Schreiber
Introduction to Higher Supergeometry

Contents

Abstract. Due to the existence of a) gauge fields and b) fermion fields, the geometry of physics is higher supergeometry, i.e. super-geometric homotopy theory. This is made precise via Grothendieck‘s functorial geometry implemented in higher topos theory. We give an introduction to the higher topos of higher superspaces and how it accomodates higher Lie theory of super L-∞ algebras. We close by indicating how geometric homotopy theory reveals that the superpoint emerges “from nothing”, and that core structure of M-theory emerges out of the superpoint via a sequence of invariant universal higher central extensions. This will be discussed in more detail in other talks in the meeting, notably in

all summed up in the proceedings contribution:


Contents

Motivation

The geometry of fundamental physics is higher differential supergeometry.

AA\phantom{AA} physics AA\phantom{AA}AA\phantom{AA} mathematics AA\phantom{AA}
AA\phantom{AA} gauge principle AA\phantom{AA}AA\phantom{AA} higher geometry AA\phantom{AA}
AA\phantom{AA} (geometric homotopy theory) AA\phantom{AA}
AA\phantom{AA} Pauli exclusion principle AA\phantom{AA}AA\phantom{AA} supergeometry AA\phantom{AA}

Here:

  1. Supergeometry is geometry whose spaces may have algebras of functions that are 2\mathbb{Z}_2-graded-commutative algebras. This is the mathematical reflection of the Pauli exclusion principle, which says that a fermionic wave function Ψ\Psi on a phase space of a physical system with fermions has to have vanishing square. By linearity this implies that

    0 =(Ψ 1+Ψ 2) 2 =(Ψ 1) 2=0+Ψ 1Ψ 2+Ψ 2Ψ 1=0+(Ψ 2) 2=0 \begin{aligned} 0 & = (\Psi_1 + \Psi_2)^2 \\ & = \underset{ = 0}{\underbrace{ (\Psi_1)^2 }} + \underset{ = 0}{\underbrace{ \Psi_1 \Psi_2 + \Psi_2 \Psi_1 }} + \underset{ = 0}{\underbrace{ (\Psi_2)^2 }} \end{aligned}

    and hence that fermionic wave functions anti-commute, and hence are the odd-graded elements in a commutative superalgebra (a slightly noncommutative algebra!)

    Ever since the existence of fermionic particles was experimentally established, around the time of the Stern-Gerlach experiment in the 1920s, it is thus an experimental fact that fundamental physics is described by supergeometry. (This is not necessarily super-symmetric, though of course there is a close relation.)

  2. Higher structures is short for higher homotopy theoretic structures and reflects the gauge principle of physics: This says that, generally, it does not make invariant sense to ask if any two things xx, yy (e.g. field histories) are equal, instead one must ask for a gauge transformation between them, mathematically a homotopy:

    xγy x \stackrel{\gamma}{\longrightarrow} y

    This principle applies also to gauge transformations themselves, and thus leads to gauge-of-gauge transformations

    and so on to ever higher gauge transformations:

    mathematically reflected by higher homotopies in higher homotopy types.

    Ever since the existence of gauge fields was understood in the 1920s, it is thus an experimental fact that fundamental physics is described by higher geometry.

A striking consequence is that, both in higher geometry as well as in supergeometry and therefore in the geometry of fundamental physics, spaces generally are not sets of points, as in the traditional definition of topological spaces or differentiable manifolds.

What, then, is the geometry of fundamental physics?

The right framework to answer questions like this has been urged by Alexander Grothendieck already in Grothendieck 73 (see Lawvere 03) and has been much expanded on by William Lawvere (e.g. Lawvere 97, Lawvere 91) and has an evident lift to higher geometry (Lurie 09, S. 13), but has remained somewhat of a “public secret”:

The answer is known, alternatively, as (higher) functorial geometry (Grothendieck) or synthetic differential geometry in gros toposes (Lawvere), or variants thereof.

In this lecture series we try to give a quick but self-contained introduction to higher differential supergeometry this way, following Schreiber 13.

We begin at the beginning, with introducing relevant basics of category theory, topos theory and homotopy theory. As a running example, we incrementally construct the higher topos of higher supergeometry. Along the way we provide pointers to applications in perturbative quantum field theory and the theory of fundamental super p-branes.

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In view of M-theory we close by briefly highlighting the following phenomena, which are being expanded on in other talks at the meeting.

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See also the exposition at:

\; Super p-Brane Theory emerging from Super Homotopy Theory

\; talk at String Math 2017, Hamburg (slides, expo, video)

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1) category theory is really the theory of duality, formalized in the guise of adjunctions, hence adjoint pairs, and more generally adjoint triples for “dualities of dualities” (Lawvere 69, Lambek 82).

Moreover, this does capture duality in string theory.

Notably double dimensional reduction of fundamental super p-branes is given by the right base change adjunction along the point inclusion *BT\ast \to \mathbf{B}T into the moduli stack of torus-principal bundles. This fact gives rise to:

2) supergeometry is stratified by a system of adjoint modalities ("super-differential cohesion") that progresses from the initial object \emptyset to homotopy localization at the superpoint 0|1\mathbb{R}^{0 \vert 1}.

Such an abstract characterization of higher supergeometry provides a powerful means to reason about complicated phenomena such as torsion-free G-structures on supergeometric orbifolds by means of modal homotopy type theory (Wellen 18).

This is necessary for studying global properties of black M-branes at ADE-singularities (Huerta-Sati-S. 18, see also the talk by H. Sati (here) at the meeting).

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3) out of the superpoint emerges spacetime and the brane bouquet, by regarding the superpoint as a super L-∞ algebra and then applying the microscope of homotopy theory – the Postnikov-Whitehead tower (Fiorenza-Sati-S. 13, Huerta-S. 17, see the talks by J. Huerta (here) and H. Sati (here) at the meeting):

(graphics taken from Huerta-Sati-S. 18)

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Lecture notes

Details behind the topics of the talk are in these two chapters from the lecture notes geometry of physics:

  1. categories and toposes

  2. supergeometry

For further lecture notes on applications see these two chapters:

  1. perturbative quantum field theory

  2. fundamental super p-branes.

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Slides

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Abstract. Due to the existence of a) gauge fields and b) fermion fields, the geometry of physics is higher supergeometry, i.e. super-geometric homotopy theory. This is made precise via Grothendieck‘s functorial geometry implemented in higher topos theory. We give an introduction to the higher topos of higher superspaces and how it accomodates higher Lie theory of super L-∞ algebras. We close by indicating how geometric homotopy theory reveals that the superpoint emerges “from nothing”, and that core structure of M-theory emerges out of the superpoint, as will be discussed in more detail in other talks in the meeting.

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1) Duality theory

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We are concerned with duality.

AAA\phantom{AAA}Informally:

AAA\phantom{AAA}Two complementary perspectives whose unity reveals deeper reality.

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For instance:

  1. in the theory of everything of 19th century philosophy:

    duality of opposites in the Science of Logic;

  2. in the theory of everything of 20th and 21st century physics:

    duality in string theory.

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Public secret:

AAA\phantom{AAA}Mathematical formalization of duality is

AAA\phantom{AAA} adjunction, adjoint equivalence, adjoint modality

AAA\phantom{AAA}The theory of duality/adjunction is category theory.

AAA\phantom{AAA}(Lambek 82, Lawvere 69).

AAA\phantom{AAA}

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A category of objects is

AAA\phantom{AAA}a set (or class) of elements,

AAA\phantom{AAA}but remembering how these may be mapped to each other,

AAA\phantom{AAA}the (homo-)morphisms between them

AAA\phantom{AAA}

structuralism:

AAA\phantom{AAA}nature of objects in a category

AAA\phantom{AAA}is reflected in their sets of morphisms to all other objects.

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Examples of concrete categories

AAA\phantom{AAA}

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A functor between two categories

AAA\phantom{AAA}is a function between their objects and morphisms

AAA\phantom{AAA}which preserves the composition of morphisms.

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AAA\phantom{AAA}𝒞 F 𝒟 \array{ \mathcal{C} \\ {}^{\mathllap{F}}\Big\downarrow \\ \mathcal{D} }

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Example. Adding a compact dimension is

AAA\phantom{AAA}an endofunctor on the category of topological spaces

AAA\phantom{AAA}TopologicalSpaces AAS 1×()AA TopologicalSpaces X X×S 1 f f×(id S 1) Y Y×S 1 \array{ TopologicalSpaces & \overset{ \phantom{AA} S^1 \times (-) \phantom{AA} }{\longrightarrow} & TopologicalSpaces \\ \\ X &\mapsto& X \times S^1 \\ {}^{\mathllap{ f }}\Big\downarrow && \Big\downarrow{}^{ \mathrlap{ f \times \left( id_{S^1}\right) } } \\ Y &\mapsto& Y \times S^1 }

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AAA\phantom{AAA}This means that if

AAAAA\phantom{AAAAA}Σ pAAΦAAX\Sigma_{p} \overset{\phantom{AA}\Phi\phantom{AA}}{\longrightarrow} X

AAA\phantom{AAA}is a p-brane worldvolume in XX, then

AAAAA\phantom{AAAAA}Σ p×S 1AAΦ×(id S 1)AAX×S 1AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA \Sigma_{p} \times S^1 \overset{ \phantom{AA} \Phi \times \left(id_{S^1}\right) \phantom{AA} }{\longrightarrow} X \times S^1 \phantom{AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA}

AAA\phantom{AAA}is the corresponding (p+1)-brane worldvolume in S 1×XS^1 \times X

AAA\phantom{AAA}which wraps the compact fiber space.

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ExampleForming free loop spaces is

AAA\phantom{AAA}an endofunctor on the category of topological spaces

AAA\phantom{AAA}TopologicalSpaces AMaps(S 1,)A TopologicalSpaces X Maps(S 1,X) f ()f Y Maps(S 1,Y) \array{ TopologicalSpaces & \overset{ \phantom{A} Maps(S^1,-) \phantom{A} }{\longrightarrow} & TopologicalSpaces \\ \\ X &\mapsto& Maps(S^1,X) \\ {}^{\mathllap{ f }}\Big\downarrow && \Big\downarrow{}^{ \mathrlap{ (-) \circ f } } \\ Y &\mapsto& Maps(S^1, Y) }

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AAA\phantom{AAA}This means that if

AAAAA\phantom{AAAAA}XAAfAAYX \overset{\phantom{AA} f \phantom{AA}}{\longrightarrow} Y

AAA\phantom{AAA}is a map of target spaces

AAA\phantom{AAA}for instance a KK-compactification-projection, then

AAAAA\phantom{AAAAA}Maps(S 1,X)AAMaps(S 1,f)AAMaps(S 1,Y)Maps(S^1,X) \overset{\phantom{AA} Maps(S^1,f) \phantom{AA}}{\longrightarrow} Maps(S^1,Y)

AAA\phantom{AAA}is the corresponding map of closed string configuration spaces

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A natural transformation

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AAAAA\phantom{AAAAA} 𝒞AAAηηAAAAAAAGAAAAAAAAFAAAA𝒟 \mathcal{C} \underoverset {\underset{\phantom{AAAA}G\phantom{AAAA}}{\longrightarrow}} {\overset{\phantom{AAAA}F\phantom{AAAA}}{\longrightarrow}} {\phantom{AAA}\phantom{\eta}\Downarrow \eta\phantom{AAA}} \mathcal{D}

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AAA\phantom{AAA}is an indexed set of morphisms between images of functors

AAAAA\phantom{AAAAA}X F(X)AAη XAAG(X) \phantom{\array{X && }} F(X) \overset{\phantom{AA}\eta_X\phantom{AA}}{\longrightarrow} G(X)

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AAA\phantom{AAA}that respects composition on both sides.

