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  1. Abstract

    In this paper we propose a definition of torsion refined Gopakumar–Vafa (GV) invariants for Calabi–Yau threefolds with terminal nodal singularities that do not admit Kähler crepant resolutions. Physically, the refinement takes into account the charge of five-dimensional BPS states under a discrete gauge symmetry in M-theory. We propose a mathematical definition of the invariants in terms of the geometry of all non-Kähler crepant resolutions taken together. The invariants are encoded in the A-model topological string partition functions associated to non-commutative (nc) resolutions of the Calabi–Yau. Our main example will be a singular degeneration of the generic Calabi–Yau double cover of$${\mathbb {P}}^3$$P3and leads to an enumerative interpretation of the topological string partition function of a hybrid Landau–Ginzburg model. Our results generalize a recent physical proposal made in the context of torus fibered Calabi–Yau manifolds by one of the authors and clarify the associated enumerative geometry.

     
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  2. A bstract In this paper we generalize previous work on decomposition in three-dimensional orbifolds by 2-groups realized as analogues of central extensions, to orbifolds by more general 2-groups. We describe the computation of such orbifolds in physics, state a version of the decomposition conjecture, and then compute in numerous examples, checking that decomposition works as advertised. 
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    Free, publicly-accessible full text available August 1, 2024
  3. In this paper, we discuss decomposition in the context of three-dimensional Chern–Simons theories. Specifically, we argue that a Chern–Simons theory with a gauged noneffectively-acting one-form symmetry is equivalent to a disjoint union of Chern–Simons theories, with discrete theta angles coupling to the image under a Bockstein homomorphism of a canonical degree-two characteristic class. On three-manifolds with boundary, we show that the bulk discrete theta angles (coupling to bundle characteristic classes) are mapped to choices of discrete torsion in boundary orbifolds. We use this to verify that the bulk three-dimensional Chern–Simons decomposition reduces on the boundary to known decompositions of two-dimensional (WZW) orbifolds, providing a strong consistency test of our proposal. 
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  4. A bstract A number of finite algorithms for constructing representation theoretic data from group multiplications in a finite group G have recently been shown to be related to amplitudes for combinatoric topological strings ( G -CTST) based on Dijkgraaf-Witten theory of flat G -bundles on surfaces. We extend this result to projective representations of G using twisted Dijkgraaf-Witten theory. New algorithms for characters are described, based on handle creation operators and minimal multiplicative generating subspaces for the centers of group algebras and twisted group algebras. Such minimal generating subspaces are of interest in connection with information theoretic aspects of the AdS/CFT correspondence. For the untwisted case, we describe the integrality properties of certain character sums and character power sums which follow from these constructive G -CTST algorithms. These integer sums appear as residues of singularities in G -CTST generating functions. S -duality of the combinatoric topological strings motivates the definition of an inverse handle creation operator in the centers of group algebras and twisted group algebras. 
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  5. A bstract In this paper we study three-dimensional orbifolds by 2-groups with a trivially-acting one-form symmetry group BK . These orbifolds have a global two-form symmetry, and so one expects that they decompose into (are equivalent to) a disjoint union of other three-dimensional theories, which we demonstrate. These theories can be interpreted as sigma models on 2-gerbes, whose formal structures reflect properties of the orbifold construction. 
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  6. A bstract In this paper, we introduce a new set of modular-invariant phase factors for orbifolds with trivially-acting subgroups, analogous to discrete torsion and generalizing quantum symmetries. After describing their basic properties, we generalize decomposition to include orbifolds with these new phase factors, making a precise proposal for how such orbifolds are equivalent to disjoint unions of other orbifolds without trivially-acting subgroups or one-form symmetries, which we check in numerous examples. 
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