Schreiber
Super topological T-Duality

Contents

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\; Urs Schreiber (CAS Prague & I-Math Zurich)

\; Super Topological T-Duality (recording)

\; talk at Conf. on Duality in Homotopy theory

\; April 3-7, 2017, Regensburg

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Abstract. Twisted cohomology is maps in the tangent homotopy theory of parameterized spectra. Its rationalization has a Quillen-Sullivan-type model in terms of unbounded L-∞ algebras (SS 17). Interesting examples appear by constructing iterated maximal higher central extensions of super L-∞ algebras, invariant with respect to automorphisms modulo R-symmetries – a super-equivariant version of the Whitehead tower. Applied to the superpoint, this process turns out to discover all the twisted cohomology seen in string/M-theory (FSS 13, HS 17), rationally, in particular twisted topological K-theory with 5-brane correction and with supersymmetric Chern-forms on 10d-superspace (FSS 16a). Passage to cyclic L-∞ cohomology reflects double dimensional reduction of super p-branes. Applied to twisted K-theory on 10d superspace this yields two L-∞ cocycles on 9d superspace which both classify the tangent complex of super T-folds. There appears a super L-∞ equivalence between them exhibiting supersymmetric topological T-duality, rationally (FSS 16b).

These rational phenomena indicate that integrally the super-equivariant Whitehead tower of the superpoint is a mechanism for discovering interesting twisted super-differential cohomology theories (dcct). We close with an outlook on the relevant constructions.

Lecture notes with more details are here:

Related talks

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Contents

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Super L L_\infty

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For the bulk of this talk, the homotopy theory we consider

is that of super L-∞ algebras:

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Those of finite type are equivalently

the formal duals

of augmented dg-algebra structures

on free (×(/2))(\mathbb{Z} \times (\mathbb{Z}/2))-bi-graded commutative algebras

over chain complexes of super vector spaces,

namely their Chevalley-Eilenberg algebras:

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CE : sL fin AAAA (dgAlg ) / op (𝔤,[],[,],[,,],) AAAA ( 𝔤 *,d 𝔤[] *+[,] *+[,,] *+). \array{ CE &\colon& s L_\infty^{fin} &\phantom{AA}\hookrightarrow \phantom{AA}& (dgAlg_{\mathbb{R}})_{/\mathbb{R}}^{op} \\ && (\, \mathfrak{g}\,, \, [-], [-,-], [-,-,-] ,\cdots \,) &\phantom{AA}\mapsto \phantom{AA}& \left( \, \wedge^\bullet \mathfrak{g}^\ast ,\, d_{\mathfrak{g}} \coloneqq [-]^\ast + [-,-]^\ast + [-,-,-]^\ast + \cdots \, \right) } \,.

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Proposition

(Pridham 10, prop. 4.36)

There is the structure of a category of fibrant objects on sL Alg sL_\infty Alg_{\mathbb{R}} whose

  • fibrations are the surjections

  • weak equivalences are the quasi-isomorphisms

on the underlying chain complexes (𝔤,=[])(\,\mathfrak{g}\,,\, \partial = [-]\,).

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We will keep using the following fact:

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Proposition

(FSS 13, theorem 3.8)

Write

B p+1 B^{p+1}\mathbb{R}

for the line Lie (p+2)-algebra, given by

CE(B p+1)=(Symc p+2single generatorin deg.(p+2,even),d=0). CE(B^{p+1}\mathbb{R}) \;=\; \left( Sym \underset{\text{single generator} \atop \text{in deg.} \, (p+2,even)}{\underbrace{\langle c_{p+2} \rangle}} \;,\; d = 0 \right) \,.

A (p+2)(p+2)-cocycle on an L L_\infty-algebra is equivalently a super L L_\infty-homomorphism

μ p+2:𝔤B p+1. \mu_{p+2} \;\colon\; \mathfrak{g} \longrightarrow B^{p+1}\mathbb{R} \,.

The homotopy fiber of this map

𝔤^ hofib(μ p+2) 𝔤 μ p+2 B p+1 \array{ \widehat{\mathfrak{g}} \\ {}^{\mathllap{hofib(\mu_{p+2})}}\downarrow \\ \mathfrak{g} &\underset{\mu_{p+2}}{\longrightarrow}& B^{p+1}\mathbb{R} }

is given, up to equivalence, by adjoining to CE(𝔤)CE(\mathfrak{g}) a single generator b p+1b_{p+1}

forced to be a potential for μ p+2\mu_{p+2}:

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

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Example

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The homotopy fiber of a 2-cocycle

is the classical central extension

that it classifies.

Hence:

\;\; The higher central extensions

\;\; classified by higher cocycles

\;\; are their homotopy fibers.

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Proposition

(L L_\infty-Algebra and Rational homotopy theory)

L-∞ algebras capture a broad range of aspects

of rational homotopy theory:

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1) The rational homotopy theory of simply connected topological spaces embeds

as the homotopy theory of connected L L_\infty-algebras (Quillen 69, Hinich 98):

𝔩:Spaces ,2L Alg ,1. \mathfrak{l} \;\colon\; Spaces_{\mathbb{R},\geq 2 } \overset{\simeq}{\longrightarrow} L_\infty Alg_{\mathbb{R},\geq 1} \,.

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2) The rational stable homotopy theory of rational spectra embeds

as the abelian L L_\infty-algebras: the plain chain complexes (Schwede-Shipley 03):

𝔩:Spectra L Alg ,D=[]Ch () \mathfrak{l} \;\colon\; Spectra_{\mathbb{R}} \overset{\simeq}{\longrightarrow} L_\infty Alg_{\mathbb{R}, D = [-]} \simeq Ch_\bullet(\mathbb{R})

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3) The parameterized homotopy theory, of rational parameterized spectra

over some connected rational space XX

embeds into the slice over 𝔩(X)\mathfrak{l}(X)

on those whose augmentation ideal is a chain complex VV (Schlegel-Schreiber):

𝔩:(Spectra /X) L Alg /𝔩(X),kerϵCh ()={V 𝔩(X)V 𝔩(X)}. \mathfrak{l} \;\colon\; (Spectra_{/X})_{\mathbb{R}} \overset{\simeq}{\longrightarrow} L_\infty Alg_{/\mathfrak{l}(X), ker \epsilon \in \mathrm{Ch}_\bullet(\mathbb{R})} = \left\{ \array{ V &\longrightarrow& \mathfrak{l}(X) \ltimes V \\ && \downarrow \\ && \mathfrak{l}(X) } \right\} \,.

4) Hence super L-∞ algebras generalize all this

to the rationalization of “super-homotopy theory”.

We say more about this below.

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Remark

These parameterized spectra

are the coefficients for twisted generalized cohomology.

For example for ring spectra EE then

there is a spectrum E/GL 1(E)E/GL_1(E) parameterized over BGL 1(E)B GL_1(E)

E E/GL 1(E) BGL 1(E) \array{ E &\longrightarrow& E/GL_1(E) \\ && \downarrow \\ && B GL_1(E) }

such that maps τ:XBGL 1(E)\tau \colon X \to B GL_1(E) are twists

and the τ\tau-twisted EE-cohomology is

E +τ(X)={X E/GL 1(E) τ BGL 1(E)} E^{\bullet + \tau}(X) = \left\{ \array{ X && \longrightarrow && E/GL_1(E) \\ & {}_{\mathllap{\tau}}\searrow && \swarrow \\ && B GL_1(E) } \right\}

See at tangent (∞,1)-topos for more on this.

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Example

(the single source of all following examples)

The simplest non-trivial super L-∞ algebras

are the superpoints 0|q\mathbb{R}^{0 \vert q}

defined by

CE( 0|1)=(Sym(i=1qdθ ideg=(1,odd)),d=0) CE(\mathbb{R}^{0 \vert 1}) = \left( Sym \left( \underoverset{i = 1}{q}{\oplus} \langle \underset{deg = (1,odd)}{\underbrace{\mathbf{d}\theta_i}} \rangle \right) , d = 0 \right)

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In the following we do nothing but analyzing the

consecutive higher invariant central extensions

of the superpoint.

