Circle Compactification and T-Duality

A point particle moving on a circle has quantized momentum. A closed string moving on a circle has something more: it can also wind around the circle. This single observation is the seed of T-duality.

Compactify one target-space coordinate, which we call $Y$, on a circle of radius $R$:

$$Y \sim Y+2\pi R.$$

For a point particle, the wavefunction must be single-valued under $Y\to Y+2\pi R$, so its momentum is $p_Y=m/R$, with $m\in\mathbb Z$. For a closed string, the map from the worldsheet circle to the target circle need not be single-valued on the universal cover. The string configuration itself is single-valued on $S^1_R$, but the lifted coordinate may obey

$$Y(\tau,\sigma+2\pi)=Y(\tau, \sigma)+2\pi wR, \qquad w\in\mathbb Z.$$

The integer $w$ is the winding number. A state is therefore labelled not only by a Kaluza-Klein momentum integer $m$, but by a pair $(m,w)$. Roughly, the momentum contribution to the energy is proportional to $1/R$, while the winding contribution is proportional to $R/\alpha'$. Large and small circles are therefore not independent regimes of string theory: exchanging momentum and winding can turn one into the other.

On the universal cover of the target circle, a closed string may close only up to a shift $2\pi wR$. Momentum and winding contribute to the left- and right-moving zero modes with opposite relative signs.

Throughout most of this page we use the closed bosonic string compactified from $26$ to $25$ noncompact dimensions. This is the cleanest setting for the momentum-winding lattice. In superstrings, the zero-mode lattice and the T-duality action are the same; what changes are the oscillator intercepts, spin structures, and GSO projections.

Zero modes and left-right momenta

Let the worldsheet coordinate be periodic as $\sigma\sim\sigma+2\pi$. Split the compact coordinate into left- and right-moving pieces,

$$Y(\tau,\sigma)=Y_L(\tau+ \sigma)+Y_R(\tau- \sigma).$$

A convenient zero-mode convention is

$$Y(\tau, \sigma) = y+{\alpha'\over 2}p_L(\tau+ \sigma) +{\alpha'\over 2}p_R(\tau- \sigma) +Y_{\rm osc}(\tau, \sigma),$$

where $Y_{\rm osc}$ is strictly periodic in $\sigma$. Shifting $\sigma$ by $2\pi$ gives

$$Y(\tau, \sigma+2\pi)-Y(\tau, \sigma) =\pi\alpha'(p_L-p_R).$$

The winding condition therefore implies

$$p_L-p_R={2wR\over \alpha'}.$$

On the other hand, the center-of-mass momentum along the circle is the average of the left and right momenta,

$$p_Y={p_L+p_R\over 2}={m\over R}, \qquad m\in\mathbb Z.$$

Solving these two equations gives the basic formula

$$\boxed{ p_L={m\over R}+{wR\over \alpha'}, \qquad p_R={m\over R}-{wR\over \alpha'} }$$

with $m,w\in\mathbb Z$. Notice the two important signs: momentum enters $p_L$ and $p_R$ with the same sign, while winding enters with opposite signs. This is the microscopic reason that T-duality acts as a reflection on one chiral half of the worldsheet theory.

The oscillator expansion compatible with these conventions is

$$Y_{\rm osc} =i\sqrt{\alpha'\over 2} \sum_{\ell\ne0}{1\over \ell} \left( \alpha_\ell^Y e^{-i\ell(\tau+\sigma)}+ \widetilde\alpha_\ell^Y e^{-i\ell(\tau-\sigma)} \right).$$

The compact direction contributes to the Virasoro zero modes as

$$L_0={\alpha'\over4}(k^2+p_L^2)+N, \qquad \widetilde L_0={\alpha'\over4}(k^2+p_R^2)+\widetilde N,$$

where $k^\mu$ is the momentum in the remaining noncompact dimensions, $k^2=-M^2$, and $N,\widetilde N$ include all oscillator excitations, including those of $Y$.

