[Paper Review] Representation of small conformal algebra

D.M. Belov|ArXiv.org|Oct 21, 2002

Advanced Topics in Algebra8 references3 citations

TL;DR

This paper establishes that the small conformal algebra generators $L_{-1}, L_0, L_1$ in the $κ$-basis of string field theory are realized as difference operators rather than kernel operators, resolving a prior issue with ill-definedness. The key result is explicit integral representations for $L_0$ and $L_{\pm1}$ acting on Schwartz functions, involving complex shifts and square roots of $\kappa$, which preserve the Virasoro algebra structure in the continuous basis.

ABSTRACT

In hep-th/0202087 it was argued that the operator L_0 is bad defined in kappa-basis as a kernel operator. Indeed, we show that L_0 is a difference operator. We also find a representation of L_1 and L_{-1} in a class of difference operators.

Motivation & Objective

• To resolve the issue of $L_0$ being ill-defined as a kernel operator in the $\kappa$-basis, as previously argued in hep-th/0202087.
• To construct a well-defined representation of the small conformal algebra $\{L_{-1}, L_0, L_1\}$ in the $\kappa$-basis using difference operators.
• To provide explicit functional expressions for $L_0$, $L_1$, and $L_{-1}$ acting on Schwartz functions, ensuring consistency with the Virasoro algebra relations.
• To demonstrate that the operators are defined on holomorphic functions in the strip $-2 < \Im\kappa < 2$, with controlled approach to poles via $i2^{-}$ notation.

Proposed method

• Derive the action of $L_0$ on one-particle states via the relation $[L_0, a^\dagger(f)] = a^\dagger(L_0[f])$, translating the Fock space action to the Schwartz space.
• Invert the operator $L_0$ to define a Green's function $G_0$ whose kernel is computed via contour integration of the eigenvector generating functions.
• Evaluate the kernel $G_0(\kappa, \kappa')$ using complex analysis, deforming contours around logarithmic branch cuts at $\pm i$, and applying known integral identities involving $\text{csch}$ and $\tanh$.
• Use the identity $\int_{-\infty}^\infty \mathscr{P}\frac{1}{\sinh 2u} e^{i\beta u} du = \pi i \tanh(\pi\beta/4)$ to evaluate the resulting integrals.
• Construct $L_1$ and $L_{-1}$ using the same strategy, leveraging their single-creation/single-annihilation structure to define their action via $[L_{\pm1}, a^\dagger(f)] = a^\dagger(L_{\pm1}[f])$.
• Verify that the resulting operators satisfy the correct $[L_{-1}, L_1] = 2L_0$ and $[L_0, L_{\pm1}] = \pm L_{\pm1}$ commutation relations in the continuous basis.

Experimental results

Research questions

• RQ1 Why is $L_0$ ill-defined as a kernel operator in the $\kappa$-basis, and can it be redefined in a consistent way?
• RQ2 Can the small conformal algebra $\{L_{-1}, L_0, L_1\}$ be consistently represented in the $\kappa$-basis using difference operators instead of integral kernels?
• RQ3 What is the explicit functional form of $L_0$, $L_1$, and $L_{-1}$ acting on Schwartz functions in the $\kappa$-basis?
• RQ4 How do the operators handle singularities and branch cuts in the complex $\kappa$-plane, particularly near $\kappa \pm 2i$?
• RQ5 Do the proposed difference operators preserve the standard Virasoro algebra relations in the continuous $\kappa$-basis?

Key findings

• The operator $L_0$ acts as a difference operator: $L_0[f](\kappa) = \frac{1}{4}\left[\sqrt{\kappa(\kappa + i2^{-})}f(\kappa + i2^{-}) + \sqrt{\kappa(\kappa - i2^{-})}f(\kappa - i2^{-})\right]$, defined on holomorphic functions in the strip $-2 < \Im\kappa < 2$.
• The operator $L_1$ is given by $L_1[f](\kappa) = -\frac{\kappa}{2}f(\kappa) + \frac{i}{4}\left[\sqrt{\kappa(\kappa + i2^{-})}f(\kappa + i2^{-}) - \sqrt{\kappa(\kappa - i2^{-})}f(\kappa - i2^{-})\right]$.
• The operator $L_{-1}$ is $L_{-1}[f](\kappa) = -\frac{\kappa}{2}f(\kappa) - \frac{i}{4}\left[\sqrt{\kappa(\kappa + i2^{-})}f(\kappa + i2^{-}) - \sqrt{\kappa(\kappa - i2^{-})}f(\kappa - i2^{-})\right]$.
• The Green's function $G_0$ for $L_0$ is explicitly computed as $G_0(\kappa, \kappa') = \left[\frac{\theta(\kappa)}{\kappa}\right]^{1/2}\left[\frac{\theta(\kappa')}{\kappa'}\right]^{1/2}\frac{1}{4\cosh\left[\frac{\pi}{4}(\kappa - \kappa')\right]}$, where $\theta(\kappa) = 2\tanh(\pi\kappa/4)$.
• The operators $L_0$, $L_1$, and $L_{-1}$ satisfy the correct Virasoro algebra commutation relations when applied to Schwartz functions, confirming consistency.
• The derivation confirms that $L_0$ cannot be represented as a distributional kernel but is instead a well-defined difference operator with controlled analytic behavior near poles at $\kappa \pm 2i$.