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\begin{document}

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\section*{zeta function of an elliptic differential operator}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{functional_analysis}{}\paragraph*{{Functional analysis}}\label{functional_analysis}

[[!include functional analysis - contents]]

\hypertarget{arithmetic_geometry}{}\paragraph*{{Arithmetic geometry}}\label{arithmetic_geometry}

[[!include arithmetic geometry - contents]]

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{the_zeta_function}{The zeta function}\dotfill \pageref*{the_zeta_function} \linebreak
\noindent\hyperlink{FunctionalDeterminant}{Functional determinant and zeta-function regularization}\dotfill \pageref*{FunctionalDeterminant} \linebreak
\noindent\hyperlink{AnalogyWithNumberTheoreticZetaFunctions}{Relation to partition functions and number-theoretic zeta/theta functions}\dotfill \pageref*{AnalogyWithNumberTheoreticZetaFunctions} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{of_laplace_operator_on_complex_torus_and_dedekind_eta_function}{Of Laplace operator on complex torus and Dedekind eta function}\dotfill \pageref*{of_laplace_operator_on_complex_torus_and_dedekind_eta_function} \linebreak
\noindent\hyperlink{analytic_torsion}{Analytic torsion}\dotfill \pageref*{analytic_torsion} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{zeta_function_regularization}{Zeta function regularization}\dotfill \pageref*{zeta_function_regularization} \linebreak
\noindent\hyperlink{functional_determinant}{Functional determinant}\dotfill \pageref*{functional_determinant} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

For a suitable [[linear operator]] $H$ (say on [[section]] of a [[line bundle]] over a [[Riemann surface]]), its \emph{zeta function} is the [[analytic continuation]] of the [[trace]]

\begin{displaymath}
ze
  \zeta_H(s) \coloneqq Tr(H^{-s})
\end{displaymath}
of the $-s$ power of $H$, which, if $H$ is suitably self-adjoint, is the [[sum]] of the $-s$-powers of all its [[eigenvalues]], as a function of $s$. This is analogous to the \emph{[[Riemann zeta function]]} and the [[Dedekind zeta function]] (or would be if there were something like a [[Laplace operator]] on [[Spec(Z)]] or more generally on an [[arithmetic curve]], see at \emph{[[function field analogy]]}).

The exponential of the derivative of the zeta function at $n = 0$ also encodes the [[functional determinant]] of $H$, a [[regularization (physics)|regularized]] version (``[[zeta function regularization]]'') of the naive and generally ill-defined product of all eigenvalues. As such, zeta functions play a central role in [[quantum field theory]].

Generally, the values of $\zeta_H(s)$ of interest in [[physics]] (when regarding $H$ as a [[Hamilton operator]]) are those for (low) integral $s$. These are just the \emph{[[special values of L-functions]]}.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{the_zeta_function}{}\subsubsection*{{The zeta function}}\label{the_zeta_function}

Given an [[elliptic differential operator]] with positive lower bound $c$, write $H$ for its [[self-adjoint extension]] and write

\begin{displaymath}
0 \lt \lambda_1 \leq \lambda_2 \leq \cdots
\end{displaymath}
for its [[eigenvalues]].

\begin{defn}
\label{ZetaBySeries}\hypertarget{ZetaBySeries}{}
The \emph{zeta function} of $H$ is the [[holomorphic function]] defined by the [[series]]

\begin{displaymath}
\begin{aligned}
    \zeta_H(s) 
    & \coloneqq 
    tr( H^{-s} )
    \\
    & \coloneqq
    \underoverset{n = 1}{\infty}{\sum} \frac{1}{(\lambda_n)^s}
  \end{aligned}
  \,.
\end{displaymath}
where this [[convergence|converges]] and then extended by [[analytic continuation]].

\end{defn}
(e.g. (\hyperlink{DuistermaatGuillemin75}{Duistermaat-Guillemin 75 (2.13)}, \hyperlink{BerlineGetzlerVergne04}{Berline-Getzler-Vergne 04, section 9.6} ) ).

