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determinant line bundle

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Definition

Let VV and WW be two vector spaces of dimension n=dimV=dimWn = dim V = dim W and

T:VW T : V \to W

a linear map.

Write nV\wedge^n V and nW\wedge^n W for the top exterior power of these vector spaces, the skew-symmetrized nnth tensor power of VV and WW. These are 1-dimensional vector spaces, hence lines over the ground field. The linear map TT induces a linear map

detT: nV nW det T : \wedge^n V\to \wedge^n W

between these lines. This is the determinant of TT. More specifically, if V=WV = W (being of the same finite dimension, both are necessarily isomorphic but not necessarily canonically so) then detT: nV nVdet T : \wedge^n V \to \wedge^n V is a linear endomorphism of a 1-dimensional vector space and by the equivalence

End( nV)k End(\wedge^n V) \simeq k

of such endomorphisms with the ground field kk is identified with an element in kk

detTk. det T \in k \,.

This is the standard meaning of the determinant of a linear endomorphism.

Notice that the determinant construction:

det:(VTW)( nVdetT nW) det : (V \stackrel{T}{\to} W) \mapsto (\wedge^n V \stackrel{det T}{\to} \wedge^n W)

is a functor from the category Vect to itself

det:VectVect. det : Vect \to Vect \,.

Any such functor F:VectVectF : Vect \to Vect with certain continuity assumptions induces an endo-functor on the category of vector bundles VectBund(X)VectBund(X) over an arbitrary manifold XX.

Concretely, if a vector bundle EXE \to X is given by a Cech cocycle

C(U i) (g ij) BGL(n) Vect X \array{ C(U_i) &\stackrel{(g_{i j})}{\to}& \mathbf{B} GL(n) &\to& Vect \\ \downarrow^{\mathrlap{\simeq}} \\ X }

with respect to an open cover {U iX}\{U_i \to X\} (see principal bundle and associated bundle for details), hence by transition functions

(g ijC(U iU j,GL(n))) (g_{i j} \in C(U_i \cap U_j, GL(n)))

with values in the general linear group, its image under det:VectBund(X)VectBund(X)det : VectBund(X) \to VectBund(X) is the bundle with transition functions the determinants of these transition functions

(detg ijC(U iU j,GL(1)k ×)). (det g_{i j} \in C(U_i \cap U_j, GL(1) \simeq k^\times)) \,.

This are the transition functions for the bundle EX\wedge^\bullet E \to X which is fiberwise the top exterior power of EXE \to X. This is the determinant line bundle of EE.

Properties

Proposition

Let

E:XC(U i)g ijBU(n) E : X \stackrel{\simeq}{\leftarrow} C(U_i) \stackrel{g_{i j}}{\to} \mathbf{B} U(n)

be a unitary group-principal bundle (to which is canonically associated a rank-nn complex vector bundle). Then the single characteristic class

[detE]H 2(X,) [det E] \in H^2(X, \mathbb{Z})

of its determinant circle bundle

detE:XC(U i)detg ijBU(1) det E : X \stackrel{\simeq}{\leftarrow} C(U_i) \stackrel{det g_{i j}}{\to} \mathbf{B} U(1)

is the first Chern class of EE

[detE]=c 1(E). [det E] = c_1(E) \,.

Moreover, if XX is a smooth manifold and (g ij,A i)(g_{i j}, A_i) is the data of a connection on a bundle (E,)(E, \nabla) on EE then (detg ij,trA i)(det g_{i j}, tr A_i) (where we take the trace tr:𝔲(n)𝔲(1)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,)(E, \nabla) induced by the canonical unary invariant polynomial on 𝔲(n)\mathfrak{u}(n).

An explicit version of this statement is for instance in (GriffithsHarris, p. 414).

One can now look at operators T:EFT:E\to F where E,FE,F are vector bundles of rank nn and the induced operators Λ nT:Λ nEΛ nF\Lambda^n T : \Lambda^n E\to \Lambda^n F which can be considered as elements detT(Λ nE) *Λ nFdet T\in (\Lambda^n E)^*\otimes\Lambda^n F.

Even more important is the case of when XX 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.

Examples

Quillen’s determinant line bundle

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 EE over a fixed Riemann surface MM. A complex structure on the bundle corresponds to an operator which in local coordinates looks as D=dz¯( z+α(z))D = d\bar{z}(\partial_z+\alpha(z)) where α(z)\alpha(z) is a smooth matrix valued function. The set of such operators is an affine space 𝒜\mathcal{A} whose underlying vector space is the space of (0,1)(0,1)-End-valued forms Ω 0,1(EndM)\Omega^{0,1} (End M). Then again a determinant is an element of a line D=λ(KerD) *λ(CokerD)\mathcal{L}_D = \lambda(Ker D)^*\otimes \lambda(Coker D) where λ\lambda is taking the top exterior power. Now one has a family D\mathcal{L}_D depending on DD what determines a holomorphic line bundle over 𝒜\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 *DD^* D and its determinant related to the zeta function. (This is related to the analytic torsion).

Determinant bundle on the Grassmanian

Let Gr k(V)Gr_k(V) be the Grassmanian? of kk-dimensional subspaces of a finite dimensional vector space VV. Let WVW\subset V be a point in Gr k(V)Gr_k(V) and Λ k(W)\Lambda^k(W) its top exterior power; it is a fiber of the bundle DetDet over Gr k(V)Gr_k(V). The determinant bundle DetDet has no non-zero holomorphic global sections. Consider its dual Det *Det^* with fiber Λ k(W) *\Lambda^k(W)^* over WW. Then the space of of global holomorphic sections Γ hol(Det *)Λ k(V *)\Gamma_{hol}(Det^*) \cong \Lambda^k(V^*). This construction can be suitably extended for the Segal Grassmanian, where V=V +V 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.

Comparing Quillen’s and Segal’s determinant line bundles

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)Gr_{cpt}(H) which is the set (space eventually) of closed supspaces WHW\subset H where the projection WH +W\to H_+ is Fredholm and WH W\to H_- is compact; then one follows the Segal’s prescription to define DetDet on Gr cpt(H)Gr_{cpt}(H). Notice that Gr cpt(H)Gr_{cpt}(H) is not a homogeneous space. Now there is a span of maps with contractible fibers

Gr cpt(H)Fred(H +). Gr_{cpt}(H)\leftarrow \mathcal{B}\to Fred(H_+).

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

Pfaffian line bundle

In dimensîon 8k+28k+2 for kk \in \mathbb{N} the determinant line bundle has a canonîcal square root line bundle, the Pfaffian line bundle.

From fermionic path integrals

See fermionic path integral.

The following table lists classes of examples of square roots of line bundles

line bundlesquare rootchoice corresponds to
canonical bundleTheta characteristicover Riemann surface and Hermitian manifold (e.g.Kähler manifold): spin structure
density bundlehalf-density bundle
canonical bundle of Lagrangian submanifoldmetalinear structuremetaplectic correction
determinant line bundlePfaffian line bundle
quadratic secondary intersection pairingpartition function of self-dual higher gauge theoryintegral Wu structure

References

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

  • Griffiths and Harris, Principles of algebraic geometry

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

Revised on November 10, 2013 03:43:15 by Anonymous Coward (74.69.73.189)