Worldsheet

A string worldsheet $\Sigma$ is the two-dimensional surface swept out by a string in spacetime. Local coordinates are commonly denoted ${\sigma }^{\alpha }=(\tau ,\sigma )$.

In flat noncompact spacetime a fixed circular loop's cylindrical worldsheet is not a stationary point of the Nambu–Goto action. A compact coordinate changes the possibilities. The displayed embedding in ${\mathbb{R}}^{2,1}\times S_R^1$ has unit winding, satisfies the wave equation, and obeys $\dot{X}\cdot X^{\prime }=0$ and ${\dot{X}}^2+X^{\prime 2}=0$. Its projection is a circle with static shape, while its motion in the compact direction supplies the second chiral null tangent. The induced worldsheet metric is $2R^2\mathrm{diag}(-1,1)$, so the full worldsheet is nondegenerate.

On a spacelike constant-target-time slice, integrate the induced spatial line element to obtain the geometric string length. It depends on the embedding and is distinct from the fundamental string length ${\ell }_s=\sqrt{{\alpha }^{\prime }}$. For transverse local motion, energy additionally includes the local Lorentz factor; the geometric length alone does not determine the total energy.

The intrinsic Lorentzian metric in the Polyakov action is independent of the string embedding map. Its equation relates it conformally to the induced worldsheet metric. A Weyl transformation changes the intrinsic metric without changing the classical action.

The area of a string worldsheet is computed from its induced worldsheet metric. The Nambu–Goto action is minus string tension times Lorentzian worldsheet area.

The string embedding map $X:\Sigma \to \mathcal{M}$ assigns each worldsheet point its position in target spacetime.

A worldsheet zero mode is a field component annihilated by the worldsheet kinetic operator. For a noncompact string embedding coordinate, its constant zero mode is the target-space center-of-mass position; integrating over it enforces target-space momentum conservation.

The induced worldsheet metric is the pullback of the target-space metric:

The Nambu–Goto actionis minus the string tension times the Lorentzian area of the worldsheet.

A circular closed string contracts and re-expands with $X^0=Rt$ and $X^1+i{X}^2=R\cos t{e}^{i\sigma }$. Its proper length is $2\pi R|\cos t|$, while its energy remains $2\pi TR$. At collapse the induced worldsheet metric degenerates, but the Nambu-Goto phase-space action supplies a regular continuation.

For a nondegenerate induced worldsheet metric $g_{ab}={\partial }_{a}X\cdot {\partial }_{b}X$, variation gives ${\partial }_a(\sqrt{-g}{g}^{ab}{\partial }_b{X}^m)=0$. In conformal gauge this is the two-dimensional wave equation, supplemented by the Virasoro constraints.

The first-order string action isThe multipliers impose the two Virasoro constraints. The canonical equations give a spatial momentum flux $J_m=e{T}^2{X}_m^{\prime }+u{P}_m$.

Translations and Lorentz transformations give Noether charges ${\mathcal{P}}_m=\int P_{m}d\sigma$ and ${\mathcal{J}}_{mn}=\int (X_m{P}_n-X_n{P}_m)d\sigma$. Their time derivatives are boundary fluxes, which vanish for a closed string or a free-ended open string.

The two constraints are $P^2+T^2{X}^{\prime 2}=0$ and $P\cdot X^{\prime }=0$. They are first-class constraints associated with worldsheet diffeomorphisms. Their Lagrange multipliers are arbitrary gauge functions rather than propagating fields.

For a closed string, the currents $J_{\pm }=P\pm T{X}^{\prime }$ have vanishing mixed Poisson brackets. The densities ${\mathcal{H}}_{\pm }=J_{\pm }^2/(4T)$ therefore generate commuting constraint algebras. With opposite Fourier orientations their modes satisfy $\{L_m,L_n\}=-i(m-n)L_{m+n}$ and the identical relation for ${\tilde{L}}_n$. This is a direct sum of Lie algebras consisting of two Witt algebras.

