Article
Mathematics
Tasuki Kinjo
Summary: In this paper, we study perverse sheaves in oriented -1-shifted symplectic derived Artin stacks and prove the isomorphism between the hypercohomology of the perverse sheaf and the Borel-Moore homology of the base stack. We provide two applications of our main theorem.
COMPOSITIO MATHEMATICA
(2022)
Article
Mathematics
Hulya Arguz, Pierrick Bousseau
Summary: This study proves the flow tree formula conjectured by Alexandrov and Pioline, which allows the computation of Donaldson-Thomas invariants of quivers with potentials using a smaller set of attractor invariants.
COMPOSITIO MATHEMATICA
(2022)
Article
Mathematics
Valery Lunts, Spela Spenko, Michel Van den Bergh
Summary: This article provides a brief review of the cohomological Hall algebra and K-theoretical Hall algebra associated with quivers. It shows a homomorphism between them in the case of symmetric quivers. Additionally, the equivalence of categories of graded modules is established.
JOURNAL OF ALGEBRA
(2024)
Article
Mathematics
Alberto Cazzaniga, Andrea T. Ricolfi
Summary: In this paper, we compute r-framed motivic DT and PT invariants of small crepant resolutions of toric Calabi-Yau 3-folds for an arbitrary integer r ≥ 1, establishing a higher rank version of the motivic DT/PT wall-crossing formula. This generalizes the work of Morrison and Nagao and shows the relationship between our formulae with r=1 theory, fitting nicely in the current development of higher rank refined DT invariants.
MATHEMATISCHE NACHRICHTEN
(2022)
Article
Mathematics
Markus Reineke, Brendon Rhoades, Vasu Tewari
Summary: We apply orbit harmonics method to break divisors and orientable divisors on graphs to obtain central and external zonotopal algebras. We relate Efimov's construction in the context of cohomological Hall algebras to the central zonotopal algebra of a graph G(Q,?) constructed from a symmetric quiver Q and a dimension vector ?. This provides a combinatorial perspective on the quantum Donaldson-Thomas invariants as the Hilbert series of S?-invariants of the Postnikov-Shapiro slim subgraph space attached to G(Q,?). The connection with orbit harmonics allows us to interpret numerical DT invariants as the number of S?-orbits under the permutation action on the set of break divisors on G. We conclude with several representation-theoretic consequences, which may have independent interest in combinatorics.
INTERNATIONAL MATHEMATICS RESEARCH NOTICES
(2023)
Article
Mathematics, Applied
Nadir Fasola, Sergej Monavari, Andrea T. Ricolfi
Summary: By utilizing the critical structure and obstruction theory on Quot(A3), the study computes K-theoretic DT invariants of the local Calabi-Yau 3-fold with rank r, proving conjectures proposed in string theory. The invariants are shown to be independent of equivariant parameters, leading to further reductions and proofs of additional conjectures. Additionally, a mathematical definition of the chiral elliptic genus is formulated, allowing for the definition and resolution of elliptic DT invariants in arbitrary rank.
FORUM OF MATHEMATICS SIGMA
(2021)
Article
Physics, Mathematical
Michele Cirafici
Summary: The quantum theory of M2 branes can be studied using the K-theoretic Donaldson-Thomas theory on certain M-theory backgrounds. This relation is extended to noncommutative crepant resolutions, where the classical smooth geometry is replaced by the algebra of paths of a specific quiver. K-theoretic quantities on the quiver representation moduli space can be computed using toric localization and result in rational functions of the toric parameters. We discuss the case of the conifold and certain orbifold singularities.
LETTERS IN MATHEMATICAL PHYSICS
(2022)
Article
Physics, Mathematical
Vladimir Dotsenko, Evgeny Feigin, Markus Reineke
Summary: We propose a new method for computing motivic Donaldson-Thomas invariants of a symmetric quiver using Koszul duality between supercommutative algebras and Lie superalgebras, bypassing cohomological Hall algebras. The method involves defining a supercommutative quadratic algebra and studying the corresponding Lie superalgebra. The positivity of the invariants is proven using the Poincare series of a certain operator's kernel.
LETTERS IN MATHEMATICAL PHYSICS
(2022)
Article
Mathematics
Bolun Tong, Wan Wu
Summary: We present a group of quiver Hecke algebras that offer a categorization of the quantum Borcherds algebra related to any Borcherds-Cartan datum.
JOURNAL OF ALGEBRA
(2023)
Article
Mathematics
Ming Lu, Weiqiang Wang
Summary: The paper explores the applications of iota quiver algebras and Dynkin miniver algebras in the Nakajima-Keller-Scherotzke categories, and provides a geometric construction of universal iota quantum groups for quantum symmetric pairs.
ADVANCES IN MATHEMATICS
(2021)
Article
Mathematics
Ying Ma, Toshiaki Shoji, Zhiping Zhou
Summary: This paper considers the construction of canonical basis and canonical signed basis for Cartan data of symmetric type, and the natural bijection between them, and provides a construction method in the case where the order of sigma is odd.
JOURNAL OF ALGEBRA
(2023)
Article
Mathematics, Applied
Shuhan Jiang
Summary: A mathematical framework for cohomological field theories (CohFTs) is presented in the language of bigraded manifolds. The algebraic properties of operators in CohFTs are studied, and methods for constructing CohFTs with or without gauge symmetries are discussed. The Mathai-Quillen formalism is generalized, and this new formalism allows for a unified approach to obtaining examples such as topological quantum mechanics, topological sigma model, topological M-theory, and topological Yang-Mills theory.
JOURNAL OF GEOMETRY AND PHYSICS
(2023)
Article
Mathematics
Li Li
Summary: In this paper, the support conjecture for all skew-symmetric rank-2 cluster algebras is proven.
JOURNAL OF ALGEBRA
(2023)
Article
Mathematics, Applied
G. Bonelli, N. Fasola, A. Tanzini, Y. Zenkevich
Summary: We study the moduli space of SU(4) invariant BPS conditions in supersymmetric gauge theory on non-commutative C4. The classification of invariant solutions under the natural toric action is given in terms of solid partitions. In the orbifold case C4/G, the classification is given in terms of coloured solid partitions. We compute several terms of the partition function expansion in the instanton counting parameter on C4 and C2 x (C2/Z2), which conjecturally provides the corresponding orbifold Donaldson-Thomas invariants.
JOURNAL OF GEOMETRY AND PHYSICS
(2023)
Article
Physics, Mathematical
Henning Bostelmann, Daniela Cadamuro, Simone Del Vecchio
Summary: The study examines the relative entropy between a general quasifree state and a coherent excitation within a subalgebra of a generic CCR algebra, presenting a unified formula based on single-particle modular data. Changes in relative entropy along subalgebras arising from an increasing family of symplectic subspaces are investigated, with convexity of entropy replaced by lower estimates for the second derivative. The assumption of subspaces in differential modular position is crucial, and the results are illustrated in examples such as thermal states for the conformal U(1)-current.
COMMUNICATIONS IN MATHEMATICAL PHYSICS
(2022)