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differential-operators
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Jannes Bantje
differential-operators
Commits
a4ca0524
Commit
a4ca0524
authored
May 25, 2020
by
Jannes Bantje
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fix thom spectrum command
\M is already def’ed
parent
1071085d
Pipeline
#59709
passed with stages
in 42 seconds
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contents/elliptic-genera.tex
contents/elliptic-genera.tex
+3
-3
math.tex
math.tex
+1
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contents/elliptic-genera.tex
View file @
a4ca0524
...
...
@@ -209,11 +209,11 @@ The $\hat{A}$-genus can be generalised to the spin bordism ring, where a morphis
emerges.
In some degrees the group on the right is
$
\mathbb
{
Z
}
/
2
$
and in particular has torsion.
One could say, this
$
\alpha
$
-invariant remedies the need to tensor with
$
\mathbb
{
Q
}$
.
This map can be represented by a map of spectra
$
\
M
\Spin
\to
\KO
$
, which factors through the connective cover
$
\kO
$
since
$
\
M\Spin
$
is connective.
This map can be represented by a map of spectra
$
\
thoM
\Spin
\to
\KO
$
, which factors through the connective cover
$
\kO
$
since
$
\tho
M\Spin
$
is connective.
This generalisation process is illustrated in the following diagram
\[
\begin
{
tikzcd
}
\pi
_
*
\M\Spin
\dar
\rar
&
\pi
_
*
\kO
\dar
\\
\pi
_
*
\
tho
M\Spin
\dar
\rar
&
\pi
_
*
\kO
\dar
\\
\Omega
^{
\Spin
}_
*
\rar
[
"
\alpha
"
]
\dar
&
\KO
_
*(
\pt
)
\dar
\\
\Omega
^{
\Spin
}_
*
\otimes
\mathbb
{
Q
}
\rar
[
"
\Ahat
"
]
\dar
&
\mathbb
{
Z
}
\dar
\\
\Omega
^{
\SO
}_
*
\otimes
\mathbb
{
Q
}
\rar
[
"
\Ahat
"
]
&
\mathbb
{
Q
}
...
...
@@ -223,7 +223,7 @@ In the end we are interested in the so-called Witten genus, which will be a genu
One now asks for a generalisation of the Witten genus in the same fashion as the generalisation of the
\Ahat
-genus.
The main observation is here, that the Witten genus maps string manifolds to
\emph
{
topological modular forms
}
,
\tmf
.
By means of homotopy theory there already is a map of spectra
$
\sigma
\colon
\M\Strng
\to
\tmf
$
.
By means of homotopy theory there already is a map of spectra
$
\sigma
\colon
\
tho
M\Strng
\to
\tmf
$
.
The million dollar question is: Is there a geometrical description as for the
\Ahat
-genus?
Some of the
\emph
{
conjectures
}
in this direction are:
...
...
math.tex
View file @
a4ca0524
...
...
@@ -230,7 +230,7 @@
\newcommand
{
\CP
}{
\mathbb
{
C
}
\mathbb
{
P
}}
\newcommand
{
\HP
}{
\mathbb
{
H
}
\mathbb
{
P
}}
% \newcommand{\M}{\mathrm{M}\mkern-3mu}
\newcommand
{
\M
}{
M
\mkern
-3mu
}
\newcommand
{
\
tho
M
}{
M
\mkern
-3mu
}
\newcommand
{
\tmf
}{
\ensuremath
{
\mathrm
{
tmf
}}}
\DeclareMathOperator
{
\Map
}{
Map
}
\newcommand
{
\sa
}{
\mathrm
{
sa
}}
...
...
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