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Let C be a smooth, connected, complex, projective curve of genus g and let D_{1},D_{2} be divisors of degree k_{1},k_{2} respectively. Let S be the decomposable ruled surface given by the total space of the following P^{1}-bundle over C:
p_{C}: P(O_{C} ⊕ O_{C}(-D_{1}))→C.
Let C_{0} be the locus of (1:0) in S ≅ P(O_{C} ⊕ O_{C}(-D_{1})). Then let X be the threefold given by the total space of the following P^{1}-bundle over S:
p_{S}:P(O_{s}⊕O_{s}(-E)) → S
where E=aC_{0}+p_{C}^{-1}(D_{2}). This determines an H_{a}-bundle over C where H_{a} is a Hirzebruch surface.
In this thesis we determine the equivariant Gromov-Witten partition function for all "section classes'' of the form s+m_{1}f_{1} + m_{2}f_{2 }where s is a section of the map X → C and f_{1},f_{2} are fiber classes, in the case that a=0,-1. A class is Calabi-Yau if K_{X}⋅β=0. For a=0, the partition function of Calabi-Yau section classes is given by
[Special characters omitted]
where v_{1},v_{2} 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|k_{1},k_{2})=Z(g|k_{1}-2,k_{2}+1)
Z(g|k_{1},k_{2})=I_{1}^{2} Z(g|k_{1}-3,k_{2})+v^{2}v_{2} Z(g|k_{1}-4,k_{2})
Z(g|k_{1},k_{2})=-φ^{2}v_{1}^{2}v_{2}^{-2}Z(g-1|k_{1},k_{2})
+6φ^{4}v_{1}^{2}v_{2}^{-1}Z(g-2|k_{1},k_{2})+(256φ^{8}v_{1}^{2}v_{2}+27φ^{8}v_{1}^{4}v_{2}^{-2})Z(g-4|k_{1},k_{2})
which allow us to compute all the Calabi-Yau section class invariants from the following base cases:
g=0 1 2 3
k_{1}=0 0 4 -φ^{2}v_{1}^{2}v_{2}^{-2} 12φ^{4}v_{1}^{2}v_{2}^{-1}+φ^{4}v_{1}^{4}v_{2}^{-4}
1 φ^{-2} 0 φ^{2}v_{1}^{2}v_{2}^{-1 }16φ^{4}v_{1}^{2}-φ^{4}v_{1}^{4}v_{2}^{-3}
2 0 0 8φ^{2}v_{1}^{2} 64φ^{4}v_{1}^{2}v_{2}+φ^{4}v_{1}^{4}v_{2}^{-2}
3 0 3v_{1}^{2} 16φ^{2}v_{1}^{2}v_{2} -φ^{4}v_{1}^{4}v_{2}^{-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.
Advisor: | Gholampour, Amin |
Commitee: | Tamvakis, Harry, Brosnan, Patrick, Ramachandran, Niranjan, Gasarch, William |
School: | University of Maryland, College Park |
Department: | Mathematics |
School Location: | United States -- Maryland |
Source: | DAI-B 81/1(E), Dissertation Abstracts International |
Source Type: | DISSERTATION |
Subjects: | Mathematics |
Keywords: | Donaldson-Thomas theory, Enumerative geometry, Gromov-Witten theory, Ruled surfaces, Stable pairs |
Publication Number: | 13812667 |
ISBN: | 9781085567459 |