Determinant line bundle

Short description: Construction for vector bundles

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

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

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

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

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

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

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 $E=g^{*}{\gamma }_{\mathbb{ℝ}}^n$, then:

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

For complex vector bundles, the proof is completely analogous.

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

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

Literature

• Bott, Raoul; Tu, Loring W. (1982) (in en). Differential Forms in Algebraic Topology. Springer. doi:10.1007/978-1-4757-3951-0. ISBN 978-1-4757-3951-0. 
• Freed, Daniel (1987-03-10). "On determinant line bundles". https://web.ma.utexas.edu/users/dafr/detsur.pdf. 
• Nicolaescu, Liviu I. (2000), Notes on Seiberg-Witten theory, Graduate Studies in Mathematics, 28, Providence, RI: American Mathematical Society, doi:10.1090/gsm/028, ISBN 978-0-8218-2145-9, http://www.nd.edu/~lnicolae/swnotes.pdf 
• Hatcher, Allen (2003). "Vector Bundles & K-Theory". https://pi.math.cornell.edu/~hatcher/VBKT/VBpage.html. 

External links

• determinant line bundle at the nLab