\hypertarget{FunctionalDeterminant}{}\subsubsection*{{Functional determinant and zeta-function regularization}}\label{FunctionalDeterminant}

Notice that the first [[derivative]] $\zeta^\prime_H$ of this zeta function is, where the original series converges, given by

\begin{displaymath}
\zeta_H^\prime(s)
  =
  \sum_{n = 1}^\infty \frac{- \ln \lambda_n}{ (\lambda_n)^s}
  \,.
\end{displaymath}
Therefore one says (\hyperlink{RaySinger71}{Ray-Singer 71}) that the \emph{[[functional determinant]]} of $H$ is the exponential of the derivative of zeta function of $H$ at 0:

\begin{displaymath}
det_{reg} H
  \coloneqq
  \exp(- \zeta_H^\prime(0))
  \,.
\end{displaymath}
Via the [[analytic continuation]] involved in defining $\zeta_H(0)$ in the first place, this may be thought of as a \emph{[[regularization (physics)|regularization]]} of the ill-defined naive definition ``$\prod_n \lambda_n$'' of the [[determinant]] of $H$. As such functional determinants often appear in [[quantum field theory]] as what is called \emph{[[zeta function regularization]]}.

Conversely, the [[logarithm]]

\begin{displaymath}
Z \coloneqq - \frac{1}{2}\zeta_H^\prime(0) = \tfrac{1}{2} log\,det_{reg} H
\end{displaymath}
is what is called the \emph{[[vacuum energy]]} in [[quantum field theory]] (for $H^{-1}$ the [[Feynman propagator]]).

If $H = D^2$ has a square root $D$ (a [[Dirac operator]]-type square root as in [[supersymmetric quantum mechanics]]) then under some conditions on the growth of the eigenvalues, then the functional determinant may also be expressed in terms of the [[eta function]] of $D$ as

\begin{displaymath}
det H 
  =
  det (D^2) 
   = 
  \exp( \frac{\partial}{\partial s}\frac{\partial}{\partial c} \eta_{D}(0))
  \,.
\end{displaymath}
See at \emph{\href{eta+invariant#RelationToTheZetaFunction}{eta invariant -- Relation to zeta function}} for more on this.

\hypertarget{AnalogyWithNumberTheoreticZetaFunctions}{}\subsubsection*{{Relation to partition functions and number-theoretic zeta/theta functions}}\label{AnalogyWithNumberTheoreticZetaFunctions}

By basic [[integration]] identities we have that

\begin{prop}
\label{IntegralKernelExpression}\hypertarget{IntegralKernelExpression}{}
The series expression in def. \ref{ZetaBySeries} is equal to the [[Mellin transform]] of the [[partition function]]

\begin{displaymath}
\zeta_H(s)
     =
     \int_{(0,\infty)}
     t^{s-1}
     \left(
        \underset{\lambda_k \neq 0}{\sum}
        \exp(-t \lambda_k)
     \right)
     d t
  \,.
\end{displaymath}
\end{prop}
(see e.g. \hyperlink{QuineHeydariSong93}{Quine-Heydari-Song 93 (8)}, \hyperlink{Richardson}{Richardson, pages 8-9}, \hyperlink{BCEMZ03}{BCEMZ 03, section A.2}, \hyperlink{ConnesMarcolli06}{Connes-Marcolli 06, theorem 13.11}).

\begin{remark}
\label{}\hypertarget{}{}
If one thinks of the operation $H$ as a [[Hamiltonian]] of a [[quantum mechanical system]], then the term

\begin{displaymath}
Tr(\exp(-\beta H))
  \coloneqq
   \underset{\lambda_k \neq 0}{\sum}
        \exp(-\beta \lambda_k)
   + 1
\end{displaymath}
is the \emph{[[partition function]]} of this system. Accordingly, prop. \ref{IntegralKernelExpression} says that the zeta function of $H$ is obtained from its partition function by

\begin{displaymath}
\zeta_H(s)
  =
   \int_{(0,\infty)}
     \beta^{s-1}
     \;
     \left(Tr(\exp(-\beta H)) - 1 \right)
     \;
   d \beta 
   \,.
\end{displaymath}
\end{remark}
Further, by a change of integration variable $t\coloneqq x^2$ in the expression in prop. \ref{IntegralKernelExpression} one obtains

\begin{prop}
\label{IntegralKernelExpressionInSquares}\hypertarget{IntegralKernelExpressionInSquares}{}
The series expression in def. \ref{ZetaBySeries} is equal to