For the closed bosonic string, the physical-state conditions are

$$L_0=1, \qquad \widetilde L_0=1.$$

Thus

$$M^2=p_L^2+{4\over\alpha'}(N-1) =p_R^2+{4\over\alpha'}(\widetilde N-1).$$

Adding and subtracting the two expressions gives the mass formula and level-matching condition:

$$\boxed{ M^2={m^2\over R^2}+{w^2R^2\over \alpha'^2} +{2\over\alpha'}(N+\widetilde N-2) }$$

and

$$\boxed{ N-\widetilde N+mw=0. }$$

This level-matching condition is sometimes the easiest place to see that momentum and winding are genuinely coupled. A state with nonzero $mw$ cannot choose arbitrary left- and right-moving oscillator levels; the mismatch $N-\widetilde N$ must compensate the worldsheet momentum carried by the zero modes.

Physical interpretation of the spectrum

At a generic radius, the lightest states come from the smallest values of $m$, $w$, $N$, and $\widetilde N$ compatible with level matching. The two zero-mode contributions behave very differently as $R$ varies:

$${m^2\over R^2} \quad\hbox{is light for large }R, \qquad {w^2R^2\over \alpha'^2} \quad\hbox{is light for small }R.$$

A field theorist compactifying on a very small circle sees Kaluza-Klein momenta becoming very heavy and might conclude that the circle disappears. A string theorist cannot stop there. Winding modes become light at small radius, and they reconstruct a dual large-radius geometry.

This is the first major conceptual lesson of compactification in string theory:

$$\hbox{short distances in target space need not be physically shorter distances in string theory.}$$

The statement is not a vague minimal-length slogan. It is an exact equivalence of conformal field theories for a free compact boson.

T-duality of the spectrum

Define the dual radius

$$R'={\alpha'\over R},$$

and exchange the two integers

$$m' = w, \qquad w' = m.$$

Then

$$p_L'={m'\over R'}+{w'R'\over\alpha'} ={wR\over\alpha'}+{m\over R}=p_L,$$

while

$$p_R'={m'\over R'}-{w'R'\over\alpha'} ={wR\over\alpha'}-{m\over R}=-p_R.$$

Thus T-duality acts as

$$\boxed{ R\longleftrightarrow {\alpha'\over R}, \qquad m\longleftrightarrow w, \qquad p_L\longmapsto p_L, \qquad p_R\longmapsto -p_R. }$$

The mass formula is invariant because it depends on $p_R^2$, not $p_R$ itself. The level-matching condition is also invariant because $mw$ is symmetric under $m\leftrightarrow w$.

On the worldsheet fields, the same transformation is

$$Y_L'(\tau+ \sigma)=Y_L(\tau+ \sigma), \qquad Y_R'(\tau- \sigma)=-Y_R(\tau- \sigma).$$

Equivalently,

$$Y'=Y_L-Y_R.$$

Since the original coordinate is $Y=Y_L+Y_R$, the duality is a reflection of the right-moving coordinate. In terms of derivatives,

$$\partial_\tau Y'=\partial_\sigma Y, \qquad \partial_\sigma Y'=\partial_\tau Y.$$

This is the canonical transformation form of T-duality. It exchanges momentum density and winding density on the string.

In dimensionless variables $P_{L,R}=\sqrt{\alpha'}p_{L,R}$ and $r=R/\sqrt{\alpha'}$, the compactification data form a Lorentzian lattice $\Gamma^{1,1}$. T-duality is the lattice isometry $P_L\to P_L$, $P_R\to -P_R$, together with $m\leftrightarrow w$.

The one-circle moduli space is therefore not the half-line $R>0$. It is the half-line modulo the identification $R\sim\alpha'/R$. One may choose a fundamental domain

$$R\ge \sqrt{\alpha'}.$$

The point

$$R_{\rm sd}=\sqrt{\alpha'}$$

is fixed by the duality and is called the self-dual radius.

The compact-boson torus partition function

The circle compactification is also a good laboratory for modular invariance. There are two equivalent ways to write the one-loop partition function of the compact boson.

The Hamiltonian form is a trace over momentum and winding states:

$$Z_R(\tau) ={1\over |\eta(\tau)|^2} \sum_{m,w\in\mathbb Z} q^{{\alpha'\over4}p_L^2} \bar q^{{\alpha'\over4}p_R^2}, \qquad q=e^{2\pi i\tau}.$$

Using the explicit left- and right-moving momenta, the zero-mode part is

$$q^{{\alpha'\over4}p_L^2}\bar q^{{\alpha'\over4}p_R^2} = \exp\left[ 2\pi i\tau_1 mw - \pi\tau_2\left({\alpha'm^2\over R^2}+{w^2R^2\over\alpha'}\right) \right].$$

The phase $e^{2\pi i\tau_1 mw}$ is the torus trace of level matching. The factor $|\eta|^{-2}$ is the oscillator determinant of one compact free boson.

