Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 64 / 4 / Solution

Consider the closed bosonic string on a rectangular product of spatial circles with no background mixing between them. Let $Y^j$ have period $2\pi R_j$, take $0\le \sigma <2\pi$, and choose $Y_L(\tau +\sigma )$ to be the tilded oscillator sector. Closed-string topology permitsSingle-valued center-of-mass wavefunctions give quantized momentum $p^j=n_j/R_j$, $n_j\in \mathbb{Z}$. These are the momentum and winding modes. The oscillator modes remain integer moded; the zero modes are split asThe two intercept-one Virasoro constraints, with noncompact $k^2=-M^2$, giveThus the mass formula and compact-circle closed-string level matching condition areHere $N$ labels right movers and $\tilde{N}$ left movers; exchanging the labels changes the displayed sign convention, not the spectrum. Noncompact oscillator polarizations are still restricted by positive-mode Virasoro conditions and null-state identifications. T-duality exchanges $n_j,w_j$ and $R_j,{\alpha }^{\prime }/R_j$, preserving masses and the physical state conditions.

At generic radius, neutral $(N,\tilde{N})=(1,1)$ states with one noncompact oscillator and one compact oscillator give two massless vectors per circle. They are the two Abelian gauge fields associated with the compact metric and two-form components. Additional charged vectors can occur at oscillator level sum one. For one circle, set all other compact charges to zero. Take $n=w=\pm 1$ and $(N,\tilde{N})=(1,0)$, with a right-moving noncompact oscillator ${\alpha }_{-1}^{\mu }$; or take $n=-w=\pm 1$ and $(N,\tilde{N})=(0,1)$, with a left-moving noncompact oscillator ${\tilde{\alpha }}_{-1}^{\mu }$. These obey level matching. Their common squared mass isTherefore four extra charged vector particles become massless at $R=\sqrt{{\alpha }^{\prime }}$, the self-dual circle. Away from that radius they are massive. They are genuine vector states: the noncompact level-one oscillator carries a vector polarization, and the Virasoro constraint plus its null identification impose transversality and remove the gauge polarization at zero mass.

The resulting self-dual circle current algebra makes the gauge enhancement explicit. At the special radius, the left-moving charges above have $p_L=\pm 2/\sqrt{{\alpha }^{\prime }}$, $p_R=0$. With $Y_L(z)Y_L(w)\sim -{\alpha }^{\prime }\log (z-w)/2$, the currentshave conformal weight one. The operator products give $J^3{J}^3\sim 1/[2(z-w{)}^2]$, $J^3{J}^{\pm }\sim \pm J^{\pm }/(z-w)$ and $J^{+}{J}^{-}\sim 1/(z-w{)}^2+2J^3/(z-w)$. They form a level-one SU(2) current algebra; the right-moving copy supplies another SU(2). Pairing each chiral current with the opposite-sector noncompact derivative yields a weight-$(1,1)$ vector vertex. Thus the generic $U(1{)}_L\times U(1{)}_R$ becomes $SU(2{)}_L\times SU(2{)}_R$, with four charged generators in addition to the two Cartan vectors. Several self-dual circles give the corresponding product enhancement.

Compactification also creates scalar states and moduli; not every additional massless state is a vector. The mechanism just shown uses the intercept-one bosonic spectrum. A superstring projection changes the allowed oscillator states, so the same mass formula cannot simply be applied to a GSO-projected superstring without modification.