nLab

Contents Context -Chern-Weil theory Differential cohomology differential cohomology Ingredients Connections on bundles Higher abelian differential cohomology Higher nonabelian differential cohomology Fiber integration Application to gauge theory Contents Idea A Chern-Simons form is a differential form naturally associated to a differential form with values in a Lie algebra : it is the form trivia…

Sergio L. Cacciatori, Bianca Cerchiai, Alessio Marrani, Squaring the Magic [arXiv:1208.6153] Sergio L. Cacciatori, Bianca Cerchiai, Alessio Marrani, Magic coset decompositions, Adv. Theor. Math. Phys. 17 5 (2013) 1077-1128 [doi:10.4310/ATMP.2013.v17.n5.a4] On generalizations of exceptional structures, including E8, octonions and the exceptional Jordan algebra: On 12-dimensional supergravity, D=14…

superalgebra and (synthetic ) supergeometry The notion of Jordan superalgebra is the analog in superalgebra/supergeometry of that of Jordan algebra: A Jordan superalgebra is a supercommutative superalgebra with underlying -graded algebra , where: is an ordinary Jordan algebra, is a -bimodule with a “Lie bracket-like” product into , satisfying a super Jordan identity. Simple Jordan superalgebras o…

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical) quantum mechanical system, quantum probability interacting field quantization For classes of gauge theories, such as (super) Yang-Mills theory or Chern-Simons theory or various matrix models, whose gauge groups may be square matrices for any natural number , notably in the special unitary group , the special orthogon…

representation, 2-representation, ∞-representation Grothendieck group, lambda-ring, symmetric function, formal group principal bundle, torsor, vector bundle, Atiyah Lie algebroid Eilenberg-Moore category, algebra over an operad, actegory, crossed module The possible actions of well-behaved topological groups (such as compact Lie groups) on topological or smooth n-spheres display various interesti…

Introducing discussion of D=2 Yang-Mills theory: Early discussion of flux tubes/Wilson lines as effective strings in Yang-Mills theory (Gauge/String duality): “So the world sheet of string should be interpreted as the color magnetic dipole sheet. The string itself should be interpreted as the electric flux tube in the monopole plasma.” On the large-N limit of QCD: Alexander A. Migdal, Loop equati…

A Malcev operation on a set is a ternary operation, a function which satisfies the identities and . An important motivating example is the operation of a heap, for example the operation on a group defined by . An algebraic theory is a Malcev theory when contains a Malcev operation. An algebraic theory is Malcev iff one of the following equivalent statements is true: in the category of -algebras, …

vector bundle, 2-vector bundle, (∞,1)-vector bundle real, complex/holomorphic, quaternionic Special and general types group cohomology, nonabelian group cohomology, Lie group cohomology cohomology with constant coefficients / with a local system of coefficients Special notions Variants differential cohomology Extra structure Operations Theorems A bibundle is a (groupoid-)principal bundle which is…

On Lie groupoids, Lie 2-groups and differentiable stacks: On Lie theory for diffeological groupoids: On Cartan calculus for diffeological spaces: On bornological groupoid convolution algebras:

superalgebra and (synthetic ) supergeometry supergravity in dimension 6 Background on spin representations and supersymmetry: 221 2 (1983) 331-348 [doi:10.1016/0550-3213(83)90582-5] Formulation of supergravity on superspace: Moustafa A. Awada, Paul Townsend, Germán Sierra: Six-dimensional simple and extended chiral supergravity in superspace, Class. Quantum Grav. 2 (1985) L85 [doi:10.1088/0264-93…

On generalized Spin(7)-manifolds and M-theory on G₂-manifolds: On G-structures in M-theory: On D=6 supergravity: and in relation to contact geometry: Discussion of Killing spinors on globally hyperbolic Lorentzian manifolds: On duality-symmetric abelian Yang-Mills theory (“premetric electromagnetism”) in the generality allowing “U-duality-twists” among several abelian gauge fields, motivated by a…

Jeff Giansiracusa is a professor at Swansea University in the Department of Mathematics. He has worked on homotopy theoretic aspects of moduli spaces, operads, topological field theory, and diffeomorphism groups, using topological techniques from algebraic K-theory to study the homotopy theory of moduli spaces arising in algebraic geometry and now includes aspects of topological data analysis, tr…

symmetric monoidal (∞,1)-category of spectra The space of functions on the space of morphisms of a small category (with coefficients in some ring ) naturally inherits a convolution algebra structure from the composition operation on morphisms. This is called its category convolution algebra or just category algebra for short. Often this is considered specifically for groupoids, and hence accordin…

Special and general types group cohomology, nonabelian group cohomology, Lie group cohomology cohomology with constant coefficients / with a local system of coefficients Special notions Variants differential cohomology Extra structure Operations Theorems higher geometry / derived geometry Ingredients Concepts geometric little (∞,1)-toposes geometric big (∞,1)-toposes Constructions Examples derive…

Interview of Mikhail Shifman by David Zierler on July 7, 2021, Niels Bohr Library & Archives, American Institute of Physics, College Park, MD USA [aip:oral-histories/47523] On the Chern-Simons level renormalization (shift by the dual Coxeter number): On superalgebra, supergeometry, supersymmetry, supergravity and superstrings: Mikhail Shifman (ed.) Yuri Golfand memorial volume World Scientific, 2…

These are solutions to Einstein's equations which are black holes in spacetimes that far away from the black hole are asymptotic (not to Minkowski spacetime but) to anti de Sitter spacetime. Under the AdS-CFT correspondence black holes in the anti-de Sitter spacetime translate to the conformal field theory on the asymptotic boundary being at positive temperature (e.g. Duff 99, section 6, Natsuume…

On conformal boundaries in the AdS/CFT correspondence: On glueballs via holographic QCD: On the quark-gluon plasma via holographic QCD: On black holes in anti-de Sitter spacetime in relation to topological charges and holographic QCD:

A coreflective subcategory is a full subcategory whose inclusion functor has a right adjoint (a cofree functor): The dual concept is that of a reflective subcategory. See there for more details. (equivalent characterizations) Given any pair of adjoint functors the following are equivalent: The left adjoint is fully faithful. (In this case is equivalent to its essential image in under , a full cor…

Inclusion of heavy flavors (strange quarks/kaons, etc.) into the Skyrme model: Curtis Callan, Igor Klebanov, Bound-state approach to strangeness in the Skyrme model, Nuclear Physics B Volume 262, Issue 2, 16 December 1985, Pages 365-382 (doi10.1016/0550-3213(85)90292-5) Curtis Callan, K. Hornbostel, Igor Klebanov, Baryon masses in the bound state approach to strangeness in the skyrme model, Physi…

Alexei Yurievich Kitaev On quantum computation and quantum error correction (and stating the Solovay-Kitaev theorem): On quantum circuits with mixed quantum states/density matrices: On quantum error correcting codes associated with planar bulk/boundary systems (precursor to holographic tensor networks): On computation in general and quantum computation in particular: On quantum measurement of Wil…