Compact boson

A compact boson takes values on a circle, $X\sim X+2\pi R$. Its states carry quantized momentum $n/R$ and integer winding $wR$.

For a compact boson of radius $R$, momentum number $n$ and winding number $w$ combine into left- and right-moving zero-mode momenta $p_L=n/R+wR$ and $p_R=n/R-wR$ in a common normalization.

In the critical closed bosonic string, compactification of one coordinate gives $M^2=n^2/R^2+w^2{R}^2/{\alpha }^{\prime 2}+2(N+\tilde{N}-2)/{\alpha }^{\prime }$ and $\tilde{N}-N+nw=0$. At oscillator level zero, level matching forces $nw=0$. Thus a pure momentum ground state becomes a massless scalar field at $R=|n|\sqrt{{\alpha }^{\prime }}/2$, while a pure winding ground state does so at $R=2\sqrt{{\alpha }^{\prime }}/|w|$. These are threshold states descended from the bosonic tachyon; they are additional to the neutral graviton, two-form, vectors, dilaton and radius modulus. At the self-dual circle, the four charges $(\pm 2,0),(0,\pm 2)$ contribute to the enhanced massless scalar field spectrum.

For a compact string coordinate of radius $R$, choose $p_L=n/R+wR/{\alpha }^{\prime }$ and $p_R=n/R-wR/{\alpha }^{\prime }$, where the left-moving oscillators are denoted by tildes. The two classical zero-mode Virasoro constraints give $M^2=p_L^2+4\tilde{N}/{\alpha }^{\prime }=p_R^2+4N/{\alpha }^{\prime }$. Their difference yields the displayed shifted closed-string level matching condition. Their average is $M^2=(n/R{)}^2+(wR/{\alpha }^{\prime }{)}^2+2(N+\tilde{N})/{\alpha }^{\prime }$. Classically the center-of-mass momentum can be continuous; the integer momentum number belongs to compact-coordinate quantization. Quantum intercepts shift the mass formula but cancel out of level matching.

The self-dual circle mass equation and closed-string level matching exhaust the possibilities at level sum zero, one or two. Besides the usual neutral $(N,\tilde{N})=(1,1)$ states, charges $(n,w)=(\pm 1,\pm 1)$ supply singly excited chiral states, with the sector fixed by level matching. Ground-state charges $(\pm 2,0)$ and $(0,\pm 2)$ also become massless. Their compact energy cancels the uncompactified tachyon mass. With $d$ physical transverse components, these families have $d^2+4d+4$ independent polarizations.

T-duality identifies a compact boson of radius $R$ with one of reciprocal radius by exchanging momentum and winding numbers. Factors of ${\alpha }^{\prime }$ restore the dimensional relation $R\leftrightarrow {\alpha }^{\prime }/R$.

For a compact string embedding map coordinate, T-duality preserves its left-moving part and reverses its right-moving part. Consequently ${\partial }_{\tau }\tilde{X}={\partial }_{\sigma }X$ and ${\partial }_{\sigma }\tilde{X}={\partial }_{\tau }X$. A Neumann boundary condition becomes a Dirichlet boundary condition and conversely. A D-brane wrapping the circle loses one spatial worldvolume dimension; a D-brane transverse to it gains one in the dual description.

Under T-duality, the positions $y_i$ of D-branes transverse to a circle of radius $R$ become diagonal Wilson lines on branes wrapping the dual circle $R^{\prime }={\alpha }^{\prime }/R$. Their holonomies are $\exp (i2\pi R^{\prime }{A}_i)=\exp (i{y}_i/R)$. An Open string stretched between D-branes with displacement $y_j-y_i+2\pi Rw$ becomes a string with shifted compact momentum $w/R^{\prime }+A_j-A_i$. The stretching contribution and the shifted-momentum contribution to its squared mass are identical.

A local Buscher procedure sets a flat auxiliary connection equal to a coordinate derivative only on a simply connected patch. A full compact-circle Polyakov path integral must also sum flat-connection holonomies, match the period of the dual multiplier, and include momentum and winding modes. A free circle isometry then gives the radius relation $R^{\prime }={\alpha }^{\prime }/R$. Fixed points, noncompact orbits and global identifications require additional analysis; a local spacelike Killing vector does not specify these data.

A circle compactification is self-dual when the radius is unchanged by the T-duality map $R\mapsto {\alpha }^{\prime }/R$. Momentum and winding then enter the bosonic string mass spectrum with equal weights. Special charge sectors can become massless at this radius.

