# Virasoro operators in bosonic String Theory

• Orbb
In summary, in a recent lecture on String Theory, a divergent sum of 1+2+3+... was encountered when calculating the zero mode Virasoro operator in bosonic String Theory. This sum was set equal to a finite negative constant based on a comparison with the definition of the Zeta function. However, there may be further justifications for taking this step. A safe way to compute the central charge is to check how the Virasoro generators act on the vacuum, using a normal-ordered expression that results in finite sums and eliminates the need for regularization.
Orbb
In a recent lecture on String Theory, we encountered the divergent sum 1+2+3+... when calculating the zero mode Virasoro operator in bosonic String Theory. This divergent sum is then set equal to a finite negative constant - the argument for doing so was a comparison with the definition of the Zeta function. However, is still have trouble with this argument, and my question is, wether there are any further justifications for taking this step. Thank you for your answers!

A safe way to compute the central charge is to check how the Virasoro generators act on the vacuum. The normal-ordered expression

L_m = sum_{n=-infty}^infty : a_{m-n} a_n :

is equivalent to the following relations (perhaps modulo some signs that I don't have the energy to check) (and m > 0 in the first three lines):

L_m |0> = sum_{n=0}^m a_{m-n} a_n |0>

L_0 |0> = h |0>

L_-m |0> = 0

[L_m, a_n] = (m-n) a_{m+n}

The point is that in the second formulation, all sums are finite, so you don't have to worry about regularization.

## 1. What are Virasoro operators in bosonic String Theory?

Virasoro operators are mathematical objects used in bosonic string theory to describe the symmetries of the theory. They are generators of the conformal transformations which preserve the shape and structure of the string. They play a crucial role in understanding the dynamics of the string and its interactions.

## 2. How are Virasoro operators derived?

Virasoro operators are derived from the Polyakov action, which is the classical action for the bosonic string theory. By varying this action with respect to the string coordinates, one can obtain the equations of motion for the string. These equations can then be rewritten in terms of the Virasoro operators, which are defined as the coefficients of the expansion of the energy-momentum tensor.

## 3. What is the significance of Virasoro operators in bosonic String Theory?

Virasoro operators are important because they generate the symmetries of the theory, which are essential for understanding the properties and behavior of the string. They also play a crucial role in the quantization of the string, as they correspond to the physical states of the theory and their algebra determines the spectrum of the string.

## 4. How do Virasoro operators relate to conformal symmetry?

Virasoro operators are closely related to conformal symmetry in string theory. Conformal transformations are those that preserve the shape and structure of the string, and the Virasoro operators generate these transformations. In fact, the Virasoro algebra is isomorphic to the algebra of conformal transformations, making them an important tool for studying the conformal symmetry of the theory.

## 5. Are Virasoro operators only applicable to bosonic String Theory?

No, Virasoro operators can also be used in other string theories, such as superstring theory and heterotic string theory. In these theories, the Virasoro algebra is extended to include fermionic operators as well, but the basic principles and techniques for using Virasoro operators remain the same. They are also applicable in other areas of theoretical physics, such as conformal field theory and two-dimensional quantum field theory.

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