Constraints

The gauge-fixed world-sheet metric is $h^{ab}=\left(\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right)$

By imposing the equations of motion, we obtain $T_{ab}\stackrel{!}{=}0$ as a constraint in the canonical formalism. Then,

$T_{00}=T_{11}=\frac{1}{2}({\dot{X}}^2+X^{\prime 2})$

$T_{01}=T_{10}=\dot{X}.X^{\prime }$

In light-cone coordinates,

$T_{+-}=T_{-+}=0$

$T_{++}={\dot{X}}_L^2\text{ and }{T}_{--}={\dot{X}}_R^2$

Closed String

We define the Fourier modes of the closed strings constraints by

$L_m:=\frac{T}{2}{\int }_0^{\pi }d\sigma e^{-2im\sigma }{T}_{--}=\frac{1}{2}{\sum }_{n\in \mathbb{Z}}{\alpha }_{m-n}.{\alpha }_n$

${\tilde{L}}_n:=\frac{T}{2}{\int }_0^{\pi }d\sigma e^{2im\sigma }{T}_{++}=\frac{1}{2}{\sum }_{n\in \mathbb{Z}}{\tilde{\alpha }}_{m-n}.{\tilde{\alpha }}_n$

where ${\alpha }_0^{\mu }={\tilde{\alpha }}_0^{\mu }=\frac{1}{2}l{p}^{\mu }$.

The constraints are then $L_m={\tilde{L}}_m=0\forall m\in \mathbb{Z}$. These are known as the classical Virasoro constraints. They have the following Poisson brackets (Virasoro Algebra):

• $\{L_m,L_n\}=-i(m-n)L_{m+n}$
• $\{{\tilde{L}}_m,{\tilde{L}}_n\}=-i(m-n){\tilde{L}}_{m+n}$
• $\{L_m,{\tilde{L}}_n\}=0$.

The combinations of $L_0$ and ${\tilde{L}}_0$ are

• $L_0+{\tilde{L}}_0=\frac{{\alpha }^{\prime }}{2}{p}^2+{\sum }_{n=1}^{\infty }({\alpha }_{-n}.{\alpha }_n+{\tilde{\alpha }}_n.\widetilde{{\alpha }_{-n}})=H$
• $L_0-{\tilde{L}}_0={\sum }_{n=1}^{\infty }({\alpha }_{-n}.{\alpha }_n-{\tilde{\alpha }}_n.\widetilde{{\alpha }_{-n}})$

Mass-Shell Constraint

$H=L_0+{\tilde{L}}_0\stackrel{!}{=}0$

Then, $\frac{{\alpha }^{\prime }{M}^2}{2}={\sum }_{n=1}^{\infty }({\alpha }_{-n}{\alpha }_n+{\tilde{\alpha }}_n.{\tilde{\alpha }}_{-n})$

where we have considered the relativistic definition of mass $M^2:=-p^2$.

Level-Matching Constraint

$L_0-{\tilde{L}}_0\stackrel{!}{=}0$ is called level-matching. In terms of the mode expansion,

${\sum }_{n=1}^{\infty }{\alpha }_{-n}{\alpha }_n={\sum }_{n=1}^{\infty }{\tilde{\alpha }}_n{\tilde{\alpha }}_{-n}$.

Open String

The Neumann boundary conditions relate the left-moving and right-moving modes. Hence, there will be no level-matching constraint.

We have a single set of Virasoro generators

$L_m:=\frac{1}{2}{\sum }_{n\in \mathbb{Z}}{\alpha }_{m-n}.{\alpha }_n;{\alpha }_0^{\mu }:=l{p}^{\mu }$.

These generators satisfy the Poisson brackets $\{L_m,L_n\}=-i(m-n)L_{m+n}$.

Mass-Shell Constraint

${\alpha }^{\prime }{M}^2={\sum }_{n=1}^{\infty }{\alpha }_{-n}.{\alpha }_n$ with the same relativistic definition for mass.