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Spacetime

Proposition

The maximal central extension of the superpoint 0|1\mathbb{R}^{0\vert 1}

is the super Lie algebra 0,1|1\mathbb{R}^{0,1\vert 1}

called the N=1N = 1 super-worldline of the superparticle:

  • whose even part is spanned by one generator HH

  • whose odd part is spanned by one generator QQ

  • the only non-trivial bracket is

    [Q,Q]=H [Q, Q] = H

Next, the maximal central extension of 0|2\mathbb{R}^{0\vert 2}

is the d=3d = 3, N=1N = 1 super Minkowski spacetime

2,1|2 0|2. \array{ \mathbb{R}^{2,1\vert \mathbf{2}} \\ \downarrow \\ \mathbb{R}^{0\vert 2} } \,.
  • whose even part is 3\mathbb{R}^3, spanned by generators P 0,P 1,P 2P_0, P_1, P_2

  • whose odd part is 2\mathbb{R}^2, regarded as

    the Majorana spinor representation 2\mathbf{2}

    of Spin(2,1)SL(2,)Spin(2,1) \simeq SL(2,\mathbb{R})

  • the only non-trivial bracket is the spinor bilinear pairing

    [Q α,Q β]=C ααΓ a α βP a [Q_\alpha, Q'_\beta] = C_{\alpha \alpha'} \Gamma_a{}^{\alpha'}{}_\beta \,P^a

where C αβC_{\alpha \beta} is the charge conjugation matrix

(and we use Einstein summation convention and leaves sums over repeated indices implicit).

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Proof

(skip proof)

Recall that

dd-dimensional central extensions of super Lie algebras 𝔤\mathfrak{g}

are classified by 2-cocycles.

These are super-skew symmetric bilinear maps

μ 2:𝔤𝔤 d \mu_2 \;\colon\; \mathfrak{g} \wedge\mathfrak{g} \longrightarrow \mathbb{R}^d

satisfying a cocycle condition.

The extension 𝔤^\widehat{\mathfrak{g}} that this classifies

has underlying super vector space

the direct sum

𝔤^𝔤 d \widehat{\mathfrak{g}} \coloneqq \mathfrak{g} \oplus \mathbb{R}^d

an the new super Lie bracket is given

on pairs (x,c)𝔤 d(x,c) \in \mathfrak{g} \oplus \mathbb{R}^d

by

[(x 1,c 1),(x 2,c 2)] μ 2=([x 1,x 2],μ 2(c 1,c 2)). [\; (x_1,c_1), (x_2,c_2)\;]_{\mu_2} \;=\; (\, [x_1,x_2]\,,\, \mu_2(c_1,c_2) \,) \,.

The condition that the new bracket [,] μ 2[-,-]_{\mu_2} satisfies the super Jacobi identity

is equivalent to the cocycle condition on μ 2\mu_2.

Now

in the case that 𝔤= 0|q\mathfrak{g} = \mathbb{R}^{0\vert q},

then the cocycle condition is trivial

and a 2-cocycle is just a symmetric bilinear form

on the qq fermionic dimensions.

So

in the case 𝔤= 0|1\mathfrak{g} = \mathbb{R}^{0\vert 1}

there is a unique such, up to scale, namely

μ 2(aQ,bQ)=abP. \mu_2(a Q,b Q) = a b P \,.

But

in the case 𝔤= 0|2\mathfrak{g} = \mathbb{R}^{0\vert 2}

there is a 3-dimensional space of 2-cocycles, namely

μ 2((Q 1 Q 2),(Q 1 Q 2))={Q 1Q 1, 12(Q 1Q 2+Q 2Q 1), Q 2Q 2 \mu_2 \left( \left( \array{ Q_1 \\ Q_2 }\right), \left( \array{ Q'_1 \\ Q'_2 } \right) \right) = \left\{ \array{ Q_1 Q'_1, & \tfrac{1}{2}\left( Q_1 Q'_2 + Q_2 Q'_1 \right), \\ & Q_2 Q'_2 } \right.

If this is identified with the three coordinates

of 3d Minkowski spacetime

2,1(t+x y tx) \mathbb{R}^{2,1} \;\simeq\; \left( \array{ t + x & y \\ & t - x } \right)

then the pairing is the claimed one

(see at supersymmetry – in dimensions 3,4,6,10).

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This phenomenon continues:

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Definition

Given a super Lie algebra 𝔤\mathfrak{g}, write

  1. 𝔞𝔲𝔱 even(𝔤)\mathfrak{aut}_{even}(\mathfrak{g}) for the even part of its super Lie algebra of even derivations;

  2. 𝔦𝔫𝔱(𝔤)stab 𝔞𝔲𝔱 even(𝔤)(𝔤 even)\mathfrak{int}(\mathfrak{g}) \coloneqq stab_{ \mathfrak{aut}_{even}(\mathfrak{g}) }(\mathfrak{g}_{even}) for the stabilizer of the even part

    – called the infinitesimal R-symmetries.

  3. 𝔢𝔵𝔱(𝔤)=𝔞𝔲𝔱 even(𝔤)/𝔦𝔫𝔱(𝔤)\mathfrak{ext}(\mathfrak{g}) = \mathfrak{aut}_{even}(\mathfrak{g})/\mathfrak{int}(\mathfrak{g}) the external automorphisms, making a short exact sequence

    0int(𝔤)𝔞𝔲𝔱 even(𝔤)𝔢𝔵𝔱(𝔤)0 0 \to \mathrm{int}(\mathfrak{g}) \hookrightarrow \mathfrak{aut}_{even}(\mathfrak{g}) \longrightarrow \mathfrak{ext}(\mathfrak{g}) \to 0

If that sequence splits and if 𝔞𝔲𝔱 even(𝔤)\mathfrak{aut}_{even}(\mathfrak{g}) is reductive

then we consider the semisimple direct summand

ext ssimple(𝔤)𝔢𝔵𝔱(𝔤)𝔞𝔲𝔱 even(𝔤). \mathrm{ext}_{ssimple}(\mathfrak{g}) \hookrightarrow \mathfrak{ext}(\mathfrak{g}) \hookrightarrow \mathfrak{aut}_{even}(\mathfrak{g}) \,.

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Theorem

(Huerta-Schreiber 17)

The diagram of super Lie algebras shown on the right

is obtained by consecutively forming

maximal central extensions

invariant with respect to

the semisimple external automorphisms (def. )

at the given stage.

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

is the dd-dimensional, N\mathbf{N}-supersymmetric

super-translation algebra (see at geometry of physics – supersymmetry).

CE( d1,1|N)=(Sym(e a a=0 d1deg=(1,even)ψ α α=1 Ndef=(1,odd)), dψ α=0,de a=ψ¯Γ aψ=(CΓ a) αβψ αψ βspinor-to-vector-pairing in real spin repN) CE(\mathbb{R}^{d-1,1\vert \mathbf{N}}) = \left( \array{ Sym\left( \underset{deg = (1,even)}{\underbrace{\langle e^a \rangle_{a = 0}^{d-1}}} \oplus \underset{def = (1,odd)}{\underbrace{\langle \psi^\alpha \rangle_{\alpha = 1}^N}} \right) , \\ { d \psi^\alpha = 0 \,, d e^a = \underset{ \text{spinor-to-vector-pairing in real spin rep}\, \mathbf{N} }{\underbrace{\overline{\psi} \wedge \Gamma^a \psi = (C \Gamma^a)_{\alpha \beta} \psi^\alpha \wedge \psi^\beta}} } } \right)

And these semisimple subgroups are in each case

the spin group covers Spin(d1,1)Spin(d-1,1)

of the proper orthochronous Lorentz groups SO +(d1,1)SO^+(d-1,1).

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Conclusion:

Just from studying iterated invariant central extensions

of super Lie algebras,

starting with the superpoint,

we (re-)discover

  1. Lorentzian geometry,

  2. spin geometry.

  3. super spacetimes.

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May we extend further?

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Branes

Proposition

(Achúcarro-Evans-Townsend-Wiltshire 87, Brandt 12-13)

The space of maximal invariant indecomposable cocycles

  1. on 10d super Minkowski spacetime 9,1|N\mathbb{R}^{9,1 \vert \mathbf{N}}

    is spanned by a single 3-cocycle

    μ F1=(ψ¯Γ aψ)e a \mu_{F1} = \left(\overline{\psi} \wedge \Gamma_a \psi\right) \wedge e^a
  2. on 11d super Minkowski spacetime 10,1|32\mathbb{R}^{10,1\vert\mathbf{32}} is spanned by the 4-cocycle

    μ M2=i(ψ¯Γ abψ)e ae b \mu_{M2} = i \left(\overline{\psi} \wedge \Gamma_{a b} \psi \right) \wedge e^a \wedge e^b

This classification is also known as

the old brane scan.

