Elliptic Selberg Integrals
G. Felder $^{\,\star}$
L. Stevens$^{\, \diamond}$
and A. Varchenko$^{\,\diamond}$
%
math0103227.tex
\documentclass[12pt]{amsart}
\usepackage{amssymb}
%\usepackage[dvips]{graphics}
\textwidth 6.2truein
\oddsidemargin 0truein
\evensidemargin 0truein
\ifx\mathrm\undefined\let\mathrm\rm\fi
\ifx\mathbf\undefined\let\mathbf\bf\fi
\ifx\mathfrak\undefined\let\mathfrak\frak\fi
\ifx\mathcal\undefined\let\mathcal\cal\fi
\ifx\mathbb\undefined\let\mathbb\Bbb\fi
\ifx\emph\undefined\let\emph\it\fi
\newcommand{\g}{{{\mathfrak g}\,}}
\newcommand{\bor}{{{\mathfrak b}}}
\newcommand{\n}{{{\mathfrak n}}}
\newcommand{\h}{{{\mathfrak h\,}}}
\newcommand{\Id}{{\operatorname{Id}}}
\newcommand{\Z}{{\mathbb Z}}
\newcommand{\N}{{\mathbb N}}
\newcommand{\R}{{\mathbb R}}
\newcommand{\C}{{\mathbb C}}
\newcommand{\Q}{{\mathbb Q}}
\newcommand{\Pee}{{\mathbb P}}
\newcommand{\Ref}[1]{{(\ref{#1})}}
\newcommand{\be}{\begin{displaymath}}
\newcommand{\ee}{\end{displaymath}}
\newcommand{\bea}{\begin{eqnarray*}}
\newcommand{\eea}{\end{eqnarray*}}
\newcommand{\tr}{{\mathrm{tr}}}
\newcommand{\T}{{\mathcal{T}}}
\newcommand{\Mu}{{\mathrm{M}}}
\newenvironment{prf}{\noindent{\it Proof\/}:}{$\;\Box$\par\medskip}
\newenvironment{example}
{\noindent{\bf Example\/}:}{\par}

%%%%%%%%%%%%%%%%

\let\dl\delta
\let\Dl\Delta
\let\epe\epsilon \let\eps\varepsilon \let\epsilon\eps
\let\io\iota
\let\al\alpha
\let\bt\beta
\let\gm\gamma
\let\Gm\Gamma

\let\ka\kappa
\let\la\lambda \let\La\Lambda
\let\om\omega \let\Om\Omega
\let\pho\phi \let\phi\varphi \let\Pho\varPhi
\let\pio\varpi
\let\Pss\varPsi
\let\si\sigma \let\Si\Sigma \let\sig\varsigma
\let\tht\theta \let\Tht\Theta \let\thi\vartheta
\let\ups\upsilon
\let\zt\zeta

\newcommand{\half}{\frac12}
%\newcommand{\Z}{{\Bbb Z}}
%\newcommand{\N}{{\Bbb N}}
%\newcommand{\R}{{\Bbb R}}
%\newcommand{\C}{{\Bbb C}}
%\newcommand{\Q}{{\Bbb Q}}
%\newcommand{\Ref}[1]{{$($\ref{#1}$)$}}
\newcommand{\bean}{\begin{eqnarray}}
\newcommand{\eean}{\end{eqnarray}}
%\newcommand{\be}{\begin{displaymath}}
%\newcommand{\ee}{\end{displaymath}}
%\newcommand{\bea}{\begin{eqnarray*}}
%\newcommand{\eea}{\end{eqnarray*}}
%\newcommand{\g}{{{\frak g}\,}}
%\newcommand{\bor}{{{\frak b}}}
%\newcommand{\n}{{{\frak n}}}
%\newcommand{\h}{{{\frak h\,}}}
%\newcommand{\Id}{{\operatorname{Id}}}
%\newcommand{\ad}{{\operatorname{ad}}}
%\newcommand{\res}{{\operatorname{res}}}
%\newcommand{\Ker}{{\operatorname{Ker}}}
\newcommand{\sea}{\searrow}
\newcommand{\swa}{\swarrow}
\newcommand{\nea}{\nearrow}
\newcommand{\nwa}{\nwarrow}
\newcommand{\ra}{\rightarrow}
\newcommand{\lar}{\leftarrow}
%\newcommand{\T}{\!\otimes\!}
\newcommand{\noi}{\noindent}
\newcommand{\vs}{\vspace{.5\baselineskip}}
%\newcommand{\Mu}{{\mathrm M}}
%\newenvironment{proof}{\noindent{\it Proof\/}:}{$\;\Box$\par\vs}
\newenvironment{definition}
{\noindent{\bf Definition\/}:}{\par\vs}
%\newenvironment{example}
%{\noindent{\bf Example\/}:}{\par\vs}
\newtheorem%
{thm}{Theorem}%[section]
\newtheorem%
{proposition}[thm]{Proposition}
\newtheorem%
{lemma}[thm]{Lemma}
\newtheorem%
{lemmadef}[thm]{Lemma-Definition}
\newtheorem%
{corollary}[thm]{Corollary}
\newtheorem%
{conjecture}[thm]{Conjecture}
%\newcommand{\th}{\theta}
%\newcommand{\End}{{\operatorname{End\,}}}
%\newcommand{\Hom}{{\operatorname{Hom\,}}}
%\newcommand{\sign}{{\operatorname{sign}}}
%\newcommand{\tr}{{\operatorname{tr}}}
%\newcommand{\Sym}{{\operatorname{Sym\,}}}
%\newcommand{\Fun}{{\operatorname{Fun\,}}}
%\renewcommand{\Im}{{\operatorname{Im\,}}}

