Closed-String T-Duality and Enhanced Symmetry

Compactification is where strings first look unmistakably different from point particles. A point particle moving on a circle of radius $R$ has quantized momentum $n/R$, so by measuring the spectrum one can infer the size of the circle. A closed string has a second integer quantum number: it can wind around the circle. T-duality is the statement that the theory on a circle of radius $R$ is equivalent to the theory on a circle of radius $\widetilde R=\alpha'/R$, provided momentum and winding are exchanged.

This is not merely an approximate symmetry of the low-energy spectrum. It is an exact perturbative string equivalence. It also foreshadows many later dualities: variables that look elementary in one description may be nonlocal, solitonic, or topological in another.

Closed strings on $S^1_R$

Let one target-space coordinate be compact,

$$Y\equiv X^{25}\sim Y+2\pi R,$$

and let the remaining coordinates be noncompact. For a closed string, $\sigma\sim\sigma+2\pi$, but the map into the compact target-space circle need only close up to an integer winding:

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

The integer $w$ is the winding number. It has no point-particle analog. It is topological: as long as the string stays closed and the target circle remains intact, a string wound $w$ times cannot be continuously deformed into a string wound $w+1$ times.

The center-of-mass wavefunction on the circle is periodic, so the momentum is quantized:

$$p_Y=\frac{n}{R}, \qquad n\in\mathbb Z.$$

A closed string on $S^1_R$ has both point-particle-like momentum $n/R$ and intrinsically stringy winding $w$.

The compact coordinate separates into left- and right-moving parts,

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

With the conventions used throughout these notes,

$$\begin{aligned} Y(\tau,\sigma) =&\ y +\frac{\alpha'}{2}p_L(\tau+\sigma) +\frac{\alpha'}{2}p_R(\tau-\sigma)\\ &+i\sqrt{\frac{\alpha'}{2}}\sum_{m\neq 0}\frac{1}{m} \left( \alpha_m^Y e^{-im(\tau+\sigma)} +\widetilde\alpha_m^Y e^{-im(\tau-\sigma)} \right). \end{aligned}$$

The average of $p_L$ and $p_R$ is the physical momentum:

$$\frac{p_L+p_R}{2}=\frac{n}{R}.$$

The difference is fixed by the winding condition. Increasing $\sigma$ by $2\pi$ changes the zero-mode part by

$$\Delta Y=\pi\alpha'(p_L-p_R)=2\pi wR.$$

Therefore

$$\boxed{ p_L=\frac{n}{R}+\frac{wR}{\alpha'}, \qquad p_R=\frac{n}{R}-\frac{wR}{\alpha'}. }$$

The pair $(p_L,p_R)$ is the compactification-lattice vector associated with the integers $(n,w)$.

The integers $(n,w)$ determine $(p_L,p_R)$. The sum $p_L^2+p_R^2$ contributes to the mass, while the difference $p_L^2-p_R^2$ enters level matching.

Mass formula and level matching

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

$$(L_0-1)|\psi\rangle=0, \qquad (\widetilde L_0-1)|\psi\rangle=0.$$

Let $M^2=-p_\mu p^\mu$ be the mass squared measured in the noncompact spacetime. Separating the compact zero modes gives

$$-\frac{\alpha'}{4}M^2 +\frac{\alpha'}{4}p_L^2 +N-1=0,$$

and

$$-\frac{\alpha'}{4}M^2 +\frac{\alpha'}{4}p_R^2 +\widetilde N-1=0.$$

Adding the two equations gives

$$\boxed{ M^2= \frac{n^2}{R^2} +\frac{w^2R^2}{\alpha'^2} +\frac{2}{\alpha'}(N+\widetilde N-2). }$$

Subtracting them gives

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

Since

$$p_L^2-p_R^2=\frac{4nw}{\alpha'},$$

we obtain the modified level-matching condition

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

In noncompact flat space, level matching says $N=\widetilde N$. On a circle, compact zero modes can compensate a mismatch between the left and right oscillator levels.

The radius-inversion symmetry

The mass formula is invariant under

$$\boxed{ R\longrightarrow R'=\frac{\alpha'}{R}, \qquad n\longleftrightarrow w. }$$

Indeed,

$$\frac{n^2}{R^2}+\frac{w^2R^2}{\alpha'^2} \longrightarrow \frac{w^2}{(\alpha'/R)^2} + \frac{n^2(\alpha'/R)^2}{\alpha'^2} = \frac{w^2R^2}{\alpha'^2}+\frac{n^2}{R^2}.$$

The oscillator term is unchanged, and $nw$ is invariant, so level matching is unchanged.

