It is kind of awkward to answer my own question: It seems the answer is related to partition function of certain quadratic functional under gauge invariance. The relevant literature is:

The Partition Function of a Degenerate Functional

and

The partition function of degenerate quadratic functional and Ray-Singer invariants

by A. S. Schwarz. I found pointer to this reference via reading John Lott's lecture notes and personal communication with Pavel Mnev. The paper is not hard to read but skipped a lot detail in the proofs. There has also been some recent important work by Phillip Andreae.

It is kind of awkward to answer my own question: It seems the answer is related to partition function of certain quadratic functional under gauge invariance. The relevant literature is:

The Partition Function of a Degenerate Functional

and

The partition function of degenerate quadratic functional and Ray-Singer invariants

by A. S. Schwarz. I found pointer to this reference via reading John Lott's lecture notes and personal communication with Pavel Mnev. The paper is not hard to read but skipped a lot detail in the proofs. So if someone can give a sketch of the main argument I will be grateful.

There has also been some recent important work by Phillip Andreae, who pointed out in his thesis that it is impossible to have a "Mckean-Singer" type of formula for analytic torsion. So in particular this rules out possible construction of arithemetic cohomology groups in Arakelov theory via integration of local terms arising from heat kernel asymptotics.

It is kind of awkward to answer my own question: It seems the answer is related to partition function of certain quadratic functional under gauge invariance. The relevant literature is:

The Partition Function of a Degenerate Functional

by A. S. Schwarz. I found pointer to this reference via reading John Lott's lecture notes and personal communication with Pavel Mnev. The paper is not hard to read but skipped a lot detail in the proofs. There has also been some recent important work by Phillip Andreae.

It is kind of awkward to answer my own question: It seems the answer is related to partition function of certain quadratic functional under gauge invariance. The relevant literature is:

The Partition Function of a Degenerate Functional

and

The partition function of degenerate quadratic functional and Ray-Singer invariants

by A. S. Schwarz. I found pointer to this reference via reading John Lott's lecture notes and personal communication with Pavel Mnev. The paper is not hard to read but skipped a lot detail in the proofs. There has also been some recent important work by Phillip Andreae.