%%%%%

%\newcommand{\zn}{{{z_1,...,z_n}\,}}
%\newcommand{\Vn}{{{V_1\otimes...\otimes V_n}\,}}

%\newcommand{\g}{{{\frak g}\,}}
\newcommand{\F}{{\mathbb F}}




\title[Elliptic Selberg Integrals]
{Elliptic Selberg Integrals}
\author[G. Felder, L. Stevens, and A. Varchenko]
{G. Felder $^{\,\star}$,
L. Stevens$^{\, \diamond}$, and
A. Varchenko$^{\,\diamond}$}
%\thanks{$^1$ Supported in part by NSF grant DMS-9801582}

\begin{document}
\maketitle

\begin{center}
{\it
$^\star$ Departement Mathematik, ETH-Zentrum, 8092 Z\"urich, Switzerland,

felder@math.ethz.ch


\medskip

$^\diamond$ Department of Mathematics, University of North Carolina
at Chapel Hill,

Chapel Hill, NC 27599-3250, USA,

stevens@math.unc.edu, av@math.unc.edu}
\end{center}


%\vsk1.5>
\centerline{March, 2001}
%\vsk1.8>

%\begin{abstract}

%\end{abstract}

%\vsk>
%\vsk0>
The Selberg integral is
\bean
B_{p}(\al,\bt,\gm)=
\int_{\Delta_p}\prod_{j=1}^{p}t_{j}\,^{\al-1}(1-t_{j})^{\bt-1}
\prod_{0\leq j<k\leq 1}(t_{j}-t_{k})^{2\gm},
\notag
\eean
where $\Delta_{p}=\{t\in\R^{p}\,|\,0\leq t_{p}\leq\cdots\leq t_{1}\leq 1\}.$
The Selberg integral is a generalization of the beta function.  It can
be calculated explicitly,
\bean
B_{p}(\al,\bt,\gm)=
{1\over p!}
\prod_{j=0}^{p-1}
\frac{\Gm(1+\gm+j\gm)
\Gm(\al+j\gm)
\Gm(\bt+j\gm)}
{\Gm(1+\gm)
\Gm(\al+\bt+(p+j-1)\gm)}.
\notag
\eean
The Selberg integral has many applications, see \cite{A1,A2,As,D,DF1,DF2,M,S}.  In this paper, we
present an elliptic version of the Selberg integral.

Let $\thi_{1}(t,\tau)$ be the first Jacobi theta function \cite{WW},
\bean
\thi_{1}(t,\tau)=-\sum_{j\in\Z}
e^{\pi i(j+\half)^2\tau+2\pi i(j+\half)(t+\half)}.
\notag
\eean

Introduce special functions 
\bean
\si_{\la}(t,\tau)=
\frac{\thi_{1}(\la-t,\tau)\thi_{1}'(0,\tau)}
{\thi_{1}(\la,\tau)\thi_{1}(t,\tau)},
\qquad
\rho(t,\tau)=\frac{\thi_{1}'(t,\tau)}{\thi_{1}(t,\tau)},
\qquad
E(t,\tau)=\frac{\thi_{1}(t,\tau)}{\thi_{1}'(0,\tau)}.
\notag
\eean
Here $'$ denotes the derivative with respect to the first argument.
%The functions $E(t,\tau)^a$ and $\si_{\la}(t,\tau)$ are
%elliptic analogs of the functions $t^a$ and $1/t$, respectively.