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Definition

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Name the homotopy fibers of these cocycles

as follows

𝔪2𝔟𝔯𝔞𝔫𝔢 hofib(μ M2) 10,1|32 μ M2 B 3 \array{ \mathfrak{m}2\mathfrak{brane} \\ {}^{\mathllap{hofib}(\mu_{M2})}\downarrow \\ \mathbb{R}^{10,1\vert \mathbf{32}} &\underset{\mu_{M2}}{\longrightarrow}& B^3 \mathbb{R} }

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𝔰𝔱𝔯𝔦𝔫𝔤 IIB hofib(μ F1 B) 9,1|16+16 μ F1 B B 2𝔰𝔱𝔯𝔦𝔫𝔤 het hofib(μ F1 het) 9,1|16 μ F1 het B 2𝔰𝔱𝔯𝔦𝔫𝔤 IIA hofib(μ F1 A) 9,1|16+16¯ μ F1 A B 2 \array{ \mathfrak{string}_{IIB} \\ {}^{\mathllap{hofib}(\mu_{F1}^B)}\downarrow \\ \mathbb{R}^{9,1\vert \mathbf{16}+ \mathbf{16}} &\underset{\mu_{F1}^B}{\longrightarrow}& B^2 \mathbb{R} } \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \array{ \mathfrak{string}_{het} \\ {}^{\mathllap{hofib}(\mu_{F1}^{het})}\downarrow \\ \mathbb{R}^{9,1\vert \mathbf{16}} &\underset{\mu_{F1}^{het}}{\longrightarrow}& B^2 \mathbb{R} } \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \array{ \mathfrak{string}_{IIA} \\ {}^{\mathllap{hofib}(\mu_{F1}^A)}\downarrow \\ \mathbb{R}^{9,1\vert \mathbf{16}+ \overline{\mathbf{16}}} &\underset{\mu_{F1}^A}{\longrightarrow}& B^2 \mathbb{R} }

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By prop.

the superstring super Lie 2-algebra 𝔰𝔱𝔯𝔦𝔫𝔤 IIA/B\mathfrak{string}_{IIA/B} is given by

CE(𝔰𝔱𝔯𝔦𝔫𝔤 IIA/B)={de a=ψ¯Γ aψ,dψ α=0 df 2 IIA/B=μ F1 IIA/B=(ψ¯Γ aψ)e a} CE(\mathfrak{string}_{IIA/B}) = \left\{ \array{ d e^a = \overline{\psi} \Gamma^a \psi, \; d \psi^\alpha = 0 \\ d f_2^{IIA/B} = \mu_{F1}^{IIA/B} = (\overline{\psi} \wedge \Gamma_a \psi)\wedge e^a } \right\}

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The membrane super Lie 3-algebra 𝔪2𝔟𝔯𝔞𝔫𝔢\mathfrak{m}2\mathfrak{brane} is given by

CE(𝔪2𝔟𝔯𝔞𝔫𝔢)={de a=ψ¯Γ aψ,dψ α=0 dh 3=μ M2=i(ψ¯Γ abψ)e ae b} CE(\mathfrak{m}2\mathfrak{brane}) = \left\{ \array{ d e^a = \overline{\psi} \wedge \Gamma^a \psi, \; d \psi^\alpha = 0 \\ d h_3 = \mu_{M2} = i (\overline{\psi} \wedge \Gamma_{a b} \psi) \wedge e^a \wedge e^b } \right\}

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This dg-algebra was first considered in D’Auria-Fré 82 (3.15)

as a tool for constructing 11-dimensional supergravity.

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Hence the progression

of maximal invariant extensions of the superpoint

continues as a diagram

of super L-∞ algebras like so:

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(While every extension displayed is a maximal invariant higher central extension, not all invariant higher central extensions are displayed. For instance there are string and membrane cocycles also on the lower dimensional super-Minkowski spacetimes (“non-critical”), e.g. the super 1-brane in 3d and the super 2-brane in 4d.)

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The progression of invariant higher central extensions continues:

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Proposition

(D’Auria-Fré 82, (3.27, 3.28))

The space of indecomposable invariant higher cocylces on 𝔪2𝔟𝔯𝔞𝔫𝔢\mathfrak{m}2\mathfrak{brane} is spanned by

μ M5 15!(ψ¯Γ a 1a 5ψ)e a 1e a 5+h 3μ M2 =15!(ψ¯Γ a 1a 5ψ)e a 1e a 5+h 3i2(ψ¯Γ a 1a 2ψ)e a 1e a 2. \begin{aligned} \mu_{M5} & \coloneqq \tfrac{1}{5!} \left( \overline{\psi} \wedge \Gamma_{a_1 \cdots a_5} \psi \right) \wedge e^{a_1} \wedge \cdots \wedge e^{a_5} \;+\; h_3 \wedge \mu_{M2} \\ & = \tfrac{1}{5!} \left( \overline{\psi} \wedge \Gamma_{a_1 \cdots a_5} \psi \right) \wedge e^{a_1} \wedge \cdots \wedge e^{a_5} \;+\; h_3 \, \wedge\, \tfrac{i}{2} \left( \overline{\psi} \Gamma_{a_1 a_2} \psi \right) \wedge e^{a_1} \wedge e^{a_2} \end{aligned} \,.

where h 3h_3 is the extra generator of CE(𝔪2𝔟𝔯𝔞𝔫𝔢)CE(\mathfrak{m}2\mathfrak{brane}).

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Proposition

(Chryssomalakos-Azcárraga-Izquierdo-Bueno 99, Sakaguchi 99)

Define the following elements in CE(𝔰𝔱𝔯𝔦𝔫𝔤 IIA)CE(\mathfrak{string}_{IIA}):

μ D0 ψ¯Γ 10ψ μ D2 (μ M2)| 8+1 =i2(ψ¯Γ a 1a 2ψ)e a 1e a 2, μ D4 (π 10) *(μ M5) =+14!(ψ¯Γ a 1a 4Γ 10ψ)e a 1e a 4 μ D6 i6!(ψ¯Γ a 1a 6ψ)e a 1e a 6, μ D8 18!(ψ¯Γ a 1a 8Γ 10ψ)e a 1e a 8, μ D10 i10!(ψ¯Γ a 1a 10ψ)e a 1e a 10, \begin{aligned} \mu_{{}_{D0}} &\coloneqq \overline{\psi}\Gamma^{10} \psi \\ \mu_{{}_{D2}} &\coloneqq (\mu_{{}_{M2}})|_{8+1} \\ & = \tfrac{i}{2} \left(\overline{\psi} \wedge \Gamma_{a_1 a_2} \psi \right) \wedge e^{a_1} \wedge e^{a_2}\;, \\ \mu_{{}_{D4}} &\coloneqq (\pi_{10})_\ast( \mu_{{}_{M5}} ) \\ & = +\tfrac{1}{4!} \left( \overline{\psi} \wedge \Gamma_{a_1\cdots a_4} \Gamma_{10}\psi \right) \wedge e^{a_1} \wedge \cdots \wedge e^{a_4} \\ \mu_{{}_{D6}} & \coloneqq \tfrac{i}{6!} \left(\overline{\psi} \Gamma_{a_1 \cdots a_6}\psi \right) \wedge e^{a_1} \wedge \cdots \wedge e^{a_6} , \\ \mu_{{}_{D8}} & \coloneqq \tfrac{1}{8!} \left( \overline{\psi} \Gamma_{a_1 \cdots a_8} \Gamma_{10} \psi \right) \wedge e^{a_1}\wedge \cdots \wedge e^{a_8} , \\ \mu_{{}_{D10}} & \coloneqq \tfrac{i}{10!} \left( \overline{\psi} \Gamma_{a_1 \cdots a_{10}} \psi \right) \wedge e^{a_1}\wedge \cdots \wedge e^{a_{10}} \,, \end{aligned}

Write

C IIAμ D0+μ D2++μ D10 C^{IIA} \coloneqq \mu_{D0} + \mu_{D2} + \cdots + \mu_{D10}

for their sum.

Then the indecomposable invariant cocycles on 𝔰𝔱𝔯𝔦𝔫𝔤 IIA\mathfrak{string}_{IIA} are spanned by

ω 2p[exp(f 2 IIA)C IIA] 2p, \omega_{2p} \coloneqq \left[ \exp(-f_2^{IIA}) \wedge C^{IIA} \right]_{2p} \,,

where f 2 IIAf_2^{IIA} is the extra generator of CE( 9,1|16+16¯)CE(\mathbb{R}^{9,1\vert \mathbf{16} + \overline{\mathbf{16}}}).

