scholarly journals BPS States, Crystals, and Matrices

2011 ◽  
Vol 2011 ◽  
pp. 1-52 ◽  
Author(s):  
Piotr Sułkowski
Keyword(s):  
Matrix Model ◽  
Topological String ◽  
Model Representation ◽  
Free Fermion ◽  
Wall Crossing ◽  
Melting Crystal ◽  
Bps States ◽  
String Amplitudes ◽  
Calabi Yau Manifolds

We review free fermion, melting crystal, and matrix model representations of wall-crossing phenomena on local, toric Calabi-Yau manifolds. We consider both unrefined and refined BPS counting of closed BPS states involving D2- and D0-branes bound to a D6-brane, as well as open BPS states involving open D2-branes ending on an additional D4-brane. Appropriate limit of these constructions provides, among the others, matrix model representation of refined and unrefined topological string amplitudes.

scholarly journals CRYSTAL MELTING AND WALL CROSSING PHENOMENA

2011 ◽  
Vol 26 (07n08) ◽  
pp. 1097-1228 ◽  
Author(s):  
MASAHITO YAMAZAKI
Keyword(s):  
Partition Function ◽  
Mechanical Model ◽  
Topological String ◽  
Planck Scale ◽  
Topological String Theory ◽  
Crystal Melting ◽  
Wall Crossing ◽  
Statistical Mechanical Model ◽  
Statistical Mechanical ◽  
Bps States

This paper summarizes recent developments in the theory of Bogomol'nyi–Prasad–Sommerfield (BPS) state counting and the wall crossing phenomena, emphasizing in particular the role of the statistical mechanical model of crystal melting. This paper is divided into two parts, which are closely related to each other. In the first part, we discuss the statistical mechanical model of crystal melting counting BPS states. Each of the BPS states contributing to the BPS index is in one-to-one correspondence with a configuration of a molten crystal, and the statistical partition function of the melting crystal gives the BPS partition function. We also show that smooth geometry of the Calabi–Yau manifold emerges in the thermodynamic limit of the crystal. This suggests a remarkable interpretation that an atom in the crystal is a discretization of the classical geometry, giving an important clue as such to the geometry at the Planck scale. In the second part, we discuss the wall crossing phenomena. Wall crossing phenomena states that the BPS index depends on the value of the moduli of the Calabi–Yau manifold, and jumps along real codimension one subspaces in the moduli space. We show that by using type IIA/M-theory duality, we can provide a simple and an intuitive derivation of the wall crossing phenomena, furthermore clarifying the connection with the topological string theory. This derivation is consistent with another derivation from the wall crossing formula, motivated by multicentered BPS extremal black holes. We also explain the representation of the wall crossing phenomena in terms of crystal melting, and the generalization of the counting problem and the wall crossing to the open BPS invariants.

scholarly journals Real topological string amplitudes

2017 ◽  
Vol 2017 (3) ◽  
Author(s):  
K. S. Narain ◽  
N. Piazzalunga ◽  
A. Tanzini
Keyword(s):  
Topological String ◽  
String Amplitudes

scholarly journals The null identities for boundary operators in the (2, 2p + 1) minimal gravity

2020 ◽  
Vol 2020 (2) ◽  
Author(s):  
Goro Ishiki ◽  
Hisayoshi Muraki ◽  
Chaiho Rim
Keyword(s):  
Matrix Model ◽  
Model Representation ◽  
Physical Implication ◽  
Boundary Interaction ◽  
Liouville Gravity ◽  
The Matrix ◽  
Boundary Operators

Abstract By using the matrix model representation, we show that correlation numbers of boundary-changing operators (BCOs) in $(2,2p+1)$ minimal Liouville gravity satisfy some identities, which we call the null identities. These identities enable us to express the correlation numbers of BCOs in terms of those of boundary-preserving operators. We also discuss a physical implication of the null identities as the manifestation of the boundary interaction.

scholarly journals CONSTRUCTIVE WALL-CROSSING AND SEIBERG–WITTEN

2013 ◽  
Vol 28 (03n04) ◽  
pp. 1340005
Author(s):  
PILJIN YI
Keyword(s):  
Bound State ◽  
Low Energy ◽  
Marginal Stability ◽  
Energy Dynamics ◽  
Principle Solution ◽  
Wall Crossing ◽  
Rational Invariant ◽  
Equivariant Version ◽  
Bps States ◽  
Diagonal Subgroup

We outline a comprehensive and first-principle solution to the wall-crossing problem in D = 4N = 2 Seiberg–Witten theories. We start with a brief review of the multi-centered nature of the typical BPS states and of how this allows them to disappear abruptly as parameters or vacuum moduli are continuously changed. This means that the wall-crossing problem is really a bound state formation/dissociation problem. A low energy dynamics for arbitrary collections of dyons is derived, with the proximity to the so-called marginal stability wall playing the role of the small expansion parameter. We discover that the low energy dynamics of such BPS dyons cannot be reduced to one on the classical moduli space, [Formula: see text], yet the index can be phrased in terms of [Formula: see text]. The so-called rational invariant, first seen in Kontsevich–Soibelman formalism of wall-crossing, is shown to incorporate Bose/Fermi statistics automatically. Furthermore, an equivariant version of the index is shown to compute the protected spin character of the underlying D = 4N = 2 theory, where [Formula: see text] isometry of [Formula: see text] is identified as a diagonal subgroup of rotation SU(2)L and R-symmetry SU(2)R.

scholarly journals A Matrix Model for the Topological String II: The Spectral Curve and Mirror Geometry

2012 ◽  
Vol 14 (1) ◽  
pp. 119-158 ◽  
Author(s):  
Bertrand Eynard ◽  
Amir-Kian Kashani-Poor ◽  
Olivier Marchal
Keyword(s):  
Matrix Model ◽  
Spectral Curve ◽  
Topological String

scholarly journals On link invariants and topological string amplitudes

2001 ◽  
Vol 600 (3) ◽  
pp. 487-511 ◽  
Author(s):  
P. Ramadevi ◽  
Tapobrata Sarkar
Keyword(s):  
Topological String ◽  
Link Invariants ◽  
String Amplitudes

scholarly journals All Loop Topological String Amplitudes from Chern-Simons Theory

2004 ◽  
Vol 247 (2) ◽  
pp. 467-512 ◽  
Author(s):  
Mina Aganagic ◽  
Marcos Mariño ◽  
Cumrun Vafa
Keyword(s):  
Topological String ◽  
Simons Theory ◽  
String Amplitudes ◽  
Chern Simons ◽  
Chern Simons Theory
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