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

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

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

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

Alternatively, the determinant line bundle can be defined as the last non-trivial exterior product. Let ${\displaystyle E\twoheadrightarrow X}$ be a vector bundle, then:^[2]

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

Properties

• The real deteminant 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.^[3] Since in this case the first Stiefel–Whitney class vanishes if and only if a real line bundle is orientable,^[4] both conditions are then equivalent to a trivial determinant line bundle.^[5]
• 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.^[3]
• The pullback bundle commutes with the determinant line bundle. For a continuous map ${\displaystyle f\colon X\rightarrow Y}$ between paracompact spaces ${\displaystyle X}$ and ${\displaystyle Y}$ as well as a vector bundle ${\displaystyle E\twoheadrightarrow Y}$, one has:

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

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

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

For complex vector bundles, the proof is completely analogous.

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

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

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).{{cite web}}: CS1 maint: year (link)
• 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".