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\section*{determinant line bundle}

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

\hypertarget{bundles}{}\paragraph*{{Bundles}}\label{bundles}

[[!include bundles - contents]]

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

\noindent\hyperlink{DeterminantLineBundles}{Definition}\dotfill \pageref*{DeterminantLineBundles} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{quillens_determinant_line_bundle}{Quillen's determinant line bundle}\dotfill \pageref*{quillens_determinant_line_bundle} \linebreak
\noindent\hyperlink{determinant_bundle_on_the_grassmannian}{Determinant bundle on the Grassmannian}\dotfill \pageref*{determinant_bundle_on_the_grassmannian} \linebreak
\noindent\hyperlink{comparing_quillens_and_segals_determinant_line_bundles}{Comparing Quillen's and Segal's determinant line bundles}\dotfill \pageref*{comparing_quillens_and_segals_determinant_line_bundles} \linebreak
\noindent\hyperlink{pfaffian_line_bundle}{Pfaffian line bundle}\dotfill \pageref*{pfaffian_line_bundle} \linebreak
\noindent\hyperlink{from_fermionic_path_integrals}{From fermionic path integrals}\dotfill \pageref*{from_fermionic_path_integrals} \linebreak
\noindent\hyperlink{relation_to_theta_function}{Relation to theta function}\dotfill \pageref*{relation_to_theta_function} \linebreak
\noindent\hyperlink{relation_to_vacuum_energy_partition_function}{Relation to vacuum energy, partition function}\dotfill \pageref*{relation_to_vacuum_energy_partition_function} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

Let $V$ and $W$ be two [[vector space]]s of [[dimension]] $n = dim V = dim W$ and

\begin{displaymath}
T : V \to W
\end{displaymath}
a [[linear map]].

Write $\wedge^n V$ and $\wedge^n W$ for the \emph{top exterior power} of these vector spaces, the skew-symmetrized $n$th [[tensor algebra|tensor power]] of $V$ and $W$. These are 1-dimensional vector spaces, hence [[line]]s over the ground field. The linear map $T$ induces a linear map

\begin{displaymath}
det T : \wedge^n V\to \wedge^n W
\end{displaymath}
between these lines. This is the \textbf{[[determinant]]} of $T$. More specifically, if $V = W$ (being of the same finite dimension, both are necessarily [[isomorphic]] but not necessarily canonically so) then $det T : \wedge^n V \to \wedge^n V$ is a linear endomorphism of a 1-dimensional vector space and by the equivalence

\begin{displaymath}
End(\wedge^n V) \simeq k
\end{displaymath}
of such endomorphisms with the ground [[field]] $k$ is identified with an element in $k$

\begin{displaymath}
det T \in k
  \,.
\end{displaymath}
This is the standard meaning of the \textbf{determinant} of a linear endomorphism.

Notice that the determinant construction:

\begin{displaymath}
det : (V \stackrel{T}{\to} W) \mapsto (\wedge^n V \stackrel{det T}{\to} \wedge^n W)
\end{displaymath}
is a [[functor]] from the [[category]] [[Vect]] to itself

\begin{displaymath}
det : Vect \to Vect
  \,.
\end{displaymath}
Any such functor $F : Vect \to Vect$ with certain continuity assumptions induces an endo-functor on the category of [[vector bundle]]s $VectBund(X)$ over an arbitrary [[manifold]] $X$.

Concretely, if a vector bundle $E \to X$ is given by a [[Cech cohomology|Cech cocycle]]

\begin{displaymath}
\itexarray{
    C(U_i) &\stackrel{(g_{i j})}{\to}& \mathbf{B} GL(n) &\to& Vect
    \\
    \downarrow^{\mathrlap{\simeq}}
    \\
    X
  }
\end{displaymath}
with respect to an [[open cover]] $\{U_i \to X\}$ (see [[principal bundle]] and [[associated bundle]] for details), hence by transition functions

\begin{displaymath}
(g_{i j} \in C(U_i \cap U_j, GL(n)))
\end{displaymath}
with values in the [[general linear group]], its image under $det : VectBund(X) \to VectBund(X)$ is the bundle with transition functions the determinants of these transition functions

