Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
read the original abstract
We define new invariants of 3d Calabi-Yau categories endowed with a stability structure. Intuitively, they count the number of semistable objects with fixed class in the K-theory of the category ("number of BPS states with given charge" in physics language). Formally, our motivic DT-invariants are elements of quantum tori over a version of the Grothendieck ring of varieties over the ground field. Via the quasi-classical limit "as the motive of affine line approaches to 1" we obtain numerical DT-invariants which are closely related to those introduced by Behrend. We study some properties of both motivic and numerical DT-invariants including the wall-crossing formulas and integrality. We discuss the relationship with the mathematical works (in the non-triangulated case) of Joyce, Bridgeland and Toledano-Laredo, as well as with works of physicists on Seiberg-Witten model (string junctions), classification of N=2 supersymmetric theories (Cecotti-Vafa) and structure of the moduli space of vector multiplets. Relating the theory of 3d Calabi-Yau categories with distinguished set of generators (called cluster collection) with the theory of quivers with potential we found the connection with cluster transformations and cluster varieties (both classical and quantum).
This paper has not been read by Pith yet.
Forward citations
Cited by 17 Pith papers
-
The non-perturbative topological string: from resurgence to wall-crossing of DT invariants
An isomorphism is shown between the algebra of alien derivatives acting on the topological string partition function and the Kontsevich-Soibelman Lie algebra, linking resurgence to DT wall-crossing with numerical matc...
-
The non-perturbative topological string: from resurgence to wall-crossing of DT invariants
Links resurgence of the topological string partition function to DT wall-crossing via an isomorphism of alien derivative algebras to the Kontsevich-Soibelman Lie algebra, with Borel singularities matched to specific D...
-
Birational Weyl Group Action on the Symplectic Groupoid and Cluster Algebras
Introduces birational Weyl group action on symplectic groupoid of A_n matrices via cluster transformations and proves invariants form finite central extension of matrix entry algebra, with applications to Teichmuller ...
-
Shading A-polynomials via huge representations of $U_q(\mathfrak{su}_N)$
Authors propose shaded A-polynomials A_a(ℓ_b, m_c) for SU(N) via CG chords from huge representations of U_q(su_N) in the classical limit, with examples for knots 3_1, 4_1, 5_1 in su_3.
-
Ishii's conjecture and Bridgeland stability conditions for dihedral reflection groups
New proof of Ishii's conjecture for dihedral reflection groups via Bridgeland stability conditions on root stacks of maximal resolutions.
-
Large Order Enumerative Geometry, Black Holes and Black Rings
Numerical study of high-genus GV invariants reveals 5D indices matching BMPV black-hole entropy below a critical angular momentum and black-ring dominance above, with additional phase transitions and growth laws in PT...
-
Classical correlation functions at strong coupling from hexagonalization
In the classical strong-coupling regime, half-BPS correlation functions in planar N=4 SYM exponentiate under the hexagon formalism and are governed by TBA equations structurally equivalent to Gaiotto-Moore-Neitzke equ...
-
Defect Triangles and Intersection-Space Hodge Atom Shadows for Calabi--Yau Conifolds
A projection of the variation morphism defines an intersection-space Hodge atom shadow package for Calabi-Yau conifolds, yielding a middle-degree IC-intersection-space defect of rank 202 for the 125-node quintic.
-
A Deligne-Malgrange Riemann-Hilbert correspondence for closed 1-forms
Establishes a Deligne-Malgrange Riemann-Hilbert correspondence for closed 1-forms and a variant comparison of isomorphisms theorem for simple algebraic 1-forms on complex curves.
-
The Pandharipande-Thomas rationality conjecture for superpositive curve classes on projective complex 3-manifolds
Proves that generating functions of Pandharipande-Thomas invariants with descendent insertions are rational with controlled poles for superpositive curve classes on projective complex 3-manifolds.
-
Macdonald Index From Refined Kontsevich-Soibelman Operator
A refined Kontsevich-Soibelman operator is conjectured to have trace equal to the Macdonald index for special 4d N=2 SCFTs, yielding closed forms for (A1, g) Argyres-Douglas theories.
-
Quiver Yangians as Coulomb branch algebras
Conjectures that quantum Coulomb branch algebras of 3D N=4 unitary quiver gauge theories equal truncated shifted quiver Yangians Y(ˆQ, ˆW), verified explicitly for tree-type quivers via monopole actions on 1/2-BPS vortices.
-
BPS Dendroscopy on Local $\mathbb{P}^1\times \mathbb{P}^1$
Construction of the scattering diagram for BPS indices on local P1 x P1 and sketch of the Split Attractor Flow Tree Conjecture for restricted central charge phase.
-
$g$-vectors and $DT$-$F$-polynomials for Grassmannians
Using Hom-infinite Frobenius categorification of the Grassmannian, the authors determine g-vectors of Plücker coordinates for the triangular seed and express DT F-polynomials in terms of 3D Young diagrams, giving a ne...
-
Non-Perturbative Real Topological Strings
Extends operator formalism of closed topological strings to derive all-order trans-series solutions for real topological strings, with disk invariants as Stokes constants and numerical checks on local P2.
-
Hyper-K\"ahler varieties: Lagrangian fibrations, atomic sheaves, and categories
Lecture notes summarizing recent progress on hyper-Kähler varieties via Lagrangian fibrations, atomic sheaves, and derived categories.
-
Spectral Networks: Bridging higher-rank Teichm\"uller theory and BPS states
A comprehensive introduction to spectral networks that develops higher-rank Teichmüller theory in parallel with class S gauge theory and BPS spectra.
discussion (0)
Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.