Orientiholes

Denef, Frederik; Esole, Mboyo; Padi, Megha

doi:10.1007/jhep03(2010)045

Cited by 5 publications

(10 citation statements)

References 78 publications

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“…We denote the projection map γ →γ. The low energy Seiberg-Witten effective IR theory is a self-dual abelian gauge theory with an invariant fieldstrength F ∈ Ω 2 (R [83].…”

Section: Fixed-point Equations In the Low Energy Effective Theorymentioning

confidence: 99%

Framed BPS states

Gaiotto

Moore

Neitzke

2013

Advances in Theoretical and Mathematical Physics

319

1,161

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We consider a class of line operators in d = 4, N = 2 supersymmetric field theories, which leave four supersymmetries unbroken. Such line operators support a new class of BPS states which we call "framed BPS states." These include halo bound states similar to those of d = 4, N = 2 supergravity, where (ordinary) BPS particles are loosely bound to the line operator. Using this construction, we give a new proof of the KontsevichSoibelman wall-crossing formula (WCF) for the ordinary BPS particles, by reducing it to the semiprimitive WCF. After reducing on S 1 , the expansion of the vevs of the line operators in the IR provides a new physical interpretation of the "Darboux coordinates" on the moduli space M of the theory. Moreover, we introduce a "protected spin character" (PSC) that keeps track of the spin degrees of freedom of the framed BPS states. We show that the generating functions of PSCs admit a multiplication, which defines a deformation of the algebra of holomorphic functions on M. As an illustration of these ideas, we consider the sixdimensional (2, 0) field theory of A 1 type compactified on a Riemann surface C. Here, we show (extending previous results) that line operators e-print archive: http://lanl.arXiv.org/abs/1006.0146v2 GAIOTTO ET AL.are classified by certain laminations on a suitably decorated version of C, and we compute the spectrum of framed BPS states in several explicit examples. Finally, we indicate some interesting connections to the theory of cluster algebras.

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“…We denote the projection map γ →γ. The low energy Seiberg-Witten effective IR theory is a self-dual abelian gauge theory with an invariant fieldstrength F ∈ Ω 2 (R [83].…”

Section: Fixed-point Equations In the Low Energy Effective Theorymentioning

confidence: 99%

Framed BPS states

Gaiotto

Moore

Neitzke

2013

Advances in Theoretical and Mathematical Physics

319

1,161

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“…• Derivation of open BPS invariants from supergravity viewpoint [43] is another interesting problem. See [44] for the related discussion in the case of orientifolds. Another related question is the connection of the crystal melting expansion of Z o BPS with open version of OSV conjecture [45].…”

Section: Resolved Conifoldmentioning

confidence: 99%

The Non-commutative Topological Vertex and Wall Crossing Phenomena

Nagao¹,

Yamazaki²

2010

Advances in Theoretical and Mathematical Physics

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We propose a generalization of the topological vertex, which we call the "non-commutative topological vertex". This gives open BPS invariants for a toric Calabi-Yau manifold without compact 4-cycles, where we have D0/D2/D6-branes wrapping holomorphic 0/2/6-cycles, as well as D2-branes wrapping disks whose boundaries are on D4-branes wrapping non-compact Lagrangian 3-cycles. The vertex is defined combinatorially using the crystal melting model proposed recently, and depends on the value of closed string moduli at infinity. The vertex in one special chamber gives the same answer as that computed by the ordinary topological vertex. We prove an identify expressing the non-commutative topological vertex of a toric Calabi-Yau manifold X as a specialization of the closed BPS partition function of an orbifold of X, thus giving a closed expression for our vertex. We also clarify the action of the Weyl group of an affine A L Lie algebra on chambers, and comment on the generalization of our results to the case of refined BPS invariants.Recently, there has been significant progress in the counting problem of BPS states in type IIA string theory on a toric Calabi-Yau 3-fold 1 . In the literature, the Calabi-Yau manifold (which we denote by X) is assumed to have no compact 4-cycles, and we consider a BPS configuration of D0/D2-branes wrapping compact holomorphic 0/2-cycles, as well as a single D6-brane filling the entire Calabi-Yau manifold. The question is to count the degeneracy of such BPS bound states of D-branes.One subtlety in this counting problem is the wall crossing phenomena, stating that the degeneracy of BPS bound states depends on the value of moduli at infinity. Indeed, the closedwhich is defined in [3] as the generation function of the degeneracy of D-brane BPS bound states 3 , depends on maps σ ′ , θ ′ specifying a chamber in the Kähler moduli space 4 . What isinteresting is that in one special chamberC top of the Kähler moduli space, the BPS partition function is equivalent the topological string partition function 5 (up to the change of variables, which we do not explicitly show here for simplicity): , 2, 3, 4, 5, 6, 7]. See also [8,9,10,11,12,13,14] for mathematical discussions 2 The upper index c stands for 'closed'. 3 The definition of the partition function Z BPS is the same as the partition function Z BH in [15]. 4 See Appendix A for details. 5 Actually, the topological string partition function depends on the choice of the resolution of the singular Calabi-Yau manifold X. This is related to the choice of the limit, as will be explained in the main text. 6 The upper index o stands for 'open'. 7 The word 'non-commutative' stems from the mathematical terminologies such as "non-commutative crepant resolution" [19] and "non-commutative Donaldson-Thomas invariant" [8]. The non-commutativity here refers to that of the path algebra of the quiver. The quiver (together with a superpotential) determines a quiver quantum mechanics, which is the low-energy effective theory on the D-brane worldvolume [3]. 9 See [6] f...

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“…Chow theoretic wall-crossing formulas. In [28] a motivic orientifold wallcrossing formula was proved using finite field Hall algebras and their representations, generalizing and explaining formulas which appeared previously in the string theory literature [7]. In this section we lift the wall-crossing formula to Chow theoretic and cohomological Hall modules.…”

Section: 3mentioning

confidence: 70%

Cohomological orientifold Donaldson–Thomas invariants as Chow groups

Franzen

Young

2018

Sel. Math. New Ser.

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We establish a geometric interpretation of orientifold Donaldson-Thomas invariants of σ-symmetric quivers with involution. More precisely, we prove that the cohomological orientifold Donaldson-Thomas invariant is isomorphic to the rational Chow group of the moduli space of σ-stable selfdual quiver representations. As an application we prove that the Chow Betti numbers of moduli spaces of stable m-tuples in classical Lie algebras can be computed numerically. We also prove a cohomological wall-crossing formula relating semistable Hall modules for different stabilities.

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Orientiholes

Cited by 5 publications

References 78 publications

Framed BPS states

Framed BPS states

The Non-commutative Topological Vertex and Wall Crossing Phenomena

Cohomological orientifold Donaldson–Thomas invariants as Chow groups

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