AAAAA\phantom{AAAAA}X F(X) AAη XAA G(X) f AAA F(f) G(f) Y F(Y) AAη YAA G(Y) \array{ X && F(X) &\overset{ \phantom{AA}\eta_X \phantom{AA} }{\longrightarrow}& G(X) \\ {}^{\mathllap{f}}\Big\downarrow &\phantom{AAA}& {}^{\mathllap{F(f)}}\Big\downarrow && \Big\downarrow{}^{\mathrlap{ G(f) }} \\ Y && F(Y) &\underset{ \phantom{AA}\eta_Y\phantom{AA} }{\longrightarrow}& G(Y) }

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Example.Brane wrapping circle fiber

AAA\phantom{AAA}Every topological space maps into

AAA\phantom{AAA}the free loop space of its Cartesian product with the circle

AAAAA\phantom{AAAAA}X AAη XAA Maps(S 1,S 1×X) x (s(s,x)) \array{ X &\overset{ \phantom{AA}\eta_X\phantom{AA} }{\longrightarrow}& Maps\big( S^1, \; S^1 \times X \big) \\ x &\mapsto& (s \mapsto (s,x)) }

AAA\phantom{AAA}by sending each point to the loop which wraps the compact fiber space S 1S^1 above it.

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AAA\phantom{AAA}Conversely, there is the evaluation map

AAAAA\phantom{AAAAA}S 1×Maps(S 1,X) ϵ X X (s,γ) γ(s) \array{ S^1 \times Maps(S^1, X) &\overset{ \epsilon_X }{\longrightarrow}& X \\ (s,\gamma) &\mapsto& \gamma(s) }

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AAA\phantom{AAA}These are natural transformations

AAAAA\phantom{AAAAA}TopologicalSpacesAAAAηAAAMaps(S 1,S 1×())AAAidAAATopologicalSpacesAAAAϵAAAAAAidAAAS 1×Maps(S 1,)TopologicalSpaces TopologicalSpaces \underoverset {\underset{Maps(S^1,S^1 \times (-))}{\longrightarrow}} {\overset{\phantom{AAA}id\phantom{AAA}}{\longrightarrow}} {\phantom{AAAA}\Downarrow \eta\phantom{AAA}} TopologicalSpaces \underoverset {\underset{\phantom{AAA}id\phantom{AAA}}{\longrightarrow}} {\overset{ S^1 \times Maps(S^1, -) }{\longrightarrow}} {\phantom{AAAA}\Downarrow \epsilon\phantom{AAA}} TopologicalSpaces

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An adjunction of categories

AAA\phantom{AAA}is a pair of functors back and forth

AAAA𝒟AAAARL𝒞 \phantom{AAAA} \mathcal{D} \underoverset {\underset{R}{\longrightarrow}} {\overset{L}{\longleftarrow}} {\phantom{AA}\bot\phantom{AA}} \mathcal{C}

AAA\phantom{AAA}equipped with
AAA\phantom{AAA}natural transformations

  • cAAη cAARL(c)c \overset{\phantom{AA}\eta_c\phantom{AA}}{\longrightarrow} R L(c) (“unit”)

  • LR(d)AAϵ dAAdL R(d) \overset{ \phantom{AA}\epsilon_d\phantom{AA} }{\longrightarrow} d (“counit”)

AAA\phantom{AAA}satisfying “zig-zag identity
AAA\phantom{AAA}as shown on the right.

AAA\phantom{AAA}This is equivalent to

AAA\phantom{AAA}hom-set natural bijection:

AAAAA\phantom{AAAAA} Hom 𝒟(L(c),d) "forming adjuncts"AA()˜AA Hom 𝒞(c,R(d)) (L(c)fd) (cη cRL(c)R(f)R(d)) \array{ Hom_{\mathcal{D}}\big(L(c), d\big) &\;\underoverset{\text{<a href="https://ncatlab.org/nlab/show/adjunct">"forming adjuncts"</a>}}{\phantom{AA}\widetilde{(-)}\phantom{AA}}{\leftrightarrow}\;& Hom_{\mathcal{C}}\big(c, R(d)\big) \\ \\ \big(L(c)\overset{f}{\to}d\big) &\mapsto& \big( c \overset{\eta_c}{\to} R L(c) \overset{R(f)}{\to}R(d) \big) }

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Example - Cartesian product/mapping space adjunction

AAAAA\phantom{AAAAA}TopologicalSpaces cgAAAAAAMaps(S 1,)S 1×()TopologicalSpaces cg TopologicalSpaces_{cg} \;\; \underoverset {\underset{Maps(S^1,-)}{\longrightarrow}} {\overset{S^1 \times (-)}{\longleftarrow}} {\phantom{AAA}\bot\phantom{AAA}} \;\; TopologicalSpaces_{cg}

AAA\phantom{AAA}The adjunct of a wrapped brane worldvolume

AAAAA\phantom{AAAAA}{Σ×S 1AAΦAAY} ()˜ {Ση ΣMaps(S 1,Σ×S 1)Maps(S 1,Φ)Maps(S 1,Y)} \array{ \Big\{ \Sigma \times S^1 \overset{ \phantom{AA} \Phi \phantom{AA} }{\longrightarrow} Y \Big\} \\ \\ \Big\updownarrow{}^{ \mathrlap{ \widetilde{ (-) } } } \\ \\ \Big\{ \Sigma \overset{\eta_{\Sigma}}{\longrightarrow} Maps(S^1 , \Sigma \times S^1 ) \overset{ Maps(S^1, \Phi) }{\longrightarrow} Maps(S^1, Y) \Big\} }

AAA\phantom{AAA}is the double dimensional reduction

From this follows (more here )

  1. duality between M-theory and type IIA string theory

  2. T-duality between type IIA and IIB string theory

on brane charges in rational super homotopy theory (Fiorenza-Sati-S. 16, BraunackMayer-Sati-S. 18).


see the talks by D. Fiorenza (here) and by V. Braunack-Mayer (here) at the meeting.

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ExampleFunctors assigning algebras of functions

Given any kind of space and a coefficient space

there is a functor which

  1. sends spaces to their algebras of functions;

  2. sends maps between spaces to the precomposition with these maps:

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AAAAA\phantom{AAAAA}Spaces algebras of functions Algebras op X Fcts(X) f ()f Y Fcts(Y) \array{ Spaces &\overset{\text{algebras of functions}}{\longrightarrow}& Algebras^{op} \\ \\ X &\mapsto& Fcts(X) \\ {}^{\mathllap{f}}\Big\downarrow && \Big\uparrow{}^{ \mathrlap{ (-)\circ f } } \\ Y &\mapsto& Fcts(Y) }

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on the right the opposite category of algebras:

AAA\phantom{AAA}direction of morphisms reversed.

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In good cases this is a fully faithful functor

AAAAA\phantom{AAAAA}Spacesalgebras of functionsAlgebras op Spaces \overset{\text{algebras of functions}}{\hookrightarrow} Algebras^{op}

AAA\phantom{AAA}meaning that there is a natural bijection

AAA\phantom{AAA}from sets of morphisms on the left to those on the right:

AAAAA\phantom{AAAAA}Hom Spaces(X,Y)Hom Algebras(Fcts(Y),Fcts(X)) Hom_{Spaces}\big( X,\; Y\big) \;\simeq\; Hom_{Algebras}\Big( Fcts(Y), \; Fcts(X) \Big)

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By the “structuralism” of category theory,

AAA\phantom{AAA}this means that such spaces

AAA\phantom{AAA}may just as well be regarded in the dual algebraic picture.

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A fully faithful functor becomes an equivalence of categories

AAA\phantom{AAA}after corestriction to its essential image

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duality between geometry and algebra:

AAAAA\phantom{AAAAA}Spacesalgebras of functionsGoodAlgebras opAAAAlgebras op Spaces \underoverset{\simeq}{\text{algebras of functions}}{\longrightarrow} GoodAlgebras^{op} \overset{\phantom{AAA}}{\hookrightarrow} Algebras^{op}

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archetypical example: Gelfand duality:

AAAAA\phantom{AAAAA}TopologicalSpaces Hausd.,compact Hom Top(,) C *Algebras comm op TopologicalAlgebras op \array{ TopologicalSpaces_{Hausd., compact} &\underoverset{\simeq}{Hom_{Top}(-,\mathbb{C})}{\longrightarrow}& C^\ast Algebras_{comm}^{op} &\hookrightarrow& TopologicalAlgebras^{op} }

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More Examples of Isbell duality between geometry and algebra

A\phantom{A}geometryA\phantom{A}A\phantom{A}categoryA\phantom{A}A\phantom{A}dual categoryA\phantom{A}A\phantom{A}algebraA\phantom{A}
A\phantom{A}topologyA\phantom{A}A\phantom{A}NCTopSpaces H,cpt\phantom{NC}TopSpaces_{H,cpt}A\phantom{A}A\phantom{A}Gelfand-KolmogorovAlg comm op\overset{\text{<a href="https://ncatlab.org/nlab/show/Gelfand-Kolmogorov+theorem">Gelfand-Kolmogorov</a>}}{\hookrightarrow} Alg^{op}_{comm}A\phantom{A}A\phantom{A}commutative algebraA\phantom{A}
A\phantom{A}topologyA\phantom{A}A\phantom{A}NCTopSpaces H,cpt\phantom{NC}TopSpaces_{H,cpt}A\phantom{A}A\phantom{A}Gelfand dualityTopAlg C *,comm op\overset{\text{<a href="https://ncatlab.org/nlab/show/Gelfand+duality">Gelfand duality</a>}}{\simeq} TopAlg^{op}_{C^\ast, comm}A\phantom{A}A\phantom{A}comm. C*-algebraA\phantom{A}
A\phantom{A}noncomm. topologyA\phantom{A}A\phantom{A}NCTopSpaces H,cptNCTopSpaces_{H,cpt}A\phantom{A}A\phantom{A}Gelfand dualityTopAlg C * op\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} TopAlg^{op}_{C^\ast}A\phantom{A}A\phantom{A}general C*-algebraA\phantom{A}
A\phantom{A}algebraic geometryA\phantom{A}A\phantom{A}NCSchemes Aff\phantom{NC}Schemes_{Aff}A\phantom{A}A\phantom{A}almost by def.TopAlg fin op\overset{\text{<a href="https://ncatlab.org/nlab/show/affine+scheme#AffineSchemesFullSubcategoryOfOppositeOfRings">almost by def.</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{fin} A\phantom{A}A\phantom{A}fin. gen.A\phantom{A}
A\phantom{A}commutative algebraA\phantom{A}
A\phantom{A}noncomm. algebraicA\phantom{A}
A\phantom{A}geometryA\phantom{A}
A\phantom{A}NCSchemes AffNCSchemes_{Aff}A\phantom{A}A\phantom{A}Gelfand dualityTopAlg fin,red op\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} \phantom{Top}Alg^{op}_{fin, red}A\phantom{A}A\phantom{A}fin. gen.
A\phantom{A}associative algebraA\phantom{A}A\phantom{A}
A\phantom{A}differential geometryA\phantom{A}A\phantom{A}SmoothManifoldsSmoothManifoldsA\phantom{A}A\phantom{A}Milnor's exerciseTopAlg comm op\overset{\text{<a href="https://ncatlab.org/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">Milnor's exercise</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{comm}A\phantom{A}A\phantom{A}commutative algebraA\phantom{A}
A\phantom{A}supergeometryA\phantom{A}A\phantom{A}SuperSpaces Cart n|q\array{SuperSpaces_{Cart} \\ \\ \mathbb{R}^{n\vert q}}A\phantom{A}A\phantom{A}Milnor's exercise Alg 2AAAA op C ( n) q\array{ \overset{\phantom{\text{Milnor's exercise}}}{\hookrightarrow} & Alg^{op}_{\mathbb{Z}_2 \phantom{AAAA}} \\ \mapsto & C^\infty(\mathbb{R}^n) \otimes \wedge^\bullet \mathbb{R}^q }A\phantom{A}A\phantom{A}supercommutativeA\phantom{A}
A\phantom{A}superalgebraA\phantom{A}
A\phantom{A}formal higherA\phantom{A}
A\phantom{A}supergeometryA\phantom{A}
A\phantom{A}(super Lie theory)A\phantom{A}
ASuperL Alg fin 𝔤A\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}AALada-MarklA sdgcAlg op CE(𝔤)A\phantom{A}\array{ \overset{ \phantom{A}\text{<a href="https://ncatlab.org/nlab/show/L-infinity-algebra#ReformulationInTermsOfSemifreeDGAlgebra">Lada-Markl</a>}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}A\phantom{A}differential graded-commutativeA\phantom{A}
A\phantom{A}superalgebra
A\phantom{A} (“FDAs”)

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2) Superalgebra

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A differential graded-commutative superalgebra

AAA\phantom{AAA}is a ×(/2)\mathbb{Z} \times (\mathbb{Z}/2)-graded algebra

AAAAA\phantom{AAAAA}(AnAcohomologicaldegree,AσAsuperdegree)×(/2) \left( \underset{ cohomological \atop degree }\underbrace{\phantom{A}n\phantom{A}} , \underset{ super \atop degree }{\underbrace{\phantom{A}\sigma\phantom{A}}} \right) \;\in\; \mathbb{Z} \times (\mathbb{Z}/2)

AAA\phantom{AAA}equipped with a derivation differential dd of degree (1,even)(1,even).

