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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 K\"ahler 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-K\"ahler 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 $\mathbb{P}^3$ and 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., Comment: 78+30 pages. Fixed acknowledgements and minor typos

We prove a `Whitney' presentation, and a `Coulomb branch' presentation, for the torus equivariant quantum K theory of the Grassmann manifold $\mathrm{Gr}(k;n)$, inspired from physics, and stated in an earlier paper. The first presentation is obtained by quantum deforming the product of the Hirzebruch $\lambda_y$ classes of the tautological bundles. In physics, the $\lambda_y$ classes arise as certain Wilson line operators. The second presentation is obtained from the Coulomb branch equations involving the partial derivatives of a twisted superpotential from supersymmetric gauge theory. This is closest to a presentation obtained by Gorbounov and Korff, utilizing integrable systems techniques. Algebraically, we relate the Coulomb and Whitney presentations utilizing transition matrices from the (equivariant) Grothendieck polynomials to the (equivariant) complete homogeneous symmetric polynomials. The calculations of K-theoretic Gromov-Witten invariants of wedge powers of the tautological subbundles on the Grassmannian utilize the `quantum=classical' statement., Comment: 39 pages; v2: rewrote Appendix A by utilizing the hypothesis that R is complete in the I-adic topology. This corrects a missing hypothesis in the Appendix from v1 of the paper. Fixed several minor typos

We review work on `decomposition,' a property of two-dimensional theories with 1-form symmetries and, more generally, d-dimensional theories with (d-1)-form symmetries. Decomposition is the observation that such quantum field theories are equivalent to (`decompose into') disjoint unions of other QFTs, known in this context as "universes." Examples include two-dimensional gauge theories and orbifolds with matter invariant under a subgroup of the gauge group. Decomposition explains and relates several physical properties of these theories -- for example, restrictions on allowed instantons arise as a "multiverse interference effect" between contributions from constituent universes. First worked out in 2006 as part of efforts to resolve technical questions in string propagation on stacks, decomposition has been the driver of a number of developments since. We give a general overview of decomposition, describe features of decomposition arising in gauge theories, then dive into specifics for orbifolds. We conclude with a discussion of the recent application to anomaly resolution of Wang-Wen-Witten in two-dimensional orbifolds. This is a contribution to the proceedings of the conference Two-dimensional supersymmetric theories and related topics (Matrix Institute, Australia, January 2022), giving an overview of a talk given there and elsewhere., 24 pages, LaTeX

In this article we review how categorical equivalences are realized by renormalization group flow in physical realizations of stacks, derived categories, and derived schemes. We begin by reviewing the physical realization of sigma models on stacks, as (universality classes of) gauged sigma models, and look in particular at properties of sigma models on gerbes (equivalently, sigma models with restrictions on nonperturbative sectors), and decomposition, in which two-dimensional sigma models on gerbes decompose into disjoint unions of ordinary theories. We also discuss stack structures on examples of moduli spaces of SCFTs, focusing on elliptic curves, and implications of subtleties there for string dualities in other dimensions. In the second part of this article, we review the physical realization of derived categories in terms of renormalization group flow (time evolution) of combinations of D-branes, antibranes, and tachyons. In the third part of this article, we review how Landau-Ginzburg models provide a physical realization of derived schemes, and also outline an example of a derived structure on a moduli spaces of SCFTs., 17 pages, Contribution to Proceedings of LMS/EPSRC Durham Symposium Higher Structures in M-Theory, August 2018

In this work we study genus one fibrations in Calabi-Yau three-folds with a non-trivial first fundamental group. The manifolds under consideration are constructed as smooth quotients of complete intersection Calabi-Yau three-folds (CICYs) by a freely acting, discrete automorphism. By probing the compatibility of symmetries with genus one fibrations (that is, discrete group actions which preserve a local decomposition of the manifold into fiber and base) we find fibrations that are inherited from fibrations on the covering spaces. Of the 7,890 CICY three-folds, 195 exhibit known discrete symmetries, leading to a total of 1,695 quotient manifolds. By scanning over 20,700 fiber/symmetry pairs on the covering spaces we find 17,161 fibrations on the quotient Calabi-Yau manifolds. It is found that the vast majority of the non-simply connected manifolds studied exhibit multiple different genus one fibrations - echoing a similar ubiquity of such structures that has been observed in other data sets. The results are available at [1]. The possible base manifolds are all singular and are catalogued. These Calabi-Yau fibrations generically exhibit multiple fibers and are of interest in F-theory as backgrounds leading to theories with superconformal loci and discretely charged matter. NSF [PHY-1720321] The work of L.A. and J.G. is supported in part by NSF grant PHY-1720321. This research is part of the working group activities of the the 4-VA initiative "A Synthesis of Two Approaches to String Phenomenology".

We study 2d N = (0, 2) supersymmetric quiver gauge theories that describe the low-energy dynamics of D1-branes at Calabi-Yau fourfold (CY4) singularities. On general grounds, the holomorphic sector of these theories - matter content and (classical) superpotential interactions - should be fully captured by the topological B-model on the CY4. By studying a number of examples, we confirm this expectation and flesh out the dictionary between B-brane category and supersymmetric quiver, the matter content of the supersymmetric quiver is encoded in morphisms between B-branes (that is, Ext groups of coherent sheaves), while the superpotential interactions are encoded in the A(infinity) algebra satisfied by the morphisms. This provides us with a derivation of the supersymmetric quiver directly from the CY4 geometry. We also suggest a relation between triality of N = (0, 2) gauge theories and certain mutations of exceptional collections of sheaves. 0d N = 1 supersymmetric quivers, corresponding to D-instantons probing CY5 singularities, can be discussed similarly. NSF [PHY-1417410, PHY-1720321] We would like to thank P. Aspinwall, S. Franco, D. Ghim, C. Herzog, S. Katz, W. Lerche, I. Melnikov, T. Pantev, and R.K. Seong for useful conversations and comments. E.S. was partially supported by NSF grants PHY-1417410 and PHY-1720321.