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










5












$begingroup$


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







share|cite|improve this question











$endgroup$
















    5












    $begingroup$


    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







    share|cite|improve this question











    $endgroup$














      5












      5








      5


      1



      $begingroup$


      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







      share|cite|improve this question











      $endgroup$




      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






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited 1 hour ago









      Ali Taghavi

      23652085




      23652085










      asked 2 hours ago









      Rohil PrasadRohil Prasad

      445411




      445411




















          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






          share|cite|improve this 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






            share|cite|improve this answer









            $endgroup$

















              5












              $begingroup$

              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






              share|cite|improve this answer









              $endgroup$















                5












                5








                5





                $begingroup$

                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






                share|cite|improve this answer









                $endgroup$



                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







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered 1 hour ago









                John PardonJohn Pardon

                9,361331106




                9,361331106



























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