Similarly define the following elements in CE(𝔰𝔱𝔯𝔦𝔫𝔤 IIB)CE(\mathfrak{string}_{IIB})

c 2 IIB ψ¯Γ 9 IIBψ=ψ¯Γ B 9ψ, μ F1 IIB :=i(ψ¯Γ a IIBΓ 10ψ)e a μ D1 i(ψ¯Γ a IIBΓ 9ψ)e a, μ D3 13!(ψ¯Γ a 1a 3 IIB(Γ 9Γ 10)ψ)e a 1e a 3, μ D5 i5!(ψ¯Γ a 1a 5 IIBΓ 9ψ)e a 1e a 5, μ D7 =17!(ψ¯Γ a 1a 7 IIB(Γ 9Γ 10)ψ)e a 1e a 7, μ D9 =i9!(ψ¯Γ a 1a 9 IIBΓ 9ψ)e a 1e a 9. \begin{aligned} c_2^{\mathrm{IIB}} & \coloneqq \overline{\psi} \Gamma_9^{\mathrm{IIB}} \psi = \overline{\psi} \Gamma^9_B \psi\;, \\ \mu_{F1}^{\mathrm{IIB}} &:= i \left(\overline{\psi} \Gamma_a^{\mathrm{IIB}} \Gamma_{10} \psi \right) \wedge e^a \\ \mu_{{}_{D1}} & \coloneqq i \left(\overline{\psi} \Gamma_a^{\mathrm{IIB}} \Gamma_9 \psi\right) \wedge e^{a}\;, \\ \mu_{{}_{D3}} & \coloneqq \tfrac{1}{3!} \left( \overline{\psi} \Gamma^{\mathrm{IIB}}_{a_1 \cdots a_3} (\Gamma_9 \Gamma_{10}) \psi \right) \wedge e^{a_1} \wedge \cdots \wedge e^{a_3}\;, \\ \mu_{{}_{D5}} & \coloneqq \tfrac{i}{5!} \left( \overline{\psi} \Gamma^{\mathrm{IIB}}_{a_1 \cdots a_5} \Gamma_9 \psi \right) \wedge e^{a_1}\wedge \cdots \wedge e^{a_5}\;, \\ \mu_{{}_{D7}} & = \tfrac{1}{7!} \left( \overline{\psi} \Gamma^{\mathrm{IIB}}_{a_1 \cdots a_7} (\Gamma_9 \Gamma_{10}) \psi \right) \wedge e^{a_1} \wedge \cdots \wedge e^{a_7}\;, \\ \mu_{{}_{D9}} & = \tfrac{i}{9!} \left( \overline{\psi} \Gamma^{\mathrm{IIB}}_{a_1\cdots a_9}\Gamma_9 \psi \right) \wedge e^{a_1} \wedge \cdots \wedge e^{a_9}\;. \end{aligned}

Then the indecomposable invariant cocycles on 𝔰𝔱𝔯𝔦𝔫𝔤 IIB\mathfrak{string}_{IIB} are spanned by

ω 2p+2[exp(f 2 IIB)C IIB] 2p+1. \omega_{2p+2} \coloneqq \left[ \exp(-f_2^{IIB}) \wedge C^{IIB} \right]_{2p+1} \,.

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As above, these cocycles classify

further higher super L L_\infty-algebra extensions

𝔡p𝔟𝔯𝔞𝔫𝔢 hofib(μ Dp) 𝔰𝔱𝔯𝔦𝔫𝔤 IIA/B μ Dp B p+1𝔪5𝔟𝔯𝔞𝔫𝔢 hofib(μ M5) 𝔪2𝔟𝔯𝔞𝔫𝔢 μ M5 B 6 \array{ \mathfrak{d}p\mathfrak{brane} \\ {}^{\mathllap{hofib(\mu_{D p})}}\downarrow \\ \mathfrak{string}_{IIA/B} &\underset{\mu_{D p}}{\longrightarrow}& B^{p+1}\mathbb{R} } \;\;\;\;\;\;\;\;\,\;\;\;\;\;\;\;\;\;\;\; \array{ \mathfrak{m}5\mathfrak{brane} \\ {}^{\mathllap{hofib(\mu_{M5})}}\downarrow \\ \mathfrak{m}2\mathfrak{brane} &\underset{\mu_{M5}}{\longrightarrow}& B^6 \mathbb{R} }

\,

In conclusion:

by forming

iterated (maximal) invariant higher central super L L_\infty-extensions

of the superpoint,

we obtain the following brane bouquet-diagram:

\,

\,

Fields

Now given one stage in the brane bouquet

𝔤^^ hofib(μ p 2) 𝔤^ μ p 2 B𝔥 2 hofib(μ p 1) 𝔤 μ p 1 B𝔥 1 \array{ \widehat{\widehat{\mathfrak{g}}} \\ {}^{\mathllap{hofib(\mu_{p_2})}}\downarrow \\ \hat \mathfrak{g} & \stackrel{\mu_{p_2}}{\longrightarrow} & B\mathfrak{h}_2 \\ {}^{\mathllap{hofib(\mu_{p_1})}}\downarrow \\ \mathfrak{g} &\overset{\mu_{p_1}}{\longrightarrow}& B\mathfrak{h}_1 }

we want to descend μ p 2\mu_{p_2} to 𝔤\mathfrak{g}.

\,

By the general theory of principal ∞-bundles (Nikolaus-Schreiber-Stevenson 12):

  1. 𝔤^\widehat{\mathfrak{g}} has a 𝔥 1\mathfrak{h}_1-∞-action

  2. equipping B𝔥 2B \mathfrak{h}_2 with an 𝔥 1\mathfrak{h}_1-∞-action

    is equivalent to finding a homotopy fiber sequence as on the right here:

    𝔤^ μ 2 B𝔥 2 hofib(μ 1) hofib(p ρ) 𝔤 (B𝔥 2)/𝔥 1 μ 1 p ρ B𝔥 1. \array{ \hat \mathfrak{g} && \stackrel{\mu_2}{\longrightarrow} && B\mathfrak{h}_2 \\ {}^{\mathllap{hofib(\mu_1)}}\downarrow && && \downarrow^{\mathrlap{hofib(p_\rho)}} \\ \mathfrak{g} && && (B\mathfrak{h}_2)/\mathfrak{h}_1 \\ & {}_{\mathllap{\mu_1}}\searrow && \swarrow_{\mathrlap{p_\rho}} \\ && B\mathfrak{h}_1 } \,.
  3. μ 2\mu_2 is 𝔥 1\mathfrak{h}_1-equivariant precisely if it descends to a morphism

    μ 2/𝔥 1:𝔤(B𝔥 2)/𝔥 1 \mu_2/\mathfrak{h}_1 \;\colon\; \mathfrak{g} \longrightarrow (B\mathfrak{h}_2)/\mathfrak{h}_1

    such that this diagram commutes up to homotopy:

    𝔤^ μ 2 B𝔥 2 hofib(μ 1) hofib(p ρ) 𝔤 μ 2/𝔥 1 (B𝔥 2)/𝔥 1 μ 1 p ρ B𝔥 1. \array{ \hat \mathfrak{g} && \stackrel{\mu_2}{\longrightarrow} && B\mathfrak{h}_2 \\ {}^{\mathllap{hofib(\mu_1)}}\downarrow && && \downarrow^{\mathrlap{hofib(p_\rho)}} \\ \mathfrak{g} && \stackrel{\mu_2/\mathfrak{h}_1}{\longrightarrow} && (B\mathfrak{h}_2)/\mathfrak{h}_1 \\ & {}_{\mathllap{\mu_1}}\searrow && \swarrow_{\mathrlap{p_\rho}} \\ && B\mathfrak{h}_1 } \,.
  4. if so, then resulting triangle diagram

    𝔤 μ 2/𝔥 1 (B𝔥 2)/𝔥 1 μ 1 p ρ B𝔥 1 \array{ \mathfrak{g} && \stackrel{\mu_2/\mathfrak{h}_1}{\longrightarrow} && (B\mathfrak{h}_2)/\mathfrak{h}_1 \\ & {}_{\mathllap{\mu_1}}\searrow && \swarrow_{\mathrlap{p_\rho}} \\ && B\mathfrak{h}_1 }

    exhibits μ 2/𝔥 1\mu_2/\mathfrak{h}_1 as a cocycle in (rational) μ 1\mu_1-twisted cohomology

    with respect to the local coefficient bundle p ρp_\rho.

\,

We now work out this general prescription

for the cocycles in the brane bouquet.

\,

RR-fields

By the brane bouquet above

the type IIA D-branes

are given by super L L_\infty cocycles of the form

𝔰𝔱𝔯𝔦𝔫𝔤 IIAμ DpB p+1 \mathfrak{string}_{IIA } \overset{\mu_{Dp}}{\longrightarrow} B^{p+1}\mathbb{R}

for p{0,2,4,6,8,10}p \in \{0,2,4,6,8,10\}.

\,

Notice that

H (BU,) H^\bullet(B U, \mathbb{Z})

has one generator in each even degree, the universal Chern classes.

