a linear map.
Write and for the top exterior power of these vector spaces, the skew-symmetrized th tensor power of and . These are 1-dimensional vector spaces, hence lines over the ground field. The linear map induces a linear map
between these lines. This is the determinant of . More specifically, if (being of the same finite dimension, both are necessarily isomorphic but not necessarily canonically so) then is a linear endomorphism of a 1-dimensional vector space and by the equivalence
of such endomorphisms with the ground field is identified with an element in
This is the standard meaning of the determinant of a linear endomorphism.
Notice that the determinant construction:
Concretely, if a vector bundle is given by a Cech cocycle
with values in the general linear group, its image under is the bundle with transition functions the determinants of these transition functions
This are the transition functions for the bundle which is fiberwise the top exterior power of . This is the determinant line bundle of .
of its determinant circle bundle
is the first Chern class of
Moreover, if is a smooth manifold and is the data of a connection on a bundle on then (where we take the trace 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 induced by the canonical unary invariant polynomial on .
An explicit version of this statement is for instance in (GriffithsHarris, p. 414).
One can now look at operators where are vector bundles of rank and the induced operators which can be considered as elements .
Even more important is the case of when 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 over a fixed Riemann surface . A complex structure on the bundle corresponds to an operator which in local coordinates looks as where is a smooth matrix valued function. The set of such operators is an affine space whose underlying vector space is the space of -End-valued forms . Then again a determinant is an element of a line where is taking the top exterior power. Now one has a family depending on what determines a holomorphic line bundle over . 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 and its determinant related to the zeta function. (This is related to the analytic torsion).
Let be the Grassmanian? of -dimensional subspaces of a finite dimensional vector space . Let be a point in and its top exterior power; it is a fiber of the bundle over . The determinant bundle has no non-zero holomorphic global sections. Consider its dual with fiber over . Then the space of of global holomorphic sections . This construction can be suitably extended for the Segal Grassmanian, where 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 which is the set (space eventually) of closed supspaces where the projection is Fredholm and is compact; then one follows the Segal’s prescription to define on . Notice that 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 and its pullback to is isomorphic to the pullback of the determinant bundle on ; in fact the Quillen’s version can be reconstructed from this pullback by certain quotienting construction.
In dimensîon for the determinant line bundle has a canonîcal square root line bundle, the Pfaffian line bundle.
|line bundle||square root||choice corresponds to|
|canonical bundle||Theta characteristic||over Riemann surface and Hermitian manifold (e.g.Kähler manifold): spin structure|
|density bundle||half-density bundle|
|canonical bundle of Lagrangian submanifold||metalinear structure||metaplectic correction|
|determinant line bundle||Pfaffian line bundle|
|quadratic secondary intersection pairing||partition function of self-dual higher gauge theory||integral Wu structure|
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:
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)
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.
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