A book that I am writing:
Differential cohomology in a cohesive $\infty$-topos
v2: pdf
v1: arXiv:1310.7930
For exposition and survey see at Higher Prequantum Geometry.
We formulate differential cohomology and Chern-Weil theory – the theory of connections on fiber bundles and of gauge fields – abstractly in homotopy toposes that we call cohesive. Cocycles in this differential cohomology classify higher principal bundles equipped with cohesive structure (topological, smooth, complex-analytic, formal, supergeometric, etc.) and equipped with connections, hence higher gauge fields. Furthermore we formulate differential geometry abstractly in a homotopy toposes that we call differentially cohesive. The manifolds in this theory are higher étale stacks (orbifolds) equipped with higher Cartan geometry (higher Riemannian-, complex, symplectic, conformal-, geometry) together with partial differential equations on spaces of sections of higher bundles over them, and equipped with higher pre-quantization of the resulting covariant phase spaces. We also formulate supergeometry abstractly in homotopy toposes and lift all these constructions to include fermionic degrees of freedom. Finally we indicate an abstract formulation of non-perturbative quantization of prequantum local field theory by fiber integration in twisted generalized cohomology of spectral linearizations of higher prequantum bundles.
We then construct models of the abstract theory in which traditional differential super-geometry is recovered and promoted to higher (derived) differential super-geometry.
We show that the cohesive and differential refinement of universal characteristic cocycles constitutes a higher Chern-Weil homomorphism refined from secondary characteristic classes to morphisms of higher moduli stacks of higher gauge fields, and at the same time constitutes extended geometric prequantization – in the sense of extended/multi-tiered quantum field theory – of hierarchies of higher dimensional Chern-Simons-type field theories, their higher Wess-Zumino-Witten-type boundary field theories and all further higher codimension defect field theories.
We find that in the Whitehead tower of superpoints in higher supergeometry one finds god given such cocycles on higher supersymmetry-groups, reflecting the completed brane scan of string/M-theory. We show that the induced higher super Cartan geometry is higher dimensional supergravity with super $p$-brane charge corrections included. For the maximal case of 11-dimensional supergravity we find the Einstein equations of motion and the higher extension of their super-isometry groups by M2/M5-brane charges in twisted generalized cohomology.
We close with an outlook on the cohomological quantization of these higher boundary prequantum field theories by a kind of cohesive motives.
The discussion here builds on and subsumes various sub-projects that appear separately.
differential cohomology in a cohesive topos
In
the case of stable cohesion for smooth cohesion is discussed in more detail – smooth spectra – and the “differential cohomology diagram” is derived from stable cohesion.
In
global equivariant homotopy theory is shown to be cohesive.
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the cohesive homotopy type theory of (Schreiber-Shulman 12) provides an example (p. 10) of homotopy type theory with higher modalities.
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the (∞,n)-category of correspondences in a slice (∞,1)-topos as appearing in Local prequantum field theory is analyzed.
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nonabelian Hodge theory is generalized to twisted bundles via the theory of cohesive principal infinity-bundles.
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further aspects of the smooth shape modality of cohesion are worked out (the etale homotopy type operation in the context of smooth infinity-stacks) as applied to orbifolds and étale groupoids and generally étale ∞-groupoids.
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the formalization in cohesive homotopy type theory is developed via adjoint logic.
List of invited talks and lectures, in chronological order.
