A typo in equations (2.6) and (4.68) of "Proof of Character-Valued Index Theorems" by Mark W. Goodman

Question

In the article "Proof of Character-Valued Index Theorems" by Mark  W.  Goodman (https://projecteuclid.org/download/pdf_1/euclid.cmp/1104116140); on page  407 appears the equation (4.68) with the form

My claim is that there is a typo in the equation (4.68) and then the correct form must be

Do you agree?

Other typo:  on page  393  appears the equation (2.6) with the form

There is a typo in the equation (2.6) and then the correct form must be

It is corroborated by the equation (4.67) which reads as

Comments

yes, as stated in 2.10 for the Lefschetz number ... but I have still to check why 4.67 + 2.6 => 4.68

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Hi @igael, thanks for your comment.  You are right (2.10) shows that there is a typo in (4.68). I do not understand what do you mean with "4.67 + 2.6 => 4.68".  The equation (4.67)  is precisely the equation (4,61) but restated in the language of forms. Similarly, the equation (4..68) is precisely the equation (4.62) but restated in the language of forms.  Then according with these facts it is nor possible to derive (4.68) from (4.67) and (2.6).  Do you agree?  All the best.

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Hi :) sorry, not well awake the last night. I did a bad substitution

Answer 1

A possible derivation of (4.68) is as follows.

Then we have that

$$(-1)^{F}|\omega_p>= (-1)^{p}|\omega_p>$$

$$<\omega_p|(-1)^F = (-1)^p<\omega_p|$$

$$H|\omega_p> = E_p|\omega_p>$$

$${e}^{-\beta\,H}|\omega_p>={e}^{-\beta E_{p}}|\omega_p>$$

with all these facts we make the following computation:

$$Tr((-1)^F b{e}^{-\beta\,H})=\sum _ p  <\omega_p| (-1)^F b{e}^{-\beta\,H}|\omega_p> =\sum _ p  (-1)^p<\omega_p|  b|\omega_p>{e}^{-\beta\,E_p} ,$$

$$Tr((-1)^F b{e}^{-\beta\,H})=\sum _ p  (-1)^p<\omega_p|  b|\omega_p>{e}^{-\beta\,E_p}|_{E_p=0} =\sum _ p  (-1)^p<\omega_p|  b|\omega_p>,$$

$$Tr((-1)^F b{e}^{-\beta\,H})=\sum _ p  (-1)^pTr_{H^p}(b) =\sum _ p  (-1)^p \chi_{{p}} \left( b \right) = \chi \left( M_{{b}} \right) .$$

Answer 2

Other derivation of (4.68) which could be useful in quantum computing is as follows.

Let $H^p$ the Hilbert space with orthonormal basis $\left\{ |\omega_p> \right\}$.  We define a unitary transformation

$$U :  H^p \rightarrow H^p$$

by the formula·

$$U = (-1)^F b {e}^{-\beta\,H}$$

with

$$U |\omega_p>= (-1)^p  {e}^{-\beta\,E_p} b |\omega_p>$$

Let a quantum state given by

$$|\Psi>= \sum _p|\omega_p>.$$

Will all these ingredients we make the following computation.

$$Tr((-1)^F b {e}^{-\beta\,H} ) = Tr(U)= < \Psi |U|\Psi>= \sum _q\sum _p<\omega_q|U|\omega_p>,$$

$$Tr((-1)^F b {e}^{-\beta\,H} ) = Tr(U)= < \Psi |U|\Psi>=\sum _q\sum _p<\omega_q|(-1)^p  {e}^{-\beta\,E_p} b |\omega_p>,$$

$$Tr((-1)^F b {e}^{-\beta\,H} ) = Tr(U)= < \Psi |U|\Psi>=\sum _q\sum _p(-1)^p  {e}^{-\beta\,E_p} <\omega_q|b |\omega_p>,$$

$$Tr((-1)^F b {e}^{-\beta\,H} ) = Tr(U)= < \Psi |U|\Psi>=\sum _q\sum _p(-1)^p  {e}^{-\beta\,E_p} <\omega_p|b |\omega_p>\delta_{{q,p}},$$

$$Tr((-1)^F b {e}^{-\beta\,H} ) = Tr(U)= < \Psi |U|\Psi>=\sum _p(-1)^p  {e}^{-\beta\,E_p} <\omega_p|b |\omega_p>,$$

$$Tr((-1)^F b {e}^{-\beta\,H} ) = Tr(U)= < \Psi |U|\Psi>= \sum _p(-1)^p \chi_{{p}} \left( b \right) = \chi \left( M_{{b}} \right) .$$

Reference :  Louis Kauffman, arXiv:1001.0354v3  [math.GT]  31 Jan 2010 Topological Quantum Information, Khovanov Homology and the Jones Polynomial  (http://lanl.arxiv.org/pdf/1001.0354)