The Lagrangian form instead sums over classical maps from the worldsheet torus into the target circle. A Euclidean torus has two cycles, and the compact coordinate may wind independently around both. If those two winding numbers are $a,b\in\mathbb Z$, then

$$Z_R(\tau) ={R\over\sqrt{\alpha'\tau_2}}{1\over |\eta(\tau)|^2} \sum_{a,b\in\mathbb Z} \exp\left[ -{\pi R^2\over\alpha'\tau_2}|a-b\tau|^2 \right].$$

The two forms are related by Poisson resummation. This relation is worth understanding. In the Hamiltonian expression, $m$ is a momentum quantum number and $w$ is a spatial winding number. In the Lagrangian expression, $a$ and $b$ are topological winding numbers around the two cycles of the Euclidean worldsheet torus. Poisson resummation transforms one of the topological sums into a momentum sum.

The Lagrangian expression makes modular invariance transparent. Under a modular transformation, the two cycles of the torus are recombined by an $SL(2,\mathbb Z)$ matrix, and the pair $(a,b)$ is correspondingly recombined. Since the sum includes all integer pairs, the partition function is invariant.

The Hamiltonian expression makes T-duality transparent. Under

$$R\to {\alpha'\over R}, \qquad m\leftrightarrow w,$$

one has $p_L\to p_L$ and $p_R\to -p_R$, so

$$Z_R(\tau)=Z_{\alpha'/R}(\tau).$$

For a single free compact boson, T-duality is therefore not merely a symmetry of the mass spectrum. It is an exact equality of the full torus partition function, and indeed of the full conformal field theory.

Generic massless fields after circle compactification

At a generic radius, the massless closed-string states with $m=w=0$ and $N=\widetilde N=1$ include the usual graviton, antisymmetric tensor, and dilaton in the noncompact directions:

$$\alpha_{-1}^{\mu}\widetilde\alpha_{-1}^{\nu}|0;k\rangle, \qquad \mu, \nu=0,\ldots,24.$$

They decompose into

$$G_{\mu\nu}, \qquad B_{\mu\nu}, \qquad \Phi.$$

There are also states with one oscillator in the noncompact direction and one in the compact direction:

$$\alpha_{-1}^{\mu}\widetilde\alpha_{-1}^{Y}|0;k\rangle, \qquad \alpha_{-1}^{Y}\widetilde\alpha_{-1}^{\mu}|0;k\rangle.$$

Equivalently, one may form the symmetric and antisymmetric combinations associated with

$$G_{\mu Y}, \qquad B_{\mu Y}.$$

From the lower-dimensional point of view, these are two $U(1)$ gauge fields. It is often cleaner to call them the left and right gauge fields,

$$A^L_\mu\sim G_{\mu Y}+B_{\mu Y}, \qquad A^R_\mu\sim G_{\mu Y}-B_{\mu Y},$$

up to normalization conventions. Thus the generic gauge symmetry is

$$U(1)_L\times U(1)_R.$$

Finally, the state

$$\alpha_{-1}^{Y}\widetilde\alpha_{-1}^{Y}|0;k\rangle$$

is a scalar in the lower-dimensional theory. It is the radion, the fluctuation of the circle radius. In a one-circle compactification there is no independent $B_{YY}$ field, because $B$ is antisymmetric.

T-duality acts nontrivially on these lower-dimensional fields. Since it flips $Y_R$ but not $Y_L$, it exchanges the geometric and antisymmetric-tensor origins of the two gauge bosons. This is the first example of a general phenomenon: in string theory, geometry and gauge fields can mix under duality.

The self-dual radius and enhanced gauge symmetry

At a generic radius, only the Cartan gauge bosons of $U(1)_L\times U(1)_R$ are massless. At the self-dual radius $R=\sqrt{\alpha'}$, additional momentum-winding states become massless.