Hence the L L_\infty-algebra

𝔩(KU) \mathfrak{l}(KU)

is given by

CE(𝔩(KU)){dω 2p+2=0|p}. CE(\mathfrak{l}(KU)) \;\simeq\; \left\{ d \omega_{2p+2} = 0 \;\vert\; p \in \mathbb{Z} \right\} \,.

This allows to unify the D-brane cocycles

into a single morphism of super L L_\infty-algebras of the form

𝔰𝔱𝔯𝔦𝔫𝔤 IIA AAμ DAA 𝔩(KU) hofib(μ F1) 9,1|16+16¯ μ F1 B 2. \array{ \mathfrak{string}_{IIA} && \stackrel{\phantom{AA}\mu_D\phantom{AA}}{\longrightarrow} && \mathfrak{l}(KU) \\ {}^{\mathllap{hofib(\mu_{F1})}}\downarrow \\ \mathbb{R}^{9,1\vert \mathbf{16}+ \overline{\mathbf{16}}} \\ & {}_{\mathllap{\mu_{F1}}}\searrow \\ && B^2 \mathbb{R} } \,.

\,

By the above prescription, descending μ D\mu_D is equivalent

to finding a commuting diagram in the homotopy category of super L L_\infty-algebras

of the form

𝔰𝔱𝔯𝔦𝔫𝔤 IIA AAμ DAA 𝔩(KU) hofib(μ F1) hofib(ϕ) 9,1|16+16¯ AAμ D/BAA something μ F1 ϕ B 2. \array{ \mathfrak{string}_{IIA} && \stackrel{\phantom{AA}\mu_D\phantom{AA}}{\longrightarrow} && \mathfrak{l}(KU) \\ {}^{\mathllap{hofib(\mu_{F1})}}\downarrow && && \downarrow^{\mathrlap{hofib(\phi)}} \\ \mathbb{R}^{9,1\vert \mathbf{16}+ \overline{\mathbf{16}}} &&\overset{\phantom{AA}\mu_{D}/B \mathbb{R}\phantom{AA} }{\longrightarrow}&& \text{something} \\ & {}_{\mathllap{\mu_{F1}}}\searrow && \swarrow_{\mathrlap{\phi}} \\ && B^2 \mathbb{R} } \,.

\,

This turns out to exist as follows (Fiorenza-Sati-Schreiber 16a, section 5):

Definition

Following prop. define the L L_\infty-algebra

𝔩(KU/BU(1)) \mathfrak{l}(\,KU / BU(1)\,)

by

CE(𝔩(KU/BU(1)))={dh 3=0, dω 2p+2=h 3ω 2p}. CE\left(\,\mathfrak{l}(\,KU / BU(1)\,)\,\right) \;=\; \left\{ \array{ d h_3 = 0\;,\; \\ d \omega_{2p+2} = h_3 \wedge \omega_{2p} } \right\} \,.

Moreover write

res 9,1|16+16¯ \mathbb{R}^{9,1\vert \mathbf{16} + \overline{\mathbf{16}}}_{res}

for the super L L_\infty-algebra whose Chevalley-Eilenberg algebra is

CE( res 9,1|16+16¯)[f 2,h 3]/(df 2=μ F1+h 3) CE\left( \mathbb{R}^{9,1\vert \mathbf{16} + \overline{\mathbf{16}}}_{res} \right)[f_2,h_3]/(d f_2 = \mu_{F1} + h_3)

\,

Proposition

(Fiorenza-Sati-Schreiber 16a, theorem 4.16)

The super L L_\infty-algebra res 9,1|16+16¯\mathbb{R}^{9,1\vert \mathbf{16} + \overline{\mathbf{16}}}_{res}

is a resolution of type IIA super-Minkowski spacetime.

in that there is a weak equivalence

res 9,1|16+16¯AAAA 9,1|16+16¯. \mathbb{R}^{9,1\vert \mathbf{16} + \overline{\mathbf{16}}}_{res} \stackrel{\phantom{AA}\simeq\phantom{AA}}{\longrightarrow} \mathbb{R}^{9,1\vert \mathbf{16} + \overline{\mathbf{16}}} \,.

This fits into a commuting diagram of the form

{de a=ψ¯Γ aψdψ α=0df 2=μ F1} 𝔰𝔱𝔯𝔦𝔫𝔤 IIA μ D 𝔩(KU) {dω 2p+2=0} hofib(μ F1) ϕ {de a=ψ¯Γ aψdψ α=0df 2=μ F1+h 3} res 9,1|16+16¯ μ F1/D 𝔩(KU/BU(1)) {h 3=0 dω 2p+2=h 3ω 2p} μ F1 ϕ B 2 {dh 3=0}. \array{ \left\{ {{d e^a = \overline{\psi}\Gamma^a \wedge \psi } \atop {d \psi^\alpha = 0}} \atop { d f_2 = \mu_{F1}} \right\} && \mathfrak{string}_{IIA} && \stackrel{ \mu_D }{\longrightarrow} && \mathfrak{l}(KU) && \left\{ d \omega_{2 p+2} = 0 \right\} \\ && \downarrow^{\mathrlap{hofib(\mu_{F1})}} && && \downarrow^{\mathrlap{\phi}} \\ \left\{ {{d e^a = \overline{\psi}\Gamma^a \wedge \psi } \atop {d \psi^\alpha = 0}} \atop { d f_2 = \mu_{F1} + h_3 } \right\} & & \mathbb{R}_{res}^{9,1\vert \mathbf{16}+ \overline{\mathbf{16}}} && \stackrel{ \mu_{F1/D} }{\longrightarrow} && \mathfrak{l}(KU / B U(1)) && \left\{ \array{ h_3 = 0 \\ d\omega_{2p+2} = h_3\wedge \omega_{2p} } \right\} \\ && & {}_{\mathllap{\mu_{F 1}}}\searrow && \swarrow_{\mathrlap{\phi}} \\ && && B^2 \mathbb{R} \\ && && \left\{ d h_3 = 0 \right\} } \,.

\,

In conclusion

\;\;the type IIA F1-brane and D-brane cocycles with \mathbb{R}-coefficients

\;\;do descend to super-Minkowski spacetime

\;\;as one single cocycle with coefficients

\;\;in rationalized twisted K-theory.

\,

M-flux fields

(skip)

The part of the brane bouquet giving the M-branes is

𝔪2𝔟𝔯𝔞𝔫𝔢 μ M5 B 6 hofib(μ M2) 10,1|32 μ M2 B 3 hofib(μ D0) 9,1|16+16¯ \array{ \mathfrak{m}2\mathfrak{brane} &\stackrel{\mu_{M5}}{\longrightarrow}& B^6 \mathbb{R} \\ {}^{\mathllap{hofib(\mu_{M2})}}\downarrow \\ \mathbb{R}^{10,1\vert \mathbf{32}} & \stackrel{\mu_{M2}}{\longrightarrow} & B^3 \mathbb{R} \\ {}^{\mathllap{hofib(\mu_{D 0})}}\downarrow \\ \mathbb{R}^{9,1\vert \mathbf{16} + \overline{\mathbf{16}}} }

\,

In order to descend this, consider the L L_\infty-algebra corresponding to the 4-sphere

𝔩(S 4). \mathfrak{l}(\,S^4\,) \,.

By standard facts on rational n-spheres, this is given by

CE(𝔩S 4){dg 4=0, dg 7+12g 4 4=0}. CE(\mathfrak{l}S^4) \;\simeq\; \left\{ \array{ d \, g_4 = 0\,,\, \\ d\, g_7 + \tfrac{1}{2} g_4 \wedge _4 = 0 } \right\} \,.
Proposition

(Fiorenza-Sati-Schreiber 15, section 3)

There is a homotopy fiber sequence of L L_\infty-algebras as on the right

( S 7 S 4 BSU(2) c 2 B 3U(1))AA𝔩()AA( B 6 hofib(𝔩(c 2)) 𝔩(S 4) 𝔩(c 2) B 3) \left( \array{ && S^7 \\ && \downarrow \\ && S^4 \\ & \swarrow \\ B SU(2) \\ \downarrow^{\mathrlap{c_2}} \\ B^3 U(1) } \right) \;\; \stackrel{\phantom{AA}\mathfrak{l}(-)\phantom{AA} }{\mapsto} \;\; \left( \array{ && B^6 \mathbb{R} \\ && \downarrow^{\mathrlap{ hofib(\mathfrak{l}(c_2)) } } \\ &&\mathfrak{l}(S^4) \\ & \swarrow_{\mathfrak{l}(c_2)} \\ B^3 \mathbb{R} } \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \right)

which is the image under 𝔩()\mathfrak{l}(-) of the quaternionic Hopf fibration.