A straight rigidly rotating classical string has speed of light at a free endpoint. Its energy and angular momentum satisfy a Regge trajectory. A segment with one anchored and one free end has $J=E^2/(\pi T)$, twice the slope of the string with two free ends.

With string tension $T=1/(2\pi {\alpha }^{\prime })$, the free-ended open string $X^0=a\tau$, $X^1+i{X}^2=a\cos \sigma e^{i\tau }$, $0\le \sigma \le \pi$, is straight at every target time. Its Nambu–Goto equations of motion and Virasoro constraints hold, and its ends move at light speed. Integrating the target-space Noether charges of a string gives $M=\pi Ta$ and $J=\pi T{a}^2/2$. Thus $J={\alpha }^{\prime }{M}^2$. Extending the same map to $0\le \sigma \le 2\pi$ describes a double-covered folded closed string, with twice the mass and twice the angular momentum, hence $J={\alpha }^{\prime }{M}^2/2$. The different slopes record the different boundary topology.

If $A(t)$ and $B(t)$ are perpendicular four-dimensional spatial vectors of equal fixed magnitude, every constant-time slice is a planar circle. Opposite rotations in two orthogonal coordinate planes can produce a Nambu–Goto action solution with constant radius and a rotating plane. The induced worldsheet metric can be constant even though the embedding moves. Rotating Cartesian axes clarify its shape without removing its kinetic target-space energy.

A Nambu–Goto action solution with $X^0=t$ is $\mathbf{X}=L\sin (\sigma /L)(\cos (t/L),\sin (t/L))$, for $0\le \sigma \le \pi L/2$. It solves the wave equation, has a fixed endpoint at the origin, and obeys the free-end string boundary condition at the other end. Its proper length is $L$ and its free endpoint has angular speed $1/L$ and spatial speed one.

A two-dimensional conformal field theory is invariant under local holomorphic and antiholomorphic conformal transformations.

For primary operators of holomorphic weights $h_i$ and a Möbius transformation invariant vacuum, infinitesimal translations, dilations and special conformal transformations give ${\sum }_i{\partial }_{i}G=0$, ${\sum }_i(z_i{\partial }_i+h_i)G=0$, and ${\sum }_i(z_i^2{\partial }_i+2h_i{z}_i)G=0$. Solving gives a constant one-point function only at $h=0$, a two-point function $C{z}_{12}^{-2h}$ only for equal weights, and a three-point function $C{z}_{12}^{h_3-h_1-h_2}{z}_{13}^{h_2-h_1-h_3}{z}_{23}^{h_1-h_2-h_3}$. Three-point covariance imposes no further equality of weights; dynamical selection rules may still force $C=0$. For more points, arbitrary functions of conformal cross-ratios remain.

Liouville field theory is a two-dimensional scalar theory with a curvature background charge and an exponential interaction. In the usual spacelike normalization, quantum conformal invariance gives $Q=b+b^{-1}$ and central charge $c_L=1+6Q^2$. When a noncritical string worldsheet conformal factor is retained as a dynamical field, its effective action has this form and its central charge combines with the matter and worldsheet ghost field charges. It is closely related to a background-charge scalar field, but its exponential potential supplies an additional interaction.

Radial quantization treats circles around the origin as equal-time slices and logarithmic radius as Euclidean time. A local insertion at the origin prepares a state on a surrounding circle. Contours of the stress tensor act on that insertion, turning its singular operator product expansion into the Virasoro algebra action on the corresponding state.

A quasi-primary operator transforms with weights $(h,\tilde{h})$ under global conformal, or Möbius, transformations. A primary operator has this covariance under every locally invertible holomorphic and antiholomorphic coordinate change. In radial quantization, quasi-primary states are annihilated by $L_1,{\tilde{L}}_1$, whereas primary states are annihilated by every positive Virasoro mode. The stress tensor at nonzero central charge is quasi-primary but not primary because its transformation under a general coordinate change includes a Schwarzian term.

