Relationship between Gromov-Witten and Taubes' Gromov invariant The 2019 Stack Overflow Developer Survey Results Are InNegative Gromov-Witten invariantsGromov-Witten invariants counting curves passing through two pointsQuestion on Ionel and Parker's paper: Relative Gromov Witten InvariantsAre genus zero Gromov Witten Invariants on Del-Pezzo surfaces enumerative?How does one define Moduli spaces in Symplectic Geometry and naively interpret higher genus GW Invariants?Is the complex structure on a del-Pezzo surface a regular complex structure?How to understand Taubes' moduli space of holomorphic curves?What is the mirror of symplectic field theory?Is there any known relationship between sutured contact homology and Legendrian contact homology?Gromov-Witten invariants and the mod 2 spectral flow
Relationship between Gromov-Witten and Taubes' Gromov invariant
The 2019 Stack Overflow Developer Survey Results Are InNegative Gromov-Witten invariantsGromov-Witten invariants counting curves passing through two pointsQuestion on Ionel and Parker's paper: Relative Gromov Witten InvariantsAre genus zero Gromov Witten Invariants on Del-Pezzo surfaces enumerative?How does one define Moduli spaces in Symplectic Geometry and naively interpret higher genus GW Invariants?Is the complex structure on a del-Pezzo surface a regular complex structure?How to understand Taubes' moduli space of holomorphic curves?What is the mirror of symplectic field theory?Is there any known relationship between sutured contact homology and Legendrian contact homology?Gromov-Witten invariants and the mod 2 spectral flow
$begingroup$
Fix a compact, symplectic four-manifold ($X$, $omega$).
Recall Taubes' Gromov invariant is a certain integer-valued function on $H^2(X; mathbbZ)$ defined by weighted counts of pseudoholomorphic curves in $X$. In particular, this count combines curves of any genus.
On the other hand, the Gromov-Witten invariants for a fixed genus $g$ and homology class $A in H_2(X; mathbbZ)$ are (very roughly) integers derived from the "fundamental class" of the moduli space $mathcalM_g^A(X)$ of pseudoholomorphic maps from a surface of genus $g$ into $X$ representing the class $A$.
Is there any sort of relationship, conjectural or otherwise, between these two invariants that is stronger than "they both count holomorphic curves"? Of course, this would require looking at Gromov-Witten invariants for any genus $g$. I personally don't understand the best way to define these for genus $g > 0$. I do understand that Zinger has a construction for $g = 1$ that is a bit more refined than looking at the entire moduli space.
sg.symplectic-geometry symplectic-topology
edited 6 hours ago
Ali Taghavi
23852085
asked 6 hours ago
Rohil PrasadRohil Prasad
445411
$endgroup$
add a comment |
$begingroup$
Fix a compact, symplectic four-manifold ($X$, $omega$).
Recall Taubes' Gromov invariant is a certain integer-valued function on $H^2(X; mathbbZ)$ defined by weighted counts of pseudoholomorphic curves in $X$. In particular, this count combines curves of any genus.
On the other hand, the Gromov-Witten invariants for a fixed genus $g$ and homology class $A in H_2(X; mathbbZ)$ are (very roughly) integers derived from the "fundamental class" of the moduli space $mathcalM_g^A(X)$ of pseudoholomorphic maps from a surface of genus $g$ into $X$ representing the class $A$.
Is there any sort of relationship, conjectural or otherwise, between these two invariants that is stronger than "they both count holomorphic curves"? Of course, this would require looking at Gromov-Witten invariants for any genus $g$. I personally don't understand the best way to define these for genus $g > 0$. I do understand that Zinger has a construction for $g = 1$ that is a bit more refined than looking at the entire moduli space.
sg.symplectic-geometry symplectic-topology
edited 6 hours ago
Ali Taghavi
23852085
asked 6 hours ago
Rohil PrasadRohil Prasad
445411
$endgroup$
$begingroup$
With respect to Ionel-Parker's paper (which is the answer), note that GT allows a single curve to be disconnected with "whatever" genus whereas GW considers a connected curve of fixed genus, so you should expect that their generating series are the same (roughly speaking).
$endgroup$
– Chris Gerig
29 mins ago
add a comment |
$begingroup$
Fix a compact, symplectic four-manifold ($X$, $omega$).
Recall Taubes' Gromov invariant is a certain integer-valued function on $H^2(X; mathbbZ)$ defined by weighted counts of pseudoholomorphic curves in $X$. In particular, this count combines curves of any genus.
On the other hand, the Gromov-Witten invariants for a fixed genus $g$ and homology class $A in H_2(X; mathbbZ)$ are (very roughly) integers derived from the "fundamental class" of the moduli space $mathcalM_g^A(X)$ of pseudoholomorphic maps from a surface of genus $g$ into $X$ representing the class $A$.
Is there any sort of relationship, conjectural or otherwise, between these two invariants that is stronger than "they both count holomorphic curves"? Of course, this would require looking at Gromov-Witten invariants for any genus $g$. I personally don't understand the best way to define these for genus $g > 0$. I do understand that Zinger has a construction for $g = 1$ that is a bit more refined than looking at the entire moduli space.
sg.symplectic-geometry symplectic-topology
edited 6 hours ago
Ali Taghavi
23852085
asked 6 hours ago
Rohil PrasadRohil Prasad
445411
$endgroup$
Fix a compact, symplectic four-manifold ($X$, $omega$).