Let $\ka\geq2$ be an integer.  The theta functions
\bean
\tht_{\ka,m}(\la,\tau)=
\sum_{j\in\Z}e^{2\pi i\ka(j+\frac{m}{2\ka})^{2}\tau+2\pi
  i\ka(j+\frac{m}{2\ka})\la},\qquad m\in\Z/2\ka\Z,
\notag
\eean
form a basis of the theta functions of level $\ka$. 

For a positive integer $p$, the elliptic Selberg integral
$I_{p}(\la,\tau)$ is the integral, 
\bean
I_{p}(\la,\tau)=J_{p}(\la,\tau)+(-1)^{p+1}J_{p}(-\la,\tau),
\notag
\eean
where
\bean
J_{p}(\la,\tau)
=
\int_{\Delta_{p}} 
\prod_{j=1}^p E(t_{j},\tau)^{-\frac{p}{p+1}}
\prod_{1\leq j<k\leq
  p}E(t_{j}-t_{k},\tau)^{\frac{1}{p+1}}\,&\times\notag
\\
\prod_{j=1}^p\si_{\la}(t_{j},\tau)
\tht_{2(p+1),p+1}\left(\la+\frac{1}{p+1}\sum_{j=1}^p
t_{j},\tau\right)&dt_{1}\dots dt_{p}\,\,.  
\notag
\eean
The branch of the logarithm is chosen in such a way that
arg $(E(t,\tau))\to 0$ as $t\to 0^{+}$, and the integral is understood as a
natural analytic continuation.\footnote{To define the analytic
  continuation, we replace the exponential $-{p\over p+1}$
  by $a$, and consider the analytic continuation with respect to $a$
  from the region where $a$ is positive.} 

\begin{thm}
We have
\bean\label{TM}
 I_{p}(\la,\tau)=c_{p}\,B_p\left(\half+{1\over2(p+1)},-{p\over p+1}, {1\over
   2(p+1)}\right)\,
\thi_{1}(\la,\tau)^{p+1}
\eean
where
\bean
c_{p}=-(2\pi)^{{p\over2}}\,e^{\pi i{p\over p+1}}
\,e^{-\pi i{p+2\over4}}\,\prod_{j=1}^{p}\left(1-e^{-\pi i{j\over p+1}}\right).  
\notag
\eean
\end{thm}
The theorem is a generalization of theorem $13$ in \cite{FV1}.  The
proof is based on the following remarks.  Consider the heat equation
\bean
4\pi i(p+1)\frac{\partial u}{\partial\tau}(\la,\tau)=
\frac{\partial^{2}u}{\partial\la^{2}}(\la,\tau)+
p(p+1)\rho'(\la,\tau)u(\la,\tau).
\notag
\eean
It is known that this equation has a one dimensional space of
solutions $u(\la,\tau)$ which are holomorphic theta functions of level
$2(p+1)$,
\bean
u(\la+2,\tau)=u(\la,\tau), \qquad
u(\la+2\tau,\tau)=e^{-  4\pi i(p+1)(\la+\tau)}u(\la,\tau)
\notag
\eean
and Weyl skew-symmetric, $u(-\la,\tau)=(-1)^{p+1}u(\la,\tau)$, see
\cite{FV1,FV2}.  The space is called the space of conformal blocks.
Clearly the right hand side of \Ref{TM} has these properties.  According to
\cite{FV1}, the left hand side of \Ref{TM} also has these properties.  Thus the
two functions are proportional.  The coefficient of proportionality is
easily calculated in the limit $\tau\to i\infty$.

\begin{thebibliography}{TV2}
\normalsize
\bibitem [A1]{A1} K.~Aomoto, {\it
Ja\-co\-bi polynomials associated with Sel\-berg in\-te\-grals}, SIAM
J.~Math.~Anal.~, 1987, N.~18, 545-549.
\bibitem [A2]{A2} K.~Aomoto, {\it
On the complex Selberg integral}, Q.~J.~Math.~Oxford, 1987, N.~38,
385-399.
\bibitem [As]{As} R.~S.~Askey, {\it
Some basic hypergeometric extensions of integrals of Selberg and
Andrews}, SIAM J.~Math.~, 1980, N.~11, 938-951. 
\bibitem [D]{D} V.~Dotsenko, {\it Solving the $SU(2)$ Conformal
    Field Theory with the Wakimoto Free-Field Representation}, (in
  Russian), Moscow, 1990, 1-31.