A real chiral Majorana fermion with contraction $\psi (z)\psi (w)\sim 1/(z-w)$ has stress tensor $T=-:\psi \partial \psi :/2$. Fermionic Wick contractions give $T(z)\psi (w)\sim \psi (w)/(2(z-w{)}^2)+\partial \psi (w)/(z-w)$. Double contractions in $TT$ give $1/(4(z-w{)}^4)$, so the central charge is $1/2$. Fermion bilinears of different species are weight-one currents; equal-species undifferentiated bilinears vanish by antisymmetry.

A chiral Majorana fermion with modes $\{b_r,b_s\}={\delta }_{r+s,0}$ and $b_r^{†}=b_{-r}$ has $T=-:\psi \partial \psi :/2$ and central charge $1/2$. The even and odd fermion-number parts of its Neveu–Schwarz sector give unitary modules with lowest weights $0$ and $1/2$. In its Ramond sector, a choice of ground-state zero-mode eigenvalue gives the module of weight $1/16$; the other choice gives an isomorphic Virasoro algebra module. Their cyclic highest-weight submodules are irreducible: a positive-energy invariant proper submodule would have a lowest vector orthogonal to every descendant of the original highest vector, which is impossible in a positive inner-product cyclic module.

A beta-gamma system is a pair of commuting chiral fields with first-order action $\int \beta \bar{\partial }\gamma$. If $\gamma$ has weight $h$ and $\beta$ has weight $1-h$, its central charge is $2(6h^2-6h+1)$.

Using the radial-ordering sign in commuting beta-gamma contractions, single contractions give $T(z)\beta (w)\sim \lambda \beta (w)/(z-w{)}^2+\partial \beta (w)/(z-w)$ and the corresponding weight $1-\lambda$ for $\gamma$. Put $A=:\partial \beta \gamma :$ and $B=:\beta \partial \gamma :$. Their double-contraction coefficients in $AA,BB,AB,BA$ are respectively $1,1,2,2$. The fourth-order pole of $TT$ is therefore $(1-\lambda {)}^2+{\lambda }^2-4\lambda (1-\lambda )=6{\lambda }^2-6\lambda +1$, which equals $c/2$. There is no anticommuting Wick sign.

Commutation allows exchanging the factors at their existing arguments; it does not allow exchanging their arguments while retaining the same pole sign. With the displayed convention, differentiation produces $\partial \beta (z)\gamma (w)\sim (z-w{)}^{-2}$ and $\partial \gamma (z)\beta (w)\sim -(z-w{)}^{-2}$. These signs are essential for the conformal weights and central charge of a commuting beta-gamma system.

A bc system is the anticommuting counterpart of a beta-gamma system. For weights $h$ and $1-h$, its central charge is $-2(6h^2-6h+1)$.

The worldsheet ghost fields form a bc system with conformal weights $2$ and $-1$. Setting $J=2$ in $c=-2(6J^2-6J+1)$ gives central charge $-26$ in each chirality. The worldsheet Weyl anomaly of $D$ free embedding bosons cancels at $D=26$.

For reparameterization ghosts of weights two and minus one, the canonical anticommutators imply $b(z)c(w)\sim 1/(z-w)$. This operator-product expansion fixes the ghost contractions and the ghost stress tensor.

A primary operator of weights $(h,\tilde{h})$ transforms covariantly under local conformal maps and has a universal operator-product expansion with the stress-energy tensor.

A primary state is a simultaneous eigenvector of $L_0,{\tilde{L}}_0$ with eigenvalues $h,\tilde{h}$ that is annihilated by all positive modes of both Virasoro algebras. Negative modes generate descendants. The state–operator correspondence identifies it with a primary operator inserted at the origin.