Recall Taubes' Gromov invariant is a certain integer-valued function on $H^2(X; mathbbZ)$ defined by weighted counts of pseudoholomorphic curves in $X$. In particular, this count combines curves of any genus.
On the other hand, the Gromov-Witten invariants for a fixed genus $g$ and homology class $A in H_2(X; mathbbZ)$ are (very roughly) integers derived from the "fundamental class" of the moduli space $mathcalM_g^A(X)$ of pseudoholomorphic maps from a surface of genus $g$ into $X$ representing the class $A$.
Is there any sort of relationship, conjectural or otherwise, between these two invariants that is stronger than "they both count holomorphic curves"? Of course, this would require looking at Gromov-Witten invariants for any genus $g$. I personally don't understand the best way to define these for genus $g > 0$. I do understand that Zinger has a construction for $g = 1$ that is a bit more refined than looking at the entire moduli space.
sg.symplectic-geometry symplectic-topology
sg.symplectic-geometry symplectic-topology
edited 6 hours ago
Ali Taghavi
23852085
asked 6 hours ago
Rohil PrasadRohil Prasad
445411
edited 6 hours ago
Ali Taghavi
23852085
asked 6 hours ago
Rohil PrasadRohil Prasad
445411
edited 6 hours ago
Ali Taghavi
23852085
edited 6 hours ago
Ali Taghavi
23852085
edited 6 hours ago
Ali Taghavi
23852085
23852085
asked 6 hours ago
Rohil PrasadRohil Prasad
445411
asked 6 hours ago
Rohil PrasadRohil Prasad
445411
asked 6 hours ago
Rohil PrasadRohil Prasad
445411
445411
$begingroup$
With respect to Ionel-Parker's paper (which is the answer), note that GT allows a single curve to be disconnected with "whatever" genus whereas GW considers a connected curve of fixed genus, so you should expect that their generating series are the same (roughly speaking).
$endgroup$
– Chris Gerig
29 mins ago
add a comment |
$begingroup$
With respect to Ionel-Parker's paper (which is the answer), note that GT allows a single curve to be disconnected with "whatever" genus whereas GW considers a connected curve of fixed genus, so you should expect that their generating series are the same (roughly speaking).
$endgroup$
– Chris Gerig
29 mins ago
$begingroup$
With respect to Ionel-Parker's paper (which is the answer), note that GT allows a single curve to be disconnected with "whatever" genus whereas GW considers a connected curve of fixed genus, so you should expect that their generating series are the same (roughly speaking).
$endgroup$
– Chris Gerig
29 mins ago
$begingroup$
With respect to Ionel-Parker's paper (which is the answer), note that GT allows a single curve to be disconnected with "whatever" genus whereas GW considers a connected curve of fixed genus, so you should expect that their generating series are the same (roughly speaking).
$endgroup$
– Chris Gerig
29 mins ago
add a comment |
1 Answer
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$begingroup$
Ionel and Parker worked out the relationship between Taubes' Gromov invariant and the usual Gromov--Witten invariants (which they refer to as "Ruan--Tian invariants") in this paper:
https://arxiv.org/abs/alg-geom/9702008
answered 5 hours ago
John PardonJohn Pardon
9,361331106
$endgroup$
add a comment |
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$begingroup$
Ionel and Parker worked out the relationship between Taubes' Gromov invariant and the usual Gromov--Witten invariants (which they refer to as "Ruan--Tian invariants") in this paper:
https://arxiv.org/abs/alg-geom/9702008
answered 5 hours ago
John PardonJohn Pardon
9,361331106
$endgroup$
add a comment |
$begingroup$
Ionel and Parker worked out the relationship between Taubes' Gromov invariant and the usual Gromov--Witten invariants (which they refer to as "Ruan--Tian invariants") in this paper:
https://arxiv.org/abs/alg-geom/9702008
answered 5 hours ago
John PardonJohn Pardon
9,361331106
$endgroup$
add a comment |
$begingroup$
Ionel and Parker worked out the relationship between Taubes' Gromov invariant and the usual Gromov--Witten invariants (which they refer to as "Ruan--Tian invariants") in this paper:
https://arxiv.org/abs/alg-geom/9702008
answered 5 hours ago
John PardonJohn Pardon
9,361331106
$endgroup$
Ionel and Parker worked out the relationship between Taubes' Gromov invariant and the usual Gromov--Witten invariants (which they refer to as "Ruan--Tian invariants") in this paper:
https://arxiv.org/abs/alg-geom/9702008
answered 5 hours ago
John PardonJohn Pardon
9,361331106
answered 5 hours ago
John PardonJohn Pardon
9,361331106
answered 5 hours ago
John PardonJohn Pardon
9,361331106
answered 5 hours ago
John PardonJohn Pardon
9,361331106
9,361331106
add a comment |
add a comment |
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$begingroup$
With respect to Ionel-Parker's paper (which is the answer), note that GT allows a single curve to be disconnected with "whatever" genus whereas GW considers a connected curve of fixed genus, so you should expect that their generating series are the same (roughly speaking).
$endgroup$
– Chris Gerig
29 mins ago