The left- and right-moving momenta transform in a particularly simple way. Under $R\to\alpha'/R$ and $n\leftrightarrow w$,

$$p_L\longrightarrow p_L, \qquad p_R\longrightarrow -p_R.$$

Thus T-duality acts on the compact coordinate as

$$Y=Y_L+Y_R \quad\longrightarrow\quad \widetilde Y=Y_L-Y_R.$$

The dual coordinate $\widetilde Y$ parametrizes a circle of radius

$$\widetilde R=\frac{\alpha'}{R}.$$

T-duality identifies the radius $R$ with the inverse radius $\alpha'/R$. The fixed point is $R_{\rm sd}=\sqrt{\alpha'}$.

A very small circle is therefore equivalent to a very large dual circle. This is the first sign that string geometry is not ordinary Riemannian geometry plus small corrections. The set of physical probes is enlarged by winding strings, and those probes identify geometries that a point particle would distinguish.

Normalization checkpoint. With $\alpha'$ explicit, the self-dual radius is $R_{\rm sd}=\sqrt{\alpha'}$. If one temporarily sets $\alpha'=2$, then $R_{\rm sd}=\sqrt 2$.

The self-dual radius and extra massless states

At a generic radius, compactification on a circle gives two abelian gauge fields in the lower-dimensional spacetime. They come from the metric and antisymmetric tensor components

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

Equivalently, the worldsheet compact boson has a holomorphic $U(1)_L$ current and an antiholomorphic $U(1)_R$ current:

$$J_L^3=\frac{i}{\sqrt{\alpha'}}\partial Y_L, \qquad \overline J_R^3=\frac{i}{\sqrt{\alpha'}}\bar\partial Y_R.$$

At the self-dual radius, new string states become massless. For example, choose

$$R=\sqrt{\alpha'}, \qquad n=w=1, \qquad N=0, \qquad \widetilde N=1.$$

Level matching is satisfied:

$$N-\widetilde N+nw=0-1+1=0.$$

The mass is

$$M^2= \frac{1}{\alpha'}+\frac{1}{\alpha'}+\frac{2}{\alpha'}(0+1-2)=0.$$

The compact momenta are

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

Thus the new state is purely left-moving in the compact direction. The state with $(n,w)=(-1,-1)$ gives the opposite left-moving charge. Together with $J_L^3$, these become the currents of $SU(2)_L$:

$$J_L^3=\frac{i}{\sqrt{\alpha'}}\partial Y_L, \qquad J_L^\pm=: \exp\left(\pm\frac{2i}{\sqrt{\alpha'}}Y_L\right):.$$

The chiral boson normalization gives

$$h[e^{iqY_L}]=\frac{\alpha' q^2}{4}.$$

For $q=2/\sqrt{\alpha'}$, this gives $h=1$, so $J_L^\pm$ are dimension-one currents.

Similarly, the states with

$$(n,w)=(1,-1),\ (-1,1)$$

are purely right-moving and enhance $U(1)_R$ to $SU(2)_R$.

Therefore

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

At $R=\sqrt{\alpha'}$, additional momentum-winding states become dimension-one currents. The compact boson realizes $SU(2)_L\times SU(2)_R$ at level one.

This is a quintessential string effect. The gauge symmetry is enhanced not because of Kaluza-Klein momentum modes alone, but because momentum and winding states become equally light.

T-duality as worldsheet Hodge duality

There is also a local worldsheet way to understand T-duality. For a free compact scalar,

$$S_Y=\frac{1}{4\pi\alpha'}\int d^2\sigma\,\partial_\alpha Y\partial^\alpha Y.$$

The equation of motion is

$$\partial_\alpha\partial^\alpha Y=0.$$

Define a dual field $\widetilde Y$ locally by

$$\boxed{ \partial_\alpha\widetilde Y = \epsilon_{\alpha\beta}\partial^\beta Y. }$$

In Lorentzian worldsheet coordinates this may be written, up to a sign convention for $\epsilon_{\tau\sigma}$, as

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

The equation of motion of $Y$ becomes the Bianchi identity for $\widetilde Y$:

$$\partial_\alpha\partial^\alpha Y=0 \quad\Longleftrightarrow\quad \epsilon^{\alpha\beta}\partial_\alpha\partial_\beta\widetilde Y=0.$$

Conversely, the Bianchi identity for $Y$ becomes the equation of motion for $\widetilde Y$. This is why T-duality is often compared to Kramers—Wannier duality: the dual variable is not the same local field, but its derivative is related to the Hodge dual of the original derivative.

Locally, T-duality is a Hodge-duality relation on the worldsheet. Globally, it exchanges momentum and winding.

The local relation alone does not know the compactification radius. The global information is supplied by the periodicity of $Y$ and $\widetilde Y$, which turns winding of one field into momentum of the other.

