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 5 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 5 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
23 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 5 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 5 hours ago
Ali Taghavi
23852085
asked 6 hours ago
Rohil PrasadRohil Prasad
445411
edited 5 hours ago
Ali Taghavi
23852085
asked 6 hours ago
Rohil PrasadRohil Prasad
445411
edited 5 hours ago
Ali Taghavi
23852085
edited 5 hours ago
Ali Taghavi
23852085
edited 5 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
23 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
23 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
23 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
23 mins ago
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
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 |
Your Answer
StackExchange.ifUsing("editor", function ()
return StackExchange.using("mathjaxEditing", function ()
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
);
);
, "mathjax-editing");
StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "504"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);
else
createEditor();
);
function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);
);
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f327802%2frelationship-between-gromov-witten-and-taubes-gromov-invariant%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
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 |
Thanks for contributing an answer to MathOverflow!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f327802%2frelationship-between-gromov-witten-and-taubes-gromov-invariant%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
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
23 mins ago