Polyakov Action & Conformal Gauge

1. The Action for a Relativistic String

How do we describe the dynamics of a string? For a point particle, the action is proportional to the length of its worldline—the path of least proper time. The natural generalization for a string is that its action should be proportional to the area of the two-dimensional surface it sweeps out in spacetime, called the worldsheet.

This principle gives rise to the Nambu-Goto action. While geometrically intuitive, it contains a square root, which is notoriously difficult to work with in the path integral formalism you're familiar with from QFT.

To address this, we use a classically equivalent but more tractable formulation: the Polyakov action. This action introduces an auxiliary dynamical metric, $g_{\alpha \beta }(\tau ,\sigma )$, on the worldsheet itself. From the worldsheet perspective, this looks like a theory of $D$ scalar fields $X^{\mu }$ (representing the string's position in $D$-dimensional spacetime) coupled to 2D gravity.

This idea of introducing an auxiliary field to create a polynomial action is a powerful trick. Your experience with GR makes the concept of a dynamical metric, even on a 2D worldsheet, a familiar one. Here, however, $g_{\alpha \beta }$ is not the metric of spacetime but a tool for describing the string's dynamics within spacetime.

2. The Symmetries of the Polyakov Action

The Polyakov action has a rich symmetry structure, which is the key to unlocking its connection to CFT. There are three fundamental symmetries.

1. Poincaré Invariance: The action is invariant under translations $X^{\mu }\to X^{\mu }+c^{\mu }$ and Lorentz transformations $X^{\mu }\to {\Lambda }_{\nu }^{\mu }{X}^{\nu }$ of the target spacetime. From the 2D worldsheet's perspective, this is a global symmetry.

2. Reparameterization Invariance (Diffeomorphisms): The physics cannot depend on our choice of coordinates $(\tau ,\sigma )$ on the worldsheet. This is a gauge symmetry, allowing us to make arbitrary coordinate transformations ${\sigma }^{\alpha }\to {\tilde{\sigma }}^{\alpha }(\sigma )$.

3. Weyl Invariance: This is a new and crucial gauge symmetry, specific to the 2D Polyakov action. The action is invariant under a local rescaling of the worldsheet metric:

   $$g_{\alpha \beta }(\sigma )\to {\Omega }^2(\sigma )g_{\alpha \beta }(\sigma )$$

   where $\Omega (\sigma )$ is an arbitrary positive function on the worldsheet. The $X^{\mu }$ fields are inert under this transformation. This symmetry holds because the $\sqrt{-g}$ term scales as ${\Omega }^2$ while the inverse metric $g^{\alpha \beta }$ scales as ${\Omega }^{-2}$, leading to a perfect cancellation. This local scale invariance is the hallmark of a conformal theory.

To solidify your understanding of these symmetries, the following video provides an excellent and detailed discussion.

3. Fixing the Gauge: The Emergence of a CFT

We now use the powerful gauge symmetries to simplify the Polyakov action. The worldsheet metric $g_{\alpha \beta }$ is a 2x2 symmetric matrix, so it has 3 independent components ($g_{00},g_{01},g_{11}$). We have:

• 2 degrees of freedom from reparameterization invariance (we can choose two coordinate conditions).
• 1 degree of freedom from Weyl invariance (we can rescale the metric locally).

Since we have 3 gauge freedoms and 3 metric components, we can completely fix the worldsheet metric to a convenient form. This process is called gauge fixing.

1. Using Reparameterization Invariance: We can always find a coordinate system $(\tau ,\sigma )$ such that the metric is conformally flat, meaning it is proportional to the flat Minkowski metric ${\eta }_{\alpha \beta }$:

   $$g_{\alpha \beta }(\tau ,\sigma )=e^{2\varphi (\tau ,\sigma )}{\eta }_{\alpha \beta }$$

   where ${\eta }_{\alpha \beta }=\text{diag}(-1,1)$. This choice is known as conformal gauge.

2. Using Weyl Invariance: We still have the freedom to perform a Weyl transformation. We can choose ${\Omega }^2=e^{-2\varphi }$ to completely remove the conformal factor, leaving us with:

   $$g_{\alpha \beta }(\tau ,\sigma )={\eta }_{\alpha \beta }$$

After fixing the gauge, the Polyakov action simplifies dramatically.

By substituting $g_{\alpha \beta }={\eta }_{\alpha \beta }$ into the Polyakov action, we arrive at the action for $D$ free, massless 2D scalar fields:

The equation of motion for $X^{\mu }$ is the simple 2D wave equation: ${\partial }_{\alpha }{\partial }^{\alpha }{X}^{\mu }=({\partial }_{\sigma }^2-{\partial }_{\tau }^2)X^{\mu }=0$.

This looks almost too simple. The non-trivial physics is hidden in the constraints that come from the equation of motion for the metric itself. Varying the Polyakov action with respect to $g^{\alpha \beta }$ gives the worldsheet stress-energy tensor, $T_{\alpha \beta }$. The equation of motion for the metric is simply $T_{\alpha \beta }=0$. In the conformal gauge, this tensor is:

The constraint $T_{\alpha \beta }=0$ yields two independent equations, known as the Virasoro constraints:

1. $T_{01}=\dot{X}\cdot X^{\prime }=0$
2. $T_{00}=T_{11}=\frac{1}{2}({\dot{X}}^2+X^{\prime 2})=0$

Here, $\dot{X}={\partial }_{\tau }X$ and $X^{\prime }={\partial }_{\sigma }X$. The fact that $T_{00}=T_{11}$ is a consequence of the tracelessness of the stress-energy tensor, $T_{\alpha }^{\alpha }=0$, which is a signature of conformal invariance in 2D.

Conclusion

We have achieved our goal for today. We have shown that the physics of a classical bosonic string is described by a simple-looking 2D quantum field theory—specifically, a theory of $D$ free bosons—subject to a crucial set of constraints.

Key Takeaways:

• The Polyakov action provides a way to describe string dynamics that is classically equivalent to the Nambu-Goto (worldsheet area) action but is better suited for quantization.
• The action possesses three key symmetries: Poincaré invariance (global), reparameterization invariance (gauge), and Weyl invariance (gauge).
• The combination of reparameterization and Weyl invariance allows us to fix the conformal gauge, where the worldsheet metric becomes the flat Minkowski metric $g_{\alpha \beta }={\eta }_{\alpha \beta }$.
• In this gauge, the action simplifies to that of D free, massless scalar fields, whose equations of motion are the simple wave equation.
• The original equations of motion for the worldsheet metric survive as the Virasoro constraints ($T_{\alpha \beta }=0$), which must be imposed on the system.

This result is profound: the worldsheet of a string is a natural home for a 2D Conformal Field Theory. The fields of this CFT are the spacetime coordinates of the string itself, and the physical states are restricted by the condition that the worldsheet stress-energy tensor must vanish.

Preview of the Next Lesson:

The constraint $T_{\alpha \beta }=0$ is a classical statement. When we move to the quantum theory, these constraints cannot be imposed so simply. Instead, they will become conditions on the physical states of the string. Imposing these constraints at the quantum level will lead to one of the most famous results in string theory: the requirement that the theory must live in a critical dimension, $D=26$. In our next lesson, we will begin the process of quantizing this system and see how this remarkable conclusion arises from the structure of the Virasoro algebra.