Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 306 / 2

The term ${\hat{p}}^{,2}/2$ is the target-space zero-mode kinetic energy. The constant $-1/12$ is the zero-point energy of the left- and right-moving oscillators: each chiral boson contributes $-1/24$. Finally,count the energies of the two independent sets of string oscillators; one excitation of mode $n$ has energy $n$.

For $n>0$, define occupation numbers $N_n,{\tilde{N}}_n\in {\mathbb{N}}_0$. A basis iswhere $\hat{p}|p;0,0⟩=p|p;0,0⟩$ and positive oscillator modes annihilate the vacuum. Since ${\alpha }_{-n}{\alpha }_n$ has eigenvalue $n{N}_n$, the simultaneous eigenvalues are

Put $\tau ={\tau }_1+i{\tau }_2$ and $q=e^{2\pi i\tau }$. The torus partition function is the traceThe continuum momentum states have density $V/(2\pi )$, so their contribution isEach left-moving oscillator contributes ${\sum }_{N\ge 0}{q}^{nN}=(1-q^n{)}^{-1}$, with the complex conjugate for the right mover. The zero-point factor combines these products into the Dedekind eta functionTherefore the torus partition function of a free boson is

For a compact boson $X\sim X+2\pi R$, a closed spatial circuit of the worldsheet may wind around the target circle:Single-valued target-space wavefunctions quantize the zero-mode momentum as $p=n/R$, $n\in \mathbb{Z}$. In the conventionthe Hamiltonian and worldsheet momentum becomeThe $nw$ term is the zero-mode contribution to closed-string level matching.

The momentum integral is replaced by the sum over momentum and winding modes. Combining the zero modes with the oscillator trace givesEquivalently,with the left/right convention of part (iv). The answer is invariant under the T-duality $R\leftrightarrow 1/R$ accompanied by $n\leftrightarrow w$ in these units.