Determinant 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 {\displaystyle X} be a paracompact space, then there is a bijection [ X , BO ⁡ ( n ) ] → ≅ Vect R n ⁡ ( X ) , [ f ] ↦ f ∗ γ R n {\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 γ R n {\displaystyle \gamma _{\mathbb {R} }^{n}} .^[1] The real determinant det : O ⁡ ( n ) → O ⁡ ( 1 ) {\displaystyle \det \colon \operatorname {O} (n)\rightarrow \operatorname {O} (1)} is a group homomorphism and hence induces a continuous map B det : BO ⁡ ( n ) → BO ⁡ ( 1 ) ≅ R P ∞ {\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:

det : Vect R n ⁡ ( X ) ≅ [ X , BO ⁡ ( n ) ] → B det ∗ [ X , BO ⁡ ( 1 ) ] ≅ Vect R 1 ⁡ ( X ) . {\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 X {\displaystyle X} be a paracompact space, then there is a bijection [ X , BU ⁡ ( n ) ] → ≅ Vect C n ⁡ ( X ) , [ f ] ↦ f ∗ γ C n {\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 γ C n {\displaystyle \gamma _{\mathbb {C} }^{n}} .^[1] The complex determinant det : U ⁡ ( n ) → U ⁡ ( 1 ) {\displaystyle \det \colon \operatorname {U} (n)\rightarrow \operatorname {U} (1)} is a group homomorphism and hence induces a continuous map B det : BU ⁡ ( n ) → BU ⁡ ( 1 ) ≅ C P ∞ {\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:

det : Vect C n ⁡ ( X ) ≅ [ X , BU ⁡ ( n ) ] → B det ∗ [ X , BU ⁡ ( 1 ) ] ≅ Vect C 1 ⁡ ( X ) . {\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 E ↠ X {\displaystyle E\twoheadrightarrow X} be a vector bundle, then:^[2]

det ( E ) := Λ rk ⁡ ( E ) ( E ) . {\displaystyle \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.^[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 f : X → Y {\displaystyle f\colon X\rightarrow Y} between paracompact spaces X {\displaystyle X} and Y {\displaystyle Y} as well as a vector bundle E ↠ Y {\displaystyle E\twoheadrightarrow Y} , one has:

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

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

det ( f ∗ E ) ≅ det ( f ∗ g ∗ γ R n ) ≅ det ( ( g ∘ f ) ∗ γ R n ) ≅ ( B det ∘ g ∘ f ) ∗ γ R 1 ≅ f ∗ ( B det ∘ g ) ∗ γ R 1 ≅ f ∗ det ( g ∗ γ R n ) ≅ f ∗ det ( E ) . {\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 E , F ↠ X {\displaystyle E,F\twoheadrightarrow X} (with the same fields as fibers), one has:

  det ( E ⊗ F ) ≅ det ( E ) rk ⁡ ( F ) ⊗ det ( F ) rk ⁡ ( E ) . {\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).
• 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".

References

External links

• determinant line bundle at the nLab