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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  Gholampour, Amin 
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  GromovWitten; surfaces; partition functions 
Genre  Thesis/Dissertation 
Type  Text 
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  Doctor of Philosophy  PhD 
Program  Mathematics 
Affiliation  Science, Faculty of; Mathematics, Department of 
Degree Grantor  University of British Columbia 
Graduation Date  200805 
Campus  UBCV 
Scholarly Level  Graduate 
Rights URI  
Aggregated Source Repository  DSpace 
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AttributionNonCommercialNoDerivatives 4.0 International