Background fields in twisted differential nonabelian cohomology, talk at Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology, Oberwolfach Report No. 28/2009 (pdf)
Gauge fields in an $\infty$-topos, talk at Categories, Logic and Foundations of Physics VI, Oxford 2010 (video)
WZW terms in a cohesive $\infty$-topos, talk in Erlangen (2011) (pdf)
Chern-Simons terms on higher moduli stacks, talk at Hausdorff Institute Bonn (2011) (pdf)
Preuqantum physics in a cohesive topos, Talk at Quantum Physics and Logic 2011, (pdf)
$\infty$-Chern-Simons functionals, Talk at Higher Structures Göttingen (2011) (pdf)
Stacks, differential geometry and action functionals, course notes for the 8th meeting of the Polish Category theory seminar as well as the 4th Odense winter school on geometry and theoretical physics (2011)
Twisted smooth cohomology in string theory, expanded notes for a lecture series at ESI Program on K-Theory and Quantum Fields (2012)
Quantum gauge field theory in Cohesive homotopy type theory, talk at Quantum Physics and Logic 2012
Quantomorphism n-groups on n-plectic higher stacks at From Poisson geometry to String geometry, Erlangen (2012)
Higher Chern-Simons theory, lecture series at Workshop on Topological Aspects of Quantum Field Theories, Singapore (14 - 18 Jan 2013) on the first sections at geometry of physics, II) Physics
Higher Chern-Simons theory Introduction, lectures at the workshop Chern-Simons Theory: Geometry, Topology and Physics University of Pittsburgh (May 4-6, 2013)
Higher geometric prequantum theory and the Brane Bouquet, talk at Bayrischzell workshop 2013 (May 24-27 2013)
Synthetic Quantum Field Theory at 2013 CMS summer meeting, session Progress in higher categories, Halifax (June 5-7, 2013)
Super L-infinity algebras and the Brane Bouquet, talk at Higher Algebras and Lie-infinity Homotopy Theory, Luxembourg, (June 25-28, 2013)
Motivic quantization of prequantum field theory, talk at GAP XI – Higher Geometry and Quantum Field Theory, Pittsburgh, (August 19-23, 2013)
Classical field theory via Cohesive homotopy types, talk at Conference on Type Theory, Homotopy Theory and Univalent Foundations 2013, Barcelona (September 23-27, 2013)
Super Gerbes, talks and seminar at Third String Geometry Network meeting at MPI, Bonn, (October 7-11, 2013)
Tangent cohesion, talk at Twists, generalised cohomology and applications, Münster (October 14-18, 2013)
Higher toposes of laws of motion, talk at 8th Scottish Category Theory Seminar, (November 29, 2013)
(canceled last minute for private reasons) Higher WZW terms on higher smooth groups, talk at Higher Lie theory workshop, Luxembourg, (December 9–11) 2013
Quantization via Linear homotopy types, talk at Workshop “Philosophy of Mechanics: Mathematical Foundations”, February 12-14, 2014 - Université Paris Diderot
Quantization via Linear homotopy types, talk at Modern trends in topological quantum field theory, ESI, Vienna Febuary 2014
Differential generalized cohomology in Cohesive homotopy type theory, talk at Formalization of Mathematics, IHP, Paris, 5-9 May 2014
Differential cohomology is Cohesive homotopy theory, talk at Higher Geometric Structures along the Lower Rhine June 2014, 19-20 June 2014
lecture series Introduction to higher geometric quantum theory at Workshop on Higher Gauge Theory and Higher Quantization, Edinburgh 25/26 June 2014
Correspondences of cohesive linear homotopy types and Quantization, talk at Symmetries and correspondences: intra-disciplinary trends, Oxford July 5-8 2014
differential cohesion and idelic structure, talk at String geometry and loop spaces in Greifswald, 28. July - 1. August 2014
Fractures, Ideles and the Differential Hexagon, talk at Workshop on differential cohomologies, CUNY Graduate Center, August 12-14 2014 (video recording)
seminar talks at String Geometry Network Meeting, Münster, 18-22 August 2014
Higher field bundles for gauge fields, talk at Operator and geometric analysis on quantum theory, Levico Terme (Trento), Italy 15-19 September 2014
Lecture series on Higher gometric quantization following the notes What, and for what is Higher geometric quantization at SPHERE, Paris-Diderot, November 2014 (webpage)
talk Linear Homotopy Types and Integration in the seminar of the Type theory and Realizability Group at Paris-Diderot
talk The Gauge Principle and Local Quantum Field Theory (pdf slides), at the workshop Sur la notion d'identification en physique et en mathématiques, Paris-Diderot, November 13
talk at the seminar of the Fields, Strings and Geometry Group of University of Surrey, 18 November 2014
Higher super-WZW models, talk at seminar of the Eduard Čech Institute for Algebra, Geometry and Physics – Seminar on harmonic analysis, Prague 4 Dec 2014
visit Minhyong Kim in Oxford 9-12 December 2014
(postponed) talk at Purdue math colloquium, Jan 2015
talk on Obstruction theory for parameterized higher WZW terms at 35th Winter School GEOMETRY AND PHYSICS Srni, Jan 17-24 2015
talk on Obstruction theory for parameterized higher WZW terms in Göttingen Kolloquium 29 Jan 2015
lecture series Higher Cartan Geometry in the Harmonic Analysis Seminar at Prague, spring 2015
talk on The reflective categories of supergravity, in the Algebra & Category Theory Seminar of Brno, organized by Jiří Rosický, March 12, 2015
talk on Some super-Cartan geometry at the Central European Seminar in Brno, March 13, 2015
talk on Modern Physics formalized in Modal Homotopy Type Theory at workshop on Applying Homotopy Type Theory to Physics, Bristol, 7-8 April 2015
Infinitesimal symmetries of Deligne cocycles, talk in the Central European Seminar, Brno, May 29 2015
Structure Theory for Higher WZW Terms, series of talks at Flavors of Cohomology, Pittsburgh June 3-5, 2015
Obstruction theory for parameterized higher WZW terms, talk at Higher dimensional algebra in geometry and quantum field theory, AMS-EMS-SPM meeting in Porto, 10-13 June 2015
(canceled, since I did not pay over 200 EUR of conference “fee”) talk The M-Theory BPS charge super Lie 6-algebra (on 1.2.15.3.3 in dcct) at XXIIIth International Conference on Integrable Systems and Quantum symmetries (ISQS-23), Prague, June 23 - 27, 2015.
