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Geometric engineering of (framed) BPS states
"... Abstract. BPS quivers for N = 2 SU(N) gauge theories are derived via geometric engineering from derived categories of toric CalabiYau threefolds. While the outcome is in agreement of previous low energy constructions, the geometric approach leads to several new results. An absence of walls conjectu ..."
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Abstract. BPS quivers for N = 2 SU(N) gauge theories are derived via geometric engineering from derived categories of toric CalabiYau threefolds. While the outcome is in agreement of previous low energy constructions, the geometric approach leads to several new results. An absence of walls conjecture is formulated for all values of N, relating the field theory BPS spectrum to large radius Dbrane bound states. Supporting evidence is presented as explicit computations of BPS degeneracies in some examples. These computations also prove the existence of BPS states of arbitrarily high spin and infinitely many marginal stability walls at weak coupling. Moreover, framed quiver models for framed BPS states are naturally derived from this formalism, as well as a mathematical formulation of framed and unframed BPS degeneracies in terms of motivic and cohomological DonaldsonThomas invariants. We verify the conjectured absence of BPS states with “exotic ” SU(2)R quantum numbers using motivic DT invariants. This application is based in particular on a complete recursive algorithm which determines the unframed BPS spectrum at any point on the Coulomb branch in terms of noncommutative DonaldsonThomas invariants for framed quiver representations.
Nekrasov’s Partition Function and Refined Donaldson–Thomas Theory: the Rank One Case ⋆
"... Abstract. This paper studies geometric engineering, in the simplest possible case of rank one (Abelian) gauge theory on the affine plane and the resolved conifold. We recall the identification between Nekrasov’s partition function and a version of refined Donaldson– Thomas theory, and study the rela ..."
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Abstract. This paper studies geometric engineering, in the simplest possible case of rank one (Abelian) gauge theory on the affine plane and the resolved conifold. We recall the identification between Nekrasov’s partition function and a version of refined Donaldson– Thomas theory, and study the relationship between the underlying vector spaces. Using a purity result, we identify the vector space underlying refined Donaldson–Thomas theory on the conifold geometry as the exterior space of the space of polynomial functions on the affine plane, with the (Lefschetz) SL(2)action on the threefold side being dual to the geometric SL(2)action on the affine plane. We suggest that the exterior space should be a module for the (explicitly not yet known) cohomological Hall algebra (algebra of BPS states) of the conifold. Key words: geometric engineering; Donaldson–Thomas theory; resolved conifold 2010 Mathematics Subject Classification: 14J32 1
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"... Abstract. In a recent paper Iyama and Yoshino consider two interesting examples of isolated singularities over which it is possible to classify the indecomposable maximal CohenMacaulay modules in terms of linear algebra data. In this paper we present two new approaches to these examples. In the fir ..."
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Abstract. In a recent paper Iyama and Yoshino consider two interesting examples of isolated singularities over which it is possible to classify the indecomposable maximal CohenMacaulay modules in terms of linear algebra data. In this paper we present two new approaches to these examples. In the first approach we give a relation with cluster categories. In the second