Explicit Riemann Hilbert correspondenceTerminology of “covariant derivative” and various...
Explicit Riemann Hilbert correspondence
Terminology of “covariant derivative” and various “connections”regular singularitiesKernel of an integrable holomorphic dee-bar connection is a holomorphic vector bundleAnalogy between connections and $ell$-adic sheaves: what happens with the residue?Existence of flat connections via characteristic classes, for nice groupsDo we have classical Riemann-Hilbert correspondence for infinite dimensional flat vector bundles?Kernel of a non-integrable connectionRiemann-Hilbert correspondence for non-flat connectionsConnection 1-form of the frame bundle associated to a vector bundle with a connectionInducing linear connections via functors
$begingroup$
For simplicity, we assume that $X=mathbb P_{mathbb C}^1-{s_1, s_2, dots, s_k}$ and $infty in X$.
Consider the trivial bundle $E=mathcal O_X^r$ with the connection $nabla$ induced by a Fushcian system on $X$, i.e.
$$nabla:= d+sum_{i=1}^k frac{A_i}{z-s_i},$$
where $A_i$'s are $rtimes r$ constant matrices over $mathbb C$.
Then we have a flat bundle with connection (or a logarithmic connection).
My question is that what is the corresponding monodromy representation to $(E,nabla)$ via the Riemann Hilbert correspondence?
The known part is that locally at each $z=s_i$, the monodromy $T_i$ can be obtained by the residue map: $T_i=exp(2pi i A_i)$, but this seems not to admit a unique representation up to isomorphism.
Is there any more properties/facts around the question?
connections d-modules monodromy local-systems
asked 5 hours ago
LongmaLongma
332
$endgroup$
add a comment |
$begingroup$
For simplicity, we assume that $X=mathbb P_{mathbb C}^1-{s_1, s_2, dots, s_k}$ and $infty in X$.
Consider the trivial bundle $E=mathcal O_X^r$ with the connection $nabla$ induced by a Fushcian system on $X$, i.e.
$$nabla:= d+sum_{i=1}^k frac{A_i}{z-s_i},$$
where $A_i$'s are $rtimes r$ constant matrices over $mathbb C$.
Then we have a flat bundle with connection (or a logarithmic connection).
My question is that what is the corresponding monodromy representation to $(E,nabla)$ via the Riemann Hilbert correspondence?
The known part is that locally at each $z=s_i$, the monodromy $T_i$ can be obtained by the residue map: $T_i=exp(2pi i A_i)$, but this seems not to admit a unique representation up to isomorphism.
Is there any more properties/facts around the question?
connections d-modules monodromy local-systems
asked 5 hours ago
LongmaLongma
332
$endgroup$
1
$begingroup$
You cannot solve ode "explicitly" - so cannot find explicit monodromies. Rare exceptions like Knizhnik-Zamolodchikov equation are due some hidden Lie group symmetry inside.
$endgroup$
– Alexander Chervov
3 hours ago
add a comment |
$begingroup$
For simplicity, we assume that $X=mathbb P_{mathbb C}^1-{s_1, s_2, dots, s_k}$ and $infty in X$.
Consider the trivial bundle $E=mathcal O_X^r$ with the connection $nabla$ induced by a Fushcian system on $X$, i.e.
$$nabla:= d+sum_{i=1}^k frac{A_i}{z-s_i},$$
where $A_i$'s are $rtimes r$ constant matrices over $mathbb C$.
Then we have a flat bundle with connection (or a logarithmic connection).
My question is that what is the corresponding monodromy representation to $(E,nabla)$ via the Riemann Hilbert correspondence?
The known part is that locally at each $z=s_i$, the monodromy $T_i$ can be obtained by the residue map: $T_i=exp(2pi i A_i)$, but this seems not to admit a unique representation up to isomorphism.
Is there any more properties/facts around the question?
connections d-modules monodromy local-systems
asked 5 hours ago
LongmaLongma
332
$endgroup$
For simplicity, we assume that $X=mathbb P_{mathbb C}^1-{s_1, s_2, dots, s_k}$ and $infty in X$.
Consider the trivial bundle $E=mathcal O_X^r$ with the connection $nabla$ induced by a Fushcian system on $X$, i.e.
$$nabla:= d+sum_{i=1}^k frac{A_i}{z-s_i},$$
where $A_i$'s are $rtimes r$ constant matrices over $mathbb C$.
Then we have a flat bundle with connection (or a logarithmic connection).
My question is that what is the corresponding monodromy representation to $(E,nabla)$ via the Riemann Hilbert correspondence?
The known part is that locally at each $z=s_i$, the monodromy $T_i$ can be obtained by the residue map: $T_i=exp(2pi i A_i)$, but this seems not to admit a unique representation up to isomorphism.
Is there any more properties/facts around the question?
connections d-modules monodromy local-systems
connections d-modules monodromy local-systems
asked 5 hours ago
LongmaLongma
332
asked 5 hours ago
LongmaLongma
332
asked 5 hours ago
LongmaLongma
332
asked 5 hours ago
LongmaLongma
332
asked 5 hours ago
LongmaLongma
332
332
1
$begingroup$
You cannot solve ode "explicitly" - so cannot find explicit monodromies. Rare exceptions like Knizhnik-Zamolodchikov equation are due some hidden Lie group symmetry inside.
$endgroup$
– Alexander Chervov
3 hours ago
add a comment |
1
$begingroup$
You cannot solve ode "explicitly" - so cannot find explicit monodromies. Rare exceptions like Knizhnik-Zamolodchikov equation are due some hidden Lie group symmetry inside.
$endgroup$
– Alexander Chervov
3 hours ago
1
1
$begingroup$
You cannot solve ode "explicitly" - so cannot find explicit monodromies. Rare exceptions like Knizhnik-Zamolodchikov equation are due some hidden Lie group symmetry inside.
$endgroup$
– Alexander Chervov
3 hours ago
$begingroup$
You cannot solve ode "explicitly" - so cannot find explicit monodromies. Rare exceptions like Knizhnik-Zamolodchikov equation are due some hidden Lie group symmetry inside.
$endgroup$
– Alexander Chervov
3 hours ago
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
It depends on what is called "explicit". If $k>2$, monodromy representation is a transcendental function of the $A_j$ and $s_j$. When $d=0$, it was expressed as an everywhere convergent power series in
the $A_j$ whose coefficients are explicit (rational) functions in $s_j$ by Lappo-Danilevski:
Lappo-Danilevsky, J. A. Mémoires sur la théorie des systèmes des équations différentielles linéaires. (French) Chelsea Publishing Co., New York, N. Y., 1953.
If $dneq 0$, the system is not Fuchsian: singularity at $infty$ is irregular. If $k=2, dneq 0$ your system already contains the
"prolate/oblate spheroid equations", which were studied much and no reasonable explicit formula for the monodromy
is known. There are asymptotics, of course. A recent paper about this special case, with a good reference list is
Richard-Jung, F.; Ramis, J.-P.; Thomann, J.; Fauvet, F. New characterizations for the eigenvalues of the prolate spheroidal wave equation. Stud. Appl. Math. 138 (2017), no. 1, 3–42.
The case $d=0$, $k=3$ was also much studied. The papers usually refer to "Heun's equation".
answered 1 hour ago
Alexandre EremenkoAlexandre Eremenko
50.5k6139257
$endgroup$
1
$begingroup$
In physics they call it 'time ordered exponent' en.m.wikipedia.org/wiki/Ordered_exponential
$endgroup$
– Alexander Chervov
1 hour ago
$begingroup$
@Alexander Chervov: good remark! This gives many references which are more modern than the original papers of Lappo-Danilevsky.
$endgroup$
– Alexandre Eremenko
1 hour ago
add a comment |
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1 Answer
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1 Answer
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$begingroup$
It depends on what is called "explicit". If $k>2$, monodromy representation is a transcendental function of the $A_j$ and $s_j$. When $d=0$, it was expressed as an everywhere convergent power series in
the $A_j$ whose coefficients are explicit (rational) functions in $s_j$ by Lappo-Danilevski:
Lappo-Danilevsky, J. A. Mémoires sur la théorie des systèmes des équations différentielles linéaires. (French) Chelsea Publishing Co., New York, N. Y., 1953.
If $dneq 0$, the system is not Fuchsian: singularity at $infty$ is irregular. If $k=2, dneq 0$ your system already contains the
"prolate/oblate spheroid equations", which were studied much and no reasonable explicit formula for the monodromy
is known. There are asymptotics, of course. A recent paper about this special case, with a good reference list is
Richard-Jung, F.; Ramis, J.-P.; Thomann, J.; Fauvet, F. New characterizations for the eigenvalues of the prolate spheroidal wave equation. Stud. Appl. Math. 138 (2017), no. 1, 3–42.
The case $d=0$, $k=3$ was also much studied. The papers usually refer to "Heun's equation".
answered 1 hour ago
Alexandre EremenkoAlexandre Eremenko
50.5k6139257
$endgroup$
1
$begingroup$
In physics they call it 'time ordered exponent' en.m.wikipedia.org/wiki/Ordered_exponential
$endgroup$
– Alexander Chervov
1 hour ago
$begingroup$
@Alexander Chervov: good remark! This gives many references which are more modern than the original papers of Lappo-Danilevsky.
$endgroup$
– Alexandre Eremenko
1 hour ago
add a comment |
$begingroup$
It depends on what is called "explicit". If $k>2$, monodromy representation is a transcendental function of the $A_j$ and $s_j$. When $d=0$, it was expressed as an everywhere convergent power series in
the $A_j$ whose coefficients are explicit (rational) functions in $s_j$ by Lappo-Danilevski:
Lappo-Danilevsky, J. A. Mémoires sur la théorie des systèmes des équations différentielles linéaires. (French) Chelsea Publishing Co., New York, N. Y., 1953.
If $dneq 0$, the system is not Fuchsian: singularity at $infty$ is irregular. If $k=2, dneq 0$ your system already contains the
"prolate/oblate spheroid equations", which were studied much and no reasonable explicit formula for the monodromy
is known. There are asymptotics, of course. A recent paper about this special case, with a good reference list is
Richard-Jung, F.; Ramis, J.-P.; Thomann, J.; Fauvet, F. New characterizations for the eigenvalues of the prolate spheroidal wave equation. Stud. Appl. Math. 138 (2017), no. 1, 3–42.
The case $d=0$, $k=3$ was also much studied. The papers usually refer to "Heun's equation".
answered 1 hour ago
Alexandre EremenkoAlexandre Eremenko
50.5k6139257
$endgroup$
1
$begingroup$
In physics they call it 'time ordered exponent' en.m.wikipedia.org/wiki/Ordered_exponential
$endgroup$
– Alexander Chervov
1 hour ago
$begingroup$
@Alexander Chervov: good remark! This gives many references which are more modern than the original papers of Lappo-Danilevsky.
$endgroup$
– Alexandre Eremenko
1 hour ago
add a comment |
$begingroup$
It depends on what is called "explicit". If $k>2$, monodromy representation is a transcendental function of the $A_j$ and $s_j$. When $d=0$, it was expressed as an everywhere convergent power series in
the $A_j$ whose coefficients are explicit (rational) functions in $s_j$ by Lappo-Danilevski:
Lappo-Danilevsky, J. A. Mémoires sur la théorie des systèmes des équations différentielles linéaires. (French) Chelsea Publishing Co., New York, N. Y., 1953.