\begin{displaymath}
(det g_{i j} \in C(U_i \cap U_j, GL(1) \simeq k^\times))
  \,.
\end{displaymath}
This are the transition functions for the bundle $\wedge^\bullet E \to X$ which is fiberwise the top exterior power of $E \to X$. This is the \textbf{[[determinant line]] bundle} of $E$.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\begin{uprop}
Let

\begin{displaymath}
E : X \stackrel{\simeq}{\leftarrow} C(U_i) \stackrel{g_{i j}}{\to} \mathbf{B} U(n)
\end{displaymath}
be a [[unitary group]]-[[principal bundle]] (to which is canonically [[associated bundle|associated]] a rank-$n$ complex [[vector bundle]]). Then the single [[characteristic class]]

\begin{displaymath}
[det E] \in H^2(X, \mathbb{Z})
\end{displaymath}
of its determinant [[circle bundle]]

\begin{displaymath}
det E : X \stackrel{\simeq}{\leftarrow} C(U_i) \stackrel{det g_{i j}}{\to} \mathbf{B} U(1)
\end{displaymath}
is the first [[Chern class]] of $E$

\begin{displaymath}
[det E] = c_1(E)
  \,.
\end{displaymath}
Moreover, if $X$ is a [[smooth manifold]] and $(g_{i j}, A_i)$ is the data of a [[connection on a bundle]] $(E, \nabla)$ on $E$ then $(det g_{i j}, tr A_i)$ (where we take the [[trace]] $tr : \mathfrak{u}(n) \to \mathfrak{u}(1)$ on the [[Lie algebra]] of the [[unitary group]]) is a [[circle n-bundle with connection|line bundle with connection]] that refines the first Chern-class to [[ordinary differential cohomology]]. In other words, this is the image under the refined [[Chern-Weil homomorphism]] of $(E, \nabla)$ induced by the canonical unary [[invariant polynomial]] on $\mathfrak{u}(n)$.

\end{uprop}
An explicit version of this statement is for instance in (\hyperlink{GriffithsHarris}{GriffithsHarris, p. 414}).

One can now look at operators $T:E\to F$ where $E,F$ are vector bundles of rank $n$ and the induced operators $\Lambda^n T : \Lambda^n E\to \Lambda^n F$ which can be considered as elements $det T\in (\Lambda^n E)^*\otimes\Lambda^n F$.

Even more important is the case of when $X$ is replaced by an appropriate [[moduli space of connections]], instantons, holomorphic structures or some other objects related to Fredholm operators for which the determinants can be defined.

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

\hypertarget{quillens_determinant_line_bundle}{}\subsubsection*{{Quillen's determinant line bundle}}\label{quillens_determinant_line_bundle}

There is a specific version called \textbf{Quillen's determinant line bundle} which is certain [[line bundle]] over the [[moduli space of complex structures]] on a fixed smooth vector bundle $E$ over a fixed [[Riemann surface]] $M$. A complex structure on the bundle corresponds to an operator which in local coordinates looks as $D = d\bar{z}(\partial_z+\alpha(z))$ where $\alpha(z)$ is a smooth matrix valued function. The set of such operators is an affine space $\mathcal{A}$ whose underlying vector space is the space of $(0,1)$-End-valued forms $\Omega^{0,1} (End M)$. Then again a determinant is an element of a line $\mathcal{L}_D = \lambda(Ker D)^*\otimes \lambda(Coker D)$ where $\lambda$ is taking the top exterior power. Now one has a family $\mathcal{L}_D$ depending on $D$, which determines a holomorphic line bundle over $\mathcal{A}$. This is the \textbf{determinant line bundle}.

If we had a trivialization of the Quillen's determinant line bundle, then we could identify every section with a holomorphic function on the base space, hence a holomorphic rule giving a number to a Cauchy-Riemann operator. For this one restricts first to the component consisting of the operators with the zero Fredholm index. Next, one considers the corresponding [[Laplace operator]] $D^* D$ and its [[functional determinant]] related to the [[zeta function of an elliptic differential operator]]. (This is related to the [[analytic torsion]]).