AAA\phantom{AAA}with either of these sign rules:

AAA\phantom{AAA}

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Special cases of dgc-superalgebras

A\phantom{A}bi-degreeA\phantom{A}
A\phantom{A}(n,σ)×𝔽 2(n,\sigma) \in \mathbb{Z} \times \mathbb{F}_2A\phantom{A}
A\phantom{A}n=0n = 0A\phantom{A}A\phantom{A}nn\; arbitraryA\phantom{A}
A\phantom{A}σ=even\sigma = evenA\phantom{A}A\phantom{A}commutative algebraA\phantom{A}A\phantom{A}differential graded-commutative algebraA\phantom{A}
A\phantom{A}σ\sigma\; arbitraryA\phantom{A}A\phantom{A}e.g. Grassmann algebraA\phantom{A}A\phantom{A}differential graded-commutative superalgebraA\phantom{A}

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Example – The Grassmann algebra (Grassmann 1844)

AAA\phantom{AAA}[θ α α=1 qdeg=(0,odd)] \mathbb{R}\left[ \underset{deg = (0,odd)}{\underbrace{\langle \theta^\alpha \rangle_{\alpha = 1}^q}} \right]

AAA\phantom{AAA}is the free supercommutative superalgebra over \mathbb{R}

AAA\phantom{AAA}on qq odd-graded generators: θ αθ β=θ βθ α\theta^\alpha \theta^\beta = - \theta^\beta \theta^\alpha

Example – A super Cartesian space n|q\mathbb{R}^{n\vert q}

AAA\phantom{AAA}is the formal dual space of the tensor product of algebras

AAA\phantom{AAA}C ( n|q)C ( n)-algebra ofsmooth functions qGrassmann algebra C^\infty\left( \mathbb{R}^{n\vert q}\right) \;\coloneqq\; \underset{ {\mathbb{R}\text{-algebra of}} \atop \text{smooth functions} }{ \underbrace{ C^\infty(\mathbb{R}^n) }} \otimes_{\mathbb{R}} \underset{ \text{Grassmann algebra} }{ \underbrace{ \wedge^\bullet_{\mathbb{R}} \mathbb{R}^q }}

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differential forms on super Cartesian space n|q\mathbb{R}^{n\vert q}

AAA\phantom{AAA}is the differential graded-commutative superalgebra

AAA\phantom{AAA}free over C ( n|q) C^\infty\left(\mathbb{R}^{n\vert q}\right)

AAA\phantom{AAA}on

  1. nn generators dx a\mathbf{d} x^a in bi-degree (1,even)(1,even)

  2. qq generators dθ α\mathbf{d} \theta^\alpha in bi-degree (1,odd)(1,odd)

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hence

AAAAA\phantom{AAAAA}Ω ( n|q)C ( n|q)[dx a a=1 ndeg=(1,even),dθ α α=1 qdeg=(1,odd)] \Omega^\bullet( \mathbb{R}^{n\vert q} ) \;\coloneqq\; C^\infty(\mathbb{R}^{n\vert q}) \left[ \underset{deg = (1,even)}{\underbrace{ \left\langle \mathbf{d} x^a \right\rangle_{a = 1}^{n} }},\;\;\;\;\; \underset{ deg = (1,odd) }{ \underbrace{ \left\langle \mathbf{d} \theta^\alpha \right\rangle_{\alpha = 1}^q }} \right]

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Bi-Degrees and signs for super differential forms

A\phantom{A}generatorA\phantom{A}A\phantom{A}bi-degreeA\phantom{A}
A\phantom{A}x ax^aA\phantom{A}A\phantom{A}(0,even)A\phantom{A}
A\phantom{A}θ α\theta^\alphaA\phantom{A}A\phantom{A}(0,odd)A\phantom{A}
A\phantom{A}dx a\mathbf{d}x^aA\phantom{A}A\phantom{A}(1,even)A\phantom{A}
A\phantom{A}dθ α\mathbf{d}\theta^\alphaA\phantom{A}A\phantom{A}(1,odd)A\phantom{A}

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A\phantom{A}sign ruleA\phantom{A}A\phantom{A}Deligne’sA\phantom{A}Bernstein’sA\phantom{A}
A\phantom{A}x ax b=x^{a} \; x^{b} =A\phantom{A}+x bx a+ x^{b} \; x^{a}A\phantom{A}A\phantom{A}+x bx a+ x^{b} \; x^{a}A\phantom{A}
A\phantom{A}x aθ α=x^a \;\theta^\alpha =A\phantom{A}A\phantom{A}+θ αx a+ \theta^\alpha \; x^aA\phantom{A}A\phantom{A}+θ αx a+ \theta^\alpha \; x^aA\phantom{A}
A\phantom{A}θ αθ β=\theta^{\alpha} \; \theta^{\beta} =A\phantom{A}A\phantom{A}θ βθ α- \theta^{\beta} \; \theta^{\alpha}A\phantom{A}A\phantom{A}θ βθ α - \theta^{\beta} \; \theta^{\alpha}A\phantom{A}
A\phantom{A}x a(dx a)=x^{a} (\mathbf{d}x^{a}) =A\phantom{A}A\phantom{A}+(dx b)x a+ (\mathbf{d}x^{b}) x^{a}A\phantom{A}A\phantom{A}+(dx b)x a+ (\mathbf{d}x^{b}) x^{a}A\phantom{A}
A\phantom{A}θ α(dx a)=\theta^\alpha (\mathbf{d}x^a) =A\phantom{A}A\phantom{A}+(dx a)θ α+ (\mathbf{d}x^a) \theta^\alphaA\phantom{A}A\phantom{A}(dx a)θ α{\color{blue}{-}} (\mathbf{d}x^a) \theta^\alphaA\phantom{A}
A\phantom{A}θ α(dθ β)=\theta^{\alpha} (\mathbf{d}\theta^{\beta}) = A\phantom{A}A\phantom{A}(dθ β)θ α- (\mathbf{d}\theta^{\beta}) \theta^{\alpha}A\phantom{A}A\phantom{A}+(dθ β)θ α{\color{blue}{+}} (\mathbf{d}\theta^{\beta}) \theta^{\alpha}A\phantom{A}
A\phantom{A}(dx a)(dx b)= (\mathbf{d}x^{a}) (\mathbf{d} x^{b}) =A\phantom{A}A\phantom{A}(dx b)(dx a)- (\mathbf{d} x^{b}) (\mathbf{d} x^{a})A\phantom{A}A\phantom{A}(dx b)(dx a) - (\mathbf{d} x^{b}) (\mathbf{d} x^{a})A\phantom{A}
A\phantom{A}(dx a)(dθ α)= (\mathbf{d}x^a) (\mathbf{d} \theta^{\alpha}) =A\phantom{A}A\phantom{A}(dθ α)(dx a) - (\mathbf{d}\theta^{\alpha}) (\mathbf{d} x^a) A\phantom{A}A\phantom{A}+(dθ α)(dx a) {\color{blue}{+}} (\mathbf{d}\theta^{\alpha}) (\mathbf{d} x^a) A\phantom{A}
A\phantom{A}(dθ α)(dθ β)=(\mathbf{d}\theta^{\alpha}) (\mathbf{d} \theta^{\beta}) =A\phantom{A}+(dθ β)(dθ α) + (\mathbf{d}\theta^{\beta}) (\mathbf{d} \theta^{\alpha})A\phantom{A}A\phantom{A}+(dθ β)(dθ α) + (\mathbf{d}\theta^{\beta}) (\mathbf{d} \theta^{\alpha})A\phantom{A}

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pullback of super differential forms

AAA\phantom{AAA}Each morphism

AAA\phantom{AAA} n 1|q 1 AfA n 2|q 2 C ( n 1|q 1) Af *A C ( n 2|q 2) \array{ \mathbb{R}^{n_1 \vert q_1} &\overset{\phantom{A}f\phantom{A}}{\longrightarrow}& \mathbb{R}^{n_2 \vert q_2} \\ C^\infty(\mathbb{R}^{n_1\vert q_1}) &\overset{\phantom{A}f^\ast\phantom{A}}{\longleftarrow}& C^\infty(\mathbb{R}^{n_2\vert q_2}) }

AAA\phantom{AAA}has unique dg-extension

AAAAA\phantom{AAAAA}f *d 2=d 1f *AAAΩ ( n 1|q 1) AAf *AA Ω ( n 2|q 2) d 1 d 2 Ω ( n 1|q 1) AAf *AA Ω ( n 2|q 2) f^\ast \circ \mathbf{d}_2 = \mathbf{d}_1 \circ f^\ast \phantom{AAA} \array{ \Omega^\bullet(\mathbb{R}^{n_1\vert q_1}) &\overset{\phantom{AA}f^\ast\phantom{AA}}{\longleftarrow}& \Omega^\bullet(\mathbb{R}^{n_2\vert q_2}) \\ {}^{\mathbf{d}_1}\Big\downarrow && \Big\downarrow{}^{\mathbf{d}_2} \\ \Omega^\bullet(\mathbb{R}^{n_1\vert q_1}) &\overset{\phantom{AA}f^\ast\phantom{AA}}{\longleftarrow}& \Omega^\bullet(\mathbb{R}^{n_2\vert q_2}) }

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AAA\phantom{AAA}This defines another fully faithful functor

AAAAA\phantom{AAAAA}AAA\phantom{AAA}Ω :SuperCartSpAAAAdgcSuperAlg op \mathbf{\Omega}^\bullet \;\colon\; SuperCartSp \overset{\phantom{AAAA}}{\hookrightarrow} dgcSuperAlg^{op}

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For 𝔤\mathfrak{g} a super Lie algebra of finite dimension,

its Chevalley-Eilenberg algebra

AAA\phantom{AAA}is the differential graded-commutative superalgebra

AAAAA\phantom{AAAAA}CE(𝔤)[𝔤 *(1,)] CE(\mathfrak{g}) \;\coloneqq\; \mathbb{R} \left[ \underset{ (1,\bullet) }{\underbrace{\mathfrak{g}^\ast}} \right]

AAA\phantom{AAA}equipped with the differential d 𝔤d_{\mathfrak{g}}

AAA\phantom{AAA}which is the linear dual of the super Lie bracket

AAA\phantom{AAA}d 𝔤[,] *:𝔤 *𝔤 *𝔤 * d_{\mathfrak{g}} \coloneqq [-,-]^\ast \;\colon\; \mathfrak{g}^\ast \to \mathfrak{g}^\ast \wedge \mathfrak{g}^\ast

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Equivalently:

AAAAA\phantom{AAAAA}AA\phantom{AA}CE(𝔤)Ω li (G)CE(\mathfrak{g}) \;\simeq\; \Omega^\bullet_{li}( G )

AAA\phantom{AAA}is the dgc-superalgebra of left-invariant super differential forms

AAA\phantom{AAA}on the corresponding simply connected super Lie group.