\begin{displaymath}
\begin{aligned}
     \zeta_H(s)
     & =
     2
     \int_{(0,\infty)}
     x^{2s-1}
     \left(
        \underset{\lambda_k \neq 0}{\sum}
        \exp(- x^2 \lambda_k)
     \right)
      d x      
  \end{aligned}
 \,.
\end{displaymath}
In particular if $H = D^2$ is the square of a [[Dirac operator]]/[[supersymmetric quantum mechanics]]-type square root operator $D$ with [[eigenvalues]] $\pm \alpha_k$,then $\lambda_k = \alpha_k^2$ and hence in this case the series is

\begin{displaymath}
\begin{aligned}
     \zeta_H(s)
     & =
     2
     \int_{(0,\infty)}
     x^{2s-1}
     \left(
        \underset{\alpha_k \neq 0}{\sum}
        \exp(- (x \alpha_k)^2)
     \right)
      d x      
  \end{aligned}
 \,.
\end{displaymath}
\end{prop}
By comparison one observes:

\begin{remark}
\label{}\hypertarget{}{}
The integral expression in prop. \ref{IntegralKernelExpressionInSquares} is analogous to the expression of [[zeta functions]] in [[number theory]]/[[arithmetic geometry]] as integrals of a [[theta function]] (for instance discussed \href{Riemann%20zeta%20function#RelationToThetaFunctions}{here} for the [[Riemann zeta function]])

\begin{displaymath}
\hat\zeta_f(2 s)
  = 
  \int_{(0,\infty)} (\theta(x^2) - 1) x^{2s-1} d x
  \,.
\end{displaymath}
Under this analogy the [[theta function]] in the case of the differential operator $H$ is

\begin{displaymath}
\theta_H(x)
   \coloneqq  
   \underset{\lambda_k \neq 0}{\sum}
        \exp(- x \lambda_l)
   \,.
\end{displaymath}
This is formally the same definition as that of adelic theta functions (e.g.\href{Iwasawa-Tate%20theory#Garrett11}{Garrett 11, section 1.8})

\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
The [[determinant line bundle]] of the [[functional determinant]] of the [[Dirac operator]] on a [[complex torus]] is a complex-analytic [[theta function]] as above, quotiented by the [[Dedekind eta function]].

Early references explaining this include \hyperlink{AlvaresGaumeMooreVafa86}{Alvarez-Gaum\'e{} \& Moore \& Vafa 86}, \hyperlink{AlvaresGaumeBostMooreNelsonVafa87}{Alvarez-Gaum\'e{} \& Bost \& Moore \& Nelson \& Vafa 87}. In a bigger perspective, this relation plays a central role in the general discussion of [[self-dual higher gauge theory]] (\href{self-dual%20higher%20gauge%20theory#Witten96}{Witten 96}).

\end{remark}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{of_laplace_operator_on_complex_torus_and_dedekind_eta_function}{}\subsubsection*{{Of Laplace operator on complex torus and Dedekind eta function}}\label{of_laplace_operator_on_complex_torus_and_dedekind_eta_function}

For $\mathbb{C}/(\mathbb{Z}\oplus \tau \mathbb{Z})$ a [[complex torus]] (complex [[elliptic curve]]) equipped with its standard flat [[Riemannian metric]], then the [[zeta function of an elliptic differential operator|zeta function]] of the corresponding [[Laplace operator]] $\Delta$ is

\begin{displaymath}
\zeta_{\Delta} = (2\pi)^{-2 s} E(s)
   \coloneqq
   (2\pi)^{-2 s} \underset{(k,l)\in \mathbb{Z}\times\mathbb{Z}-(0,0)}{\sum}
  \frac{1}{{\vert k +\tau l\vert}^{2s}}
  \,.
\end{displaymath}
The corresponding [[functional determinant]] is

\begin{displaymath}
\exp(
    E^\prime_{\Delta}(0)
  )
  = (Im \tau)^2 {\vert \eta(\tau)\vert}^4
  \,,
\end{displaymath}
where $\eta$ is the [[Dedekind eta function]].

(recalled e.g. in \hyperlink{Todorov03}{Todorov 03, page 3})

For more see also at \emph{[[zeta function of a Riemann surface]]}.