\hypertarget{determinant_bundle_on_the_grassmannian}{}\subsubsection*{{Determinant bundle on the Grassmannian}}\label{determinant_bundle_on_the_grassmannian}

Let $Gr_k(V)$ be the [[Grassmannian]] of $k$-dimensional subspaces of a finite dimensional vector space $V$. Let $W\subset V$ be a point in $Gr_k(V)$ and $\Lambda^k(W)$ its top exterior power; it is a fiber of the bundle $Det$ over $Gr_k(V)$. The determinant bundle $Det$ has no non-zero holomorphic global sections. Consider its dual $Det^*$ with fiber $\Lambda^k(W)^*$ over $W$. Then the space of of global holomorphic sections $\Gamma_{hol}(Det^*) \cong \Lambda^k(V^*)$. This construction can be suitably extended for the Segal Grassmannian, where $V= V_+\oplus V_-$ is a separable Hilbert space equipped with a polarization, see chapter 7 and especially 7.7 in the Pressley-Segal book listed below.

\hypertarget{comparing_quillens_and_segals_determinant_line_bundles}{}\subsubsection*{{Comparing Quillen's and Segal's determinant line bundles}}\label{comparing_quillens_and_segals_determinant_line_bundles}

The determinant line bundle of Quillen is in fact related to a variant of Segal's determinant bundle on the ``semiinfinite'' Grassmannian. Namely one considers instead $Gr_{cpt}(H)$ which is the set (space eventually) of closed supspaces $W\subset H$ where the projection $W\to H_+$ is [[Fredholm operator|Fredholm]] and $W\to H_-$ is [[compact operator|compact]]; then one follows the Segal's prescription to define $Det$ on $Gr_{cpt}(H)$. Notice that $Gr_{cpt}(H)$ is not a homogeneous space. Now there is a span of maps with contractible fibers

\begin{displaymath}
Gr_{cpt}(H)\leftarrow \mathcal{B}\to Fred(H_+).
\end{displaymath}
The Quillen's determinant line bundle is defined in general on the whole $Fred(H_+)$ and its pullback to $\mathcal{B}$ is isomorphic to the pullback of the determinant bundle on $Gr_{cpt}(H)$; in fact the Quillen's version can be reconstructed from this pullback by certain quotienting construction.

\hypertarget{pfaffian_line_bundle}{}\subsubsection*{{Pfaffian line bundle}}\label{pfaffian_line_bundle}

In dimens\^i{}on $8k+2$ for $k \in \mathbb{N}$ the determinant line bundle has a canon\^i{}cal square root line bundle, the [[Pfaffian line bundle]].

\hypertarget{from_fermionic_path_integrals}{}\subsubsection*{{From fermionic path integrals}}\label{from_fermionic_path_integrals}

See at \emph{[[fermionic path integral]]}.

\hypertarget{relation_to_theta_function}{}\subsubsection*{{Relation to theta function}}\label{relation_to_theta_function}

the determinant of the Dirac operator is, up to choice of isomorphism, the [[theta function]]-section of the determinant line bundle (\hyperlink{Freed87}{Freed 87, pages 30-31}).

\hypertarget{relation_to_vacuum_energy_partition_function}{}\subsubsection*{{Relation to vacuum energy, partition function}}\label{relation_to_vacuum_energy_partition_function}

See at \emph{[[vacuum energy]]}

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

\begin{itemize}%
\item [[density bundle]]


\item [[theta function]]



\end{itemize}
[[!include square roots of line bundles - table]]

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

The relation between determinant line bundles and the first Chern class is stated explicitly for instance on p. 414 of

\begin{itemize}%
\item Griffiths and Harris, \emph{Principles of algebraic geometry}

\end{itemize}
Literature on determinant line bundles of infinite-dimensional bundles includes the following:

\begin{itemize}%
\item [[Daniel Quillen|D.G. Quillen]], \emph{Determinants of Cauchy-Riemann operators over a Riemann surface}, Funkcionalnii Analiz i ego Prilozhenija \textbf{19} (1985), 37-41, (\href{http://www.mathnet.ru/php/getFT.phtml?jrnid=faa&paperid=1334&volume=19&year=1985&issue=1&fpage=37&what=fullt&option_lang=eng}{pdf} of Russian version).

