Let C be a smooth, connected, complex, projective curve of genus g and let D1,D2 be divisors of degree k1,k2 respectively. Let S be the decomposable ruled surface given by the total space of the following P1-bundle over C:
pC: P(OC ⊕ OC(-D1))→C.
Let C0 be the locus of (1:0) in S ≅ P(OC ⊕ OC(-D1)). Then let X be the threefold given by the total space of the following P1-bundle over S:
pS:P(Os⊕Os(-E)) → S
where E=aC0+pC-1(D2). This determines an Ha-bundle over C where Ha is a Hirzebruch surface.
In this thesis we determine the equivariant Gromov-Witten partition function for all "section classes'' of the form s+m1f1 + m2f2 where s is a section of the map X → C and f1,f2 are fiber classes, in the case that a=0,-1. A class is Calabi-Yau if KX⋅β=0. For a=0, the partition function of Calabi-Yau section classes is given by
[Special characters omitted]
where v1,v2 count the number of fibers and φ=2sinu/2. In the case that a=-1 the partition functions of Calabi-Yau section classes satisfy the following relations
Z(g|k1,k2)=I12 Z(g|k1-3,k2)+v2v2 Z(g|k1-4,k2)
which allow us to compute all the Calabi-Yau section class invariants from the following base cases:
g=0 1 2 3
k1=0 0 4 -φ2v12v2-2 12φ4v12v2-1+φ4v14v2-4
1 φ-2 0 φ2v12v2-1 16φ4v12-φ4v14v2-3
2 0 0 8φ2v12 64φ4v12v2+φ4v14v2-2
3 0 3v12 16φ2v12v2 -φ4v14v2-1
As a corollary, we establish the Gromov-Witten/Donaldson-Thomas/Stable Pairs correspondence for the Calabi-Yau section class partition functions for these families of non-toric threefolds.
|Commitee:||Tamvakis, Harry, Brosnan, Patrick, Ramachandran, Niranjan, Gasarch, William|
|School:||University of Maryland, College Park|
|School Location:||United States -- Maryland|
|Source:||DAI-B 81/1(E), Dissertation Abstracts International|
|Keywords:||Donaldson-Thomas theory, Enumerative geometry, Gromov-Witten theory, Ruled surfaces, Stable pairs|
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