This makes a commuting diagram

in the homotopy category of super L L_\infty-algebas

of the form

{de a=ψ¯Γ aψdψ α=0dh 3=μ M2} 𝔪2𝔟𝔯𝔞𝔫𝔢 μ M5 B 6 {dg 7=0} hofib(μ M2) hofib(𝔩(c 2)) {de a=ψ¯Γ aψdψ α=0dh 3=g 4μ M2} res 10,1|32 μ M2/M5 𝔩S 4 {dg 4=0dg 7+12g 4g 4=0} μ M2 𝔩(c 2) B 3 {dg 4=0} \array{ \left\{ { { d e^a = \overline{\psi}\wedge \Gamma^a \wedge \psi } \atop { d \psi^\alpha = 0 } } \atop d h_3 = - \mu_{M2} \right\} && \mathfrak{m}2\mathfrak{brane} && \stackrel{ \mu_{M5} }{\longrightarrow} && B^6 \mathbb{R} && \left\{ d g_7 = 0 \right\} \\ && \downarrow^{\mathrlap{hofib(\mu_{M2})}} && && \downarrow^{\mathrlap{hofib(\mathfrak{l}(c_2))}} \\ \left\{ { { d e^a = \overline{\psi}\wedge \Gamma^a \wedge \psi } \atop { d \psi^\alpha = 0 } } \atop d h_3 = g_4 - \mu_{M2} \right\} && \mathbb{R}_{res}^{10,1\vert\mathbf{32}} && \stackrel{ \mu_{M2/M5} }{\longrightarrow} && \mathfrak{l} S^4 && \left\{ {d g_4 = 0} \atop {d g_7 + \tfrac{1}{2} g_4 \wedge g_4 = 0} \right\} \\ && & {}_{\mathllap{\mu_{M2}}}\searrow && \swarrow_{\mathrlap{\mathfrak{l}(c_2)}} \\ && && B^3 \mathbb{R} \\ && && \left\{ d g_4 = 0\right\} }

In conclusion

\;\;this says that, rationally,

\;\;M2-brane charge is in degree-4 ordinary cohomology

\;\;and it twists M5-brane charge

\;\;which is, rationally, in unstable degree-4 cohomotopy.

\,

Now that we have found

the descended L L_\infty-cocycles

for all super pp-branes

in twisted cohomology, rationally,

we may analyze their behaviour under double dimensional reduction

and discover the super L L_\infty-algebraic incarnation

of various dualities in string theory.

\,

\,

Reduction

\,

We have discovered higher cocycles

on higher dimensional super-spacetimes.

Now we discuss that there is double dimensional reduction

which takes these to lower-degree cocycle on lower-dimensional space.

This is given by passage to cyclic loop spaces:

\,

Proposition

Let H\mathbf{H} be any (∞,1)-topos

such as

and let GGrp(H)G \in Grp(\mathbf{H}) be an ∞-group in H\mathbf{H},

then right base change along *BG\ast \to \mathbf{B}G is a pair of adjoint ∞-functors of the form

HAA[G,]/GAAAAhofibAAH /BG, \mathbf{H} \underoverset {\underset{\phantom{AA}[G,-]/G \phantom{AA}}{\longrightarrow}} {\overset{\phantom{AA} hofib \phantom{AA} }{\longleftarrow}} {\bot} \mathbf{H}_{/\mathbf{B}G} \,,

where

∞-action for GG equipped with its canonical ∞-action by left multiplication and the argument

regarded as equipped with its trivial GG-\infty-action;

for G=S 1G = S^1 the circle group, then this is the cyclic loop space ()/S 1\mathcal{L}(-)/S^1.

Hence for

then there is a natural equivalence

H(X^,A)originalfluxesoxidationreductionH(X,[G,A]/G)doublydimensionally reducedfluxes \underset{ \text{original} \atop \text{fluxes} }{ \underbrace{ \mathbf{H}(\hat X\;,\; A) } } \;\; \underoverset {\underset{oxidation}{\longleftarrow}} {\overset{reduction}{\longrightarrow}} {\simeq} \;\; \underset{ \text{doubly} \atop { \text{dimensionally reduced} \atop \text{fluxes} } }{ \underbrace{ \mathbf{H}(X \;,\; [G,A]/G) } }

given by

{X^ ϕ A hofib(c) X c BG}{X ϕ˜ [G,A]/G c BG} \left\{ \array{ \hat X &\overset{\phi}{\longrightarrow}& A \\ \downarrow^{{\mathrlap{hofib(c)}}} \\ X \\ & {}_{\mathllap{c}}\searrow \\ && \mathbf{B} G } \right\} \;\;\; \leftrightarrow \;\;\; \left\{ \array{ X && \overset{\tilde \phi}{\longrightarrow} && [G,A]/G \\ & {}_{\mathllap{c}}\searrow && \swarrow \\ && \mathbf{B}G } \right\}
Proof

(skip proof)

First observe that the conjugation action on [G,X][G,X] is the internal hom in the (∞,1)-category of GG-∞-actions Act G(H)Act_G(\mathbf{H}). Under the equivalence of (∞,1)-categories

Act G(H)H /BG Act_G(\mathbf{H}) \simeq \mathbf{H}_{/\mathbf{B}G}

(from Nikolaus-Schreiber-Stevenson 12) then GG with its canonical ∞-action is (*BG)(\ast \to \mathbf{B}G) and XX with the trivial action is (X×BGBG)(X \times \mathbf{B}G \to \mathbf{B}G).

Hence

[G,X]/G[*,X×BG] BGH /BG. [G,X]/G \simeq [\ast, X \times \mathbf{B}G]_{\mathbf{B}G} \;\;\;\;\; \in \mathbf{H}_{/\mathbf{B}G} \,.

Actually, this is the very definition of what [G,X]/GH /BG[G,X]/G \in \mathbf{H}_{/\mathbf{B}G} is to mean in the first place, abstractly.

But now since the slice (∞,1)-topos H /BG\mathbf{H}_{/\mathbf{B}G} is itself cartesian closed, via

E× BG()[E,] BG E \times_{\mathbf{B}G}(-) \;\;\; \dashv \;\;\; [E,-]_{\mathbf{B}G}

it is immediate that there is the following sequence of natural equivalences

H /BG(Y,[G,X]/G) H /BG(Y,[*,X×BG] BG) H /BG(Y× BG*,X×BGp *X) H(p !(Y× BG*)hofib(Y),X) H(hofib(Y),X) \begin{aligned} \mathbf{H}_{/\mathbf{B}G}(Y, [G,X]/G) & \simeq \mathbf{H}_{/\mathbf{B}G}(Y, [\ast, X \times \mathbf{B}G]_{\mathbf{B}G}) \\ & \simeq \mathbf{H}_{/\mathbf{B}G}( Y \times_{\mathbf{B}G} \ast, \underset{p^\ast X}{\underbrace{X \times \mathbf{B}G }} ) \\ & \simeq \mathbf{H}( \underset{hofib(Y)}{\underbrace{p_!(Y \times_{\mathbf{B}G} \ast)}}, X ) \\ & \simeq \mathbf{H}(hofib(Y),X) \end{aligned}

Here p:BG*p \colon \mathbf{B}G \to \ast denotes the terminal morphism and p !p *p_! \dashv p^\ast denotes the base change along it.

\,

We now apply this general mechanism to the brane bouquet.

\,

By the discussion of rational homotopy theory above

we may think of L-∞ algebras as rational topological spaces

and more generally as rational parameterized spectra.

For instance above we found that the coefficient space

for RR-fields in rational twisted K-theory is the

L-∞ algebra𝔩(KU/BU(1))\mathfrak{l}(KU/BU(1)).

\,

Hence in order to apply double dimensional reduction

to super p-branes

we now specialize the above formalization to

cyclification of super L-∞ algebras (FSS 16b)

\,

Definition

For 𝔤\mathfrak{g} any super L-∞ algebra of finite type, its cyclification

𝔏𝔤/sL Alg \mathfrak{L}\mathfrak{g}/\mathbb{R} \in s L_\infty Alg_{\mathbb{R}}

is defined by having Chevalley-Eilenberg algebra of the form

CE(𝔏𝔤/)( (𝔤 *originals𝔤 *shifted copyω 2new generatorin degree 2),d 𝔡𝔤/:{ω 2 0 α d 𝔤α+ω 2sα sα sd 𝔤α) CE(\mathfrak{L}\mathfrak{g}/\mathbb{R}) \coloneqq \left( \wedge^\bullet \left( \underset{\text{original}}{\underbrace{\mathfrak{g}^\ast}} \oplus \underset{\text{shifted copy}}{\underbrace{s\mathfrak{g}^\ast}} \oplus \underset{\text{new generator} \atop \text{in degree 2}}{\underbrace{\langle \omega_2 \rangle}} \right) \;,\; d_{\mathfrak{d}\mathfrak{g}/\mathbb{R}} \;\colon\; \left\{ \array{ \omega_2 &\mapsto& 0 \\ \alpha &\mapsto& d_{\mathfrak{g}} \alpha + \omega_2 \wedge s \alpha \\ s \alpha &\mapsto& - s d_{\mathfrak{g}} \alpha } \right. \right)

where

s𝔤 * s \mathfrak{g}^\ast

is a copy of 𝔤 *\mathfrak{g}^\ast with cohomological degrees shifted down by one, and where ω\omega is a new generator in degree 2.

