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 spinᶜ structures and are therefore of central importance for Seiberg–Witten theory.

Definition

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

$\det \colon \operatorname {Vect} _{\mathbb {R} }^{n}(X)\cong [X,\operatorname {BO} (n)]\xrightarrow {{\mathcal {B}}\det _{*}} [X,\operatorname {BO} (1)]\cong \operatorname {Vect} _{\mathbb {R} }^{1}(X).$

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

$\det \colon \operatorname {Vect} _{\mathbb {C} }^{n}(X)\cong [X,\operatorname {BU} (n)]\xrightarrow {{\mathcal {B}}\det _{*}} [X,\operatorname {BU} (1)]\cong \operatorname {Vect} _{\mathbb {C} }^{1}(X).$

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

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

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 $f\colon X\rightarrow Y$ 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 $g\colon Y\rightarrow \operatorname {BO} (n)$ be its classifying map with $E=g^{*}\gamma _{\mathbb {R} }^{n}$, then:

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

For complex vector bundles, the proof is completely analogous.

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

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

Literature

• Bott, Raoul; Tu, Loring W. (1982). 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" (PDF).
• Nicolaescu, Liviu I. (2000), Notes on Seiberg-Witten theory (PDF), Graduate Studies in Mathematics, vol. 28, Providence, RI: American Mathematical Society, doi:10.1090/gsm/028, ISBN 978-0-8218-2145-9, MR 1787219
• Hatcher, Allen (2003). "Vector Bundles & K-Theory".

External links

• determinant line bundle at the nLab