Duality of Monads in Geometric Homotopy Theory, talk at Duality in contemporary mathematics – philosophical aspects, Wuppertal University, Sept. 3-5, 2015
Obstruction theory for parameterized higher WZW terms, talk in the Algebra section of DMV meeting 2015, 21. - 25. Sept 2015, Hamburg
Some thoughts on the future of modal homotopy type theory, talk at Homotopy Type Theory and Univalent Foundations - Mini-Symposium at DMV meeting 2015, 21. - 25. Sept 2015, Hamburg
Higher Prequantum Geometry, talk at New Spaces for Mathematics and Physics, at Institut Henri Poincaré, Sept. 28 - Oct 2, 2015
Prequantum field theory, at “Talks in mathematical physics”, ETH Zurich, Nov 6, 2015
Prequantum field theories from Shifted symplectic structures, talk at MPI Bonn Higher Differential Geometry Seminar, 11 Nov 2015
Synthetic prequantum field theory in a cohesive homotopy topos, talk at Topos à l’IHES, conference on topos theory applied to number theory, algebraic geometry, logic, functional analysis, topology and mathematical physics; Paris, Nov. 25-27 2015
Prequantum field theory and the Green-Schwarz WZW terms, lecture series at Higher Structures in String Theory and Quantum Field Theory, ESI Program, Vienna, in the week November 30-December 4
Generalized cohomology of M2/M5-branes, talk at main workshop of Higher Structures in String Theory and Quantum Field Theory, ESI Program, Vienna, Dec 7 - 11, 2015
Differential cohesion and Arithmetic geometry, talk at Karslruher Weihnachts-Workshop 2015
Equivariant generalized cohomology of M2/M5-branes talk in Seminar on Higher Structures in the Program on Higher Structures in Geometry and Physics, MPI Bonn, 8 Jan 2016
Structure Theory for Higher WZW Terms, lecture at Seminário de Teoria Quântica do Campo Topológica (part I on 26 Jan 2016, part II on 29 Jan 2016), Lisbon
Structure Theory for Higher WZW Terms, lecture at Seminário de Teoria Quântica do Campo Topológica (part III on 23 Feb, part IV on 26 Feb 2016), Lisbon
Structure Theory for Higher WZW Terms, lecture at Seminário de Teoria Quântica do Campo Topológica (part V on 15 Mar, part VI on 18 Mar 2016), Lisbon
talk at FOMUS - Foundations of Mathematics – Univalent foundations and set theory, July 23 2016
Generalized cohomology of M2/M5-branes, talk at parallel session for ECM Satellite meeting “Classical and quantum symmetries in mathematics and physics”, Jena, 25-29 July 2016
Formalizing Cartan geometry in Modal Homotopy Type Theory talk at Homotopy Type Theory in Logic, Metaphysics and Philosophy of Physics at Bristol, 12-16 September 2016
Higher Structures in Mathematics and Physics, talk at the meeting of the Maths Department of the Czech Academy of Sciences, 9–11 November 2016 in Brno
geometry of physics -- superalgebra, lecture in the series From the Superpoint to T-Folds, CAS Prague, Nov 24, 2016
geometry of physics -- supergeometry, lecture in the series From the Superpoint to T-Folds, CAS Prague, Dec 1, 2016
Super Lie n-algebra of fundamental super p-branes, a couple of talks at Fields, Strings and Geometry Seminar, Surrey, Dec. 6, 2016
Higher Structures in Mathematics and Physics, and Quantization via twisted generalized cohomology, talks at “New interactions between homotopical algebra and quantum field theory”, workshop at Oberwolfach, Dec. 18-23, 2016
Duality in String/M-theory from Cyclic cohomology of Super Lie n-algebras, talk at Topological Quantum Seminar (alongside Iberian Strings 2017), Lisbon, 20. Jan 2017
Super topological T-Duality, talk at Conference on invertible objects and duality in derived algebraic geometry and homotopy theory, April 3-7, 2017, Regensburg
(upcoming) Super p-Brane Theory emerging from Super Homotopy-Theory talk in parallel session at String Math 2017, 24-28 July 2017 Hamburg, Germany
(upcoming) lecture series “Categories and Differentially cohesive Infinity-Toposes” at Diffeology, Catgories and Toposes Summer School June 18-29 2018, Galatasaray Istanbul, June 2018, Nesin Mathematics Village — Şirince, Izmir, Turkey