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51
Dbranes in LandauGinzburg models and algebraic geometry
 J. High Energy Phys
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Dbranes, open string vertex operators, and Ext groups
 Adv. Theor. Math. Phys
, 2003
"... In this note we explicitly work out the precise relationship between Ext groups and massless modes of Dbranes wrapped on complex submanifolds of CalabiYau manifolds. Specifically, we explicitly compute the boundary vertex operators for massless Ramond sector states, in open string B models describ ..."
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Cited by 69 (12 self)
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In this note we explicitly work out the precise relationship between Ext groups and massless modes of Dbranes wrapped on complex submanifolds of CalabiYau manifolds. Specifically, we explicitly compute the boundary vertex operators for massless Ramond sector states, in open string B models describing CalabiYau manifolds at large radius, directly in BCFT using standard methods. Naively these vertex operators are in onetoone correspondence with certain sheaf cohomology groups (as is typical for such vertex operator calculations), which are related to the desired Ext groups via spectral sequences. However, a subtlety in the physics of the open string B model has the effect of physically realizing those spectral sequences in BRST cohomology, so that the vertex operators are actually in onetoone correspondence with Ext group elements. This gives an extremely concrete physical test of recent proposals regarding the relationship between derived categories and Dbranes. We check these results extensively in numerous examples, and comment on several related issues.
Moduli stacks and invariants of semistable objects on K3
, 2007
"... For a K3 surface X and its bounded derived category of coherent sheaves D(X), we have the notion of stability conditions on D(X) in the sense of T.Bridgeland. In this paper, we show that the moduli stack of semistable objects in D(X) with a fixed numerical class and a phase is represented by an Arti ..."
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Cited by 32 (9 self)
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For a K3 surface X and its bounded derived category of coherent sheaves D(X), we have the notion of stability conditions on D(X) in the sense of T.Bridgeland. In this paper, we show that the moduli stack of semistable objects in D(X) with a fixed numerical class and a phase is represented by an Artin stack of finite type over C. Then following D.Joyce’s work, we introduce the invariants counting semistable objects in D(X), and show that the invariants are independent of a choice of a stability condition.
Limit stable objects on CalabiYau 3folds
"... In this paper, we introduce new enumerative invariants of curves on CalabiYau 3folds via certain stable objects in the derived category of coherent sheaves. We introduce the ..."
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Cited by 28 (8 self)
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In this paper, we introduce new enumerative invariants of curves on CalabiYau 3folds via certain stable objects in the derived category of coherent sheaves. We introduce the
Spectra of Dbranes with Higgs vevs
 Adv. Theor. Math. Phys
"... In this paper we continue previous work on counting open string states between Dbranes by considering open strings between Dbranes with nonzero Higgs vevs, and in particular, nilpotent Higgs vevs, as arise, for example, when studying Dbranes in orbifolds. Ordinarily Higgs vevs can be interpreted ..."
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Cited by 26 (8 self)
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In this paper we continue previous work on counting open string states between Dbranes by considering open strings between Dbranes with nonzero Higgs vevs, and in particular, nilpotent Higgs vevs, as arise, for example, when studying Dbranes in orbifolds. Ordinarily Higgs vevs can be interpreted as moving the Dbrane, but nilpotent Higgs vevs have zero eigenvalues, and so their interpretation is more interesting – for example, they often correspond to nonreduced schemes, which furnishes an important link in understanding old results relating classical Dbrane moduli spaces in orbifolds to Hilbert schemes, resolutions of quotient spaces, and the McKay correspondence. We give a sheaftheoretic description of Dbranes with Higgs vevs, including nilpotent Higgs vevs, and check that description by noting that Ext groups between the sheaves modelling the Dbranes, do in fact correctly count open string states. In particular, our analysis expands the types of sheaves which admit onshell physical interpretations, which is an important step for making derived categories
Stability Conditions on AnSingularities
, 2006
"... We study the spaces of locallyfinite stability conditions on the derived categories of coherent sheaves on the minimal resolutions of Ansingularities supported at the exceptional sets. Our main theorem is that they are connected and simplyconnected. The proof is based on the study of spherical ob ..."
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Cited by 24 (0 self)
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We study the spaces of locallyfinite stability conditions on the derived categories of coherent sheaves on the minimal resolutions of Ansingularities supported at the exceptional sets. Our main theorem is that they are connected and simplyconnected. The proof is based on the study of spherical objects in [30] and the homological mirror symmetry for Ansingularities. 1
Generating functions of stable pair invariants via wallcrossings in derived categories
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NonBirational Twisted Derived Equivalences in Abelian GLSMs
, 2007
"... In this paper we discuss some examples of abelian gauged linear sigma models realizing twisted derived equivalences between nonbirational spaces, and realizing geometries in novel fashions. Examples of gauged linear sigma models with nonbirational Kähler phases are a relatively new phenomenon. Mos ..."
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Cited by 23 (9 self)
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In this paper we discuss some examples of abelian gauged linear sigma models realizing twisted derived equivalences between nonbirational spaces, and realizing geometries in novel fashions. Examples of gauged linear sigma models with nonbirational Kähler phases are a relatively new phenomenon. Most of our examples involve gauged linear sigma models for complete intersections of quadric hypersurfaces, though we also discuss some more general cases and their interpretation. We also propose a more general understanding of the relationship between Kähler phases of gauged linear sigma models, namely that they are related by (and realize) Kuznetsov’s ‘homological projective duality. ’ Along the way, we shall see how ‘noncommutative spaces ’ (in Kontsevich’s sense) are realized physically in gauged linear sigma models, providing examples of new types of conformal field theories. Throughout, the physical realization of stacks plays a key role in interpreting physical structures appearing in GLSMs, and we find that stacks are implicitly much more common in GLSMs than previously realized.
Stability conditions and crepant small resolutions
, 2007
"... In this paper, we describe the spaces of stability conditions on the triangulated categories associated to three dimensional crepant small resolutions. The resulting spaces have chamber structures such that each chamber corresponds to a birational model together with a special FourierMukai transfor ..."
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Cited by 22 (9 self)
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In this paper, we describe the spaces of stability conditions on the triangulated categories associated to three dimensional crepant small resolutions. The resulting spaces have chamber structures such that each chamber corresponds to a birational model together with a special FourierMukai transform. We observe that these spaces are covering spaces over certain open subsets of finite dimensional vector spaces, and determine their deck transformations.