At $R=\sqrt{{\alpha }^{\prime }}$, a chiral compact boson has operator product expansion $Y_L(z)Y_L(w)\sim -{\alpha }^{\prime }\log (z-w)/2$. The currents $J^3=i\partial Y_L/\sqrt{{\alpha }^{\prime }}$ and $J^{\pm }=:e^{\pm 2i{Y}_L/\sqrt{{\alpha }^{\prime }}}:$ have conformal weight one. Contractions give $J^3(z)J^3(w)\sim 1/[2(z-w{)}^2]$, $J^3(z)J^{\pm }(w)\sim \pm J^{\pm }(w)/(z-w)$ and $J^{+}(z)J^{-}(w)\sim 1/(z-w{)}^2+2J^3(w)/(z-w)$. These are the level-one SU(2) Lie algebra currents. The right-moving copy supplies a second factor. Combining a current with a noncompact coordinate derivative gives a massless string vertex operator for each of the six vectors; four are additional to the generic two Cartan vectors.

At the self-dual circle, left and right dimension-one currents generate $SU(2{)}_L\times SU(2{)}_R$. The compact-coordinate vertex $\partial Y_L\bar{\partial }{Y}_R$ is a Cartan–Cartan component of a scalar field in the $(3,3)$ representation. Its expectation value changes the circle radius and leaves $U(1{)}_L\times U(1{)}_R$ unbroken. The four extra charged vectors acquire the displayed mass, and four scalar field polarizations become their longitudinal modes. This realizes the Higgs mechanism geometrically. In a bosonic string the ever-present tachyon and additional scalar field instabilities prevent interpreting the whole family as a stable vacuum.

The Buscher procedure derives T-dual backgrounds by replacing the derivative of an isometric target coordinate with an independent worldsheet one-form and imposing its flatness with a Lagrange multiplier. Integrating out either field yields the original or dual sigma model.

Replace the derivative of an isometric coordinate by an independent one-form $A$ and impose $dA=0$ with a Lagrange multiplier. In signature $(-,+)$ with ${ϵ}^{\tau \sigma }=1$, its equation is $V{A}^{\nu }={ϵ}^{\mu \nu }{\partial }_{\mu }\tilde{z}$. Eliminating $A$ from the full action, including the multiplier term, gives $\frac{1}{2}{V}^{-1}{\partial }_{\mu }\tilde{z}{\partial }^{\mu }\tilde{z}$. Omitting the multiplier term during substitution would give the wrong sign. This derives the block-diagonal metric inversion in the Buscher rules.

The quantum measure in abelian T-duality changes the conventional dilaton by ${\phi }^{\prime }=\phi -\frac{1}{2}\log g_{yy}$ in alpha-prime-one units for a spacelike isometry. The regulated Gaussian integration over the auxiliary gauge field produces this curvature coupling. It is consistent with invariance of the string-frame density $e^{-2\phi }\sqrt{\det g}$ under the local Buscher rules.

The flat polar metric $d{r}^2+r^{2}d{z}^2$ has a circle isometry away from $r=0$. Its local Buscher rules give the displayed dual metric and dilaton. The dual Ricci tensor has components $R_{rr}=-2/r^2$ and $R_{\tilde{z}\tilde{z}}=-2/r^4$, while the dilaton Hessian has components $1/r^2$ and $1/r^4$. Thus the leading metric-dilaton Weyl condition $R_{ab}+2{\nabla }_a{\nabla }_b\Phi =0$ holds despite nonzero curvature. Flat spectators complete a critical target; the scalar anomaly condition also holds because $|\nabla \Phi {|}^2=1/r^2$, $\square \Phi =2/r^2$ and $R=-4/r^2$. The construction is local on $r>0$, excluding the degenerating circle.

The Buscher rules invert the metric component along an Abelian isometry and transform the other metric and B-field components. The Gaussian functional determinant also shifts the dilaton; when the metric is block diagonal with $G_{zz}=V$, $\tilde{\Phi }=\Phi -\frac{1}{2}\log V$.

For a nondegenerate isometry direction $y$ and initially zero two-form, the local T-dual fields in alpha-prime-one units are $g_{yy}^{\prime }=1/g_{yy}$, $g_{yi}^{\prime }=0$, $g_{ij}^{\prime }=g_{ij}-g_{yi}{g}_{yj}/g_{yy}$, $B_{yi}^{\prime }=g_{yi}/g_{yy}$ and $B_{ij}^{\prime }=0$. Gauging the isometry and eliminating its auxiliary connection derives these rules. The spectator metric is a Schur complement. The sign of the mixed two-form follows the multiplier and dual-coordinate orientation convention.