There seem to be two different things one must consider when representing a symmetry group in quantum mechanics:

• The universal cover: For instance, when representing the rotation group $\mathrm{SO}(3)$, it turns out that one must allow also $\mathrm{SU}(2)$ representations, since the negative sign a "$2\pi$ rotation" induces in $\mathrm{SU}(2)$ is an overall phase that doesn't change the physics. Equivalently, all representations of the Lie algebra are what we seek. ($\mathfrak{so}(3) = \mathfrak{su}(2)$, but although every representation of the algebra is one of the universal cover, not every representation of the algebra is one of $\mathrm{SO}(3)$.)

• Central extensions: In conformal field theory, one has classically the Witt algebra of infinitesimal conformal transformations. From the universal cover treatment one is used to in most other cases, one would expect nothing changes in the quantum case, since we are already seeking only representation of an algebra. Nevertheless, in the quantization process, a "central charge" appears, which is often interpreted to arise as an "ordering constant" for the now no longer commuting fields, and we have to consider the Virasoro algebra instead.

What is going on here? Is there a way to explain both the appearence of universal covers and central extensions in a unified way?

There seem to be two different things one must consider when representing a symmetry group in quantum mechanics:

• The universal cover: For instance, when representing the rotation group $\mathrm{SO}(3)$, it turns out that one must allow also $\mathrm{SU}(2)$ representations, since the negative sign a "$2\pi$ rotation" induces in $\mathrm{SU}(2)$ is an overall phase that doesn't change the physics. Equivalently, all representations of the Lie algebra are what we seek. ($\mathfrak{so}(3) = \mathfrak{su}(2)$, but although every representation of the algebra is one of the universal cover, not every representation of the algebra is one of $\mathrm{SO}(3)$.)

• Central extensions: In conformal field theory, one has classically the Witt algebra of infinitesimal conformal transformations. From the universal cover treatment one is used to in most other cases, one would expect nothing changes in the quantum case, since we are already seeking only representation of an algebra. Nevertheless, in the quantization process, a "central charge" appears, which is often interpreted to arise as an "ordering constant" for the now no longer commuting fields, and we have to consider the Virasoro algebra instead.

The question is: What is going on here? Is there a way to explain both the appearence of universal covers and central extensions in a unified way?

There seem to be two different things one must consider when representing a symmetry group in quantum mechanics:

• The universal cover: For instance, when representing the rotation group $\mathrm{SO}(3)$, it turns out that one must allow also $\mathrm{SU}(2)$ representations, since the negative sign a "$2\pi$ rotation" induces in $\mathrm{SU}(2)$ is an overall phase that doesn't change the physics. Equivalently, all representations of the Lie algebra are what we seek. ($\mathfrak{so}(3) = \mathfrak{su}(2)$, but although every representation of the algebra is one of the universal cover, not every representation of the algebra is one of $\mathrm{SO}(3)$.)

• Central extensions: In conformal field theory, one has classically the Witt algebra of infinitesimal conformal transformations. From the universal cover treatment one is used to in most other cases, one would expect nothing changes in the quantum case, since we are already seeking only representation of an algebra. Nevertheless, in the quantization process, a "central charge" appears, which is often interpreted to arise as an "ordering constant" for the now no longer commuting fields, and we have to consider the Virasoro algebra instead.

The question is: What is going on here? Is there a way to explain both the appearence of universal covers and central extensions in a unified way?

There seem to be two different things one must consider when representing a symmetry group in quantum mechanics:

• The universal cover: For instance, when representing the rotation group $\mathrm{SO}(3)$, it turns out that one must allow also $\mathrm{SU}(2)$ representations, since the negative sign a "$2\pi$ rotation" induces in $\mathrm{SU}(2)$ is an overall phase that doesn't change the physics. Equivalently, all representations of the Lie algebra are what we seek. ($\mathfrak{so}(3) = \mathfrak{su}(2)$, but although every representation of the algebra is one of the universal cover, not every representation of the algebra is one of $\mathrm{SO}(3)$.)

• Central extensions: In conformal field theory, one has classically the Witt algebra of infinitesimal conformal transformations. From the universal cover treatment one is used to in most other cases, one would expect nothing changes in the quantum case, since we are already seeking only representation of an algebra. Nevertheless, in the quantization process, a "central charge" appears, which is often interpreted to arise as an "ordering constant" for the now no longer commuting fields, and we have to consider the Virasoro algebra instead.

What is going on here? Is there a way to explain both the appearence of universal covers and central extensions in a unified way?