The differential is given for α 1𝔤 *\alpha \in \wedge^1 \mathfrak{g}^\ast by

d 𝔡𝔤/:{ω 2 0 α d 𝔤α±ω 2sα sα sd 𝔤α d_{\mathfrak{d}\mathfrak{g}/\mathbb{R}} \;\colon\; \left\{ \array{ \omega_2 &\mapsto& 0 \\ \alpha &\mapsto& d_{\mathfrak{g}} \alpha \pm \omega_2 \wedge s \alpha \\ s \alpha &\mapsto& - s d_{\mathfrak{g}} \alpha } \right.

where on the right we are extendng ss as a graded derivation.

Define

𝔏𝔤sL Alg \mathfrak{L}\mathfrak{g} \in s L_\infty Alg_{\mathbb{R}}

in the same way, but with ω 20\omega_2 \coloneqq 0.

For every 𝔤\mathfrak{g} there is a homotopy fiber sequence

𝔏𝔤 𝔏𝔤/ ω 2 B \array{ && \mathfrak{L}\mathfrak{g} \\ && \downarrow \\ && \mathfrak{L} \mathfrak{g}/\mathbb{R} \\ & \swarrow_{\mathrlap{\omega_2}} \\ B \mathbb{R} }

which hence exhibits 𝔏𝔤/\mathfrak{L} \mathfrak{g}/\mathbb{R} as the homotopy quotient of 𝔏𝔤\mathfrak{L}\mathfrak{g} by an \mathbb{R}-action.

\,

The following says that the L L_\infty-cyclification from prop.

indeed does model correspond to the topological cyclification .

\,

Proposition

(Vigué-Sullivan 76, Vigué-Burghelea 85)

If

𝔤=𝔩(X) \mathfrak{g} = \mathfrak{l}(\,X\,)

is the L L_\infty-algebra associated by rational homotopy theory to a simply connected topological space XX,

then its cyclification (def. )

corresponds to the cyclic loop space of XX

𝔏(𝔩(X))/𝔩(X/S 1). \mathfrak{L}( \;\mathfrak{l}( X )\; )/\mathbb{R} \simeq \mathfrak{l}( \;\mathcal{L}X/S^1\; ) \,.

The cochain cohomology of the Chevalley-Eilenberg algebra

CE(𝔩(X/S 1)) CE(\mathfrak{l}( \;\mathcal{L}X/S^1\; ))

computes the cyclic cohomology of XX with coefficients in \mathbb{R}.

Moreover the homotopy fiber sequence of the cyclification corresponds to that of the free loop space:

(X hofib(p) X/S 1 p BS 1)AA𝔩()AA(𝔏𝔩(X) hofib(𝔩(p)) 𝔏𝔩(X)/ 𝔩(p) B) \left( \array{ \mathcal{L}X \\ \downarrow^{\mathrlap{hofib(p)}} \\ \mathcal{L}X/S^1 \\ \downarrow^{\mathrlap{p}} \\ B S^1 } \;\;\;\;\;\;\;\; \right) \;\;\;\; \stackrel{\phantom{AA}\mathfrak{l}(-)\phantom{AA}}{\mapsto} \;\;\;\; \left( \array{ \mathfrak{L} \mathfrak{l}(X) \\ \downarrow^{ \mathrlap{ hofib( \mathfrak{l}(p) ) } } \\ \mathfrak{L}\mathfrak{l}(X)/\mathbb{R} \\ \downarrow^{\mathrlap{\mathfrak{l}(p)}} \\ B \mathbb{R} } \;\;\;\;\;\;\;\; \right)

\,

We now have the following super L L_\infty-algebraic incarnation

of the general double dimensional reduction from prop. :

\,

Proposition

(Fiorenza-Sati-Schreiber 16b, prop. 3.5)

Let

𝔤^ π 𝔤 μ 2 B \array{ \widehat{\mathfrak{g}} \\ {}^{\mathllap{\pi}}\downarrow \\ \mathfrak{g} \\ & {}_{\mathllap{\mu_2}}\searrow \\ && B \mathbb{R} }

be a central extension of super L-∞ algebras. According to prop. we have

CE(𝔤^)CE(𝔤)[e]/(de=μ 2). CE(\widehat{\mathfrak{g}}) \simeq CE(\mathfrak{g})[e]/(d e = \mu_2) \,.

and hence every generator α pCE(𝔤^)\alpha_p \in CE(\widehat{\mathfrak{g}}) has a unique decomposition

α p=β peα˜ p1 \alpha_p = \beta_p - e \wedge \tilde \alpha_{p-1}

where β p\beta_p and α˜ p1\tilde \alpha_{p-1} do not involve the generator ee. We may think of this as

π *(α p)α˜ p1α p| 𝔤β p. \pi_\ast(\alpha_p) \coloneqq \tilde \alpha_{p-1} \;\;\;\;\,\; \alpha_p|_{\mathfrak{g}} \coloneqq \beta_p \,.

Under this identification any super L L_\infty-homomorphism

ϕ:𝔤^ϕ𝔥 \phi \;\colon\; \widehat{\mathfrak{g}} \overset{\phi}{\longrightarrow} \mathfrak{h}

hence a dg-algebra homomorphism

ϕ *:CE(𝔥)CE(𝔤^) \phi^\ast \;\colon\; CE(\mathfrak{h}) \longrightarrow CE(\widehat{\mathfrak{g}})

gives rise to a homomorphism of the form

ϕ˜:𝔤𝔏𝔤/ \tilde \phi \;\colon\; \mathfrak{g} \longrightarrow \mathfrak{L}\mathfrak{g}/\mathbb{R}

which, in the notation of def. , is given dually by

ϕ˜ *:{α (ϕ *α)| 𝔤 sα π *(ϕ *α) ω 2 μ 2. \tilde \phi^\ast \colon \left\{ \array{ \alpha & \mapsto (\phi^\ast \alpha)|_{\mathfrak{g}} \\ s \alpha & \mapsto \pi_\ast(\phi^\ast \alpha) \\ \omega_2 & \mapsto \mu_2 } \right. \,.

Moreover, this construction constitutes a natural bijection

Hom(𝔤^,𝔥)originalcocycles oxidationreduction Hom /B(𝔤,𝔏𝔥/)doublydimensionally reducedcocycles given by {𝔤^ μ 𝔥 hofib(c 2) 𝔤 c 2 B} {𝔤 μ˜ 𝔏𝔥/ μ 2 ω 2 B} \array{ \underset{ \text{original} \atop \text{cocycles} }{ \underbrace{ Hom( \widehat{\mathfrak{g}}, \mathfrak{h} ) }} \;&\; \underoverset {\underset{oxidation}{\longleftarrow}} {\overset{reduction}{\longrightarrow}} {\simeq} \;&\; \underset{ \text{doubly} \atop { \text{dimensionally reduced} \atop \text{cocycles} } }{ \underbrace{ Hom_{/B\mathbb{R}}( \mathfrak{g}, \mathfrak{L}\mathfrak{h}/\mathbb{R} ) } } \\ \\ \text{given by} \\ \\ \left\{ \array{ \widehat{\mathfrak{g}} &\overset{\mu}{\longrightarrow}& \mathfrak{h} \\ \downarrow^{\mathrlap{hofib(c_2)}} \\ \mathfrak{g} \\ & {}_{\mathllap{c_2}}\searrow \\ && B \mathbb{R} } \right\} \;&\; \leftrightarrow \;&\; \left\{ \array{ \mathfrak{g} && \overset{\tilde \mu}{\longrightarrow} && \mathfrak{L}\mathfrak{h}/\mathbb{R} \\ & {}_{\mathllap{\mu_2}}\searrow && \swarrow_{\mathrlap{\omega_2}} \\ && B \mathbb{R} } \right\} }

between super L L_\infty-homomorphisms out of the exteded super L L_\infty-algebra 𝔤^\widehat{\mathfrak{g}} and homomorphism out of the base 𝔤\mathfrak{g} into the cyclification (def. ) of the original coefficients with the latter constrained so that the canonical 2-cocycle on the cyclification is taken to the 2-cocycle classifying the given extension.

