Determinant line bundle

In differential geometry, the determinant line bundle is a construction, which assigns every vector bundle over paracompact spaces a line bundle. Its name comes from using the determinant on their classifying spaces. Determinant line bundles naturally arise in four-dimensional spin^c structures and are therefore of central importance for Seiberg–Witten theory.

Definition

Let $X$ be a paracompact space, then there is a bijection $[X,\mathrm{BO}(n)]\stackrel{\cong }{\to }{\mathrm{Vect}}_{ℝ}^n(X),[f]\mapsto f^{*}{\gamma }_{ℝ}^n$ with the real universal vector bundle ${\gamma }_{ℝ}^n$. The real determinant $\det :\mathrm{O}(n)\to \mathrm{O}(1)$ is a group homomorphism and hence induces a continuous map $ℬ\det :\mathrm{BO}(n)\to \mathrm{BO}(1)\cong ℝP^{\mathrm{\infty }}$ on the classifying space for O(n). Hence there is a postcomposition:

$\det :{\mathrm{Vect}}_{ℝ}^n(X)\cong [X,\mathrm{BO}(n)]\stackrel{ℬ\underset{*}{\det }}{\to }[X,\mathrm{BO}(1)]\cong {\mathrm{Vect}}_{ℝ}^1(X).$

Let $X$ be a paracompact space, then there is a bijection $[X,\mathrm{BU}(n)]\stackrel{\cong }{\to }{\mathrm{Vect}}_{ℂ}^n(X),[f]\mapsto f^{*}{\gamma }_{ℂ}^n$ with the complex universal vector bundle ${\gamma }_{ℂ}^n$. The complex determinant $\det :\mathrm{U}(n)\to \mathrm{U}(1)$ is a group homomorphism and hence induces a continuous map $ℬ\det :\mathrm{BU}(n)\to \mathrm{BU}(1)\cong ℂP^{\mathrm{\infty }}$ on the classifying space for U(n). Hence there is a postcomposition:

$\det :{\mathrm{Vect}}_{ℂ}^n(X)\cong [X,\mathrm{BU}(n)]\stackrel{ℬ\underset{*}{\det }}{\to }[X,\mathrm{BU}(1)]\cong {\mathrm{Vect}}_{ℂ}^1(X).$

Alternatively, the determinant line bundle can be defined as the last non-trivial exterior product. Let $E\twoheadrightarrow X$ be a vector bundle, then:

$\det (E):={\Lambda }^{\mathrm{rk}(E)}(E).$

Properties

• The real determinant line bundle preserves the first Stiefel–Whitney class, which for real line bundles over topological spaces with the homotopy type of a CW complex is a group isomorphism. Since in this case the first Stiefel–Whitney class vanishes if and only if a real line bundle is orientable, both conditions are then equivalent to a trivial determinant line bundle.
• The complex determinant line bundle preserves the first Chern class, which for complex line bundles over topological spaces with the homotopy type of a CW complex is a group isomorphism.
• The pullback bundle commutes with the determinant line bundle. For a continuous map $f:X\to Y$ between paracompact spaces $X$ and $Y$ as well as a vector bundle $E\twoheadrightarrow Y$, one has:

  $\det (f^{*}E)\cong f^{*}\det (E).$

Proof: Assume $E\twoheadrightarrow Y$ is a real vector bundle and let $g:Y\to \mathrm{BO}(n)$ be its classifying map with $E=g^{*}{\gamma }_{ℝ}^n$, then:

$\det (f^{*}E)\cong \det (f^{*}{g}^{*}{\gamma }_{ℝ}^n)\cong \det ((g\circ f{)}^{*}{\gamma }_{ℝ}^n)\cong (ℬ\det \circ g\circ f{)}^{*}{\gamma }_{ℝ}^1\cong f^{*}(ℬ\det \circ g{)}^{*}{\gamma }_{ℝ}^1\cong f^{*}\det (g^{*}{\gamma }_{ℝ}^n)\cong f^{*}\det (E).$

For complex vector bundles, the proof is completely analogous.

• For vector bundles $E,F\twoheadrightarrow X$ (with the same fields as fibers), one has:

  $\det (E\otimes F)\cong \det (E{)}^{\mathrm{rk}(F)}\otimes \det (F{)}^{\mathrm{rk}(E)}.$

External links

• determinant line bundle at the nLab