Relationship between the Witt algebra and vector fields on the circle

Question

I have seen some (apparently contradictory) claims by mathematicians and physicists regarding the existence of an (infinite dimensional) Lie group whose Lie algebra is the Virasoro algebra.

The Virasoro algebra is a central extension of the Witt algebra, which is slightly simpler to describe and for which the same confusion is present, so I will formulate my question in that setting.

We define the Witt algebra to be $\mathfrak g:=\mathbb R[L_n\colon n\in\mathbb Z]$ where $L_n$ is the operator on the Laurent polynomial ring $\mathbb R[z,z^{-1}]$ given by $L_np(z)=z^{1-n}p'(z)$. When equipped with the commutator of operators as Lie bracket, $\mathfrak g$ becomes an (infinite dimensional) Lie algebra.

The Witt algebra can also be described abstractly, as the Lie algebra generated by elements $(L_n)_{n\in\mathbb Z}$ satisfying the commutation relations $$ [L_m,L_n]=(m-n)L_{m+n},\qquad m,n\in\mathbb Z. $$

There is also a complex Witt algebra, $\mathfrak g_{\mathbb C}$, obtained by tensoring with $\mathbb C$. Equivalently, this is the algebra generated as above by operators acting on the Laurent polynomial ring $\mathbb C[z,z^{-1}]$.

Question 1. Are $\mathfrak g$ and $\mathfrak g_{\mathbb C}$ dense subsets of reasonable Banach algebras?

Regardless of the answer to the previous question, both $\mathfrak g$ and $\mathfrak g_{\mathbb C}$ can be realized as dense subsets of reasonable locally convex topological vector spaces of vector fields.

For a compact smooth manifold $M$, the group $\text{Diff}(M)$ of smooth diffeomorphisms can be given a Lie group structure such that the Lie algebra $\text{Vect}(M)$ consists of the smooth vector fields on $M$.

If I am understanding several claims by physicists correctly, taking $M=\mathbb S^1$ in the previous paragraph should yield a Lie group whose Lie algebra is closely related to either $\mathfrak g$ or $\mathfrak g_{\mathbb C}$. This leads me to the following questions.

Question 2. Does $\text{Vect}(\mathbb S^1)$ contain a Lie subalgebra isomorphic to $\mathfrak g$?

Question 3. What is the precise relationship between $\text{Vect}(\mathbb S^1)$ and $\mathfrak g_{\mathbb C}$?

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References: The article Kac-Moody and Virasoro algebras in relation to quantum physics by Goddard and Olive claims that the diffeomorphism group of the circle has the infinitesimal symmetries of the Witt algebra. Schottenloher's book A Mathematical Introduction to Conformal Field Theory claims, in section 5.4, that there is no complex Virasoro group and also no complex Witt group. Remarks on infinite dimensional Lie groups by Milnor discusses infinite dimensional Lie groups modeled upon locally convex topological Lie algebras and, in particular, section 6 of that article is devoted to groups of smooth diffeomorphisms of spheres.

I have asked a related question on the physics stackexchange.

Answer 1

The following answers Question 2 and 3:

For 2, $\frak{g}$ is a dense Lie subalgebra of Vect($S^1$). Tensoring with $\mathbb{C}$ gives you 3.

Identify an infinitestimal diffeomorphism on the circle to a vector field on $S^1$. Visualize a vector field as a field of tangent vectors. Each tangent vector is a multiple of $\partial_\theta$, so a vector field can be described by $A(\theta)\partial_\theta$, where $A$ is a smooth function on $S^1$. Fourier theory lets you express $A$ in terms of its Fourier series, so the space of vector fields is generated by $\{e^{in\theta}\ | n\in \mathbb{Z}\}$. Define

$$ L_n = -ie^{in\theta}\partial_\theta $$

It's easy to check that these satisfy the generator relations you gave: for all smooth functions $f$ on $S^1$,

$$ [L_n,L_m]f = e^{i(n+m)\theta} i (n-m) f = (m-n) L_{n+m} f, $$

and thus

$$ [L_n,L_m] = (m-n) L_{n+m}. $$