1. Analytic Torsion
A cochain complex of vector spaces (or simply a cochain complex) over a k is a sequence of vector spaces {Ci : i ∈ Z} together with a sequence of k-linear maps {di : Ci → Ci+1 : i ∈ Z} such that di◦ di−1 = 0 for any i ∈ Z. A cochain complex over k is denoted by C = (Ci, di)i∈Z. The i-th cohomology of C is defined to be
Hi(C) = ker di/ Im di−1, i ∈ Z.
We denote H∗(C) =L
i∈ZHi(C) and d =L
i∈Zdi.
Example 1.1. For each i ∈ Z, let 0i be the zero map from Hi(C) to Hi+1(C). Then (Hi(C), 0i) forms a cochain complex over k and is denoted by H(C).
A cochain complex C over k is bounded above/below if there exists N ∈ N such that Ci = 0 and di = 0 for any i > N / i < N. A cochain complex is bounded if it is both bounded above and bounded below.
Definition 1.1. Let C be a bounded cochain complex of finite dimensional k-vector spaces. The determinant of a bounded cochain complex C to be the k-vector space
det C =O
n∈Z
(Det Cn)(−1)n.
Lemma 1.1. Let C be a bounded cochain complex of finite dimensional k-vector spaces. There is a canonical linear isomorphism
τC: Det C → Det H(C).
Let us assume that k = R or k = C.
Definition 1.2. A (Riemannian or Hermitian) metric hC on a cochain complex C is a sequence of functions {hi: Ci× Ci→ k} such that hi is an inner product on Ci for each i ∈ Z.
The inner product hi on Ci will be denoted by h·, ·iCi.
Let hC be a metric on a bounded cochain complex C of finite dimensional vector vector spaces.
We denote the inner product on Det C induced from hCby hDet C. From now on, all the cochain com- plexes are assumed to be bounded cochain complexes of finite dimensional vector spaces.
Let hC be a metric on the cochain complex C and hH(C) be a metric on H(C) respectively. The metric hH(C)induces an inner product hDet H(C)on Det H(C) while the canonical linear isomorphism τC: Det C → Det H(C) introduces an inner product (τC)∗hDet Con Det H(C). Thus we obtain two inner products hDet H(C) and (τC)∗hDet C on Det H(C). Since Det H(C) is one dimensional, there exists a positive real number T (C, hC, hH(C)) ∈ R such that
hDet H(C) = T (C, hC, hH(C))(τC)∗hDet C.
Definition 1.3. The real number T (C, hC, hH(C)) is called the analytic torsion with respect to the triple (C, hC, hH(C)).
Let hCbe a metric on C. We will define a metric hH(C)Hodgeon H(C) using hCand the Hodge decom- position for C. The metric hH(C)Hodgeis called the Hodge metric on H(C). We will compute the analytic torsion T (C, hC, hH(C)Hodge) with respect to (C, hC, hH(C)Hodge). The analytic torsion T (C, hC, hH(C)Hodge) is a very important example in differential geometry/topology. We will briefly review the Hodge decom- position for C.
For each i ∈ Z, we define a linear map (di+1)∗: Ci+1→ Ci by hdix, yiCi+1 = hx, (di+1)∗yiCi
1
2
for any x ∈ Ci and for any y ∈ Ci+1. We define a linear operator ∆ion Ci by
∆i= di−1◦ (di)∗+ (di+1)∗◦ di. We call ∆i the Laplace operator on C with respect to d and to hC.
Theorem 1.1. (Hodge Decomposition Theorem) Let C be a cochain complex of finite dimensional vector spaces and hC be a metric on C. For each i ∈ Z, we have the orthogonal decomposition of vector spaces
Ci= ker ∆i⊕ Im di−1⊕ Im(di+1)∗.
Corollary 1.1. Let C be a cochain complex of finite dimensional vector spaces and hCbe a metric on C. Then there is a linear isomorphism of vector spaces
Hi(C) ∼= ker ∆i for any i ∈ Z.
Since hC is a metric on C and ker ∆i is a vector subspace of Ci, the restriction of hi to ker ∆i is an inner product hiker ∆i on ker ∆i. Denote the isomorphism ker ∆i ∼= Hi(C) by fi for each i ∈ Z.
The Hodge metric hH(C)Hodge on H(C) is defined to be the sequence of inner products hH(C)Hodge= {f∗ihiker ∆i : Hi(C) × Hi(C) → k : i ∈ Z}.
Since ∆i : Ci → Ci is self-adjoint, Ci = ker ∆i⊕ Im ∆i and ∆i(Im ∆i) ⊆ Im ∆i. We define the restriction of ∆i to Im ∆i by ∆i|Im ∆i. Then we have the following result.
Theorem 1.2. The analytic torsion T (C, hC, hH(C)Hodge) with respect to the triple (C, hC, hH(C)Hodge) is given by
T (C, hC, hH(C)Hodge)2=Y
i∈Z
det(∆i|Im ∆i)(−1)nn.