Set $R=\sqrt{\alpha'}$. Then

$$\sqrt{\alpha'}p_L=m+w, \qquad \sqrt{\alpha'}p_R=m-w.$$

The massless conditions may be written as

$$(m+w)^2+4N=4, \qquad (m-w)^2+4\widetilde N=4.$$

Besides the generic $m=w=0$, $N=\widetilde N=1$ states, there are new possibilities:

$$(m,w)=(1,1),(-1,-1), \qquad N=0, \qquad \widetilde N=1,$$

for which

$$p_R=0, \qquad p_L=\pm {2\over\sqrt{\alpha'}}.$$

These give two additional left charged gauge bosons. Similarly,

$$(m,w)=(1,-1),(-1,1), \qquad N=1, \qquad \widetilde N=0,$$

for which

$$p_L=0, \qquad p_R=\pm {2\over\sqrt{\alpha'}}.$$

These give two additional right charged gauge bosons.

The CFT explanation is elegant. With the normalization

$$Y_L(z)Y_L(0)\sim -{\alpha'\over2}\log z,$$

the chiral exponential

$$:e^{ikY_L}:$$

has holomorphic dimension

$$h={\alpha' k^2\over4}.$$

At $k=\pm2/\sqrt{\alpha'}$, the dimension is $h=1$. Therefore the operators

$$J_L^\pm(z)=:e^{\pm2iY_L(z)/\sqrt{\alpha'}}:, \qquad J_L^3(z)\sim i\partial Y_L(z)$$

are dimension-one holomorphic currents. They generate an affine $SU(2)_L$ current algebra at level one. Similarly,

$$J_R^\pm(\bar z)=:e^{\pm2iY_R(\bar z)/\sqrt{\alpha'}}:, \qquad J_R^3(\bar z)\sim i\bar\partial Y_R(\bar z)$$

generate $SU(2)_R$.

Thus the gauge group is enhanced:

$$\boxed{ U(1)_L\times U(1)_R \quad\longrightarrow\quad SU(2)_L\times SU(2)_R \qquad (R=\sqrt{\alpha'}). }$$

At $R=\sqrt{\alpha'}$, the charged roots of $SU(2)_L$ and $SU(2)_R$ are realized by momentum-winding states. Away from this radius, these states become massive and only the Cartan gauge bosons remain massless.

The corresponding vertex operators for the new gauge bosons are schematically

$$J_L^\pm(z)\,\bar\partial X^\mu(\bar z)e^{ik\cdot X}, \qquad \partial X^\mu(z)\,J_R^\pm(\bar z)e^{ik\cdot X},$$

with $k^2=0$ in the noncompact spacetime directions. Together with the Cartan gauge bosons, these fill out the adjoint representations of $SU(2)_L$ and $SU(2)_R$.

There are also additional massless scalars at the self-dual radius. They can be organized as

$$J_L^a(z)J_R^b(\bar z)e^{ik\cdot X}, \qquad a,b=1,2,3,$$

and transform as $(3,3)$ under $SU(2)_L\times SU(2)_R$. The radion is one component of this set. Moving away from the self-dual radius gives a vacuum expectation value to a scalar in this multiplet and Higgses the enhanced gauge symmetry back down to $U(1)_L\times U(1)_R$.

This is a prototype for gauge enhancement in string compactification. Gauge symmetries need not be put in by hand; they can appear when the compactification lattice develops extra vectors of the right length.

Narain-lattice viewpoint

The pair $(p_L,p_R)$ is not just a bookkeeping device. It belongs to an even Lorentzian lattice of signature $(1,1)$. In dimensionless variables

$$P_L=\sqrt{\alpha'}p_L, \qquad P_R=\sqrt{\alpha'}p_R, \qquad r={R\over\sqrt{\alpha'}},$$

one has

$$P_L={m\over r}+wr, \qquad P_R={m\over r}-wr.$$

The Lorentzian norm is

$$P_L^2-P_R^2=4mw.$$

Because $m$ and $w$ are integers, this norm is integral and even in the normalization relevant for the closed-string level-matching condition. T-duality is an automorphism of this lattice.