For free embedding bosons, the operator product expansion of the stress tensor with ${\zeta }_{\mu }:\partial X^{\mu }{e}^{ipX}:$ has a cubic pole $-i{\alpha }^{\prime }(p\cdot \zeta ):e^{ipX}:/2$. Its vanishing is precisely the primary condition, with weight $1+{\alpha }^{\prime }{p}^2/4$. The two-derivative closed-string vertex has the two displayed transversality conditions from its holomorphic and antiholomorphic cubic poles. Being primary does not require $p^2=0$; that extra condition follows when a physical integrated vertex must have conformal weights $(1,1)$.

The conformal weights determine the response of a primary operator to holomorphic and antiholomorphic coordinate rescaling. Its scaling dimension is $\Delta =h+\tilde{h}$ and its spin is $h-\tilde{h}$.

A Virasoro descendant is obtained by applying negative Virasoro modes $L_{-n}$ to a primary state or operator. Its level is the sum of the mode numbers.

An operator product expansion expresses the short-distance product of local operators as a sum of local operators multiplied by singular coefficient functions.

The holomorphic stress-energy tensor generates infinitesimal holomorphic conformal transformations through its operator product expansions.

In a two-dimensional conformal field theory,The fourth-order pole is the central extension, while the remaining poles express that $T$ has conformal weight two.

The Virasoro algebra is the central extension of the algebra of holomorphic vector fields on a circle. Its generators are the Laurent modes of the holomorphic stress-energy tensor, and its central term is controlled by the central charge $c$.

A Virasoro character records the graded dimensions of a positive-energy Virasoro algebra module. A state of weight $h+N$ contributes $q^{h+N-c/24}$, and its multiplicity is the dimension of that energy eigenspace. The shift accounts for the Casimir energy of a two-dimensional conformal field theory. For a Virasoro Verma module, the character is $q^{h-c/24}{\prod }_{n\ge 1}(1-q^n{)}^{-1}$, by counting the ordered descendant basis.

The Virasoro Verma module $M(c,h)$ is the universal highest-weight representation with central charge $c$ and ground weight $h$. The Poincaré-Birkhoff-Witt theorem gives a basis $L_{-n_1}\cdots L_{-n_s}|h⟩$ with $n_1\ge \cdots \ge n_s>0$. Its invariant Hermitian form is fixed by $L_n^{†}=L_{-n}$ and the norm of the ground vector. Quotienting by its maximal proper submodule produces the irreducible highest-weight module; in the unitary cases this removes the null vectors.

For a normalized highest-weight vector with $L_0|h⟩=h|h⟩$, $L_{n>0}|h⟩=0$ and $L_n^{†}=L_{-n}$, the level-two descendants $L_{-2}|h⟩,L_{-1}^2|h⟩$ have the displayed Gram matrix. It follows from the Virasoro algebra: $⟨L_2{L}_{-2}⟩=4h+c/2$, $⟨L_2{L}_{-1}^2⟩=6h$ and $⟨L_1^2{L}_{-1}^2⟩=4h(2h+1)$. At $h=-1,c=26$ its null direction is $(1,3/2)$, giving the level-two scalar Virasoro null state. Nullity of a descendant is a metric statement; satisfying all physical constraints must be checked separately.

A unitary highest-weight Virasoro module is generated by a normalized vector satisfying the displayed relations, with a positive-definite inner product and $L_n^{†}=L_{-n}$. Its Virasoro descendants have $L_0$ eigenvalues $h+N$ for nonnegative integral levels $N$. The commutator gives $\Vert L_{-1}|h⟩{\Vert }^2=2h$, so unitarity requires $h\ge 0$. Such a module is a quotient of a Virasoro Verma module; its Unitary Virasoro discrete series describes the cases with central charge below one.

