Computing Small 1-Homological Models for Commutative Differential Graded Algebras

We use homological perturbation machinery specific for the algebra category [P. Real. Homological Perturbation Theory and Associativity. Homology, Homotopy and Applications vol. 2, n. 5 (2000) 51-88] to give an algorithm for computing the differential structure of a small 1--homological model for commutative differential graded algebras (briefly, CDGAs). The complexity of the procedure is studied and a computer package in Mathematica is described for determining such models.Comment: 17 page

Computing “Small” 1–Homological Models for Commutative Differential Graded Algebras

We use homological perturbation machinery specific for the algebra category [13] to give an algorithm for computing the differential structure of a small 1– homological model for commutative differential graded algebras (briefly, CDGAs). The complexity of the procedure is studied and a computer package in Mathematica is described for determining such models.Ministerio de Educación y Ciencia PB98–1621–C02–02Junta de Andalucía FQM–014

Mukai duality for gerbes with connection

We study gerbes with connection over an etale stack via noncommutative algebras of differential forms on a groupoid presenting the stack. We then describe a dg-category of modules over any such algebra, which we claim represents a dg-enhancement of the derived category of coherent analytic sheaves on the gerbe in question. This category can be used to phrase and prove Fourier-Mukai type dualities between gerbes and other noncommutative spaces. As an application of the theory, we show that a gerbe with flat connection on a torus is dual (in a sense analogous to Fourier-Mukai duality or T-duality) to a noncommutative holomorphic dual torus.Comment: Final version. To appear in Crelle's journa

Constructive Homological Algebra and Applications

This text was written and used for a MAP Summer School at the University of Genova, August 28 to September 2, 2006. Available since then on the web site of the second author, it has been used and referenced by several colleagues working in Commutative Algebra and Algebraic Topology. To make safer such references, it was suggested to place it on the Arxiv repository. It is a relatively detailed exposition of the use of the Basic Perturbation Lemma to make constructive Homological Algebra (standard Homological Algebra is not constructive) and how this technology can be used in Commutative Algebra (Koszul complexes) and Algebraic Topology (effective versions of spectral sequences).Comment: Version 3: Error corrected p. 111, see footnote 2

Formal Homotopy Quantum Field Theories, II : Simplicial Formal Maps

Simplicial formal maps were introduced in the first paper, (math.QA/0512032), of this series as a tool for studying Homotopy Quantum Field Theories with background a general homotopy 2-type. Here we continue their study, showing how a natural generalisation can handle much more general backgrounds. The question of the geometric interpretation of these formal maps is partially answered in terms of combinatorial bundles. This suggests new interpretations of HQFTs.Comment: 21 pages, Contribution to the Streetfest proceeding

The Calabi complex and Killing sheaf cohomology

It has recently been noticed that the degeneracies of the Poisson bracket of linearized gravity on constant curvature Lorentzian manifold can be described in terms of the cohomologies of a certain complex of differential operators. This complex was first introduced by Calabi and its cohomology is known to be isomorphic to that of the (locally constant) sheaf of Killing vectors. We review the structure of the Calabi complex in a novel way, with explicit calculations based on representation theory of GL(n), and also some tools for studying its cohomology in terms of of locally constant sheaves. We also conjecture how these tools would adapt to linearized gravity on other backgrounds and to other gauge theories. The presentation includes explicit formulas for the differential operators in the Calabi complex, arguments for its local exactness, discussion of generalized Poincar\'e duality, methods of computing the cohomology of locally constant sheaves, and example calculations of Killing sheaf cohomologies of some black hole and cosmological Lorentzian manifolds.Comment: tikz-cd diagrams, 69 page

On boundary conditions and spacetime/fibre duality in Vasiliev's higher-spin gravity

This paper discusses some aspects of the Vasiliev system, beginning with a review of a recent proposal for an alternative perturbative scheme: solutions are built by means of a convenient choice of homotopy-contraction operator and subjected to asymptotically anti-de Sitter boundary conditions by perturbatively adjusting a gauge function and integration constants. At linear level the latter are fibre elements that encode, via unfolded equations, propagating massless fields of any spin. Therefore, linearized solution spaces, distinguished by their spacetime properties (regularity and boundary conditions), have parallels in the fibre. The traditional separation of different branches of linearized solutions via their spacetime features is reviewed, and their dual fibre characterization, as well as the arrangement of the corresponding fibre elements into AdS irreps, is illustrated. This construction is first reviewed for regular and singular solutions in compact basis, thereby capturing massless particles and static higher-spin black holes, and then extended to solutions in conformal basis, capturing bulk-to-boundary propagators and certain singular solutions with vanishing scaling dimension, related to boundary Green's functions. The non-unitary transformation between the two bases is recalled at the level of their fibre representatives.Comment: 53 pages. Contribution to the proceedings of the Corfu Summer Institute 2019 "School and Workshops on Elementary Particle Physics and Gravity" (CORFU2019), 31 August - 25 September 2019, Corfu, Greec

Understanding Weil-Petersson curvature

A brief history of the investigation of the Weil-Petersson curvature and a summary of Teichm\"{u}ller theory are provided. A report is presented on the program to describe an intrinsic geometry with the Weil-Petersson metric and geodesic-length functions. Formulas for the metric, covariant derivative and formulas for the curvature tensor are presented. A discussion of methods is included. Recent and new applications are sketched, including results from the work of Liu-Sun-Yau, an examination of the Yamada model metric and a description of Jacobi fields along geodesics to the boundary

Einstein Metrics on Spheres

We prove the existence of an abundance of new Einstein metrics on odd dimensional spheres including exotic spheres, many of them depending on continuous parameters. The number of families as well as the number of parameter grows double exponentially with the dimension. Our method of proof uses Brieskorn-Pham singularities to realize spheres (and exotic spheres) as circle orbi-bundles over complex algebraic orbifolds, and lift a Kaehler-Einstein metric from the orbifold to a Sasakian-Einstein metric on the sphere.Comment: 19 pages, some references added and clarifications made. to appear in Annals of Mathematic

On the Twisted K-Homology of Simple Lie Groups

We prove that the twisted K-homology of a simply connected simple Lie group G of rank n is an exterior algebra on n-1 generators tensor a cyclic group. We give a detailed description of the order of this cyclic group in terms of the dimensions of irreducible representations of G and show that the congruences determining this cyclic order lift along the twisted index map to relations in the twisted Spin-c bordism group of G.Comment: 38 pages, 2 figures. Added table of contents, remarks in sections 1.2 and 4.1.