\hypertarget{analytic_torsion}{}\subsubsection*{{Analytic torsion}}\label{analytic_torsion}

The functional determinant of a [[Laplace operator]] of a [[Riemannian manifold]] acting on [[differential n-forms]] is up to a sign in the exponent a factor in what is called the \emph{[[analytic torsion]]} of the manifold.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[partition function]]

\end{itemize}
[[!include zeta-functions and eta-functions and theta-functions and L-functions -- table]]

\hypertarget{references}{}\subsection*{{References}}\label{references}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

An early reference is

\begin{itemize}%
\item [[Hans Duistermaat]], [[Victor Guillemin]], \emph{The Spectrum of Positive Elliptic Operators and Periodic Bicharacteristics},Inventiones mathematicae (1975) Volume: 29, page 39-80 (\href{https://eudml.org/doc/142329}{EuDML})

\end{itemize}
(see also at \emph{[[Duistermaat-Guillemin trace formula]]})

Textbook accounts include

\begin{itemize}%
\item [[Nicole Berline]], [[Ezra Getzler]], [[Michèle Vergne]], section 9.6 of \emph{Heat Kernels and Dirac Operators}

\end{itemize}
Review includes

\begin{itemize}%
\item [[Ken Richardson]], section 3 of \emph{Introduction to the Eta invariant} (\href{http://faculty.tcu.edu/richardson/Seminars/etaInvariant.pdf}{pdf})


\item J. R. Quine, S. H. Heydari, R. Y. Song, \emph{Zeta regularized products}, Transactions of the AMS volume 338, number 1, 1993 ([[QuineZetaRegularization.pdf:file]])


\item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Functional_determinant#Zeta_function_version}{Functional determinant -- Zeta function version}}


\item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Zeta_function_regularization}{Zeta function regularization}}


\item [[Alain Connes]], [[Matilde Marcolli]], \emph{A walk in the noncommutative garden} (\href{http://arxiv.org/abs/math/0601054}{arXiv:0601054})



\end{itemize}
\hypertarget{zeta_function_regularization}{}\subsubsection*{{Zeta function regularization}}\label{zeta_function_regularization}

\begin{itemize}%
\item [[Eugene Speer]], \emph{On the structure of Analytic Renormalization}, Comm. math. Phys. 23, 23-36 (1971) (\href{http://projecteuclid.org/euclid.cmp/1103857549}{Euclid})


\item A. Bytsenko, G. Cognola, [[Emilio Elizalde]], [[Valter Moretti]], S. Zerbini, section 2 of \emph{Analytic Aspects of Quantum Fields}, World Scientific Publishing, 2003, ISBN 981-238-364-6



\end{itemize}
\hypertarget{functional_determinant}{}\subsubsection*{{Functional determinant}}\label{functional_determinant}

The definition of a \emph{[[functional determinant]]} via the exponential of the derivative of the zeta function at 0 originates in

\begin{itemize}%
\item D. Ray, [[Isadore Singer]], \emph{R-torsion and the Laplacian on Riemannian manifolds}, Advances in Math. 7: 145--210, (1971) doi:10.1016/0001-8708(71)90045-4, MR 0295381

\end{itemize}
Discussion in the special case of [[2d CFT]] ([[worldsheet]] [[string theory]]) is in

\begin{itemize}%
\item [[Luis Alvarez-Gaumé]], [[Gregory Moore]], [[Cumrun Vafa]], \emph{Theta functions, modular invariance, and strings}, Communications in Mathematical Physics Volume 106, Number 1 (1986), 1-4 (\href{http://projecteuclid.org/euclid.cmp/1104115581}{Euclid})


\item [[Luis Alvarez-Gaumé]], [[Jean-Benoit Bost]], [[Gregory Moore]], Philip Nelson, [[Cumrun Vafa]], \emph{Bosonization on higher genus Riemann surfaces}, Communications in Mathematical Physics, Volume 112, Number 3 (1987), 503-552 (\href{http://projecteuclid.org/euclid.cmp/1104159982}{Euclid})


\item [[Andrey Todorov]], \emph{The analogue of the Dedekind eta function for CY threefolds}, 2003 \href{http://www.ma.huji.ac.il/conf/crelle.pdf}{pdf}



\end{itemize}
[[!redirects zeta functions of elliptic differential operators]]



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