For one circle, the T-duality group is essentially $O(1,1;\mathbb Z)$, generated by

$$R\to {\alpha'\over R}, \qquad m\leftrightarrow w,$$

together with signs of $m$ and $w$. For a compactification on a $d$-torus, the same idea becomes the Narain lattice $\Gamma^{d,d}$ and the T-duality group $O(d,d;\mathbb Z)$. The special radii where extra lattice vectors have the right length are the places where nonabelian gauge symmetries appear.

This lattice viewpoint is one of the most durable ideas in string compactification. It connects worldsheet modular invariance, target-space duality, gauge enhancement, and the geometry of moduli space.

Exercises

Exercise 1: Derive the left- and right-moving momenta

Starting from

$$Y(\tau, \sigma)=y+{\alpha'\over2}p_L(\tau+ \sigma)+{\alpha'\over2}p_R(\tau- \sigma)+Y_{\rm osc},$$

with $Y_{\rm osc}$ periodic in $\sigma$, derive

$$p_L={m\over R}+{wR\over\alpha'}, \qquad p_R={m\over R}-{wR\over\alpha'}.$$

Solution

The winding condition gives

$$Y(\tau, \sigma+2\pi)-Y(\tau, \sigma)=2\pi wR.$$

The oscillator part drops out because it is periodic. The zero-mode part changes by

$${\alpha'\over2}p_L(2\pi)+{\alpha'\over2}p_R(-2\pi) =\pi\alpha'(p_L-p_R).$$

Thus

$$p_L-p_R={2wR\over\alpha'}.$$

Single-valued wavefunctions on the target circle require the center-of-mass momentum to be

$$p_Y={m\over R}.$$

Since $p_Y=(p_L+p_R)/2$, we also have

$$p_L+p_R={2m\over R}.$$

Adding and subtracting the two equations gives

$$p_L={m\over R}+{wR\over\alpha'}, \qquad p_R={m\over R}-{wR\over\alpha'}.$$

Exercise 2: Mass formula and level matching

Use

$$L_0={\alpha'\over4}(k^2+p_L^2)+N, \qquad \widetilde L_0={\alpha'\over4}(k^2+p_R^2)+\widetilde N,$$

and $L_0=\widetilde L_0=1$ to prove

$$M^2={m^2\over R^2}+{w^2R^2\over\alpha'^2}+{2\over\alpha'}(N+\widetilde N-2),$$

and

$$N-\widetilde N+mw=0.$$

Solution

Since $k^2=-M^2$, the left-moving physical-state condition gives

$$1=-{\alpha'M^2\over4}+{\alpha'p_L^2\over4}+N,$$

so

$$M^2=p_L^2+{4\over\alpha'}(N-1).$$

Similarly,

$$M^2=p_R^2+{4\over\alpha'}(\widetilde N-1).$$

Adding the two equations and using

$${p_L^2+p_R^2\over2} ={m^2\over R^2}+{w^2R^2\over\alpha'^2}$$

gives the mass formula.

Subtracting the two equations gives

$$0=p_L^2-p_R^2+{4\over\alpha'}(N-\widetilde N).$$

But

$$p_L^2-p_R^2 =\left({m\over R}+{wR\over\alpha'}\right)^2 -\left({m\over R}-{wR\over\alpha'}\right)^2 ={4mw\over\alpha'}.$$

Therefore

$$0={4\over\alpha'}(mw+N-\widetilde N),$$

or

$$N-\widetilde N+mw=0.$$

Exercise 3: T-duality as a right-moving reflection

Show that under

$$R'={\alpha'\over R}, \qquad m'=w, \qquad w'=m,$$

the left- and right-moving momenta transform as

$$p_L'=p_L, \qquad p_R'=-p_R.$$

Then show that $Y'=Y_L-Y_R$ obeys

$$\partial_\tau Y'=\partial_\sigma Y, \qquad \partial_\sigma Y'=\partial_\tau Y.$$

Solution

Using the dual radius,

$$p_L'={m'\over R'}+{w'R'\over\alpha'} ={w\over \alpha'/R}+{m(\alpha'/R)\over\alpha'} ={wR\over\alpha'}+{m\over R}=p_L.$$

Similarly,

$$p_R'={m'\over R'}-{w'R'\over\alpha'} ={wR\over\alpha'}-{m\over R}=-p_R.$$

For the field transformation, write

$$Y=Y_L(\tau+ \sigma)+Y_R(\tau- \sigma), \qquad Y'=Y_L(\tau+ \sigma)-Y_R(\tau- \sigma).$$

