Let $V$ and $W$ be two vector spaces of dimension $n = dim V = dim W$ and
a linear map.
Write $\wedge^n V$ and $\wedge^n W$ for the top exterior power of these vector spaces, the skew-symmetrized $n$th tensor power of $V$ and $W$. These are 1-dimensional vector spaces, hence lines over the ground field. The linear map $T$ induces a linear map
between these lines. This is the 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
of such endomorphisms with the ground field $k$ is identified with an element in $k$
This is the standard meaning of the determinant of a linear endomorphism.
Notice that the determinant construction:
is a functor from the category Vect to itself
Any such functor $F : Vect \to Vect$ with certain continuity assumptions induces an endo-functor on the category of vector bundles $VectBund(X)$ over an arbitrary manifold $X$.
Concretely, if a vector bundle $E \to X$ is given by a Cech cocycle
with respect to an open cover $\{U_i \to X\}$ (see principal bundle and associated bundle for details), hence by transition functions
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
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 determinant line bundle of $E$.
Let
be a unitary group-principal bundle (to which is canonically associated a rank-$n$ complex vector bundle). Then the single characteristic class
of its determinant circle bundle
is the first Chern class of $E$
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 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 $\mathfrak{u}(n)$.
An explicit version of this statement is for instance in (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.
There is a specific version called 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 $D$ what determines a holomorphic line bundle over $\mathcal{A}$. This is the 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 determinant related to the zeta function. (This is related to the analytic torsion).
Let $Gr_k(V)$ be the Grassmanian? 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 Grassmanian, 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.
The determinant line bundle of Quillen is in fact related to a variant of Segal’s determinant bundle on the “semiinfinite” Grassmanian. 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 and $W\to H_-$ is 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
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.
In dimensîon $8k+2$ for $k \in \mathbb{N}$ the determinant line bundle has a canonîcal square root line bundle, the Pfaffian line bundle.
The following table lists classes of examples of square roots of line bundles
The relation between determinant line bundles and the first Chern class is stated explicitly for instance on p. 414 of
Literature on determinant line bundles of infinite-dimensional bundles includes the following:
D.G. Quillen, Determinants of Cauchy-Riemann operators over a Riemann surface, Funkcionalnii Analiz i ego Prilozhenija 19 (1985), 37-41, (pdf of Russian version).
M. F. Atiyah, I. M. Singer, Dirac operators coupled to vector potentials, Proc. Nat. Acad. Sci. USA 81, 2597-2600 (1984) (pdf at pnas site)
Dan Freed, On determinant line bundles, Math. aspects of string theory, ed. S. T. Yau, World Sci. Publ. 1987, (revised pdf, dg-ga/9505002)
J.-M. Bismut, Dan Freed, The analysis of elliptic families.I. Metrics and connections on determinant bundles, Comm. Math. Phys. 106, 1 (1986), 159-176, euclid, II. Dirac operators, eta invariants, and the holonomy theorem, Comm. Math. Phys. 107, 1 (1986), 103-163. euclid
J-M. Bismut, Quillen metrics and determinant bundles, 2 conference lectures in honour of A. N. Tyurin, video at link
A. Pressley, G. Segal, Loop Groups, Oxford Math. Monographs, 1986.
Kenro Furutani, On the Quillen determinant, J. Geom. Phys. 49, 4, 366-375, math.DG/0309127, doi
M. Kontsevich, S. Vishik, 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 I.M. Gelfand. Birkhäuser, Progr. Math. 131 (1993), 173-197, pdf, hep-th/9406140
R. Dijkgraaf, E. Witten, Topological gauge theories and group cohomology, Commun. Math.Phys. 129, 393–429 (1990), euclid, MR1048699