If $dneq 0$, the system is not Fuchsian: singularity at $infty$ is irregular. If $k=2, dneq 0$ your system already contains the
"prolate/oblate spheroid equations", which were studied much and no reasonable explicit formula for the monodromy
is known. There are asymptotics, of course. A recent paper about this special case, with a good reference list is
Richard-Jung, F.; Ramis, J.-P.; Thomann, J.; Fauvet, F. New characterizations for the eigenvalues of the prolate spheroidal wave equation. Stud. Appl. Math. 138 (2017), no. 1, 3–42.
The case $d=0$, $k=3$ was also much studied. The papers usually refer to "Heun's equation".
answered 1 hour ago
Alexandre EremenkoAlexandre Eremenko
50.5k6139257
$endgroup$
It depends on what is called "explicit". If $k>2$, monodromy representation is a transcendental function of the $A_j$ and $s_j$. When $d=0$, it was expressed as an everywhere convergent power series in
the $A_j$ whose coefficients are explicit (rational) functions in $s_j$ by Lappo-Danilevski:
Lappo-Danilevsky, J. A. Mémoires sur la théorie des systèmes des équations différentielles linéaires. (French) Chelsea Publishing Co., New York, N. Y., 1953.
If $dneq 0$, the system is not Fuchsian: singularity at $infty$ is irregular. If $k=2, dneq 0$ your system already contains the
"prolate/oblate spheroid equations", which were studied much and no reasonable explicit formula for the monodromy
is known. There are asymptotics, of course. A recent paper about this special case, with a good reference list is
Richard-Jung, F.; Ramis, J.-P.; Thomann, J.; Fauvet, F. New characterizations for the eigenvalues of the prolate spheroidal wave equation. Stud. Appl. Math. 138 (2017), no. 1, 3–42.
The case $d=0$, $k=3$ was also much studied. The papers usually refer to "Heun's equation".
answered 1 hour ago
Alexandre EremenkoAlexandre Eremenko
50.5k6139257
edited 1 hour ago
answered 1 hour ago
Alexandre EremenkoAlexandre Eremenko
50.5k6139257
answered 1 hour ago
Alexandre EremenkoAlexandre Eremenko
50.5k6139257
answered 1 hour ago
Alexandre EremenkoAlexandre Eremenko
50.5k6139257
50.5k6139257
1
$begingroup$
In physics they call it 'time ordered exponent' en.m.wikipedia.org/wiki/Ordered_exponential
$endgroup$
– Alexander Chervov
1 hour ago
$begingroup$
@Alexander Chervov: good remark! This gives many references which are more modern than the original papers of Lappo-Danilevsky.
$endgroup$
– Alexandre Eremenko
1 hour ago
add a comment |
1
$begingroup$
In physics they call it 'time ordered exponent' en.m.wikipedia.org/wiki/Ordered_exponential
$endgroup$
– Alexander Chervov
1 hour ago
$begingroup$
@Alexander Chervov: good remark! This gives many references which are more modern than the original papers of Lappo-Danilevsky.
$endgroup$
– Alexandre Eremenko
1 hour ago
1
1
$begingroup$
In physics they call it 'time ordered exponent' en.m.wikipedia.org/wiki/Ordered_exponential
$endgroup$
– Alexander Chervov
1 hour ago
$begingroup$
In physics they call it 'time ordered exponent' en.m.wikipedia.org/wiki/Ordered_exponential
$endgroup$
– Alexander Chervov
1 hour ago
$begingroup$
@Alexander Chervov: good remark! This gives many references which are more modern than the original papers of Lappo-Danilevsky.
$endgroup$
– Alexandre Eremenko
1 hour ago
$begingroup$
@Alexander Chervov: good remark! This gives many references which are more modern than the original papers of Lappo-Danilevsky.
$endgroup$
– Alexandre Eremenko
1 hour ago
add a comment |
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$begingroup$
You cannot solve ode "explicitly" - so cannot find explicit monodromies. Rare exceptions like Knizhnik-Zamolodchikov equation are due some hidden Lie group symmetry inside.
$endgroup$
– Alexander Chervov
3 hours ago