Worldsheet CFT and Critical Dimension of Bosonic String

1. The Worldsheet CFT: A Theory of Free Bosons

As we established, after fixing the conformal gauge, the string worldsheet theory is described by the action for $D$ free, massless scalar fields:
$S=-\frac{1}{4\pi {\alpha }^{\prime }}\int d^2\sigma {\eta }^{\alpha \beta }{\partial }_{\alpha }{X}^{\mu }{\partial }_{\beta }{X}_{\mu }$
where $(\tau ,\sigma )$ are the worldsheet coordinates and $\mu =0,\dots ,D-1$ are spacetime indices. The equation of motion is the 2D wave equation, ${\partial }_{\alpha }{\partial }^{\alpha }{X}^{\mu }=0$.

The general solution for a closed string can be decomposed into left-moving and right-moving waves, which are then expanded into Fourier modes.

This CFT is our object of study. The fields $X^{\mu }$ are the fundamental fields of the CFT. The oscillators ${\alpha }_n^{\mu }$ and ${\tilde{\alpha }}_n^{\mu }$ will become the creation and annihilation operators of our quantum theory.

2. Old Covariant Quantization and the Virasoro Algebra

We now quantize the theory using the "old covariant quantization" method, which is analogous to the Gupta-Bleuler quantization of QED you are familiar with. The key steps are:

1. Promote the Poisson brackets for the oscillator modes to commutators.
2. Represent these operators on a Fock space.
3. Impose the classical constraints $L_m={\tilde{L}}_m=0$ as conditions on the physical states of the Hilbert space.

This last step is where the subtleties arise.

The Quantum Virasoro Algebra

Upon quantization, the ${\alpha }_n^{\mu }$ become operators satisfying:
$[{\alpha }_m^{\mu },{\alpha }_n^{\nu }]=m{\delta }_{m+n,0}{\eta }^{\mu \nu }$
$[{\tilde{\alpha }}_m^{\mu },{\tilde{\alpha }}_n^{\nu }]=m{\delta }_{m+n,0}{\eta }^{\mu \nu }$
The operators ${\alpha }_n^{\mu }$ for $n>0$ are annihilation operators, and ${\alpha }_{-n}^{\mu }$ for $n>0$ are creation operators.

The classical Virasoro generators $L_m=\frac{1}{2}{\sum }_k{\alpha }_{m-k}\cdot {\alpha }_k$ are now operators. However, for $L_0$, this definition is ill-defined due to an ordering ambiguity of non-commuting operators. We must normal-order it by placing all creation operators to the left of annihilation operators. This procedure introduces an undetermined constant, $a$, called the normal-ordering constant or intercept.
$L_0=\frac{{\alpha }^{\prime }}{2}{p}^2+{\sum }_{k=1}^{\infty }{\alpha }_{-k}\cdot {\alpha }_k⟶L_0^{\text{op}}=\frac{{\alpha }^{\prime }}{2}{p}^2+{\sum }_{k=1}^{\infty }{\alpha }_{-k}\cdot {\alpha }_k-a$

This ordering issue has a profound effect on the algebra of the generators. While the classical generators form the Witt algebra, the quantum operators obey the Virasoro algebra, which is a central extension of the Witt algebra:
$[L_m,L_n]=(m-n)L_{m+n}+\frac{c}{12}m(m^2-1){\delta }_{m+n,0}$
The new term is the central charge, and for our CFT of $D$ free bosons, it turns out that $c=D$.

3. Physical States and the Critical Dimension

Because the Virasoro algebra is not trivial, we cannot impose $L_m=0$ as operator equations. Instead, we impose weaker conditions on the physical states $|\text{phys}⟩$:

• $L_m|\text{phys}⟩=0$ for $m>0$
• $(L_0-a)|\text{phys}⟩=0$
• And identical conditions for the tilded operators, ${\tilde{L}}_m$.

The condition $(L_0-a)|\text{phys}⟩=0$ gives the quantum mass-shell condition. For a state at oscillator level $N={\sum }_{k>0}{\alpha }_{-k}\cdot {\alpha }_k$, the mass-squared is:
$M^2=-p^2=\frac{2}{{\alpha }^{\prime }}(N-a)$
The condition $(L_0-{\tilde{L}}_0)|\text{phys}⟩=0$ implies $N=\tilde{N}$, which is known as the level-matching condition.

The No-Ghost Theorem

The central challenge is the presence of "ghosts"—negative-norm states created by the timelike oscillators ${\alpha }_{-n}^0$, which violate the unitarity of the theory. The Virasoro constraints must be powerful enough to ensure that for any physical process, these ghosts decouple and are never produced as final states.

A careful analysis, known as the Goddard-Thorn no-ghost theorem, reveals that the spectrum of physical states is free of ghosts if and only if:

1. The spacetime dimension is $D=26$.
2. The normal-ordering constant is $a=1$.

Let's unpack why this happens.

The core of the argument is that in $D=26$ with $a=1$, the gauge symmetry generated by the Virasoro algebra is exceptionally large. It creates a vast number of "null states"—states that are both physical ($L_n|\psi ⟩=0$) and spurious (orthogonal to all physical states). These null states have zero norm and decouple from all physical amplitudes, effectively acting as pure gauge degrees of freedom. Their existence is crucial for projecting out all the negative-norm ghost states. In any other dimension, this perfect cancellation fails, and ghosts remain in the physical spectrum.

The video below provides a more detailed, calculation-oriented derivation of the Virasoro algebra and the physical state conditions. While the notes give the conceptual overview, the video shows the machinery in action.

Conclusion

We have now fully identified the worldsheet theory of the bosonic string as a 2D CFT with central charge $c=D$, and we have seen its most dramatic consequence.

Key Takeaways:

• The worldsheet CFT for the bosonic string consists of $D$ free scalar fields, whose quantization leads to a Fock space built by oscillator operators ${\alpha }_n^{\mu }$.
• The classical Virasoro constraints become quantum conditions on physical states. Due to operator ordering, the algebra of the Virasoro generators is centrally extended, becoming the Virasoro algebra with central charge $c=D$.
• The physical states $|\text{phys}⟩$ must be annihilated by $L_n$ for $n>0$ and satisfy the mass-shell condition $(L_0-a)|\text{phys}⟩=0$.
• The requirement that the theory be unitary (free of negative-norm "ghost" states) is only satisfied if the spacetime dimension is $D=26$ and the normal ordering constant (intercept) is $a=1$.

This result is a cornerstone of string theory. It demonstrates how a seemingly simple theoretical model, when subjected to the strict demands of quantum consistency, can make a definite and surprising prediction about the structure of spacetime itself.

Preview of the Next Lesson:

Now that we have established the rules for physical states, our next step is to understand how to represent these states—the tachyon, the massless states like the graviton, etc.—within the framework of the 2D worldsheet CFT. This will lead us to the concept of vertex operators and the powerful state-operator correspondence, a central pillar of conformal field theory.