Quantum Gauge Field Theory in Cohesive Homotopy Type Theory

We implement in the formal language of homotopy type theory a new set of axioms called cohesion. Then we indicate how the resulting cohesive homotopy type theory naturally serves as a formal foundation for central concepts in quantum gauge field theory. This is a brief survey of work by the authors developed in detail elsewhere.

[1]  Jonathan Wang,et al.  The moduli stack of $G$-bundles , 2011, 1104.4828.

[2]  A. Kock.  Synthetic Differential Geometry , 1981 .

[3]  Urs Schreiber,et al.  Cech cocycles for differential characteristic classes -- An infinity-Lie theoretic construction , 2010, 1011.4735.

[4]  Marc Henneaux,et al.  Quantization of Gauge Systems , 1992 .

[5]  Urs Schreiber,et al.  Differential cohomology in a cohesive infinity-topos , 2013, 1310.7930.

[6]  G. M. Kelly,et al.  Reflective subcategories, localizations and factorizationa systems , 1985, Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics.

[7]  G. M. Kelly,et al.  On Localization and Stabilization for Factorization Systems , 1997, Appl. Categorical Struct..

[8]  Vladimir Voevodsky,et al.  Univalent Foundations of Mathematics , 2011, WoLLIC.

[9]  René Lavendhomme.  Note on toposes and models of S.D.G. , 1996 .

[10]  J. Bernstein.  Spontaneous symmetry breaking, gauge theories, the Higgs mechanism and all that , 1974 .

[11]  René Lavendhomme,et al.  Basic Concepts of Synthetic Differential Geometry , 1996 .

[12]  J. Lurie.  Higher Topos Theory , 2006, math/0608040.

[13]  Kenneth S. Brown,et al.  Abstract homotopy theory and generalized sheaf cohomology , 1973 .

[14]  Thomas Schucker,et al.  Differential geometry, gauge theories, and gravity , 1987 .

[15]  John C. Baez,et al.  Categorified Symplectic Geometry and the Classical String , 2008, 0808.0246.

[16]  Ieke Moerdijk,et al.  Orbifolds, sheaves and groupoids , 1997 .

[17]  I. Moerdijk,et al.  Models for smooth infinitesimal analysis , 1990 .

[18]  Lisa Randall,et al.  Higgs Discovery: The Power of Empty Space , 2013 .

[19]  J. Baez,et al.  Higher dimensional algebra and topological quantum field theory , 1995, q-alg/9503002.

[20]  Johannes Zierenberg,et al.  SPONTANEOUS SYMMETRY BREAKING , 2020, Supersymmetry and Supergravity.

[21]  Urs Schreiber,et al.  Survey of mathematical foundations of QFT and perturbative string theory , 2011, 1109.0955.

[22]  Chris Kapulkin,et al.  Univalence in Simplicial Sets , 2012, 1203.2553.

[23]  Peter Dybjer,et al.  The Biequivalence of Locally Cartesian Closed Categories and Martin-Löf Type Theories , 2014, Math. Struct. Comput. Sci..

[24]  M. Faizal,et al.  Chern-Simons-Matter Theory , 2013, 1301.5664.

[25]  L. O'raifeartaigh,et al.  Gauge theory: Historical origins and some modern developments , 2000 .

[26]  Bas Spitters,et al.  Deep Beauty: Bohrification , 2011 .

[27]  John C. Baez,et al.  Convenient Categories of Smooth Spaces , 2008, 0807.1704.

[28]  Tobias J. Hagge,et al.  Physics , 1929, Nature.

[29]  F. WILLIAM LAWVERE,et al.  AXIOMATIC COHESION , 2007 .

[30]  Michael Shulman,et al.  The univalence axiom for elegant Reedy presheaves , 2013, 1307.6248.

[31]  John C. Baez,et al.  Physics, Topology, Logic and Computation: A Rosetta Stone , 2009, 0903.0340.

[32]  Jean-Luc Brylinski,et al.  Loop Spaces, Characteristic Classes and Geometric Quantization , 1994 .

[33]  Urs Schreiber,et al.  Twisted Differential String and Fivebrane Structures , 2009, 0910.4001.

[34]  Jacob Lurie,et al.  On the Classification of Topological Field Theories , 2009, 0905.0465.

[35]  I. V. Kanatchikov.  Precanonical quantization of Yang-Mills fields and the functional Schrödinger representation , 2002 .

[36]  Christopher L. Rogers.  Higher Symplectic Geometry , 2011, 1106.4068.

[37]  Pierre Deligne,et al.  Quantum Fields and Strings: A Course for Mathematicians , 1999 .

[38]  Lars Birkedal,et al.  Elementary axioms for local maps of toposes , 2003 .

[39]  Jacob Lurie,et al.  Higher Topos Theory (AM-170) , 2009 .

[40]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[41]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[42]  Clark Barwick,et al.  On the Unicity of the Homotopy Theory of Higher Categories , 2011 .

[43]  Eugene Lerman,et al.  Orbifolds as stacks , 2008, 0806.4160.

[44]  I. M. Singer,et al.  Quadratic functions in geometry, topology, and M-theory , 2002 .

[45]  Steven Awodey,et al.  Type Theory and Homotopy , 2010, Epistemology versus Ontology.

[46]  Urs Schreiber,et al.  Principal ∞-bundles - General theory , 2012 .

[47]  Daniel S. Freed.  Dirac Charge Quantization and Generalized Differential Cohomology , 2000 .

[48]  Urs Schreiber,et al.  Extended higher cup-Product Chern-Simons theories , 2012, 1207.5449.