"Virasoro algebra"의 두 판 사이의 차이

2009년 7월 29일 (수) 21:18 판

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

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

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

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

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

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

Daniel Friedan, Zongan Qiu, and Stephen Shenker (1984) showed that these conditions are necessary
Peter Goddard, Adrian Kent and David Olive (1986) used the coset construction or GKO construction (identifying unitary representations of the Virasoro algebra within tensor products of unitary representations of affine Kac-Moody algebras) to show that they are sufficient.

constructed by GKO construction which uses the representation theory of affine Kac-Moody algebras

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

@@ 4번째 줄: / 4번째 줄: @@
 * 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
@@ 65번째 줄: / 69번째 줄: @@
-<h5>No-Gho</h5>
+<h5>No-Ghost theorem</h5>
+* refer to the [[3917551|No Ghost theorem]].