Mapping cone Thom forms
For the de Rham mapping cone cochain complex induced by a smooth closed 2-form, we explicitly write down the associated mapping cone Thom form in the sense of Mathai-Quillen. Our construction uses the mapping cone covariant derivative, carrying the e…
Authors: Hao Zhuang
Mapping cone Thom forms Hao Zh uang Marc h 27, 2026 Abstract F or the de Rham mapping cone co c hain complex induced by a smo oth closed 2-form, we explicitly write do wn the associated mapping cone Thom form in the sense of Mathai-Quillen. Our construction uses the mapping cone co v arian t deriv ativ e, carrying the extra information brought b y the 2-form. Our main to ol is the Berezin in tegral. As the main result, w e show that this Thom form is closed with resp ect to the mapping cone differentiation, its in tegration along the fib er is 1, and it satisfies the transgression form ula. Con ten ts 1 In tro duction 1 2 Mapping cone cov arian t deriv ativ e 4 3 Sk ew-adjoin t Bianc hi identit y 5 4 Berezin in tegral of pairs 8 5 Thom form for the mapping cone complex 9 6 T ransgression of the Thom form 13 1 In tro duction The de Rham mapping cone co c hain complex asso ciated with a smooth closed form carries the studies from v arious p ersp ectiv es in geometry and top ology . The symplectic case connects with the primitiv e cohomology and the filtered cohomology [ 9 , 10 , 11 , 12 ]. F or the de Rham mapping cone co chain complex asso ciated with a general closed form, we hav e seen the studies in characteristic classes [ 19 , 20 , 21 ], gauge theory [ 13 , 14 , 15 ], Morse theory [ 5 , 6 , 22 ], and so on. An imp ortan t mechanism supp orting the studies of c haracteristic classes, gauge theory , and Morse theory is the Thom isomorphism [ 2 , Section 6]. The analytic expression of the associated Thom form is explicitly form ulated b y Mathai and Quillen [ 7 , Theorem 4.5]. No w, in the mapping cone case, w e also exp ect to build 1 suc h a mechanism. In [ 8 , Theorem 3.13], the Thom isomorphism is formulated via the top ological approach. This reminds us of the necessity to giv e an explicit construction of the mapping cone Thom form in the sense of Mathai-Quillen. In this pap er, we give the construction. W e assume the following settings: Assumption 1.1. W e let M b e a smo oth manifold without b oundary , ω b e a smo oth closed 2-form on M , and E b e an oriented smo oth v ector bundle of rank n o v er M . Also, we equip E with a smo oth vector bundle metric g . F or the construction of the mapping cone Thom form, we follow the pattern in [ 1 , Section 1.6], [ 17 , Chapter 3], and [ 18 , Chapter 3]. F or the notion of the mapping cone co v ariant deriv ative, we follow [ 13 , Prop osition 3.8] and [ 15 , Section 2.1]. With resp ect to g , we let ∇ b e a Euclidean connection on E , and let Φ ∈ Ω 0 ( M , End( E )) b e sk ew-adjoint. They form a mapping cone cov ariant deriv ative A : Ω i ( M , E ) ⊕ Ω i − 1 ( M , E ) → Ω i +1 ( M , E ) ⊕ Ω i ( M , E ) ( α, β ) 7→ ( ∇ α + ω ∧ β , Φ α − ∇ β ) . (1.1) Let σ : E → M b e the pro jection to the base space. W e then obtain the pullbac k bundle e E = σ ∗ E and the asso ciated e ω = σ ∗ ω , e ∇ = σ ∗ ∇ , e Φ = σ ∗ Φ , e g = σ ∗ g . (1.2) Then, w e hav e the de Rham mapping cone co chain complex d e ω : Ω i ( E ) ⊕ Ω i − 1 ( E ) → Ω i +1 ( E ) ⊕ Ω i ( E ) ( α, β ) 7→ ( dα + e ω ∧ β , − dβ ) (1.3) of ( E , e ω ), and the mapping cone cov arian t deriv ative e A : Ω i ( E , Λ ∗ e E ) ⊕ Ω i − 1 ( E , Λ ∗ e E ) → Ω i +1 ( E , Λ ∗ e E ) ⊕ Ω i ( E , Λ ∗ e E ) ( α, β ) 7→ e ∇ α + e ω ∧ β , e Φ Λ α − e ∇ β (1.4) on Λ ∗ E . Here, e Φ Λ is the deriv ation (See ( 2.9 )-( 2.11 )) induced by e Φ, and e ∇ extends naturally to Λ ∗ e E . Let e 1 , · · · , e n b e an oriented lo cal orthonormal frame of E , then w e hav e its lift e e 1 , · · · , e e n to e E . Then, w e let Q e A = X 1 ⩽ i
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