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\begin{center}
	 { \Large \bf 
Topics for oral qualifying exam for Liang Kong\\
\vskip .5em
 Fall, 2001} 
\end{center}

\begin{center}
{\bf Major topic: Vertex operator algebras}
\end{center}

The general theory of vertex operator algebras, basic examples, the
basic geometry of vertex operator algebras as presented in \cite{FLM}
(Chapters 1--8), \cite{FHL}, \cite{H} and \cite{L}.  The specific
topics are:

\begin{enumerate}

\item Definitions and properties.

\begin{enumerate}

\item Formal calculus.

\item The notions of vertex algebra and of vertex operator algebra,
and basic properties.

\item Rationality, commutativity and associativity (various formulations,
including ``weak'' formulations.

\end{enumerate}


\item Examples of vertex (operator) algebras.

\begin{enumerate}

\item Vertex (operator) algebras based on
Heisenberg Lie algebras.

\item Vertex (operator) algebras based on affine Lie algebras.

\item Vertex (operator) algebras on the Virasoro
algebra.

\item Vertex (operator) algebras on even lattices.

\end{enumerate}

\item Representations of vertex (operator) algebras.

\begin{enumerate}

\item The notion of module and basic properties.

\item The relations between this notion and the topics listed
above, including the construction of modules from mutually local
operators.

\item The notions of intertwining operator and fusion rule,
basic properties.

\item Examples of modules for the vertex operator algebras listed above.

\end{enumerate}

\item The geometry of vertex operator algebras.

\begin{enumerate}

\item The moduli spaces of spheres with tubes, the sewing operation,
examples for the sewing operation.



\item The geometric interpretations of vertex operators and 
vacua, the geometric meanings of commutativity, associativity,
skew-symmetry, $L(-1)$-derivative property, vacuum property and 
creation property.

\item The notion of geometric vertex operator algebra and the 
statement of the isomorphism
theorem.

\item Determinant lines of Fredholm operators, determinant lines over
Riemann surfaces with parametrized boundaries, relation to the 
central charges of vertex operator algebras.



\end{enumerate}

\end{enumerate}

\begin{center}
{\bf Minor topic: Quantum field theory}
\end{center}

Basic quantum field theory as presented in \cite{IZ} and \cite{PS}.
The specific topics are:

\begin{enumerate}
\item Classical field theory 

\begin{enumerate}

\item Action, Lagrangian, action principle and Euler-Lagrange Equation. 

\item Canonical momentum, Hamiltonian. 

\item Symmetry, Noether current, stress-energy tensor.  

\end{enumerate}


\item Canonical quantization of free theories. 



\item Interaction quantum theory.  

\begin{enumerate}

\item $n$-point Green's functions, Wick's theorem. 

\item Feynman diagrams.  

\item Loop integral computation, dimensional regularization. 

\end{enumerate}

\item Functional integral formulation of quantum field theory.  

\begin{enumerate}

\item Partition function, Feynman-Kac formula.  

\item Feynman diagram.   

\item Equation of motion, Ward's identity.  

\end{enumerate}

\item Systematic renormalization.   

\begin{enumerate}

\item Renormalizability, counterterms.  

\item $1$-loop result of the $\phi^4$ theory.  

\item Callan-Symanzyk equation.  

\item Running coupling constant, $\beta(g)$, and its one-loop
calculation. (optional)

\end{enumerate}
\end{enumerate}








\begin{thebibliography}{FLM2}

\bibitem[FHL]{FHL}
I. Frenkel, Y.-Z. Huang and J. Lepowsky, On Axiomatic Approaches
to Vertex Operator Algebras and Modules, Memoirs Amer. Math. Soc. 104
(1993).

\bibitem[FLM]{FLM} 
I. Frenkel, J. Lepowsky and A. Meurman, Vertex Operator Algebras
and the Monster, Academic Press, 1988.


\bibitem[H]{H}
Y.-Z. Huang, Two-dimensional conformal geometry and vertex operator
algebras, Progress in Math., Vol. 148, Birkh\"{a}user Boston, 1997.


\bibitem[L]{L}
H. Li, Local systems of vertex operators, vertex superalgebras and
modules, J. Pure and Applied Algebra 109 (1996), 143-195.

\bibitem[IZ]{IZ}
C. Itzykson and J. B. Zuber, Quantum field
theory, International Series in Pure and Applied Physics.
McGraw-Hill International Book Co., New York, 1980. 

\bibitem[PS]{PS} 
M. E. Peskin and D. V. Schroeder, An introduction to quantum
field theory, edited and with a foreword by
D. Pines, Addison-Wesley Publishing Company, Advanced Book Program,
Reading, MA, 1995.

\end{thebibliography}
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