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key Example – super Minkowski spacetime

AAA\phantom{AAA}For dd \in \mathbb{N}

AAA\phantom{AAA}and N\mathbf{N} a real spin representation of Spin(d1,1)Spin(d-1,1)

AAA\phantom{AAA}the super-translation supersymmetry super Lie algebra

AAAAA\phantom{AAAAA} d1,1|N\mathbb{R}^{d-1,1\vert \mathbf{N}}

AAA\phantom{AAA}has Chevalley-Eilenberg algebra generated by

AAAAA\phantom{AAAAA}{Ae aA(1,even)} a=0 d1AAA{Aψ αA(1,odd)} α=1 N \left\{\; \underset{(1,even)}{\underbrace{ \phantom{A}e^a\phantom{A} }}\; \right\}_{a = 0}^{d-1} \phantom{AAA} \left\{\; \underset{(1,odd)}{\underbrace{ \phantom{A}\psi^\alpha \phantom{A} }} \; \right\}_{\alpha = 1}^N

AAA\phantom{AAA}with differential

AAAAA\phantom{AAAAA}dψ α =0 de a =ψ¯Γ aψspinor-to-vectorpairing \begin{aligned} d \, \psi_\alpha &\;=\; 0 \\ d\, e^a & \;=\; \underset{ \text{spinor-to-vector} \atop \text{pairing} }{ \underbrace{ \overline{\psi} \wedge \Gamma^a \psi }} \end{aligned}

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If we think of super Minkowski spacetime d1,1|N\mathbb{R}^{d-1,1\vert \mathbf{N}}

as the super Cartesian space with coordinates

AAAAA\phantom{AAAAA}{Ax aA(0,even)} a=0 d1AAA{Aθ αA(0,odd)} α=1 N \left\{\; \underset{(0,even)}{\underbrace{ \phantom{A}x^a\phantom{A} }}\; \right\}_{a = 0}^{d-1} \phantom{AAA} \left\{\; \underset{(0,odd)}{\underbrace{ \phantom{A}\theta^\alpha \phantom{A} }} \; \right\}_{\alpha = 1}^N

then these generators correspond to

the super-left invariant

super vielbein

AAAAA\phantom{AAAAA}ψ α =dθ α e a =dx aordinaryMinkowski vielbein+θ¯Γ adθcorrection termfor left super-invariance \begin{aligned} \psi^\alpha & \;=\; d \theta^\alpha \\ e^a & \;=\; \underset{ \text{ordinary} \atop \text{Minkowski vielbein} }{ \underbrace{ \;d x^a\; }} + \underset{\text{correction term} \atop \text{for left super-invariance}}{\underbrace{\;\overline{\theta} \Gamma^a d \theta\;}} \end{aligned}

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Homomorphisms of super Lie algebras

are in natural bijection with

the dual homomorphisms of dgc-superalgebras

between their Chevalley-Eilenberg algebra,

AAAAA\phantom{AAAAA}CE(𝔤 1)AAfAACE(𝔤 2) CE(𝔤 1)AAf *AACE(𝔤 2) \array{ \phantom{CE(}\mathfrak{g}_1\phantom{)} \overset{\phantom{AA}f\phantom{AA}}{\longrightarrow} \phantom{CE(}\mathfrak{g}_2\phantom{)} \\ CE(\mathfrak{g}_1) \underset{\phantom{AA}f^\ast \phantom{AA}}{\longleftarrow} CE(\mathfrak{g}_2) }

hence we have another fully faithful functor

AAAAA\phantom{AAAAA}CE:SuperLieAlgAAAAdgcSuperAlg op CE \;\colon\; SuperLieAlg \overset{\phantom{AAAA}}{\hookrightarrow} dgcSuperAlg^{op}

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Hence: A super L-∞ algebra of finite type is

  1. a \mathbb{Z}-graded super vector space 𝔤\mathfrak{g}

    degreewise of finite dimension

  2. for all n1n \geq 1 a multilinear map

    AA\phantom{AA}[,,]: n𝔤 * 1𝔤 * [-,\cdots, -] \;\colon\; \wedge^n \mathfrak{g}^\ast \longrightarrow \wedge^1 \mathfrak{g}^\ast

    of degree (1,even)(-1,even)

AAA\phantom{AAA}such that the graded derivation

AAAAA\phantom{AAAAA}d 𝔤[] *+[,] *+[,,] *+: 1𝔤 * 𝔤 * d_{\mathfrak{g}} \;\coloneqq\; [-]^\ast + [-,-]^\ast + [-,-,-]^\ast + \cdots \;\;\colon\;\; \wedge^1 \mathfrak{g}^\ast \longrightarrow \wedge^\bullet \mathfrak{g}^\ast

AAA\phantom{AAA}squares to zero: d 𝔤d 𝔤=0d_{\mathfrak{g}} d_{\mathfrak{g}} = 0

This defines a dgc-superalgebra

AAAAA\phantom{AAAAA}CE(𝔤)( 𝔤 *,d 𝔤). CE(\mathfrak{g}) \coloneqq ( \wedge^\bullet \mathfrak{g}^\ast, d_{\mathfrak{g}} ) \,.

AAA\phantom{AAA} super L-∞ algebras form the full subcategory

AAAAA\phantom{AAAAA}SuperL Alg finAAACEAAAdgcSuperAlg op Super L_\infty Alg^{fin} \overset{ \phantom{AAA}CE\phantom{AAA} }{\hookrightarrow} dgcSuperAlg^{op}

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Side remarks.

  1. We may drop the assumption of degreewise finiteness

    by regarding 𝔤\vee^\bullet \mathfrak{g} as

    a free graded co-commutative coalgebra

    and D[]+[,]+[,,]+D \coloneqq [-] + [-,-] + [-,-,-] + \cdots as a differential

    making a differential graded coalgebra.

  2. The above definition is for non-curved L-∞ algebras

    but allows for “curved sh-maps”.

    This is important in the discussion of Super topological T-Duality.

    (see the talk by D. Fiorenza at the meeting)

  3. By allowing algebras of smooth functions in degree 0

    we obtain super-L-∞ algebroids (Sati-S.-Stasheff 12, A.1)

    formally dual to super dg-manifolds.

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convoluted History:

L L_\infty-algebras were explicitly introduced in Lada-Stasheff 92

but following their discovery by Zwiebach 89, Zwiebach 92 in closed string field theory.

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The equivalence to Chevalley-Eilenberg dg-(co-)algebras

is due to Lada-Markl 94, see Sati-Schreiber-Stasheff 08, around def. 13.

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But these dual CE-algebras of super L-∞ algebras of finite type

were introduced already in van Nieuwenhuizen 82, D’Auria-Fré 82

for higher super Cartan geometry of supergravity, there called FDAs.

A\phantom{A}higher Lie theoryA\phantom{A}A\phantom{A}supergravityA\phantom{A}
A\phantom{A} super L-∞ algebras 𝔤\mathfrak{g} A\phantom{A}A\phantom{A}FDAsCE(𝔤)CE(\mathfrak{g}) A\phantom{A}

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The string theory-community began to pay attention with Hohm-Zwiebach 17

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Higher Lie algebras turn out to be the answer to the question:

What do higher Lie algebra cocycles classify?

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Recall:

A (p+2)-cocycle on a Lie algebra (𝔤,[,])(\mathfrak{g},[-,-]) is

  1. multi-linear, skew-symmetric function μ p+2:𝔤××𝔤p+2copies\mu_{p+2} \;\colon\; \underset{p+2\;\text{copies}}{\underbrace{\mathfrak{g} \times \cdots \times \mathfrak{g}}} \longrightarrow \mathbb{R};

  2. cocycle condition: for all (p+3)(p+3)-tuples (x 0,,x p+2)(x_0, \cdots, x_{p+2}) in 𝔤\mathfrak{g}

    AA\phantom{AA}σperm.(1) sgn(σ)μ p+2([x σ 0,x σ 1],x σ 2,,x σ p+2)=0. \underset{\sigma\in perm.}{\sum} (-1)^{sgn(\sigma)} \, \mu_{p+2} \left([x_{\sigma_0},x_{\sigma_1}], x_{\sigma_2}, \cdots, x_{\sigma_{p+2}}\right) = 0 \,.

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In terms of Chevalley-Eilenberg algebras this means simply

  1. μ 2CE(𝔤)\mu_2 \in CE(\mathfrak{g})

  2. dμ 2=0d \mu_2 = 0

A\phantom{A}(this is the classical use of CE-algebras)

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A Lie algebra 2-cocycle μ 2\mu_2

corresponds to a central Lie algebra extension,

AAA\phantom{AAA}𝔤^ 𝔤, A [(x 1,c 1),(x 2,c 2)]([x 1,x 2],μ 2(x 1,x 2)) p 𝔤. \array{ \widehat \mathfrak{g} & \coloneqq \mathfrak{g} \oplus \mathbb{R}, &\phantom{A}& [\, (x_1,c_1),\,(x_2,c_2) \,] \; \coloneqq \; (\, [x_1,x_2], \, \mu_2(x_1,x_2) \,) \\ {}^{\mathllap{p}}\Big\downarrow \\ \mathfrak{g} } \,.

AAA\phantom{AAA}the new Jacobi identity is the cocycle condition on μ 2\mu_2.

In terms of dual Chevalley-Eilenberg algebras this means simply:

AAA\phantom{AAA}CE(𝔤^) =CE(𝔤)[AbA(1,even)], A db=μ 2 p * CE(𝔤) \array{ CE(\widehat{\mathfrak{g}}) & = CE(\mathfrak{g})[ \underset{(1,even)}{\underbrace{\phantom{A} b \phantom{A} }} ], &\phantom{A}& d \,b = \mu_2 \\ {}^{\mathllap{p^\ast}}\Big\uparrow \\ CE(\mathfrak{g}) }

AAA\phantom{AAA}the new nilpotency d 2=0d^2 = 0 is the cocycle condition on μ 2\mu_2

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Non-classical Fact:

A (p+2)(p+2)-cocycle μ p+2\mu_{p+2} still corresponds to

a higher central extension to an L-∞ algebra

AAA\phantom{AAA}𝔤^ =[0(p,even)00𝔤], A [x 1,,x p+2]=μ p+2(x 1,,x p+2) p 𝔤 \array{ \widehat{\mathfrak{g}} & = \left[ \cdots \overset{0}{\to} \underset{ (p,even) }{\underbrace{\mathbb{R}}} \overset{0}{\to} \cdots \overset{0}{\to} \mathfrak{g} \right], &\phantom{A}& [x_1, \cdots, x_{p+2}] = \mu_{p+2}(x_1, \cdots, x_{p+2}) \\ {}^{\mathllap{p}}\Big\downarrow \\ \mathfrak{g} }

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In terms of dual Chevalley-Eilenberg algebras this means simply

AAA\phantom{AAA}CE(𝔤^) =CE(𝔤)[AbA(p+1,even)] A db=μ p+2 p * CE(𝔤) \array{ CE(\widehat{\mathfrak{g}}) & = CE(\mathfrak{g})[ \underset{(p+1,even)}{\underbrace{ \phantom{A}b\phantom{A} }} ] &\phantom{A}& d \, b = \mu_{p+2} \\ {}^{\mathllap{p^\ast}}\Big\uparrow \\ CE(\mathfrak{g}) }

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Example – string Lie 2-algebra

AAA\phantom{AAA}Every semisimple Lie algebra 𝔤\mathfrak{g} carries a Killing form pairing

AAAAA\phantom{AAAAA}AA\phantom{AA},:𝔤×𝔤𝔤 \langle -,- \rangle \;\colon\; \mathfrak{g} \times \mathfrak{g} \longrightarrow \mathfrak{g}

AAA\phantom{AAA}and the resulting trilinear map

AAA\phantom{AAA}μ 3,[,] \mu_3 \coloneqq \langle -, [-,-] \rangle

AAA\phantom{AAA}is a 3-cocycle.

