Rigidity and defect actions in Landau-Ginzburg models

Studying two-dimensional field theories in the presence of defect lines naturally gives rise to monoidal categories: their objects are the different (topological) defect conditions, their morphisms are junction fields, and their tensor product describes the fusion of defects. These categories should be equipped with a duality operation corresponding to reversing the orientation of the defect line, providing a rigid and pivotal structure. We make this structure explicit in topological Landau-Ginzburg models with potential x^d, where defects are described by matrix factorisations of x^d-y^d. The duality allows to compute an action of defects on bulk fields, which we compare to the corresponding N=2 conformal field theories. We find that the two actions differ by phases.Comment: 53 pages; v2: clarified exposition of pivotal structures, corrected proof of theorem 2.13, added remark 3.9; version to appear in CM

Landau-Ginzburg/Calabi-Yau correspondence, global mirror symmetry and Orlov equivalence

We show that the Gromov-Witten theory of Calabi-Yau hypersurfaces matches, in genus zero and after an analytic continuation, the quantum singularity theory (FJRW theory) recently introduced by Fan, Jarvis and Ruan following ideas of Witten. Moreover, on both sides, we highlight two remarkable integral local systems arising from the common formalism of Gamma-integral structures applied to the derived category of the hypersurface {W=0} and to the category of graded matrix factorizations of W. In this setup, we prove that the analytic continuation matches Orlov equivalence between the two above categories.Comment: 72pages, v2: Appendix B and references added. Typos corrected, v3: several mistakes corrected, final versio

Homological Mirror Symmetry for Calabi-Yau hypersurfaces in projective space

We prove Homological Mirror Symmetry for a smooth d-dimensional Calabi-Yau hypersurface in projective space, for any d > 2 (for example, d = 3 is the quintic three-fold). The main techniques involved in the proof are: the construction of an immersed Lagrangian sphere in the `d-dimensional pair of pants'; the introduction of the `relative Fukaya category', and an understanding of its grading structure; a description of the behaviour of this category with respect to branched covers (via an `orbifold' Fukaya category); a Morse-Bott model for the relative Fukaya category that allows one to make explicit computations; and the introduction of certain graded categories of matrix factorizations mirror to the relative Fukaya category.Comment: 133 pages, 17 figures. Changes to the argument ruling out sphere bubbling in the relative Fukaya category, and dealing with the behaviour of the symplectic form under branched covers. Other minor changes suggested by the referee. List of notation include

Women on boards of Malaysian firms: Impact on market and accounting performance

We seek to offer some reconciliation for the conflicting theoretical arguments and empirical findings regarding the impact of women’s participation in boards on firms’ performance.We suggest that this impact differs in relation to market- and accounting-performance, and it is firm-specific, and varies by firms’ ownership type and the composition of their boards.These arguments find theoretical underpinnings in agency and resource-dependency theories, combined with behavioral and discrimination theories that articulate women behavior in the workplace and market perception of gender equality.The empirical analysis is based on a dataset of 841 publicly-listed firms in Malaysia.The results show positive impact of women’s participation on accounting-performance and negative impact on market-performance, suggesting that women directors create economic value, which is undervalued by the market. We interpret the findings with reference to the perception of women’s role in society and business in Malaysia, and the nature of corporate governance and ownership types prevalent among Malaysian firms.We suggest that the relationships might be context-specific, and hence the desired level of women’s participation varies across countries.We discuss the normative implications of the findings for government authorities considering legislation of gender-quota on boards, and for firms

Compact generators in categories of matrix factorizations

We study the category of matrix factorizations associated to the germ of an isolated hypersurface singularity. This category is shown to admit a compact generator which is given by the stabilization of the residue field. We deduce a quasi-equivalence between the category of matrix factorizations and the dg derived category of an explicitly computable dg algebra. Building on this result, we employ a variant of Toen's derived Morita theory to identify continuous functors between matrix factorization categories as integral transforms. This enables us to calculate the Hochschild chain and cochain complexes of these categories. Finally, we give interpretations of the results of this work in terms of noncommutative geometry based on dg categories

Higher Segal spaces

This monograph initiates a theory of new categorical structures that generalize the simplicial Segal property to higher dimensions. The authors introduce the notion of a d-Segal space, which is a simplicial space satisfying locality conditions related to triangulations of d-dimensional cyclic polytopes. Focus here is on the 2-dimensional case. Many important constructions are shown to exhibit the 2-Segal property, including Waldhausen’s S-construction, Hecke-Waldhausen constructions, and configuration spaces of flags. The relevance of 2-Segal spaces in the study of Hall and Hecke algebras is discussed. Higher Segal Spaces marks the beginning of a program to systematically study d-Segal spaces in all dimensions d. The elementary formulation of 2-Segal spaces in the opening chapters is accessible to readers with a basic background in homotopy theory. A chapter on Bousfield localizations provides a transition to the general theory, formulated in terms of combinatorial model categories, that features in the main part of the book. Numerous examples throughout assist readers entering this exciting field to move toward active research; established researchers in the area will appreciate this work as a reference.Front Matter Pages i-xv Preliminaries Tobias Dyckerhoff, Mikhail Kapranov Pages 1-8 Topological 1-Segal and 2-Segal Spaces Tobias Dyckerhoff, Mikhail Kapranov Pages 9-30 Discrete 2-Segal Spaces Tobias Dyckerhoff, Mikhail Kapranov Pages 31-70 Model Categories and Bousfield Localization Tobias Dyckerhoff, Mikhail Kapranov Pages 71-84 The 1-Segal and 2-Segal Model Structures Tobias Dyckerhoff, Mikhail Kapranov Pages 85-94 The Path Space Criterion for 2-Segal Spaces Tobias Dyckerhoff, Mikhail Kapranov Pages 95-106 2-Segal Spaces from Higher Categories Tobias Dyckerhoff, Mikhail Kapranov Pages 107-124 Hall Algebras Associated to 2-Segal Spaces Tobias Dyckerhoff, Mikhail Kapranov Pages 125-151 Hall (∞, 2)-Categories Tobias Dyckerhoff, Mikhail Kapranov Pages 153-167 An (∞, 2)-Categorical Theory of Spans Tobias Dyckerhoff, Mikhail Kapranov Pages 169-199 2-Segal Spaces as Monads in Bispans Tobias Dyckerhoff, Mikhail Kapranov Pages 201-208 Back Matter Pages 209-21

Higher Segal spaces I

This is the first paper in a series on new higher categorical structures called higher Segal spaces. For every d > 0, we introduce the notion of a d-Segal space which is a simplicial space satisfying locality conditions related to triangulations of cyclic polytopes of dimension d. In the case d=1, we recover Rezk's theory of Segal spaces. The present paper focuses on 2-Segal spaces. The starting point of the theory is the observation that Hall algebras, as previously studied, are only the shadow of a much richer structure governed by a system of higher coherences captured in the datum of a 2-Segal space. This 2-Segal space is given by Waldhausen's S-construction, a simplicial space familiar in algebraic K-theory. Other examples of 2-Segal spaces arise naturally in classical topics such as Hecke algebras, cyclic bar constructions, configuration spaces of flags, solutions of the pentagon equation, and mapping class groups