**superalgebra** and (synthetic ) **supergeometry** ## Background * algebra * geometry * graded object ## Introductions * geometry of physics -- superalgebra * geometry of physics -- supergeometry ## Superalgebra ## * super commutative monoid * super abelian group * super ring * supercommutative ring * exterior ring * Clifford ring * super module * super vector space, SVect * super algebra * supercommutative algebra * exterior algebra * Clifford algebra * superdeterminant * super Lie algebra * super Poincare Lie algebra * chain complex of super vector spaces (model structure) * differential graded-commutative superalgebra (model structure) * super L-infinity algebra ## Supergeometry ## * superpoint * super Cartesian space * supermanifold, SDiff * NQ-supermanifold * super vector bundle * complex supermanifold * Euclidean supermanifold * super spacetime * super Minkowski spacetime * integration over supermanifolds * Berezin integral * super Lie group * super translation group * super Euclidean group * super ∞-groupoid * super formal smooth ∞-groupoid * super line 2-bundle ## Supersymmetry supersymmetry * division algebra and supersymmetry * super Poincare Lie algebra * supermultiplet * BPS state * M-theory super Lie algebra, type II super Lie algebra * supergravity Lie 3-algebra, supergravity Lie 6-algebra ## Supersymmetric field theory * superfield * supersymmetric quantum mechanics * superparticle * adinkra * super Yang-Mills theory * supergravity * gauged supergravity * superstring theory * type II string theory * heterotic string theory ## Applications * geometric model for elliptic cohomology *** **synthetic differential geometry** **Introductions** from point-set topology to differentiable manifolds geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry **Differentials** * differentiation, chain rule * differentiable function * infinitesimal space, infinitesimally thickened point, amazing right adjoint **V-manifolds** * differentiable manifold, coordinate chart, atlas * smooth manifold, smooth structure, exotic smooth structure * analytic manifold, complex manifold * formal smooth manifold, derived smooth manifold **smooth space** * diffeological space, Frölicher space * manifold structure of mapping spaces **Tangency** * tangent bundle, frame bundle * vector field, multivector field, tangent Lie algebroid; * differential forms, de Rham complex, Dolbeault complex * pullback of differential forms, invariant differential form, Maurer-Cartan form, horizontal differential form, * cogerm differential form * integration of differential forms * local diffeomorphism, formally étale morphism * submersion, formally smooth morphism, * immersion, formally unramified morphism, * de Rham space, crystal * infinitesimal disk bundle **The magic algebraic facts** * embedding of smooth manifolds into formal duals of R-algebras * smooth Serre-Swan theorem * derivations of smooth functions are vector fields **Theorems** * Hadamard lemma * Borel's theorem * Boman's theorem * Whitney extension theorem * Steenrod-Wockel approximation theorem * Whitney embedding theorem * Poincare lemma * Stokes theorem * de Rham theorem * Hochschild-Kostant-Rosenberg theorem * differential cohomology hexagon **Axiomatics** * Kock-Lawvere axiom * smooth topos, super smooth topos * microlinear space * integration axiom **cohesion** * (shape modality $\dashv$ flat modality $\dashv$ sharp modality) $(\esh \dashv \flat \dashv \sharp )$ * discrete object, codiscrete object, concrete object * points-to-pieces transform * structures in cohesion * dR-shape modality $\dashv$ dR-flat modality $\esh_{dR} \dashv \flat_{dR}$ **infinitesimal cohesion** * classical modality **tangent cohesion** * differential cohomology diagram **differential cohesion** * (reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality) $(\Re \dashv \Im \dashv \)$ * reduced object, coreduced object, formally smooth object * formally étale map * [structures in differential cohesion](cohesive+%28infinity%2C1%29-topos+--+infinitesimal+cohesion#StructuresInDifferentialCohesion) **graded differential cohesion** * fermionic modality $\dashv$ bosonic modality $\dashv$ rheonomy modality $(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$ **singular cohesion $$ \array{ id \dashv id \\ \vee \vee \\ \stackrel{fermionic}{} \rightrightarrows \dashv \rightsquigarrow \stackrel{bosonic}{} \\ \bot \bot \\ \stackrel{bosonic}{} \rightsquigarrow \dashv \mathrm{R}\!\!\mathrm{h} \stackrel{rheonomic}{} \\ \vee \vee \\ \stackrel{reduced}{} \Re \dashv \Im \stackrel{infinitesimal}{} \\ \bot \bot \\ \stackrel{infinitesimal}{} \Im \dashv \ \stackrel{\text{étale}}{} \\ \vee \vee \\ \stackrel{cohesive}{} \esh \dashv \flat \stackrel{discrete}{} \\ \bot \bot \\ \stackrel{discrete}{} \flat \dashv \sharp \stackrel{continuous}{} \\ \vee \vee \\ \emptyset \dashv \ast } $$ {#Diagram} **Models** {#models_2} * Models for Smooth Infinitesimal Analysis * smooth algebra ($C^\infty$-ring) * smooth locus * Fermat theory * Cahiers topos * smooth ∞-groupoid * formal smooth ∞-groupoid * super formal smooth ∞-groupoid **Lie theory, ∞-Lie theory** * Lie algebra, Lie n-algebra, L-∞ algebra * Lie group, Lie 2-group, smooth ∞-group **differential equations, variational calculus** * D-geometry, D-module * jet bundle * variational bicomplex, Euler-Lagrange complex * Euler-Lagrange equation, de Donder-Weyl formalism, * phase space **Chern-Weil theory, ∞-Chern-Weil theory** * connection on a bundle, connection on an ∞-bundle * differential cohomology * ordinary differential cohomology, Deligne complex * differential K-theory * differential cobordism cohomology * parallel transport, higher parallel transport, fiber integration in differential cohomology * holonomy, higher holonomy * gauge theory, higher gauge theory * Wilson line, Wilson surface **Cartan geometry (super, higher)** * Klein geometry, (higher) * G-structure, torsion of a G-structure * Euclidean geometry, hyperbolic geometry, elliptic geometry * (pseudo-)Riemannian geometry * orthogonal structure * isometry, Killing vector field, Killing spinor * spacetime, super-spacetime * complex geometry * symplectic geometry * conformal geometry