AAA\phantom{AAA}This classifies a Lie 2-algebra extension of 𝔤\mathfrak{g},

AAA\phantom{AAA}called the string Lie 2-algebra

AAA\phantom{AAA}(Baez-Crans 03, Baez-Crans-S.-Stevenson 05).

In string theory/M-theory this controls

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Generally: For 𝔤\mathfrak{g} a super L-∞ algebra

  1. an L L_\infty p+2p+2-cocycle is μ p+2CE(𝔤)\mu_{p+2} \in CE(\mathfrak{g}),

  2. the higher central extension 𝔤^\widehat \mathfrak{g} it classifies has

    CE(𝔤^)=CE(𝔤)[b(p+1,even)]/(db=μ p+2)CE(\widehat{\mathfrak{g}}) \;=\; CE(\mathfrak{g})[ \underset{(p+1,even)}\underbrace{\;b\;}] / (\; d \, b = \mu_{p+2}\;)

Hence from any super L-∞ algebra emanates a “bouquet” of consecutive higher central extension

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3) Homotopy theory

Traditionally, mathematics and physics

have been founded on set theory,

whose concept of sets is that of “bags of distinguishable points”.

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But fundamental physics is governed by the gauge principle.

This says that given any two “things”,

such as two field histories xx and yy,

it is in general wrong to ask whether they are equal or not,

instead one has to ask where there is a gauge transformation

AAA\phantom{AAA}xAAγAAy x \overset{\phantom{AA}\gamma\phantom{AA}}{\longrightarrow} y

between them.

In mathematics this is called a homotopy.

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The gauge principle applies to gauge transformations/homotopies themselves,

and thus leads to gauge-of-gauge transformations or homotopies of homotopies

AAA\phantom{AAA}

and so on to ever higher gauge transformations or higher homotopies:

AAA\phantom{AAA}

A collection of elements with higher gauge transformations/higher homotopies

is called a higher homotopy type.

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Hence the theory of homotopy types

AAA\phantom{AAA}homotopy theory

AAA\phantom{AAA}is much like set theory,

AAA\phantom{AAA}but with the concept of gauge transformation

AAA\phantom{AAA}built right into its foundations.

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Homotopy theory is gauged mathematics.

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A model for homotopy

AAA\phantom{AAA}

may be given by

AA\phantom{AA}left homotopyAA\phantom{AA}AA\phantom{AA}right homotopyAA\phantom{AA}
AAAAX (id,δ 0) f X×I η Y (id,δ 1) g XAA \phantom{AAAA} \array{ X \\ {}^{\mathllap{(id,\delta_0)}}\downarrow & \searrow^{\mathrlap{f}} \\ X \times I &\stackrel{\eta}{\longrightarrow}& Y \\ {}^{\mathllap{(id,\delta_1)}}\uparrow & \nearrow_{\mathrlap{g}} \\ X } \phantom{AA}AA Y f Maps(δ 0,Y) X η˜ Maps(I,Y) g Maps(δ 1,Y) YAAAA \phantom{AA} \array{ && Y \\ & {}^{\mathllap{f}}\nearrow & \Big\uparrow{}^{\mathrlap{ \Maps(\delta_0,Y) }} \\ X &\overset{\widetilde \eta}{\longrightarrow}& Maps(I,Y) \\ &{}_{\mathllap{g}}\searrow& \Big\downarrow{}^{ \mathrlap{Maps( \delta_1, Y ) } } \\ && Y } \phantom{AAAA}

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A category 𝒞\mathcal{C} with a good supply of

  1. cylinder objectsX×IX \times I

  2. path space objectsMaps(I,X)Maps(I,X)

allowing homotopy theory this way

is called a model category

short for “category of models for homotopy types”.

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Induced: Its homotopy category Ho(𝒞)Ho(\mathcal{C}) whose

  1. objects are good models for homotopy types;

  2. morphisms are homotopy classes of morphisms in 𝒞\mathcal{C}

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Example – topological homotopy theory

AAA\phantom{AAA}The category Top of topological spaces

AAA\phantom{AAA}becomes a model category, Top Qu{}_{Qu}

AAA\phantom{AAA}via the usual cylinder spaces and path spaces

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Example – dgc-algebraic super homotopy theory

AAA\phantom{AAA}The category of real differential graded-commutative superalgebras

AAA\phantom{AAA}carries a “projective” model category structure dgcSuperAlg projdgcSuperAlg_{proj}

AAA\phantom{AAA}with path space objects given by tensor product of algebras with Ω poly ([0,1])\Omega^\bullet_{poly}\big([0,1]\big)

(Bousfield-Gugenheim 76, Carchedi-Roytenberg 12)

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Example – simplicial homotopy theory

AAA\phantom{AAA}An n-simplex is an nn-dimension generalization of a triangle.

AAA\phantom{AAA}A simplicial complex is a space glued from simplices of any dimension.

AAA\phantom{AAA}

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Generally, a simplicial set XX is

AAA\phantom{AAA}for each nn \in \mathbb{N} a set X nX_n of would-be nn-simplices,

AAA\phantom{AAA}and functions between these sets that behave like

  1. sending (n+1)(n+1)-simplices to their nn-dimensional faces,

  2. constructing degenerate (n+1)(n+1)-simplices on given nn-simplices.

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Example – The canonical cylinder object

AAA\phantom{AAA}on the 2-simplex look as follows

AAA\phantom{AAA}

graphics grabbed from Friedman 08, p. 33

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∞-groupoids

A simplicial set is called a Kan complex if

  1. its nn-simplices may be composed

  2. each has an inverse under composition, up to (n+1)(n+1)-simplices

These are the good models for simplicial homotopy types.

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Example – a discrete group GG, is a Kan complex BGB G with

  1. a single 0-simplex AAAAAAAAAAAa\phantom{AAAAAAAAAAAa}*\ast

  2. 1-simplices the group elements AA\phantom{AA}*AAgAA*\ast \overset{\phantom{AA}g\phantom{AA}}{\longrightarrow} \ast

  3. 2-simplices their product AAAAAAa\phantom{AAAAAAa} * g 1 g 2 * g 2g 1 *\array{ && \ast\\ & {}^{\mathllap{g_1}}\nearrow && \searrow^{\mathrlap{g_2}} \\ \ast && \underset{g_2 g_1}{\longrightarrow} && \ast }

  4. 3-simplices their associativity AAA\phantom{AAA}* AAgAA * f hg h * (hg)f *=* AAgAA * f gf h * h(gf) * \array{ \ast &\overset{ \phantom{AA}g\phantom{AA} }{\longrightarrow}& \ast \\ {}^{\mathllap{f}}\Big\uparrow & \searrow^{\mathrlap{h g}} & \Big\downarrow{}^{\mathrlap{h}} \\ \ast &\underset{ (h g) f }{\longrightarrow}& \ast } \;\;\;\;=\;\;\;\; \array{ \ast &\overset{ \phantom{AA}g\phantom{AA} }{\longrightarrow}& \ast \\ {}^{\mathllap{f}}\Big\uparrow & {}^{\mathllap{ g f }}\nearrow & \Big\downarrow{}^{\mathrlap{h}} \\ \ast &\underset{ h (g f) }{\longrightarrow}& \ast }

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An adjunction between model categories

AAA\phantom{AAA}is called a Quillen adjunction

AAA\phantom{AAA}𝒞 Qu QuRL𝒟 \mathcal{C} \; \underoverset {\underset{R}{\longrightarrow}} {\overset{L}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} \; \mathcal{D}

AAA\phantom{AAA}if, very roughly (see here)

  1. the left adjoint preserves cylinder objects;

  2. the right adjoint preserves path space objects.

This induces an adjunction of derived functors on homotopy categories

AAA\phantom{AAA}Ho(𝒞)AAAAR𝕃LHo(𝒟) Ho(\mathcal{C}) \; \underoverset {\underset{\mathbb{R}R}{\longrightarrow}} {\overset{\mathbb{L}L}{\longleftarrow}} {\phantom{AA}\bot\phantom{AA}} \; Ho(\mathcal{D})

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This is a Quillen equivalence if the derived adjunction is an equivalence of categories

AAA\phantom{AAA}Ho(𝒞)AAAAR𝕃LHo(𝒟) Ho(\mathcal{C}) \; \underoverset {\underset{\mathbb{R}R}{\longrightarrow}} {\overset{\mathbb{L}L}{\longleftarrow}} {\phantom{AA}\simeq\phantom{AA}} \; Ho(\mathcal{D})

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Example

simplicial and topological homotopy theory are Quillen equivalent

AAAAA\phantom{AAAAA}Top QuA Qu QuASing||sSet Qu Top_{Qu} \; \underoverset {\underset{Sing}{\longrightarrow}} {\overset{{\vert-\vert}}{\longleftarrow}} {\phantom{A}\phantom{{}_{Qu}}\simeq_{Qu}\phantom{A}} \; sSet_{Qu}

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AAAAA\phantom{AAAAA}Ho(Top)AAAASing𝕃||Ho(sSet) Ho(Top) \; \underoverset {\underset{\mathbb{R}Sing}{\longrightarrow}} {\overset{\mathbb{L}{\vert-\vert}}{\longleftarrow}} {\phantom{AA}\simeq\phantom{AA}} \; Ho(sSet)

via forming

(Quillen 67)

Hence both represent

the homotopy theory of ∞-groupoids.

