_{R} α_{- k, R} = - α_{- k, R^{'}} T_{R}$ and $T_{R} {\tilde{α}}_{- k, R} = {\tilde{α}}_{- k, R^{'}} T_{R}$ for every $k \geq 1$ ; taking adjoints gives the same signs on annihilation modes. By [F1], $α_{0, R} = \sqrt{α^{'} / 2} p_{R}$ and ${\tilde{α}}_{0, R} = \sqrt{α^{'} / 2} p_{L}$ , so step 1.1 gives $T_{R} α_{0, R} = - α_{0, R^{'}} T_{R}$ and $T_{R} {\tilde{α}}_{0, R} = {\tilde{α}}_{0, R^{'}} T_{R}$ . It fixes the left/right occupation sequences and spectator labels, hence preserves $N$ and $\overset{ˉ}{N}$ .

3.1givenF1step 1.1step 2.1algebra∎

By step 1.1, $p_{L}^{' 2} = p_{L}^{2}$ and $p_{R}^{' 2} = p_{R}^{2}$ ; by step 2.1, $N^{'} = N$ and ${\overset{ˉ}{N}}^{'} = \overset{ˉ}{N}$ . Thus both weights in [F1] are preserved. The exchanged integers satisfy $n^{'} w^{'} = w n = n w$ , so the formula in [F1] gives the same level-matching condition on both sides.

Remarks

The radius/momentum/winding map is an involution on this kinematic sector. The cited sources use different names for their left/right oscillator modes; here the reflection is assigned by the explicitly stated $τ - σ$ / $τ + σ$ convention of [F1]. No operator-product or global worldsheet conclusion follows from this graded-state calculation.

Depends on

Compact Free String Circle Spectrum