# nLab determinant line bundle

### Context

#### Bundles

bundles

fiber bundles in physics

# Contents

## Definition

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

$T : V \to W$

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

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

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 $F : Vect \to Vect$ 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

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

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

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

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

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

## 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)

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