Virasoro algebra

Unitarity means the inner product in the space of states is positive definite (or semi-positive definite)
A state with negative norm is called a ghost.
If a ghost is found on any level the represetation cannot occur in a unitary theory

start with given c and h
construct \(M(c,h)\)
- quotients from the Universal enveloping algebra
- tensor product from the one dimensional Borel subalgebra representations
there exists a unique contravariant hermitian form
a natural grading given by the \(L_0\)-eigenvalues
contains a unique maximal submodule, and its quotient is the unique (up to isomorphism) irreducible representation with highest weight
When is \(M(c,h)\) unitary?

They are classified by c>1 and c<1 case.
- \(c> 1, h > 0\)
- \(c\geq 1, h \geq 0\)
- \(c>0, h >0\) with Kac determinant condition
  - called the discrete series representations

\(m= 2, 3, 4.\cdots\)

\(c = 1-{6\over m(m+1)} = 0,\quad 1/2,\quad 7/10,\quad 4/5,\quad 6/7,\quad 25/28, \ldots\)

\(h_{r,s}(c) = {((m+1)r-ms)^2-1 \over 4m(m+1)}\)

\(r = 1, 2, 3,\cdots,m-1\)

\(s= 1, 2, 3,\cdots, r\)

the highest weight representation V of an affine Kac-Moody algebra gives unitary representation of the Virasoro algebra
This is because V is unitary highest weigh representation of the AKMA.
Read chapter 4 of Kac-Raina on Wedge space

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