TITLE: Motivation of Virasoro algebra
QUESTION [26 upvotes]: I have a question on definition/motivation of Virasoro algebra. Recall that Virasoro algebra is an infinite Lie algebra generated by elements $L_n$ $(n\in \mathbb{Z})$ and $c$ over $\mathbb{C}$ with relations
$$
[L_m,L_n]=(m-n)L_{m+n}+\frac{c}{12}(m^3-m)\delta_{m+n,0}.
$$
A typical explanation of this definition is the following. 

Define vector fields $l_n=-z^n\frac{\partial}{\partial z}$ on $\mathbb{C}\setminus \{0\}$. They form a Lie algebra of infinitesimal conformal transformation 
  $$
[l_m,l_n]=(m-n)l_{m+n}.
$$ 
  So the Virasoro algebra is a central extension of this algebra by $c$. $c$ is called the central charge.

My questions are 

How can one see that the Lie algebra above is associate to infinitesimal conformal transformation?
What is the central charge $c$ intuitively? Why are we interested in such a central extension?

As to second question, I don't have enough physics background to check what the central charge $c$ means in physics literature. 
At this point, I don't have any intuition and have trouble in digesting the concept. I would really appreciate your help.

REPLY [3 votes]: How to see that $-z^{m+1}\frac{\partial }{\partial z}$ is related to an infinitesimal conformal transformation:
With $\omega=\frac{z^{-m}}{m}$ and $z=(m \cdot \omega)^{\frac{-1}{m}}$,  the exponential mapping gives
$exp[-t\cdot z^{m+1}\frac{\partial }{\partial z}]f(z)=exp[t\cdot \frac{\partial }{\partial \omega}]f[(m \cdot \omega)^{\frac{-1}{m}}]=f[(m \cdot (\omega+t))^{\frac{-1}{m}}]=f\left [\frac{z}{(1+\ m \cdot t \cdot z^m)^{\frac{1}{m}}}  \right ]=f(g_{m}(z,t)).$
Consistently, $g_{m}(g_{m}(z,s),t)=g_{m}(z,s+t).$
So the exponential mapping induced by the tangent vector results in composition of $f$ with $g_{m}(z,t)=z-\ t\cdot z^{m+1} +\ ....$.
Then to first order in $t$, a Taylor series expansion in $t$ about $t=0$ of the infinitesimal composition gives
$f(z-\ t\cdot z^{m+1})\approx f(z)-{f}'(z)z^{m+1}\cdot t=(1-\ t \cdot z^{m+1}\frac{\partial }{\partial z})f(z),$
and in the domain of analyticity of $g_{m}(z,t)$, the mappings are conformal. 
Some interesting associated combinatorics:
1) $g_m$ is related to the e.g.f. for planar m-ary trees and double factorials (m=1), triple factorials (m=2), quartic (m=3), etc. (Cf. OEIS A094638)
2) The compositional inverse of $h(x)= x-\ t\cdot x^{m+1}$ gives a generating function for the Fuss-Catalan numbers (e.g., OEIS A001764).
Edit (April 2018): 
More generally, let $h(h^{-1}(y))=y$ and $g(z) = 1/(dh(z)/dz)$. Then
$$ exp[t \cdot g(z) \frac{d}{dz}]f(z) = f[h^{-1}(t + h(z))] = f(W(t,z)), $$ 
which is analytic about $t=0$. Note that $W(t,W(s,z))=W(s+t,z)$ is the flow map described in OEIS A145271 related to the refined Eulerian numbers and other number arrays related to important combinatorial structures--the associahedra and noncrossing partitions among others.
About $t=0$,
$$f[W(t,z)] = f(z) + f'(z)g(z)t + \cdots = (1 + t \cdot g(z) \frac{d}{dz}) f(z) + \cdots .$$
Since the composition and product of analytic functions are analytic, conformality is locally preserved.