For a restricted representation of an affine current algebra with $[J_m^a,J_n^b]=i{f}^{abc}{J}_{m+n}^c+km{\delta }^{ab}{\delta }_{m+n,0}$ and $f^{abc}{f}^{abd}=Q{\delta }^{cd}$, the displayed normal ordering construction is defined when $2k+Q\ne 0$. Moving the currents past each other gives $[L_m,J_n^a]=-n{J}_{m+n}^a$; the additional $Q$ in the denominator is the adjoint-Casimir reordering term. The resulting Virasoro algebra has central charge $c=k\dim \mathfrak{g}/(k+Q/2)$. On a compact unitary positive-energy module, $L_m^{†}=L_{-m}$.

For a current subalgebra $\mathfrak{h}\subset \mathfrak{g}$, its currents have conformal weight one under both Sugawara construction stress tensors. Their difference therefore commutes with all $\mathfrak{h}$ currents and with the $\mathfrak{h}$ stress tensor. It obeys the Virasoro algebra with central charge $c_{\mathfrak{g}}-c_{\mathfrak{h}}$. For a positive-level compact unitary vacuum representation, $\Vert L_{-2}^{\mathfrak{g}/\mathfrak{h}}|0⟩{\Vert }^2=(c_{\mathfrak{g}}-c_{\mathfrak{h}})/2\ge 0$. A Cartan subalgebra has $c_{\mathfrak{h}}=\mathrm{rank}\mathfrak{g}$, giving the lower bound on $c_{\mathfrak{g}}$ under these hypotheses.

An irreducible unitary highest weight module of the Virasoro algebra with $c<1$ has integer $m\ge 2$, $1\le r\le m-1$, $1\le s\le m$, and the displayed weights. Labels $(r,s)$ and $(m-r,m+1-s)$ represent the same module. It is obtained from the Verma module by quotienting its zero-norm submodules; the two generating singular-vector levels are $rs$ and $(m-r)(m+1-s)$. For $m=2$ this is the trivial $c=h=0$ module. For $m=3$, the three weights are $0$, $1/2$, $1/16$.

Virasoro generators are the contour modes of the holomorphic stress tensor in a two-dimensional conformal field theory. Their commutators are $[L_m,L_n]=(m-n)L_{m+n}+c(m^3-m){\delta }_{m+n,0}/12$. A contour around an operator insertion extracts the corresponding coefficient of its stress-tensor operator product expansion. $L_{-1}$ generates translations, $L_0$ measures holomorphic weight, and all positive generators annihilate a primary state. An independent antiholomorphic copy acts on the other weight.

Shifting the zero mode changes the algebra's central representative:The linear term follows from replacing $L_{m+n}$ by ${\mathcal{L}}_{m+n}+a{\delta }_{m+n,0}$. It changes the zero-mode convention, not the cubic coefficient or central charge. This distinction separates the intercept from the Virasoro central extension.

The Virasoro central extension is the term $c(m^3-m){\delta }_{m+n,0}/12$ in $[L_m,L_n]$. It records the conformal anomaly and commutes with every Virasoro generator.

For $d$ free target coordinates, normal ordering of $L_m={\sum }_n:{\alpha }_{m-n}\cdot {\alpha }_n:/2$ gives a matter central charge $d$. On the formal zero-momentum vacuum, the double contractions yield $⟨L_m{L}_{-m}⟩=(d/2){\sum }_{r=1}^{m-1}r(m-r)=d(m^3-m)/12$. Two target-metric contractions give ${\eta }_{ab}{\eta }^{ab}=d$, so timelike target coordinates do not subtract from this central charge. Transverse-only and covariant generators must not be confused.

The central charge is the coefficient of the fourth-order pole in the stress-tensor operator-product expansion, $T(z)T(w)\sim c/[2(z-w{)}^4]+\cdots$. A free real boson contributes one and a free real chiral fermion contributes one half.

In a two-dimensional supersymmetric conformal field theory, the superconformal current has weight $3/2$ and its self-OPE closes on the stress tensor and the central charge.

