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Stability structures, motivic DonaldsonThomas invariants and cluster transformations
, 2008
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Curve counting via stable pairs in the derived category
, 2009
"... For a nonsingular projective 3fold X, we define integer invariants virtually enumerating pairs (C,D) where C ⊂ X is an embedded curve and D ⊂ C is a divisor. A virtual class is constructed on the associated moduli space by viewing a pair as an object in the derived category of X. The resulting in ..."
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For a nonsingular projective 3fold X, we define integer invariants virtually enumerating pairs (C,D) where C ⊂ X is an embedded curve and D ⊂ C is a divisor. A virtual class is constructed on the associated moduli space by viewing a pair as an object in the derived category of X. The resulting invariants are conjecturally equivalent, after universal transformations, to both the GromovWitten and DT theories of X. For CalabiYau 3folds, the latter equivalence should be viewed as a wallcrossing formula in the derived category. Several calculations of the new invariants are carried out. In the Fano case, the local contributions of nonsingular embedded curves are found. In the local toric CalabiYau case, a completely new form of the topological vertex is described. The virtual enumeration of pairs is closely related to the geometry underlying the BPS state counts of Gopakumar and Vafa. We
Refined BPS invariants, ChernSimmons theory, and the . . .
, 2010
"... In this thesis, we consider two main subjects: the refined BPS invariants of CalabiYau threefolds, and threedimensional ChernSimons theory with complex gauge group. We study the wallcrossing behavior of refined BPS invariants using a variety of techniques, including a fourdimensional supergra ..."
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In this thesis, we consider two main subjects: the refined BPS invariants of CalabiYau threefolds, and threedimensional ChernSimons theory with complex gauge group. We study the wallcrossing behavior of refined BPS invariants using a variety of techniques, including a fourdimensional supergravity analysis, statisticalmechanical melting crystal models, and relations to new mathematical invariants. We conjecture an equivalence between refined invariants and the motivic DonaldsonThomas invariants of Kontsevich and Soibelman. We then consider perturbative ChernSimons theory with complex gauge group, combining traditional and novel approaches to the theory (including a new state integral model) to obtain exact results for perturbative partition functions. We thus obtain a new
Preprint typeset in JHEP style HYPER VERSION Sparticles at the LHC
, 802
"... Abstract: Sparticle mass hierarchies will play an important role in the type of signatures that will be visible at the Large Hadron Collider. We analyze these hierarchies for the four lightest sparticles for a general class of supergravity unified models including nonuniversalities in the soft break ..."
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Abstract: Sparticle mass hierarchies will play an important role in the type of signatures that will be visible at the Large Hadron Collider. We analyze these hierarchies for the four lightest sparticles for a general class of supergravity unified models including nonuniversalities in the soft breaking sector. It is shown that out of nearly 10 4 possibilities of sparticle mass hierarchies, only a small number survives the rigorous constraints of radiative electroweak symmetry breaking, relic density and other experimental constraints. The signature space of these mass patterns at the Large Hadron Collider is investigated using a large set of final states including multileptonic states, hadronically decaying τs, tagged b jets and other hadronic jets. In all, we analyze more than 40 such lepton plus jet and missing energy signatures along with several kinematical signatures such as missing transverse momentum, effective mass, and invariant mass distributions of final state observables. It is shown that a composite analysis can produce significant discrimination among sparticle mass patterns allowing for a possible identification of the source of soft breaking. While the analysis given is for supergravity models, the techniques based on mass pattern analysis are applicable to wide class of models including string and brane models.
Symmetry, Integrability and Geometry: Methods and Applications Wall Crossing, Discrete Attractor Flow and Borcherds Algebra ⋆
"... Abstract. The appearance of a generalized (or Borcherds–) Kac–Moody algebra in the spectrum of BPS dyons in N = 4, d = 4 string theory is elucidated. From the lowenergy supergravity analysis, we identify its root lattice as the lattice of the Tduality invariants of the dyonic charges, the symmetry ..."
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Abstract. The appearance of a generalized (or Borcherds–) Kac–Moody algebra in the spectrum of BPS dyons in N = 4, d = 4 string theory is elucidated. From the lowenergy supergravity analysis, we identify its root lattice as the lattice of the Tduality invariants of the dyonic charges, the symmetry group of the root system as the extended Sduality group P GL(2, Z) of the theory, and the walls of Weyl chambers as the walls of marginal stability for the relevant twocentered solutions. This leads to an interpretation for the Weyl group as the group of wallcrossing, or the group of discrete attractor flows. Furthermore we propose an equivalence between a “secondquantized multiplicity ” of a charge and modulidependent highest weight vector and the dyon degeneracy, and show that the wallcrossing formula following from our proposal agrees with the wallcrossing formula obtained from the supergravity analysis. This can be thought of as providing a microscopic derivation of the wallcrossing formula of this theory. Key words: generalized Kac–Moody algebra; black hole; dyons 2000 Mathematics Subject Classification: 81R10; 17B67 1