\bibitem [DF1]{DF1} V.~Dotsenko and V.~Fateev, {\it
Conformal algebra and multipoint correlation functions in 2-D
statistical models}, Nucl.~Phys., 1984, N.~B240, 312-348.
\bibitem [DF2]{DF2} V.~Dotsenko and V.~Fateev, {\it
Four-point correlation functions and the operator algebra in 2-D
conformal invariant theories with central charge $C\leq 1$},
Nucl.~Phys., 1985, N.~B251, 691-734.


\bibitem [FV1]{FV1} G.Felder and A.Varchenko, {\it
Integral representation of solutions of the elliptic
Knizhnik-Zamolodchikov-Bernard equations},
Int.~Math.~Res.~notices, 1995, N.~5, 221-233.


\bibitem [FV2]{FV2} G.~Felder and A.~Varchenko, {\it The q-De\-formed
    KZB-heat Equation}, math.\-~QA/9809139.
\bibitem [M]{M} M.L.~Mehta, {\it Random Matrices}, Academic Press, 1991.
\bibitem [S]{S} A.~Selberg, {\it Bemerkninger om et multiplet
    integral}, Norsk Mat.~Tidsskr., 1944, N.~26, 71-78.
\bibitem [WW]{WW} E.T.~Whittaker and G.N.~Watson, {\it Modern
    Analysis}, Cambridge University Press, 1927.


\end{thebibliography}

\end{document}





















































































math0103227.extracted_bib
\begin{thebibliography}{TV2}
\normalsize
\bibitem [A1]{A1} K.~Aomoto, {\it
Ja\-co\-bi polynomials associated with Sel\-berg in\-te\-grals}, SIAM
J.~Math.~Anal.~, 1987, N.~18, 545-549.
\bibitem [A2]{A2} K.~Aomoto, {\it
On the complex Selberg integral}, Q.~J.~Math.~Oxford, 1987, N.~38,
385-399.
\bibitem [As]{As} R.~S.~Askey, {\it
Some basic hypergeometric extensions of integrals of Selberg and
Andrews}, SIAM J.~Math.~, 1980, N.~11, 938-951. 
\bibitem [D]{D} V.~Dotsenko, {\it Solving the $SU(2)$ Conformal
    Field Theory with the Wakimoto Free-Field Representation}, (in
  Russian), Moscow, 1990, 1-31.

\bibitem [DF1]{DF1} V.~Dotsenko and V.~Fateev, {\it
Conformal algebra and multipoint correlation functions in 2-D
statistical models}, Nucl.~Phys., 1984, N.~B240, 312-348.
\bibitem [DF2]{DF2} V.~Dotsenko and V.~Fateev, {\it
Four-point correlation functions and the operator algebra in 2-D
conformal invariant theories with central charge $C\leq 1$},
Nucl.~Phys., 1985, N.~B251, 691-734.


\bibitem [FV1]{FV1} G.Felder and A.Varchenko, {\it
Integral representation of solutions of the elliptic
Knizhnik-Zamolodchikov-Bernard equations},
Int.~Math.~Res.~notices, 1995, N.~5, 221-233.


\bibitem [FV2]{FV2} G.~Felder and A.~Varchenko, {\it The q-De\-formed
    KZB-heat Equation}, math.\-~QA/9809139.
\bibitem [M]{M} M.L.~Mehta, {\it Random Matrices}, Academic Press, 1991.
\bibitem [S]{S} A.~Selberg, {\it Bemerkninger om et multiplet
    integral}, Norsk Mat.~Tidsskr., 1944, N.~26, 71-78.
\bibitem [WW]{WW} E.T.~Whittaker and G.N.~Watson, {\it Modern
    Analysis}, Cambridge University Press, 1927.


\end{thebibliography}