nLab

superalgebra and (synthetic ) supergeometry A supermanifold is a space locally modeled on Cartesian spaces and superpoints. There are different approaches to the definition and theory of supermanifolds in the literature. The definition is popular. The definition has been argued to have advantages, see also the references at super ∞-groupoid. See at geometry of physics – supergeometry the section …

An affine connection on a smooth manifold is a connection on the frame bundle of , i.e., the principal bundle of frames in the tangent bundle . The components of the local Lie-algebra valued 1-form of an affine connection are called Christoffel symbols. A coordinate-free treatment first appeared in: See also: Wikipedia: Affine connection Formulation in synthetic differential geometry: Wolfgang Be…

On differential geometry, Lie groups and symmetric spaces using (Weil algebra) methods from synthetic differential geometry:

synthetic differential geometry Introductions geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry Differentials Tangency The magic algebraic facts Theorems Axiomatics Models smooth algebra (-ring) differential equations, variational calculus Chern-Weil theory, ∞-Chern-Weil theory Cartan geometry (super, higher) For a topological space satisfying…

This entry is about a notion in category theory. For a different notion of the same name in (stable) homotopy theory see at Goodwillie calculus. The concept of a polynomial functor is a categorification of that of a polynomial. Polynomial endo-functors are used to encode a class of inductive types called W-types, and also as the underlying data of polynomial monads. Let be a locally cartesian clo…

Classical groups Finite groups Group schemes Topological groups Lie groups Super-Lie groups Higher groups Cohomology and Extensions Related concepts synthetic differential geometry Introductions geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry Differentials Tangency The magic algebraic facts Theorems Axiomatics Models smooth algebra (-ring) d…

model category, model -category Definitions Morphisms Universal constructions Refinements Producing new model structures Presentation of -categories Model structures for -groupoids on chain complexes/model structure on cosimplicial abelian groups related by the Dold-Kan correspondence for equivariant -groupoids for rational -groupoids for rational equivariant -groupoids for -groupoids for -groups…

synthetic differential geometry Introductions geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry Differentials Tangency The magic algebraic facts Theorems Axiomatics Models smooth algebra (-ring) differential equations, variational calculus Chern-Weil theory, ∞-Chern-Weil theory Cartan geometry (super, higher) Given a vector space and an elemen…

Interview with Dmitry Volkov by Yuri Ranyuk, Imperial College, London March 1995 (aip oral histories:4392) S. I. Volkova, Aleksandr A. Zheltukhin, Glimpses of Dmitry Volkov’s life and work, Nuclear Physics B - Proceedings Supplements 101 1–3 (2001) 20-25 [doi:10.1016/S0920-5632(01)01489-X] A.S. Bakai, S. V. Peletminskii, N.F. Shul’ga, Yu.V. Slyusarenko, D.V. Uvarov, A.A. Zheltukhin, To the 90-th …

Steven Duplij, Supergravity was discovered by D.V. Volkov and V.A. Soroka in 1973, wasn’t it?, East Eur. J. Phys., v3, p. 81-82 (2019) (Journal version), (arXiv:1910.03259) Precursor discussion of supergravity (D=4 supergravity): On D=4 supergravity formulated in superspace: On the history of supergravity:

symmetric monoidal (∞,1)-category of spectra A mathematical structure is essentially algebraic if its definition involves partially defined operations satisfying equational laws, where the domain of any given operation is a subset where various other operations happen to be equal. An actual algebraic theory is one where all operations are total functions. The most familiar example may be the (str…

fields and particles in particle physics and in the standard model of particle physics: matter field fermions (spinors, Dirac fields) (also: antiparticles) hadrons (bound states of the above quarks) bosinos: dark matter candidates Exotica Hadron supersymmetry (Miyazawa 66, Miyazawa 68) is an approximate (dynamically broken) supersymmetry among the experimentally observed spectra of hadrons, hence…

superalgebra and (synthetic ) supergeometry A superpoint is an infinitesimally thickened point whose infinitesimal extension is odd in the sense of supergeometry. A super Cartesian space of vanishing ordinary dimension. A superpoint is a supermanifold of the form . For more see at geometry of physics – superalgebra. The object is also called the odd line. The category of superpoints is the full s…

On the NSR superstring: On cosmic inflation in supergravity: On the history of the concept of supersymmetry: On historical work by Yoichiro Nambu on the Ising model:

Sylvester James Gates Jr., known as Jim Gates, works on supersymmetry, supergravity and superstring theory. Formulating supergravity in superspace: Warren Siegel, S. James Gates Jr., Superfield supergravity, Nuclear Physics B 147 1–2 (1979) 77-104 [doi:10.1016/0550-3213(79)90416-4] S. James Gates Jr., Kellogg S. Stelle, Peter C. West Algebraic origins of superspace constraints in supergravity, Nu…

On cosmic inflation in supergravity: The original suggestion that the lightest supersymmetric particle (LSP) would be a natural dark matter candidate:

Keith Alison Olive On cosmic inflation in supergravity: The original suggestion that the lightest supersymmetric particle (LSP) would be a natural dark matter candidate: Review: On the Starobinsky model of cosmic inflation and its embedding into supergravity: John Ellis, Dimitri Nanopoulos, Keith Olive, Sarunas Verner, A Unified No-Scale Model of Modulus Fixing, Inflation, Supersymmetry Breaking …

This page compiles pointers to material contained in the book collection: Cosimo Bambi, Leonardo Modesto, Ilya L. Shapiro (eds.): Springer (2025) on aspects of quantum gravity. Contents: for the time being in no particular order On perturbative quantum gravity: Viewed as an effective quantum field theory: On gauge invariant renormalization of perturbative quantum gravity: On renormalization of pe…

On supergravity models of cosmic inflation: On cosmic inflation: On the Starobinsky model of cosmic inflation:

On supergravity models of cosmic inflation: On embedding the Starobinsky model of cosmic inflation into supergravity: and into string theory/higher curvature corrections: On the Starobinsky model of cosmic inflation with further higher curvature corrections: On the Starobinsky model of cosmic inflation: