nLab determinant line bundle

Definition

Let $V$ and $W$ be two vector spaces of dimension $n = dim V = dim W$ and

$$T : V \to W$$

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

$$det T \colon \wedge^n V\to \wedge^n W$$

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

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

of such endomorphisms with the ground field $k$, is identified with an element in $k$

$$det T \in k \,.$$

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

Notice that the determinant construction:

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

is a functor from the category Vect to itself

$$det : Vect \to Vect \,.$$

Any such functor $F \colon Vect \to Vect$ with certain continuity assumptions induces an endo-functor on the category of vector bundles $VectBund(X)$ over any manifold $X$ (see there).

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

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

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

$$(g_{i j} \in C(U_i \cap U_j, GL(n)))$$

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

$$(det g_{i j} \in C(U_i \cap U_j, GL(1) \simeq k^\times)) \,.$$

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 $E$.

Properties

Proposition

Let

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

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

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

of its determinant circle bundle

$$det E \colon X \stackrel{\simeq}{\leftarrow} C(U_i) \stackrel{det g_{i j}}{\to} \mathbf{B} U(1)$$

is the first Chern class of $E$

$$[det E] = c_1(E) \,.$$

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.

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 $E$ 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 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).

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.

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” 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 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

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

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.

Pfaffian line bundle

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

From fermionic path integrals

See at fermionic path integral.

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 (Freed 87, pages 30-31).

Relation to vacuum energy, partition function

See at vacuum energy

Related concepts

• density bundle

• pseudoscalar

• theta function

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

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

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).

  reviewed e.g. in

  Arlo Caine, Quillen’s construction of Determinants of Cauchy–Riemann operators over Riemann Surfaces, 2005 (pdf)

• Michael Atiyah, Isadore Singer, Dirac operators coupled to vector potentials, Proc. Nat. Acad. Sci. USA 81, 2597-2600 (1984) (pdf at pnas site)

• Daniel Freed, On determinant line bundles, Math. aspects of string theory, ed. S. T. Yau, World Sci. Publ. 1987, (dg-ga/9505002, revised pdf)

• Jean-Michel Bismut, Daniel 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

• Jean-Michel 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

• Robbert Dijkgraaf, E. Witten, Topological gauge theories and group cohomology, Commun. Math.Phys. 129, 393–429 (1990), euclid, MR1048699

Discussion in the context of the modular functor is in

• Graeme Segal, section 6 and section 5 of The definition of conformal field theory , preprint, 1988; also in Ulrike Tillmann (ed.) Topology, geometry and quantum field theory , London Math. Soc. Lect. Note Ser., Vol. 308. Cambridge University Press, Cambridge (2004) 421-577. (pdf)