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 GromovWitten theory in dimensions two and three
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GromovWitten theory in dimensions two and three Gholampour, Amin
Abstract
In this thesis, we solve for (equivariant) GromovWitten theories of some important classes of surfaces and threefolds, and study their relationships to other brances of mathematics. The first object is the class of P2bundles over a smooth curve C of genus g. Our bundles are of the form P(L0 + L1 +L2) for arbitrary line bundles L0, L1 and L2 over C. We compute the partition functions of these invariants for all classes of the form s + nf, where s is a section, f is a fiber and n is an integer. In the case where the class is CalabiYau, i.e., K • (s + nf) = 0,the partition function is given by 3g (2sin u/2) 2g2 As an application, one can obtain a series of full predictions for the equivariant Donaldson Thomas invariants for this family of nontoric threefolds. Secondly, we compute the Cequivariant quantum cohomology ring of Y, the minimal resolution of the DuVal singularity C2 /G where G is a finite subgroup of SU(2). The quantum product is expressed in terms of an ADE root system canonically associated to G. We generalize the resulting Frobenius manifold to nonsimply laced root systems to obtain an n parameter family of algebra structures on the affine root lattice of any root system. Using the Crepant Resolution Conjecture, we obtain a prediction for the orbifold GrornovWitten potential of [C2 /G]. Thirdly, for a polyhedral group G, that is a finite subgroup of S0(3), we completely determine the GromovWitten theory of Nakamura's G Hilbert scheme, which is a preferred CalabiYau resolution of the polyhedral singularity C3/G. The classical McKay correspondence determines the (classical) cohomology of this resolution in terms of the representation theory of G. We express the CromovWitten potential in terms of an ADE root system associated to G. As an application, we use the Crepant Resolution Conjecture to provide a full prediction for the orbifold GrornovWitten invariants of [C3/G]. Finally, in the case that G is the group A4 or Z2 x Z2, we compute the integral of Ag on the Hurwitz locus HG C Mg of curves admitting a degree 4 cover of P1 having monodromy group G. We compute the generating functions for these integrals and write them as a trigonometric expression summed over the positive roots of the E6 and D4 root systems respectively. As an application, we prove the Crepaut Resolution Conjecture for the orbifolds [C3/A4] and [C3/(Z2 x Z2)].
Item Metadata
Title 
GromovWitten theory in dimensions two and three

Creator  
Publisher 
University of British Columbia

Date Issued 
2007

Description 
In this thesis, we solve for (equivariant) GromovWitten theories of some important classes of surfaces and threefolds, and study their relationships to other brances of mathematics.
The first object is the class of P2bundles over a smooth curve C of genus g. Our bundles are of the form P(L0 + L1 +L2) for arbitrary line bundles L0, L1 and L2 over C. We compute the partition functions of these invariants for all classes of the form s + nf, where s is a section, f is a fiber and n is an integer. In the case where the class is CalabiYau, i.e., K • (s + nf) = 0,the partition function is given by
3g (2sin u/2) 2g2
As an application, one can obtain a series of full predictions for the equivariant Donaldson Thomas invariants for this family of nontoric threefolds.
Secondly, we compute the Cequivariant quantum cohomology ring of Y, the minimal resolution of the DuVal singularity C2 /G where G is a finite subgroup of SU(2). The quantum product is expressed in terms of an ADE root system canonically associated to G. We generalize the resulting Frobenius manifold to nonsimply laced root systems to obtain an n parameter family of algebra structures on the affine root lattice of any root system. Using the Crepant Resolution Conjecture, we obtain a prediction for the orbifold GrornovWitten potential of [C2 /G].
Thirdly, for a polyhedral group G, that is a finite subgroup of S0(3), we completely determine the GromovWitten theory of Nakamura's G Hilbert scheme, which is a preferred CalabiYau resolution of the polyhedral singularity C3/G. The classical McKay correspondence determines the (classical) cohomology of this resolution in terms of the representation theory of G. We express the CromovWitten potential in terms of an ADE root system associated to G. As an application, we use the Crepant Resolution Conjecture to provide a full prediction for the orbifold GrornovWitten invariants of [C3/G].
Finally, in the case that G is the group A4 or Z2 x Z2, we compute the integral of Ag on the Hurwitz locus HG C Mg of curves admitting a degree 4 cover of P1 having monodromy group G. We compute the generating functions for these integrals and write them as a trigonometric expression summed over the positive roots of the E6 and D4 root systems respectively. As an application, we prove the Crepaut Resolution Conjecture for the orbifolds [C3/A4] and [C3/(Z2 x Z2)].

Extent 
4771698 bytes

Subject  
Genre  
Type  
File Format 
application/pdf

Language 
eng

Date Available 
20080220

Provider 
Vancouver : University of British Columbia Library

Rights 
AttributionNonCommercialNoDerivatives 4.0 International

DOI 
10.14288/1.0066273

URI  
Degree  
Program  
Affiliation  
Degree Grantor 
University of British Columbia

Graduation Date 
200805

Campus  
Scholarly Level 
Graduate

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Aggregated Source Repository 
DSpace

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AttributionNonCommercialNoDerivatives 4.0 International