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Remarkable richness of homotopy theory:

For a pointed object *aA\ast \overset{a}{\to} A and a morphism XcAX \overset{c}{\longrightarrow} A

define the homotopy fiber X^hofib(c)X\widehat X \overset{hofib(c)}{\longrightarrow} X of cc over aa by:

  1. X^\hat X is in the fiber, up to homotopy: AA\phantom{AA}hofib(c)\phantom{hofib(c)}X^ AAcAA * hofib(c) X AAcAA A\array{ \hat X &\overset{\phantom{AA}\phantom{c}\phantom{AA}}{\longrightarrow} & \ast \\ {}^{\mathllap{hofib(c)}}\Big\downarrow &\swArrow& \Big\downarrow\\ X &\underset{\phantom{AA} c \phantom{AA}}{\longrightarrow}& A }

  2. X^\hat X is universal with this property.

Then short exact sequences become long homotopy fiber sequences:

Ω 2A ΩX^ Ωhofib(c) ΩX AΩcA ΩA connectinghomomorphism X^ loop spaceA hofib(c) X AAcAA A \array{ \cdots &\longrightarrow& \Omega^2 A &\overset{}{\longrightarrow}& \Omega \widehat X \\ && && {}^{\mathllap{ \Omega hofib(c) }}\Big\downarrow \\ && &&\Omega X &\overset{\phantom{A} \Omega c \phantom{A}}{\longrightarrow}& \Omega A &\overset{ \text{connecting} \atop \text{homomorphism} }{\longrightarrow}& \widehat X & { \text{loop space} \atop \phantom{A} } \\ && && && && {}^{\mathllap{hofib(c)}}\Big\downarrow \\ && && && && X &\underset{\phantom{AA}c\phantom{AA}}{\longrightarrow}& A }

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example:

higher central extensions 𝔤^\widehat{\mathfrak{g}} of super L-∞ algebras 𝔤\mathfrak{g}

are homotopy fibers of the corresponding cocycles μ p+2\mu_{p+2}:

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AAAAAAAAA\phantom{AAAAAAAAA}𝔤^ hofib(μ p+2) 𝔤 μ p+2 B p+1AAAAAAACEAAAAAAACE(𝔤)[b(p+1,even)]/(db=μ p+2) CE(𝔤) cμ p+2 [c(p+2,even)] \array{ \widehat {\mathfrak{g}} \\ {}^{\mathllap{hofib\left(\mu_{p+2}\right)}}\Big\downarrow \\ \mathfrak{g} &\underset{\mu_{p+2}}{\longrightarrow}& B^{p+1} \mathbb{R} } \phantom{AAAAA} \overset{\phantom{AA}CE\phantom{AA}}{\mapsto} \phantom{AAAAA} \array{ CE(\mathfrak{g})[ \underset{(p+1,even)}{\underbrace{\,b\,}} ]/(d \,b = \mu_{p+2}) \\ \Big\uparrow \\ CE(\mathfrak{g}) &\underset{c \mapsto \mu_{p+2}}{\longleftarrow}& \mathbb{R}[ \underset{(p+2, even)}{\underbrace{\;c\;}} ] }

\,

\,

\,

\,

\,

\,

4) Higher super geometry

\,

So far we have established two ingredients:

\,

  1. realization of the Pauli exclusion principle:

    affine superspaces formally dual to dgc-superalgebras,

    such as super Cartesian spaces, super L-∞ algebroids

    \,

  2. realization of the gauge principle:

    homotopy theory,

    notably of geometrically discrete ∞-groupoids (Kan complexes)

\,

Now to combine this to

supergeometric ∞-groupoids (“∞-stacks”) AA\phantom{AA}inAA\phantom{AA} higher supergeometry.

\,

\,

\,

Consider any category of affine spaces as above

AffineSpaces={AffineSchemes, CartesianSpaces, SuperCartesianSpaces, AffineSpaces \;=\; \left\{ \array{ AffineSchemes, \\ CartesianSpaces, \\ SuperCartesianSpaces, \\ \cdots } \right.

\,

We may bootstrap a notion of generalized spaces:

Geometry as seen by classical sigma-models:

AAA\phantom{AAA}To determine a generalized target space X\mathbf{X}

AAA\phantom{AAA}…consider for each brane worldvolume

AAAAA\phantom{AAAAA} ΣAffineSpaces\Sigma \in AffineSpaces

AAA\phantom{AAA}the collection of sigma-model fields

AAAAA\phantom{AAAAA}{ΣAAϕAAX}AAAA \left\{ \; \Sigma \overset{\phantom{AA}\phi\phantom{AA}}{\longrightarrow} \mathbf{X} \; \right\} \phantom{ {A \atop A} \atop {A \atop A} }

\,

\,

Minimum consistency conditions.

  1. For every ΣAffineSpaces\Sigma \;\in\; AffineSpaces, there should be a simplicial set

    “of sigma-model fields into X\mathbf{X} and their gauge-of-gauge equivalences

    AAA\phantom{AAA}Σ "Maps(Σ,X)" \array{ \Sigma &\mapsto& \text{"Maps(}\Sigma,\mathbf{X}\text{)"} }

    (see also the talk by E. Sharpe at the meeting)

    quotation marks because X\mathbf{X} is yet to exist, will be removed in a moment

  2. For every morphism ϕ\phi in AffineSpacesAffineSpaces there should be a function

    precomposition of sigma-model fields with ff”:

    AAA\phantom{AAA}Σ 1 "Maps(Σ 1,X)" ϕ "precomposition withϕ" Σ 2 "Maps(Σ 2,X)"\array{ \Sigma_1 &\mapsto& \text{"Maps(}\Sigma_1,\mathbf{X}\text{)"} \\ {}^{\mathllap{\phi}}\Big\downarrow && \Big\uparrow{}^{ \mathrlap{ \;\;\text{"precomposition with}\; \phi\text{"} } } \\ \Sigma_2 &\mapsto& \text{"Maps(}\Sigma_2,\mathbf{X}\text{)"} }

  3. Compatibility with composition of morphisms in AffineSpacesAffineSpaces.

\,

\,

\,

\,

In the language of category theory this is simply summarized by:

\,

Classicalsigma-models see generalized spaces X\mathbf{X}

AAA\phantom{AAA}as functors of the form

AAA\phantom{AAA}X:AffineSpaces opsSet, \mathbf{X} \;\colon\; AffineSpaces^{op} \longrightarrow sSet \,,

AAA\phantom{AAA}also called simplicial presheaves on AffineSpacesAffineSpaces.

\,

Hence the category of generalized spaces

AAA\phantom{AAA}should be a full subcategory of the functor category

AAAAA\phantom{AAAAA}GeneralizedSpacesFunct(AffineSpaces op,sSet) GeneralizedSpaces \;\subset\; Funct\left(AffineSpaces^{op}, sSet\right)

AAA\phantom{AAA}also called the simplical presheaf topos over AffineSpacesAffineSpaces.

\,

This is naturally a model category: the global model structure on simplicial presheaves:

AAA\phantom{AAA}AAA\phantom{AAA}Funct(AffineSpaces op,sSet Qu) glob Funct(AffineSpaces^{op}, sSet_{Qu})_{glob}

\,

\,

\,

key bootstrap theorem of functorial geometry says that

AAA\phantom{AAA} the wish becomes true; we may remove the quotation marks:

AAAAA\phantom{AAAAA}Hom GeneralSpaces(Σ,X)X(Σ)="Maps(Σ,X)" Hom_{GeneralSpaces}( \Sigma, \mathbf{X} ) \;\simeq\; \mathbf{X}(\Sigma) \;=\; \text{"Maps(}\Sigma,\mathbf{X}\text{)"}

AAA\phantom{AAA}known as the Yoneda lemma.

\,

Direct consequence is the key consistency result

AAAAA\phantom{AAAAA}Hom AffineSpaces(Σ 1,Σ 2)Hom GeneralizedSpaces(Σ 1,Σ 2) Hom_{AffineSpaces}(\Sigma_1,\Sigma_2) \;\simeq\; Hom_{GeneralizedSpaces}(\Sigma_1,\Sigma_2)

AAA\phantom{AAA}hence a full subcategory embedding

AAAAA\phantom{AAAAA}AffineSpacesAAAAGeneralizedSpacesAffineSpaces \overset{\phantom{AAAA}}{\hookrightarrow} GeneralizedSpaces

AAA\phantom{AAA}called the Yoneda embedding.

\,

\,

\,

\,

Impose one more condition: locality.

\,

AAA\phantom{AAA}If ΣAffineSpaces\Sigma \in AffineSpaces is covered AA\phantom{AA} U i 1 U i 2 U i 3 Σ\array{ U_{i_1} && U_{i_2} && U_{i_3} & \cdots \\ & \searrow & \Big\downarrow & \swarrow \\ && \Sigma }

AAA\phantom{AAA} by patches U iAffineSpacesU_i \in AffineSpaces

AAA\phantom{AAA}then a decent generalized space X\mathbf{X} should satisfy:

AAAAA\phantom{AAAAA}{ΣAAϕAAX}weakhomotopy equivalence{tuples of mapsU iX that agree on intersections, up to gauge transformations which are compatible, up to gauge-of-gauge-transformations and so on } \Big\{ \Sigma \overset{\phantom{AA}\phi\phantom{AA}}{\longrightarrow} \mathbf{X} \Big\} \;\underset{\text{weak} \atop \text{homotopy equivalence}}{\simeq}\; \left\{ \array{ \text{tuples of maps}\; U_i \to \mathbf{X} \\ \text{that agree on intersections, up to gauge transformations} \\ \text{which are compatible, up to gauge-of-gauge-transformations } \\ \text{ and so on } } \right\}

\,

If X\mathbf{X} satisfies this locality condition

it is called an ∞-stack or (∞,1)-sheaf over AffineSpacesAffineSpaces.

\,

\,

\,

There is a local model structure on simplicial presheaves

AAA\phantom{AAA}and a Quillen adjunction (“Bousfield localization”)

AAA\phantom{AAA}Funct(AffineSpaces,sSet) loclocal model structureAA Qu QuAAididFunct(AffineSpaces,sSet) proj \underset{ \text{local model structure} }{ \underbrace{ Funct(AffineSpaces,sSet)_{loc} }} \;\; \underoverset {\underset{id}{\longrightarrow}} {\overset{id}{\longleftarrow}} {\phantom{AA}\phantom{{}_{Qu}}\bot_{Qu}\phantom{AA}} \;\; Funct(AffineSpaces,sSet)_{proj}

AAA\phantom{AAA}whose derived adjunction on homotopy categories

AAA\phantom{AAA}is the full subcategory inclusion on the ∞-stacks

AAAAA\phantom{AAAAA}Ho(Funct(AffineSpaces,sSet)) locHo(Sh (AffineSpaces))∞-stackificationHo(Funct(AffineSpaces,sSet)) \overset{ Ho\left(\, Sh_\infty(AffineSpaces) \, \right) \;\coloneqq\; }{ \overbrace{ Ho\left(Funct(AffineSpaces,sSet)\right)_{loc} }} \; \; \underoverset {\underset{}{\hookrightarrow}} {\overset{\text{∞-stackification}}{\longleftarrow}} {\bot} \;\; Ho\left(Funct(AffineSpaces,sSet)\right)

\,

This homotopy theory is called the

AAA\phantom{AAA}∞-stack ∞-topos over AffineSpacesAffineSpaces

or

AAA\phantom{AAA}(∞,1)-sheaf ∞-topos over AffineSpacesAffineSpaces

\,

\,

\,

\,

Now to get higher supergeometry

AAA\phantom{AAA} specify to AffineSpacesAffineSpaces \;\coloneqq\; SuperCartesianSpaces

\,

So super geometric homotopy theory is

AAA\phantom{AAA}Ho(SuperSpaces)Ho(Sh (SuperCartesianSpaces))Ho\left( \, SuperSpaces \, \right) \;\coloneqq\; Ho\big( \, Sh_\infty( \, SuperCartesianSpaces \, ) \, \big)

\,

\,

\,

\,

\,

\,

this subsumes a rich amount of particular geometries

discussed in the literature:

\,

\,

\,

\,

5) Higher super Lie theory

\,

In particular we have a variety of notions of geometric symmetry groups

AAA\phantom{AAA}Lie groups AAAA\overset{\phantom{AAAA}}{\hookrightarrow} smooth ∞-groups

AAA\phantom{AAA}Lie groupoids AAAA\overset{\phantom{AAAA}}{\hookrightarrow} smooth ∞-groupoids

AAA\phantom{AAA}super Lie groups AAAA\overset{\phantom{AAAA}}{\hookrightarrow} super smooth ∞-groups

\,

these should be related to

the super L-∞ algebroids discussed above

by higher Lie theory

\,

\,

\,

\,

idea (due to Ševera 01,

AAA\phantom{AAA}see also Pavol Ševera‘s talk at the meeting):

\,

generalize the path method of Lie integration:

AAA\phantom{AAA}For 𝔤\mathfrak{g} a Lie algebra

AAA\phantom{AAA}and GG its simply connected Lie group

AAA\phantom{AAA}observe that parallel transport

AAA\phantom{AAA}(after a choice of base point)