reviewed e.g. in

Arlo Caine, \emph{Quillen's construction of Determinants of Cauchy--Riemann operators over Riemann Surfaces}, 2005 (\href{http://web.archive.org/web/20150326122017/http://www.cpp.edu/~jacaine/pdf/quillen_determinant_notes.pdf}{pdf})


\item [[Michael Atiyah]], [[Isadore Singer]], \emph{Dirac operators coupled to vector potentials}, Proc. Nat. Acad. Sci. USA \textbf{81}, 2597-2600 (1984) (\href{http://www.pnas.org/content/81/8/2597.full.pdf}{pdf} at pnas site)


\item [[Daniel Freed]], \emph{On determinant line bundles}, Math. aspects of [[string theory]], ed. S. T. Yau, World Sci. Publ. 1987, (\href{http://arxiv.org/abs/dg-ga/9505002}{dg-ga/9505002}, \href{http://www.math.utexas.edu/~dafr/Index/determinants.pdf}{revised pdf})


\item [[Jean-Michel Bismut]], [[Daniel Freed]], \emph{The analysis of elliptic families.I. Metrics and connections on determinant bundles}, Comm. Math. Phys. \textbf{106}, 1 (1986), 159-176, \href{http://projecteuclid.org/euclid.cmp/1104115586}{euclid}, \emph{II. Dirac operators, eta invariants, and the holonomy theorem}, Comm. Math. Phys. \textbf{107}, 1 (1986), 103-163. \href{http://projecteuclid.org/euclid.cmp/1104115934}{euclid}


\item [[Jean-Michel Bismut]], \emph{Quillen metrics and determinant bundles}, 2 conference lectures in honour of A. N. Tyurin, video at \href{http://www.mathnet.ru/php/presentation.phtml?presentid=62&option_lang=eng}{link}


\item A. Pressley, [[Graeme Segal|G. Segal]], \emph{Loop Groups}, Oxford Math. Monographs, 1986.


\item Kenro Furutani, \emph{On the Quillen determinant}, J. Geom. Phys. \textbf{49}, 4, 366-375, \href{http://arxiv.org/abs/math/0309127}{math.DG/0309127}, \href{http://dx.doi.org/10.1016/j.geomphys.2003.07.001}{doi}


\item [[M. Kontsevich]], S. Vishik, \emph{Geometry of determinants of elliptic operators}, in Functional Analysis on the Eve of the 21st Century. Vol. I (S. Gindikin, et al., eds.) In honor of the 80th birthday of [[Israel Gel'fand|I.M. Gelfand]]. Birkh\"a{}user, Progr. Math. \textbf{131} (1993), 173-197, \href{http://193.51.104.7/~maxim/TEXTS/geometry_determinants_12.pdf}{pdf}, \href{http://arxiv.org/abs/hep-th/9406140}{hep-th/9406140}


\item [[Robbert Dijkgraaf]], [[Edward Witten|E. Witten]], \emph{Topological gauge theories and group cohomology}, Commun. Math.Phys. \textbf{129}, 393--429 (1990), \href{http://projecteuclid.org/euclid.cmp/1104180750}{euclid}, \href{http://www.ams.org/mathscinet-getitem?mr=1048699}{MR1048699}



\end{itemize}
Discussion in the context of the [[modular functor]] is in

\begin{itemize}%
\item [[Graeme Segal]], section 6 and section 5 of \emph{The definition of conformal field theory} , preprint, 1988; also in [[Ulrike Tillmann]] (ed.) \emph{Topology, geometry and quantum field theory} , London Math. Soc. Lect. Note Ser., Vol. 308. Cambridge University Press, Cambridge (2004) 421-577. (\href{https://people.maths.ox.ac.uk/segalg/0521540496txt.pdf}{pdf})

\end{itemize}
[[!redirects determinant line bundles]]



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