Compact Free String Circle Spectrum

Fix a free closed-string coordinate $X^1$ with target circle $S_R^1:=\mathbb{R}\mathrm{/}(2\pi R\mathbb{Z})$, $R>0$, on the Lorentzian cylinder $\mathrm{\Sigma }={\mathbb{R}}_{\tau }\times (\mathbb{R}\mathrm{/}2\pi \mathbb{Z}{)}_{\sigma }$, with dimensionless coordinates $\tau ,\sigma$. The compact center-of-mass coordinate is $x_0\in S_R^1$. With $\mathrm{\hslash }=1$, take its Hilbert space to be $L^2(S_R^1,d{x}_0\mathrm{/}(2\pi R))$ and its momentum operator to be $P=-id\mathrm{/}d{x}_0$ on the periodic Sobolev domain $H^1(S_R^1)$. The normalized Fourier functions ${\psi }_n(x_0)=e^{in{x}_0\mathrm{/}R}$, $n\in \mathbb{Z}$, satisfy $P{\psi }_n=(n\mathrm{/}R){\psi }_n$; thus the momentum eigenvalue is $p=n\mathrm{/}R$.

For a $C^2$ closed-string map $X^1:\mathrm{\Sigma }\to S_R^1$, define its winding number $w\in \mathbb{Z}$ by lifting $\sigma \mapsto X^1(\tau ,\sigma )$ to ${\tilde{X}}_{\tau }^1:\mathbb{R}\to \mathbb{R}$ and requiring ${\tilde{X}}_{\tau }^1(\sigma +2\pi )-{\tilde{X}}_{\tau }^1(\sigma )=2\pi Rw$. Choose a real representative ${\tilde{x}}_0\in \mathbb{R}$ of $x_0\in S_R^1$. The zero-mode lift is

where ${\alpha }^{\mathrm{\prime }}>0$ has dimension length squared. This profile has center-of-mass momentum $n\mathrm{/}R$ and winding $w$.

Use the mode convention of the supplied closed-string expansion: modes ${\alpha }_m$ multiply the $\tau -\sigma$ (right-moving) coordinate and modes ${\tilde{\alpha }}_m$ multiply the $\tau +\sigma$ (left-moving) coordinate. Thus ${\alpha }_0=\sqrt{{\alpha }^{\mathrm{\prime }}\mathrm{/}2}{p}_R$ and ${\tilde{\alpha }}_0=\sqrt{{\alpha }^{\mathrm{\prime }}\mathrm{/}2}{p}_L$. In the selected free compact-boson quantum sector, adopt the oscillator algebra $[{\alpha }_m,{\alpha }_n]=m{\delta }_{m+n,0}$, $[{\tilde{\alpha }}_m,{\tilde{\alpha }}_n]=m{\delta }_{m+n,0}$, and $[{\alpha }_m,{\tilde{\alpha }}_n]=0$, with ${\alpha }_k$ and ${\tilde{\alpha }}_k$ annihilating the vacuum for every $k\ge 1$. To specify the Hilbert-space and adjoint data, put ${\mathbb{Z}}_{>0}:=\{1,2,\dots \}$ and ${\mathbb{N}}_0:=\{0,1,2,\dots \}$, and let $\mathcal{K}:=\{\mathbf{r}:{\mathbb{Z}}_{>0}\to {\mathbb{N}}_0:\{k:r_k\mathrm{\ne }0\}\text{ is finite}\}$ be the set of finite-support nonnegative integer occupation sequences, and let $\mathcal{I}={\mathbb{Z}}^2\times \mathcal{K}\times \mathcal{K}$. Define ${\mathcal{H}}_R:={\mathrm{\ell }}^2(\mathcal{I}),⟨c,d⟩={\sum }_{I\in \mathcal{I}}\overline{c_I}{d}_I\mathrm{.}$ This inner product is antilinear in the first argument. Write $e_{n,w;{\mathbf{r}}_L,{\mathbf{r}}_R}$ for the coordinate unit vectors, and ${\mathcal{D}}_R$ for their finite linear span. These vectors are an orthonormal basis by the definition of ${\mathrm{\ell }}^2$; the Hilbert space is their completion. Identify $\mathrm{\mid }n,w⟩=e_{n,w;0,0}$, with the momentum wavefunction ${\psi }_n$ and winding label $w$ already specified. Thus different momentum or winding sectors are orthogonal and every oscillator vacuum has norm one.