AAA\phantom{AAA}provides a bijection

AAAAA\phantom{AAAAA}Hom dgcAlgebras(CE(𝔤),Ω ( n))Ω flat 1( n,𝔤)AAparalleltransportAAHom SmoothManifolds( n,G) \underset{ \Omega^1_{flat}\left( \mathbb{R}^n, \mathfrak{g} \right) }{ \underbrace{ Hom_{dgcAlgebras} \left( CE(\mathfrak{g})\;,\; \Omega^\bullet(\mathbb{R}^{n}) \right) }} \underoverset {\simeq} {\phantom{AA}{\text{parallel} \atop \text{transport}}\phantom{AA}} {\longrightarrow} Hom_{SmoothManifolds} \left( \mathbb{R}^{n} \,,\, G \right)

\,

Hence we recover GG via smooth homotopy theory as

AAAAA\phantom{AAAAA}GMaps([0,1],Ω flat(,𝔤))/homotopy G \;\;\simeq\;\; Maps\big( [0,1]\,, \, \mathbf{\Omega}_{flat}(-,\mathfrak{g}) \big)\,/\,homotopy

\,

\,

\,

to generalize this

AAA\phantom{AAA}recall the functor of super differential forms

AAAAA\phantom{AAAAA}Ω :SuperCartesianSpacesdgcSuperAlgebras op\mathbf{\Omega}^\bullet \;\colon\; SuperCartesianSpaces \longrightarrow dgcSuperAlgebras^{op}

\,

Consider its enhancement to smooth differential forms on simplices:

AAAA\phantom{AAAA}Ω vert,si :( n|q,k)Ω vert,si ( n|q×Δ mfd k) \mathbf{\Omega}^\bullet_{vert, si} \;\colon\; (\mathbb{R}^{n\vert q}, k) \;\mapsto\; \mathbf{\Omega}^\bullet_{vert,si}\left( \mathbb{R}^{n\vert q} \times \Delta^k_{mfd} \right)

A\phantom{A}where

  1. Δ k\Delta^k is the standard k-simplex as a smooth manifold with boundaries and corners;

  2. “vert” = vertical differential forms: all legs along the k-simplex;

  3. “si” = forms with sitting instants: constant towards boundary of k-simplex

\,

(Fiorenza-S.-Stasheff 12, Def. 4.2.1; see Braunack-Mayer 18, Def. 3.1.3)

\,

\,

\,

higher super Lie integration via higher super path method

for 𝔤\mathfrak{g} a super L-∞ algebra

with Chevalley-Eilenberg algebra CE(𝔤)dgcSuperAlgebrasCE(\mathfrak{g}) \in dgcSuperAlgebras

Let

AAA\phantom{AAA}Spec(CE(𝔤))Ho(Sh (SuperCartSp))=Ho(SuperSpaces) Spec(CE(\mathfrak{g})) \;\in\; Ho\left( \,Sh_\infty(\,SuperCartSp\,)\, \right) \;=\; Ho\left( \,SuperSpaces\, \right)

be given by the simplicial presheaf

AAA\phantom{AAA}Spec(CE(𝔤)) k: n|qHom dgcSuperalgebras(CE(𝔤),Ω vert,si ( n|q×Δ mfd k)) Spec(CE(\mathfrak{g}))_k \;\;\colon\;\; \mathbb{R}^{n\vert q} \;\;\mapsto\;\; Hom_{dgcSuperalgebras} \left( CE(\mathfrak{g})\,,\, \Omega^\bullet_{vert,si}\left( \mathbb{R}^{n\vert q} \times \Delta^k_{mfd} \right) \right)

\,

(Fiorenza-S.-Stasheff 12)

\,

example:

For 𝔤\mathfrak{g} a Lie algebra with simply connected Lie group GG, we recover

τ 1Spec(CE(𝔤))BGHo(SuperSpaces) \tau_1 Spec(CE(\mathfrak{g})) \;\simeq\; \mathbf{B}G \;\in\; Ho\left( \, SuperSpaces \, \right)

\,

\,

\,

\,

\,

Theorem (Braunack-Mayer 18)

higher super Lie integration is part of a Quillen adjunction

AAA\phantom{AAA}dgcSuperAlgebras 0,proj opA Qu QuASpec𝒪[SuperCartSp op,sSet Qu] loc dgcSuperAlgebras^{op}_{\geq 0, proj} \; \underoverset {\underset{ Spec }{\longrightarrow}} {\overset{ \mathcal{O} }{\longleftarrow}} {\phantom{A}\phantom{{}_{Qu}}\bot_{Qu}\phantom{A}} \; [SuperCartSp^{op},sSet_{Qu}]_{loc}

\,

hence induces a derived adjunction on homotopy categories:

AAA\phantom{AAA}Ho(dgcSuperAlgebras 0 op)A QuQuASpec𝕃𝒪Ho(SuperSpaces) Ho\left(\, dgcSuperAlgebras^{op}_{\geq 0} \, \right) \; \underoverset {\underset{ \mathbb{R} Spec }{\longrightarrow}} {\overset{ \mathbb{L} \mathcal{O} }{\longleftarrow}} {\phantom{A}\phantom{{}_{Qu}}\bot\phantom{_{Qu}}\phantom{A}} \; Ho\left( \, SuperSpaces \, \right)

\,

\,

\,

\,

\,

Theorem (Braunack-Mayer 18)

higher super Lie integration restricts to the Sullivan de Rham adjunction

dgcSuperAlg ,0,proj opA Qu QuASpec𝒪AAhigher superLie integrationFunct(SuperCartSp op,sSet Qu) locA Qu QuAΓconstAAinclude geometrically discrete∞-groupoidssSet QuAASullivan de Rham adjunctionA Qu QuASing||AAregard ∞-groupoidsas topological spacesTop Qu \underset{ \color{blue} \text{Sullivan de Rham adjunction} }{ \underbrace{ \underset{\phantom{A} \atop \phantom{A}}{ dgcSuperAlg^{op}_{\mathbb{R}, \geq 0, proj} \; \underset{ \color{blue} {\text{higher super} \atop \text{Lie integration} } }{ \underbrace{ \underset{ \phantom{A} \atop \phantom{A} }{ \underoverset {\underset{ Spec }{\longrightarrow}} {\overset{ \mathcal{O} }{\longleftarrow}} {\phantom{A}\phantom{{}_{Qu}}\bot_{Qu}\phantom{A}} }}} \; Funct\left( SuperCartSp^{op},sSet_{Qu}\right)_{loc} \; \overset{ \color{blue}{ \text{include } \atop { \text{geometrically discrete} \atop \text{∞-groupoids} }} }{ \overbrace{ \overset{ \phantom{A} \atop \phantom{A} }{ \underoverset {\underset{\Gamma}{\longrightarrow}} {\overset{const}{\longleftarrow}} {\phantom{A}\phantom{{}_{Qu}}\bot_{Qu}\phantom{A}} }}} \; sSet_{Qu} }} } \; \overset{ \color{blue}{ \text{regard ∞-groupoids} \atop \text{as topological spaces} } }{ \overbrace{ \overset{ \phantom{A} \atop \phantom{A} }{ \underoverset {\underset{Sing}{\longleftarrow}} {\overset{{\vert- \vert}}{\longrightarrow}} {\phantom{A}\phantom{{}_{Qu}}\simeq_{Qu}\phantom{A}} }}} \; Top_{Qu}

\,

classical fact of rational homotopy theory:

AAA\phantom{AAA}the derived adjunction restricts to an equivalence of categories

AAAAA\phantom{AAAAA}Ho(dgcSuperAlg op) bos,conn,finbosonic, connective, finite type algebrasAAAA(ΓSpec)𝕃(𝒪const)Ho(TopologicalSpaces) ,conn,nil,finrational, connected, nilpotent, finite type spaces \underset{ \color{blue}{ \text{bosonic, connective, finite type algebras} } }{ \underbrace{ Ho\left(\, dgcSuperAlg^{op} \, \right)_{bos, conn, fin} }} \; \; \underoverset {\underset{ \mathbb{R} ( \Gamma \circ Spec ) }{\longrightarrow}} {\overset{ \mathbb{L}( \mathcal{O} \circ const ) }{\longleftarrow}} { \phantom{AA}\simeq\phantom{AA} } \; \; \underset{ \color{blue}{ \text{rational, connected, nilpotent, finite type spaces} } }{ \underbrace{ Ho\left( \, TopologicalSpaces \, \right)_{\mathbb{Q},conn, nil, fin} }}

(for more on this see D. Fiorenza‘s and V. Braunack-Mayer’s talks at the meeting)

\,

\,

This implies that

\,

higher super Lie integration preserves homotopy fiber sequences

hence in particular higher central extensions

\,

AAAA\phantom{AAAA}SuperL Algebras Spec(CE()) SuperSpaces A 𝔤^ hofib(μ p+2) 𝔤 Aμ p+2A B p+1 AAAAAAAAAAAAAAAAAAAA BG^ hofib(c p+2) BG c p+2 B p+2 \array{ Super L_\infty Algebras &\overset{ Spec\big(CE(-)\big) }{\longrightarrow}& SuperSpaces \\ \phantom{A} \\ \array{ \widehat{\mathfrak{g}} \\ {}^{\mathllap{hofib\left(\mu_{p+2}\right)}}\Big\downarrow \\ \\ \mathfrak{g} &\underset{ \phantom{A}\mu_{p+2}\phantom{A} }{\longrightarrow}& B^{p+1} \mathbb{R} } & \phantom{AAAAAAAA} \overset{ \phantom{AAAA} }{\mapsto} \phantom{AAAAAAAA} & \array{ \mathbf{B} \widehat{G} \\ {}^{\mathllap{hofib\left(\mathbf{c}_{p+2}\right)}}\Big\downarrow \\ \mathbf{B}G &\underset{ \mathbf{c}_{p+2} }{\longrightarrow}& \mathbf{B}^{p+2} \mathbb{R} } }

\,

\,

\,

\,

\,

6) Emergence of M-theory

\,

homotopy theory reveals rich inner structure of god-given objects

Already the homotopy integers – known as the sphere spectrum

exhibit endless patterns, order and chaos:

stable homotopy groups of spheres at 2

(the 2-primary components of the first few stable homotopy groups of spheres)

\,

\,

\,

The microscope of homotopy theory is the Postnikov-Whitehead tower

For example Postnikov-Whitehead tower of the orthogonal group:

Whiteheadtower BFivebrane * second fractionalPontryagin class BString 16p 2 B 8 AAA * first fractionalPontryagin class BSpin A12p 1A B 4 * secondStiefel-Whitney class BSO AAA AAw 2AA B 2 2 AAA * firstStiefel-Whitney class BO τ 8BO AAAA τ 4BO AAAA τ 2BO Aw 1A τ 1BOB 2 Postnikovtower \array{ & \mathbf{\text{Whitehead} \atop \text{tower}} \\ &\vdots \\ & B Fivebrane &\to& \cdots &\to& * \\ & \Big\downarrow && \swArrow && \Big\downarrow \\ \mathbf{\text{second fractional} \atop \text{Pontryagin class}} & B String &\to& \cdots &\stackrel{\tfrac{1}{6}p_2}{\to}& B^8 \mathbb{Z} &\overset{\phantom{AAA}}{\longrightarrow}& * \\ & \Big\downarrow && && \Big\downarrow &\swArrow& \Big\downarrow \\ \mathbf{\text{first fractional} \atop \text{Pontryagin class}} & B Spin && \swArrow && &\stackrel{\phantom{A}\tfrac{1}{2}p_1\phantom{A}}{\to}& B^4 \mathbb{Z} &\to & * \\ & \Big\downarrow && && \Big\downarrow &\swArrow& \Big\downarrow &\swArrow& \Big\downarrow \\ \mathbf{\text{second} \atop \text{Stiefel-Whitney class}} & B S O &\overset{}{\to}& \cdots &\overset{}{\to}& &\overset{\phantom{AAA}}{\longrightarrow}& & \overset{\phantom{AA}w_2\phantom{AA}}{\longrightarrow} & \mathbf{B}^2 \mathbb{Z}_2 &\overset{\phantom{AAA}}{\longrightarrow}& * \\ & \Big\downarrow && \swArrow && \Big\downarrow &\swArrow& \Big\downarrow &\swArrow& \Big\downarrow &\swArrow& \Big\downarrow \\ \mathbf{\text{first} \atop \text{Stiefel-Whitney class}} & B O &\to& \cdots &\to& \tau_{\leq 8 } B O &\underset{\phantom{AAAA}}{\longrightarrow}& \tau_{\leq 4 } B O &\underset{\phantom{AAAA}}{\longrightarrow}& \tau_{\leq 2 } B O &\overset{\phantom{A}w_1\phantom{A}}{\longrightarrow}& \tau_{\leq 1 } B O \simeq B \mathbb{Z}_2 & \mathbf{\text{Postnikov} \atop \text{tower}} }

where each rectangle is a homotopy fiber square.