Summary

Closed strings on a circle have two integer quantum numbers:

$$(n,w)=\text{momentum and winding}.$$

Their spectrum is invariant under

$$R\leftrightarrow \frac{\alpha'}{R}, \qquad n\leftrightarrow w.$$

The duality acts asymmetrically on left and right movers, leaving $Y_L$ fixed and flipping the sign of $Y_R$. At the self-dual radius, extra momentum-winding states become massless and enhance the abelian symmetry to $SU(2)_L\times SU(2)_R$.

The next page applies the same idea to open strings. There the result is even more dramatic: T-duality changes boundary conditions and produces D-branes.

Exercises

Exercise 1. Derive $p_L$ and $p_R$

Starting from

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

and $p_Y=n/R$, derive

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

Solution

The zero-mode part is

$$Y_0=y+\frac{\alpha'}{2}p_L(\tau+\sigma)+\frac{\alpha'}{2}p_R(\tau-\sigma).$$

When $\sigma\to\sigma+2\pi$,

$$\Delta Y_0=\pi\alpha'(p_L-p_R).$$

This must equal $2\pi wR$, so

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

The average momentum is

$$\frac{p_L+p_R}{2}=\frac{n}{R}.$$

Solving these two equations gives

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

Exercise 2. Check T-duality invariance

Show that

$$M^2= \frac{n^2}{R^2}+\frac{w^2R^2}{\alpha'^2} +\frac{2}{\alpha'}(N+\widetilde N-2)$$

and

$$N-\widetilde N+nw=0$$

are invariant under $R\to\alpha'/R$ and $n\leftrightarrow w$.

Solution

Under $R'=\alpha'/R$ and $n'=w$, $w'=n$, the zero-mode contribution becomes

$$\frac{(n')^2}{(R')^2}+\frac{(w')^2(R')^2}{\alpha'^2} = \frac{w^2}{(\alpha'/R)^2} +\frac{n^2(\alpha'/R)^2}{\alpha'^2} = \frac{w^2R^2}{\alpha'^2}+\frac{n^2}{R^2}.$$

The oscillator term is unchanged. Also $n'w'=wn=nw$, so the level-matching condition is unchanged.

Exercise 3. Extra massless states at $R=\sqrt{\alpha'}$

At the self-dual radius, show that

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

gives a massless state. Determine $p_L$ and $p_R$.

Solution

Level matching gives

$$N-\widetilde N+nw=0-1+1=0.$$

The mass is

$$M^2=\frac{1}{\alpha'}+\frac{1}{\alpha'}+\frac{2}{\alpha'}(0+1-2)=0.$$

The momenta are

$$p_L=\frac{1}{\sqrt{\alpha'}}+\frac{\sqrt{\alpha'}}{\alpha'} =\frac{2}{\sqrt{\alpha'}}, \qquad p_R=\frac{1}{\sqrt{\alpha'}}-\frac{\sqrt{\alpha'}}{\alpha'}=0.$$

The state is purely left-moving in the compact direction. It is one of the charged states that enhances $U(1)_L$ to $SU(2)_L$.

Exercise 4. Dimension of the enhanced currents

Using

$$h[e^{iqY_L}]=\frac{\alpha' q^2}{4},$$

show that

$$J_L^\pm=:\exp\left(\pm\frac{2i}{\sqrt{\alpha'}}Y_L\right):$$

have conformal weight $h=1$.

Solution

Here

$$q=\pm\frac{2}{\sqrt{\alpha'}}.$$

Therefore

$$h=\frac{\alpha'}{4}\frac{4}{\alpha'}=1.$$

Thus $J_L^\pm$ are dimension-one holomorphic currents. Together with $J_L^3=i\partial Y_L/\sqrt{\alpha'}$, they form the $SU(2)_L$ current algebra at the self-dual radius.

Exercise 5. Equations of motion and Bianchi identities

Assume

$$\partial_\alpha\widetilde Y=\epsilon_{\alpha\beta}\partial^\beta Y.$$

Show that the equation of motion of $Y$ implies the Bianchi identity of $\widetilde Y$.

Solution

The equation of motion is

$$\partial_\alpha\partial^\alpha Y=0.$$

Using the duality relation,

$$\epsilon^{\gamma\alpha}\partial_\gamma\partial_\alpha\widetilde Y = \epsilon^{\gamma\alpha}\partial_\gamma \left(\epsilon_{\alpha\beta}\partial^\beta Y\right).$$

Up to the sign determined by the worldsheet signature convention,

$$\epsilon^{\gamma\alpha}\epsilon_{\alpha\beta}\propto \delta^\gamma_{\ \beta},$$

so this is proportional to

$$\partial_\beta\partial^\beta Y,$$

which vanishes by the equation of motion. Hence the dual field satisfies the Bianchi identity

$$\epsilon^{\gamma\alpha}\partial_\gamma\partial_\alpha\widetilde Y=0.$$