For RNS string matter normalized by $[{\dot{X}}^{\mu }(\sigma ),X^{\nu }({\sigma }^{\prime })]=-i\pi {\eta }^{\mu \nu }\delta (\sigma -{\sigma }^{\prime })$ and $\{{\psi }_1^{\mu }(\sigma ),{\psi }_1^{\nu }({\sigma }^{\prime })\}=\pi {\eta }^{\mu \nu }\delta (\sigma -{\sigma }^{\prime })$, put $A={\partial }_{+}X$ and $J={\psi }_1\cdot A$. Then $[A^{\mu }(\sigma ),A^{\nu }({\sigma }^{\prime })]=(i\pi /2){\eta }^{\mu \nu }{\partial }_{\sigma }\delta (\sigma -{\sigma }^{\prime })$. The noncentral part of $\{J(\sigma ),J({\sigma }^{\prime })\}$ is $\pi \delta A^2+(i\pi /2){\partial }_{\sigma }\delta {\psi }_1(\sigma )\cdot {\psi }_1({\sigma }^{\prime })$. The distributional derivative identity $f(\sigma ,{\sigma }^{\prime }){\delta }^{\prime }=f(\sigma ,\sigma ){\delta }^{\prime }+({\partial }_{{\sigma }^{\prime }}f)(\sigma ,\sigma )\delta$, with a vanishing coincident Grassmann bilinear, converts this to $\pi \delta {\theta }_{++}$. Use the chiral equation of motion to identify ${\partial }_{\sigma }{\psi }_1={\partial }_{+}{\psi }_1$. Quantum normal ordering adds a central contact term proportional to ${\delta }^{\prime \prime }$, the local form of the N=1 super-Virasoro algebra central extension.

The holomorphic N=1 super-Virasoro algebra extends the Virasoro algebra by fermionic modes $G_r$ satisfying $\{G_r,G_s\}=2L_{r+s}+(c/12)(4r^2-1){\delta }_{r,-s}$.

With the standard string oscillator algebra and $G_0={\sum }_m{\alpha }_m\cdot {\psi }_{-m}$, pairing positive and negative modes gives $G_0^2={\alpha }_0^2/2+{\sum }_{m>0}({\alpha }_{-m}\cdot {\alpha }_m+m{\psi }_{-m}\cdot {\psi }_m)$. This is the vacuum-normal-ordered Ramond sector generator $L_0^R$, so $\{G_0,G_0\}=2L_0^R$. Rescaling the superconformal current rescales its square quadratically; its normalization cannot be ignored when comparing it with a fixed Virasoro generator.

The free boson conformal field theory has action quadratic in a scalar $X$ and central charge one. Its oscillator partition function is expressed through the Dedekind eta function.

For the normalization $S=(2\pi {\alpha }^{\prime }{)}^{-1}\int d^{2}z\partial X\bar{\partial }X$, the free-boson worldsheet propagator has singular partDifferentiation gives the singular operator-product expansions of the embedding currents.

A linear dilaton conformal field theory improves the free-boson stress tensor by a second derivative, $T=-{\alpha }^{\prime -1}:\partial X\partial X:-q{\partial }^{2}X$. The improvement shifts the central charge to $c=1+6{\alpha }^{\prime }{q}^2$ and corresponds to a dilaton linear in the target coordinate $X$.

A background-charge scalar field is a free scalar coupled linearly to the worldsheet Ricci scalar. This curvature coupling produces the improved stress tensor of a linear dilaton conformal field theory.

For a noncompact free boson with target volume $V$, the torus partition function is $Z=V[2\pi \sqrt{\mathrm{Im}\tau }{]}^{-1}|\eta (\tau ){|}^{-2}$ in the normalization where the zero-mode Hamiltonian is $p^2/2$.

The Dedekind eta function isIt packages the oscillator contribution of a chiral free 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.