Virasoro algebra

Unitarity and Ghost

• 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

 

 

Verma module

• start with given c and h
• construct \(M(c,h)\)
• 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? 

 

unitary representations

• 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
      

 

Discrete series unitary representations

• c<1 case

 

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

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

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

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

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

 

 

관련된 다른 주제들

• Vertex Algebras
• BRST Cohomology

 

표준적인 도서 및 추천도서

• http://search.gigapedia.com/?q=
• http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=

 

 

참고할만한 자료

• Quantum Group Structure of the q-Deformed Virasoro Algebra
  
  • Haihong Hu
• http://ko.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/Virasoro_algebra
•  
• http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q=
• http://www.ams.org/mathscinet/search/publications.html?pg4=AUCN&s4=&co4=AND&pg5=TI&s5=&co5=AND&pg6=PC&s6=&co6=AND&pg7=ALLF&co7=AND&Submit=Search&dr=all&yrop=eq&arg3=&yearRangeFirst=&yearRangeSecond=&pg8=ET&s8=All&s7=