\,

Under higher Lie integration this corresponds to

tower of universal higher central extensions of L-∞ algebras

\,

𝔫𝔦𝔫𝔢𝔟𝔯𝔞𝔫𝔢 hofib(μ 11) 𝔣𝔦𝔳𝔢𝔟𝔯𝔞𝔫𝔢 μ 11=,[,],[,],[,],[,],[,] B 10 hofib(μ 7) 𝔰𝔱𝔯𝔦𝔫𝔤 [,]μ 7=,[,],[,],[,][,] B 6 hofib(μ 3) 𝔰𝔬 [,][,]μ 3=,[,][,][,] B 2 \array{ \mathfrak{ninebrane} \\ {}^{\mathllap{ hofib\left( \mu_{11} \right) }}\Big\downarrow \\ \mathfrak{fivebrane} &\overset{ \mu_{11} =\langle -,[-,-], [-,-], [-,-], [-,-], [-,-]\rangle }{\longrightarrow}& B^{10} \mathbb{R} \\ {}^{\mathllap{ hofib\left( \mu_7 \right) }}\Big\downarrow \\ \mathfrak{string} &\overset{ \phantom{[-,-]} \mu_7 = \langle -,[-,-], [-,-], [-,-]\rangle \phantom{[-,-]} }{\longrightarrow}& B^6 \mathbb{R} \\ {}^{\mathllap{hofib\left(\mu_3\right)}}\Big\downarrow \\ \mathfrak{so} &\overset{ \phantom{[-,-]} \phantom{[-,-]} \mu_3 = \langle -,[-,-]\rangle \phantom{[-,-]} \phantom{[-,-]} }{\longrightarrow}& B^2 \mathbb{R} }

\,

As the terminology indicates (string Lie 2-algebra, fivebrane Lie 6-algebra,…),

pure homotopy theory is beginning to see string theory-structure…

(Sati-S.-Stasheff 09, Fiorenza-S.-Stasheff 10, Sati 14)

\,

\,

To get to the bottom of this

apply this microscope to the superpoint 0|1\mathbb{R}^{0\vert 1}

AAA\phantom{AAA}regarded as a super Lie algebra, hence as a higher superspace

AAA\phantom{AAA}superL Algebras AACEAA dgcSuperAlgebras op ASpecA Ho(SuperSpaces) 0|1 [dθ(1,odd)] B 0|1 \array{ super L_\infty Algebras &\overset{ \phantom{AA} CE \phantom{AA} }{\longrightarrow}& dgcSuperAlgebras^{op} &\overset{ \phantom{A} Spec \phantom{A} }{\longrightarrow}& Ho\left( \,SuperSpaces\, \right) \\ \mathbb{R}^{0\vert 1} &\mapsto& \mathbb{R} [ \underset{(1,odd) }{\underbrace{\,d \theta\,}} ] &\mapsto& \mathbf{B} \mathbb{R}^{0\vert 1} }

\,

Consider iteratively the

AAA\phantom{AAA}invariant universal higher central extensions,

AAA\phantom{AAA}at each stage invariant with respect to automorphisms modulo R-symmetries

\,

Then the following brane bouquet is revealed…

\,

(Fiorenza-Sati-S. 13, Huerta-S. 17, BraunackMayer-Sati-S. 18, Huerta-Sati-S. 18,

AAA\phantom{AAA}see the talks by H. Sati, V. Braunack-Mayer, D. Fiorenza, J. Huerta (here) at the meeting):

\,

\,

AAA\phantom{AAA}

\,

\,

Hence in super homotopy theory

AAA\phantom{AAA}core structure of M-theory emerges

AAA\phantom{AAA}out of the superpoint.

\,

Does the superpoint itself emerge by itself?

\,

\,

To see this, we invoke now a yet stronger microscope,

AAA\phantom{AAA}that of modal homotopy theory

\,

(see also Modern Physics formalized in Modal Homotopy Type Theory)

\,

A reflective subcategory is

AAA\phantom{AAA}a fully faithful functor with a left adjoint

AAA\phantom{AAA}H AAAAιLH \mathbf{H}_{\bigcirc} \; \underoverset {\underset{\iota}{\hookrightarrow}} {\overset{ L }{\longleftarrow}} {\phantom{AA}\bot\phantom{AA}} \; \mathbf{H}

A modal operator is the resulting endofunctor

AAA\phantom{AAA}ιL:HH \bigcirc \;\coloneqq\; \iota \circ L \;\colon\; \mathbf{H} \to \mathbf{H}

AAA\phantom{AAA}which projects onto H \mathbf{H}_{\bigcirc}.

dually:

A coreflective subcategory

AAA\phantom{AAA}is a fully faithful functor with a right adjoint

AAA\phantom{AAA}H AAAARιH \mathbf{H}_{\Box} \; \underoverset {\underset{ R }{\longleftarrow}} {\overset{\iota}{\hookrightarrow}} {\phantom{AA}\bot\phantom{AA}} \; \mathbf{H}

A comodal operator is the resulting endofunctor

AAA\phantom{AAA}ιR:HH \Box \;\coloneqq\; \iota \circ R \;\colon\; \mathbf{H} \to \mathbf{H}

AAA\phantom{AAA}which projects onto H \mathbf{H}_{\Box}.

\,

\,

similarly in homotopy theory:

A Quillen reflection

AAA\phantom{AAA}is Quillen adjunction whose derived adjunction is a reflective subcategory

AAA\phantom{AAA}𝒞AA Qu QuAARL𝒟 Ho(𝒞)AAAAAAR𝕃LHo(𝒟) \array{ \mathcal{C} \underoverset {\underset{R}{\longrightarrow}} {\overset{L}{\longleftarrow}} {\phantom{AA}\phantom{{}_{Qu}}\bot_{Qu}\phantom{AA}} \mathcal{D} \\ \\ Ho(\mathcal{C}) \underoverset {\underset{\mathbb{R}R}{\hookrightarrow}} {\overset{\mathbb{L}L}{\longleftarrow}} {\phantom{AAA}\bot\phantom{AAA}} Ho(\mathcal{D}) }

AAA\phantom{AAA}etc.

\,

example: the homotopy localization at an object 𝔸\mathbb{A},

AAA\phantom{AAA}is the reflective subcategory of coefficients

AAA\phantom{AAA} representing 𝔸\mathbb{A}-homotopy invariant cohomology

denote the corresponding modal operator by 𝔸 \bigcirc\!\!\!\!\!\!\!\!\mathbb{A}

\,

\,

\,

an adjoint modality

AAA\phantom{AAA}is an adjunction (duality) between

AAA\phantom{AAA}a modal operator and a comodal operator

AAA\phantom{AAA}\bigcirc \;\dashv\; \Box

AAA\phantom{AAA}or

AAA\phantom{AAA}\Box \;\dashv\; \bigcirc

this expresses two dual and opposite subcategories

\,

for example

AAA\phantom{AAA}EvenOdd: Even \dashv Odd \;\colon\; \mathbb{Z}_{\leq} \longrightarrow \mathbb{Z}_{\leq}

AAA\phantom{AAA}with

AAA\phantom{AAA}Even(n)2n/2AAAOdd(n)2n/2+1Even(n) \;\coloneqq\; 2 \lfloor n/2 \rfloor \phantom{AAA} Odd(n) \;\coloneqq\; 2 \lfloor n/2\rfloor + 1

AA\phantom{AA}is an adjoint modality,

expressing the dual opposition between even and odd numbers.

\,

\,

preorder on modal operators

AAA\phantom{AAA}by inclusion of the subcategories which they project onto

AAA\phantom{AAA} 1< 2AAifAAH 1H 2 \bigcirc_1 \lt \bigcirc_2 \phantom{AA}if\phantom{AA} \mathbf{H}_{\bigcirc_1} \subset \mathbf{H}_{\bigcirc_2}

\,

A\phantom{A} bottom adjoint modality A\phantom{A}A\phantom{A}top adjoint modalityA\phantom{A}
A\phantom{A}*\emptyset \;\dashv \ast\; A\phantom{A}A\phantom{A}ididid \dashv idA\phantom{A}

\,

everything happens in between:

AAA\phantom{AAA}id id *\array{ id &\;\dashv\;& id \\ \vee && \vee \\ \vdots && \vdots \\ \vee && \vee \\ \emptyset &\;\dashv\;& \ast }

\,

\,

\,

Given a progression of adjoint modalities

2 2 1 1 \array{ \Box_2 &\dashv& \bigcirc_2 \\ \vee && \vee \\ \Box_1 &\dashv& \bigcirc_1 }

we say it provides

sublation of 1 1\Box_1 \dashv \bigcirc_1

if in addition

  1. 1< 2\Box_1 \lt \bigcirc_2 AAA\phantom{AAA} denoted AAA\phantom{AAA} 2 2 / 1 1\array{ \Box_2 &\dashv& \bigcirc_2 \\ \vee &/& \vee \\ \Box_1 &\dashv& \bigcirc_1 }

    A\phantom{A}

    or

  2. 1< 2\bigcirc_1 \lt \Box_2 AAA\phantom{AAA} denoted AAA\phantom{AAA} 2 2 \ 1 1\array{ \Box_2 &\dashv& \bigcirc_2 \\ \vee &\backslash& \vee \\ \Box_1 &\dashv& \bigcirc_1 }

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an adjunction of adjunctions,

hence a duality of dualities

is

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AAA\phantom{AAA} \array{ \lozenge &\dashv& \Box \\ \bot && \bot \\ \Box & \dashv& \bigcirc }

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hence is an adjoint triple

AAA\phantom{AAA}\lozenge \dashv \Box \dashv \bigcirc

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Proposition

We have a progression of adjoint modalities on Ho(Ho\big(SuperSpaces)\big) as follows:

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AA\phantom{AA}id id | 0|1 \ & | ʃ / * \array{ id &\dashv& id \\ \vee &\vert& \vee \\ \rightrightarrows &\dashv& \rightsquigarrow &\dashv& {\bigcirc\!\!\!\!\!\!\!\!\mathbb{R}^{0\vert 1}} \\ && \vee &\backslash& \vee \\ && \Re &\dashv& \Im &\dashv& \& \\ && && \vee &\vert& \vee \\ && && &#643; &\dashv& \flat &\dashv& \sharp \\ && && && \vee &/& \vee \\ && && && \emptyset &\dashv& \ast }

In this way

the superpoint 0|1\mathbb{R}^{0\vert 1}

emerges from nothing \emptyset

by a progression of dual oppositions, their dualities and their sublation.

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For details see the lecture notes

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Last revised on May 17, 2019 at 09:41:03. See the history of this page for a list of all contributions to it.