Let ${\mathbf{e}}_k$ be the occupation sequence with entry one at $k$ and zero elsewhere. On ${\mathcal{D}}_R$ define the nonzero modes by $\begin{array}{rlrl}{\displaystyle {\alpha }_{-k}{e}_{n,w;{\mathbf{r}}_L,{\mathbf{r}}_R}}& {\displaystyle =\sqrt{k(r_{R,k}+1)}{e}_{n,w;{\mathbf{r}}_L,{\mathbf{r}}_R+{\mathbf{e}}_k},}& {\displaystyle {\alpha }_k{e}_{n,w;{\mathbf{r}}_L,{\mathbf{r}}_R}}& {\displaystyle =\sqrt{k{r}_{R,k}}{e}_{n,w;{\mathbf{r}}_L,{\mathbf{r}}_R-{\mathbf{e}}_k},}\\ {\displaystyle {\tilde{\alpha }}_{-k}{e}_{n,w;{\mathbf{r}}_L,{\mathbf{r}}_R}}& {\displaystyle =\sqrt{k(r_{L,k}+1)}{e}_{n,w;{\mathbf{r}}_L+{\mathbf{e}}_k,{\mathbf{r}}_R},}& {\displaystyle {\tilde{\alpha }}_k{e}_{n,w;{\mathbf{r}}_L,{\mathbf{r}}_R}}& {\displaystyle =\sqrt{k{r}_{L,k}}{e}_{n,w;{\mathbf{r}}_L-{\mathbf{e}}_k,{\mathbf{r}}_R},k\ge 1.}\end{array}$ A lowering expression is zero when its indicated occupation is zero. Take their maximal closed extensions in ${\mathcal{H}}_R$. For each fixed $k$, both right-moving operators have domain $\{c:{\sum }_{I}k{r}_{R,k}\mathrm{\mid }{c}_I{\mathrm{\mid }}^2<\mathrm{\infty }\}$ and both left-moving operators have the corresponding domain with $r_{L,k}$. The creation-domain condition with $k(r_{R,k}+1)$, or $k(r_{L,k}+1)$, is equivalent because $c\in {\mathrm{\ell }}^2$ and $k$ is fixed. On these domains their Hilbert adjoints satisfy ${\alpha }_k^{†}={\alpha }_{-k}$ and ${\tilde{\alpha }}_k^{†}={\tilde{\alpha }}_{-k}$. The zero modes act diagonally by the real eigenvalues displayed above, with their maximal square-summability domains. The space ${\mathcal{D}}_R$ is a common invariant core for these oscillator modes. This is an explicit local Hilbert realization of the source's free spacelike oscillator convention, not an assertion about a BRST quotient or a full string Hilbert space. For finite-support sequences ${\mathbf{r}}_L=(r_{L,k}{)}_{k\ge 1},{\mathbf{r}}_R=(r_{R,k}{)}_{k\ge 1}\in \mathcal{K}$, define the normalized occupation state

Here $\mathrm{\mid }n,w⟩$ is the zero-mode state with center-of-mass wavefunction ${\psi }_n$ and winding sector $w$. Thus ${\tilde{\alpha }}_{-k}$ is a left-moving creation mode and ${\alpha }_{-k}$ a right-moving creation mode. Define the compact-coordinate oscillator grades $N_L:={\sum }_{k\ge 1}k{r}_{L,k}$ and $N_R:={\sum }_{k\ge 1}k{r}_{R,k}$. If a fixed spectator sector is included, let its prescribed nonnegative grades be $N_{\mathrm{s}\mathrm{p}},{\bar{N}}_{\mathrm{s}\mathrm{p}}$ and put $N:=N_L+N_{\mathrm{s}\mathrm{p}}$ and $\bar{N}:=N_R+{\bar{N}}_{\mathrm{s}\mathrm{p}}$; for the compact boson alone take both spectator grades to be zero. The conformal-weight labels in this selected free sector are $h={\alpha }^{\mathrm{\prime }}{p}_L^2\mathrm{/}4+N$ and $\bar{h}={\alpha }^{\mathrm{\prime }}{p}_R^2\mathrm{/}4+\bar{N}$. Their difference is $h-\bar{h}=nw+N-\bar{N}$, so level matching is $nw+N-\bar{N}=0$ when the chosen other sectors are included consistently. This is a definition of the kinematic free-circle graded spectrum; it does not assert BRST/GSO constraints, operator-product or cocycle equivalence, partition-function or sewing equivalence, or an interacting-worldsheet equivalence.

Flat Bosonic String Model

String Tension Alpha Prime

Closed Classical Mode Expansion

The oscillator grade is the free-sector grading variable; this carrier does not claim that the displayed labels alone impose BRST, GSO, or full physical-state constraints. The cited sources use different $\alpha \mathrm{/}\tilde{\alpha }$ labels for their chiral sectors; this definition follows the local $\tau -\sigma$/$\tau +\sigma$ convention of Closed Classical Mode Expansion, so $\alpha$ is right-moving and $\tilde{\alpha }$ is left-moving. The zero-mode weights are unchanged by relabeling the two conventions.