Since

$$\partial_\tau Y_L=\partial_\sigma Y_L, \qquad \partial_\tau Y_R=-\partial_\sigma Y_R,$$

we find

$$\partial_\tau Y' =\partial_\tau Y_L-\partial_\tau Y_R =\partial_\sigma Y_L+\partial_\sigma Y_R =\partial_\sigma Y.$$

Likewise,

$$\partial_\sigma Y' =\partial_\sigma Y_L-\partial_\sigma Y_R =\partial_\tau Y_L+\partial_\tau Y_R =\partial_\tau Y.$$

Exercise 4: T-duality invariance of the partition function

Use

$$Z_R(\tau) ={1\over |\eta(\tau)|^2} \sum_{m,w\in\mathbb Z} q^{{\alpha'\over4}p_L^2} \bar q^{{\alpha'\over4}p_R^2}$$

to show that $Z_R(\tau)=Z_{\alpha'/R}(\tau)$.

Solution

For the radius $R'= \alpha'/R$, relabel the integers by

$$m'=w, \qquad w'=m.$$

The sum over all pairs $(m',w')\in\mathbb Z^2$ is the same as the sum over all pairs $(m,w)\in\mathbb Z^2$. By Exercise 3,

$$p_L'=p_L, \qquad p_R'=-p_R.$$

Therefore

$$q^{{\alpha'\over4}(p_L')^2}\bar q^{{\alpha'\over4}(p_R')^2} =q^{{\alpha'\over4}p_L^2}\bar q^{{\alpha'\over4}p_R^2}.$$

The oscillator factor $|\eta|^{-2}$ is independent of $R$, so the full partition function is invariant:

$$Z_{\alpha'/R}(\tau)=Z_R(\tau).$$

Exercise 5: Extra massless states at the self-dual radius

At $R=\sqrt{\alpha'}$, solve the massless conditions

$$(m+w)^2+4N=4, \qquad (m-w)^2+4\widetilde N=4$$

for states with either $N=0$, $\widetilde N=1$ or $N=1$, $\widetilde N=0$. Identify their left- or right-moving charges.

Solution

First take $N=0$, $\widetilde N=1$. The two massless equations become

$$(m+w)^2=4, \qquad (m-w)^2=0.$$

Thus $m=w$ and $2m=\pm2$, so

$$(m,w)=(1,1),(-1,-1).$$

For these states,

$$p_L=\pm {2\over\sqrt{\alpha'}}, \qquad p_R=0.$$

They are charged under the left current algebra and neutral under the right one.

Now take $N=1$, $\widetilde N=0$. The equations become

$$(m+w)^2=0, \qquad (m-w)^2=4.$$

Thus $m=-w$ and $2m=\pm2$, so

$$(m,w)=(1,-1),(-1,1).$$

For these states,

$$p_L=0, \qquad p_R=\pm {2\over\sqrt{\alpha'}}.$$

They are charged under the right current algebra and neutral under the left one. These four states are the extra charged gauge bosons completing

$$U(1)_L\times U(1)_R \to SU(2)_L\times SU(2)_R.$$

Exercise 6: Dimension-one currents

Assume the holomorphic compact boson has OPE

$$Y_L(z)Y_L(0)\sim -{\alpha'\over2}\log z.$$

Show that

$$J_L^\pm(z)=:e^{\pm2iY_L(z)/\sqrt{\alpha'}}:$$

has conformal weight $h=1$.

Solution

For a free boson with

$$Y_L(z)Y_L(0)\sim -{\alpha'\over2}\log z,$$

the holomorphic exponential

$$:e^{ikY_L(z)}:$$

has conformal weight

$$h={\alpha'k^2\over4}.$$

For

$$k=\pm {2\over\sqrt{\alpha'}},$$

we get

$$h={\alpha'\over4}{4\over\alpha'}=1.$$

Thus $J_L^\pm$ are dimension-one holomorphic currents. Together with $J_L^3\sim i\partial Y_L$, they generate the enhanced left-moving $SU(2)_L$ current algebra. The same argument applies to the antiholomorphic right-moving currents.