Relationship between Gromov-Witten and Taubes' Gromov invariant












5












$begingroup$


Fix a compact, symplectic four-manifold ($X$, $omega$).



Recall Taubes' Gromov invariant is a certain integer-valued function on $H^2(X; mathbb{Z})$ 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 $A in H_2(X; mathbb{Z})$ are (very roughly) integers derived from the "fundamental class" of the moduli space $mathcal{M}_g^A(X)$ 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







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    5












    $begingroup$


    Fix a compact, symplectic four-manifold ($X$, $omega$).



    Recall Taubes' Gromov invariant is a certain integer-valued function on $H^2(X; mathbb{Z})$ 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 $A in H_2(X; mathbb{Z})$ are (very roughly) integers derived from the "fundamental class" of the moduli space $mathcal{M}_g^A(X)$ 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 $H^2(X; mathbb{Z})$ 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 $A in H_2(X; mathbb{Z})$ are (very roughly) integers derived from the "fundamental class" of the moduli space $mathcal{M}_g^A(X)$ 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 $H^2(X; mathbb{Z})$ 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 $A in H_2(X; mathbb{Z})$ are (very roughly) integers derived from the "fundamental class" of the moduli space $mathcal{M}_g^A(X)$ 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






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      share|cite|improve this question













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      edited 3 hours ago









      Ali Taghavi

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      23852085










      asked 4 hours ago









      Rohil PrasadRohil Prasad

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      445411






















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          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






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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 3 hours ago









                John PardonJohn Pardon

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