\,

Example

(skip example)

Let

[𝔤^ 𝔤 b][ d,1|N d+1 d1,1|N d ψ¯Γ dψ b] \left[ \array{ \widehat{\mathfrak{g}} \\ \downarrow \\ \mathfrak{g} \\ & {}_{}\searrow \\ && b \mathbb{R} } \right] \;\coloneqq\; \left[ \array{ \mathbb{R}^{d,1\vert N_{d+1}} \\ \downarrow \\ \mathbb{R}^{d-1,1\vert N_d} \\ & {}_{\mathllap{\overline{\psi}\wedge \Gamma^{d}\psi}} \searrow \\ && b \mathbb{R} } \right]

be the extension of a super Minkowski spacetime from dimension dd to dimension d+1d+1.

Let moreover

𝔥b (p+1)+1 \mathfrak{h} \coloneqq b^{(p+1)+1} \mathbb{R}

be the line Lie (p+3)-algebra

and consider any super (p+1)-brane cocycle from the old brane scan in dimension d+1d+1

μ (p+1)+2a i=0d(ψ¯Γ a 1a p+1ψ)e a 1e a p+1: d,1|N d+1b p+1. \mu_{(p+1)+2} \;\coloneqq\; \underoverset{a_i = 0}{d}{\sum} \left( \overline{\psi} \wedge \Gamma_{a_1 \cdots a_{p+1}} \psi \right) \wedge e^{a_1} \wedge \cdots \wedge e^{a_{p+1}} \;\;\colon\;\; \mathbb{R}^{d,1\vert N_{d+1}} \longrightarrow b^{p+1} \mathbb{R} \,.

Then the cyclification 𝔏(b p+1)/\mathfrak{L}(b^{p+1}\mathbb{R})/\mathbb{R} of the coefficients (prop. ) is

CE(𝔏(b p+2)/)={dω 2=0 dω p+2=0 dω (p+1)+2=ω p+1ω 2} CE\left( \, \mathfrak{L}(b^{p+2}\mathbb{R})/\mathbb{R} \, \right) \;=\; \left\{ \array{ d \omega_2 = 0 \\ d \omega_{p + 2} = 0 \\ d \omega_{(p+1)+2} = \omega_{p+1} \wedge \omega_2 } \right\}

and the dimensionally reduced cocycle

d1,1|N d 𝔏(b p+1)/ b \array{ \mathbb{R}^{d-1,1\vert N_d} && \overset{}{\longrightarrow} && \mathfrak{L}(b^{p+1}\mathbb{R})/\mathbb{R} \\ & \searrow && \swarrow \\ && b \mathbb{R} }

has the following components

μ (p+1)+2 d+1=d=0d(ψ¯Γ a 1a p+1ψ)e a 1ea p+1p+1-brane wrapped notwrapped μ 0+2 d=(ψ¯Γ dψ)0-brane μ p+2 d=a i=0d1(ψ¯Γ a 1a pψ)e a 1e a pp-brane μ p+2 d=a i=0d1(ψ¯Γ a 1a p+1ψ)e a 1e a p+1p+1-brane \array{ && && \overset{ p+1\text{-brane} }{ \overbrace{ { \mu^{d+1}_{(p+1)+2} = } \atop { \underoverset{d=0}{d}{\sum} \left( \overline{\psi} \wedge \Gamma_{a_1 \cdots a_{p+1}} \psi \right) \wedge e^{a_1} \wedge \cdots e {a_{p+1}} } } } \\ && & {}^{\mathllap{\text{wrapped}}}\swarrow && \searrow^{\mathrlap{\text{not} \atop \text{wrapped}}} \\ \underset{ \text{0-brane} }{ \underbrace{ { \mu^{d}_{0+2} = } \atop { \left( \overline{\psi} \wedge \Gamma^d \psi \right) } } } && \underset{ p\text{-brane} }{ \underbrace{ { \mu^{d}_{p+2} = } \atop { \underoverset{a_i = 0}{d-1}{\sum} \left( \overline{\psi} \wedge \Gamma_{a_1 \cdots a_{p}} \psi \right) \wedge e^{a_1} \wedge \cdots e^{a_p} } } } && && \underset{ p+1\text{-brane} }{ \underbrace{ { \mu^{d}_{p+2} = } \atop { \underoverset{a_i = 0}{d-1}{\sum} \left( \overline{\psi} \wedge \Gamma_{a_1 \cdots a_{p+1}} \psi \right) \wedge e^{a_1} \wedge \cdots e^{a_{p+1}} } } } }

It follows that with

dμ (p+1)+2 d+1=0 d \,\mu^{d+1}_{(p+1)+2} = 0

also

dμ p+2 d=0. d\, \mu^d_{p+2} = 0 \,.

This is the dimensional reduction observed in the old brane scan (Achúcarro-Evans-Townsend-Wiltshire 87)

graphics grabbed from (Duff 87)

But there is more: the un-wrapped component of the dimensionally reduced cocycle satisfies the twisted cocycle condition

dμ (p+1)+2 d=μ p+2 dμ 0+2 d. d \, \mu^d_{(p+1)+2} \;=\; \mu^d_{p+2} \wedge \mu^d_{0+2} \,.

These relations are not to be ignored.

This we turn to now.

\,

\,

T-Duality

\,

We consider now the D-brane cocycles

on type IIA and IIB superspacetime

and dimensionally reduce both to their joint 9d base space.

There we discover a hidden duality: T-duality.

\,

Proposition

(Fiorenza-Sati-Schreiber 16b, prop. 5.1)

The cyclification 𝔏𝔩(KU/BU(1))/\mathfrak{L}\mathfrak{l}(\mathrm{KU}/BU(1))/\mathbb{R} (def. ) of 𝔩(KU/BU(1))\mathfrak{l}(\mathrm{KU}/BU(1)) (def. ) has CE-algebra

CE(𝔏𝔩(KU/BU(1))/)={dc 2=0,dc˜ 2=0 dh 3=c 2c˜ 2 dω 2p+2=h 3ω 2p+c 2ω 2p+1 dω 2p+1=h 3ω 2p1+c˜ 2ω 2p}. \mathrm{CE}(\mathfrak{L}\mathfrak{l}(\mathrm{KU}/BU(1))/\mathbb{R}) = \left\{ \array{ d c_2 = 0\,,\;\;\;\;d \tilde c_2 = 0 \\ d h_3 = - c_2 \wedge \tilde c_2 \\ d \omega_{2p+2} = h_3 \wedge \omega_{2p} + c_2 \wedge \omega_{2p+1} \\ d \omega_{2p+1} = h_3 \wedge \omega_{2p-1} + \tilde c_2 \wedge \omega_{2p} } \right\}\;.

The cyclification 𝔏𝔩(ΣKU/BU(1))/\mathfrak{L}\mathfrak{l}(\Sigma\mathrm{KU}/BU(1))/\mathbb{R} of 𝔩(ΣKU/BU(1))\mathfrak{l}(\Sigma\mathrm{KU}/BU(1)) has CE-algebra

CE(𝔏𝔩(ΣKU/BU(1))/)={dc 2=0,dc˜ 2=0 dh 3=c 2c˜ 2 dω 2p+2=h 3ω 2p+c˜ 2ω 2p+1 dω 2p+1=h 3ω 2p1+c 2ω 2p}. \mathrm{CE}(\mathfrak{L}\mathfrak{l}(\Sigma\mathrm{KU}/BU(1))/\mathbb{R}) = \left\{ \begin{array}{l} d c_2 = 0 \,, \;\;\;\; d \tilde c_2 = 0 \\ d h_3 = - c_2 \wedge \tilde c_2 \\ d \omega_{2p+2} = h_3 \wedge \omega_{2p} + \tilde c_2 \wedge \omega_{2p+1} \\ d \omega_{2p+1} = h_3 \wedge \omega_{2p-1} + c_2 \wedge \omega_{2p} \end{array} \right\}\;.

Hence there is an L L_\infty-isomorphism of the form

ϕ T:𝔩((KU/BU(1))/S 1)AAϕ TAA𝔩((ΣKU/BU(1))/S 1) \phi_T \;:\; \mathfrak{l}( \mathcal{L}(\mathrm{KU}/BU(1))/S^1 ) \underoverset{\simeq}{\phantom{AA}\phi_T \phantom{AA}}{\longrightarrow} \mathfrak{l}( \mathcal{L}(\Sigma\mathrm{KU}/BU(1))/S^1 )

relating the cyclifications of the rational twisted KU-coefficients, which is given by

c 2c˜ 2,h 3h 3,ω pω p. c_2 \leftrightarrow \tilde c_2\;, \qquad h_3 \mapsto h_3\;, \qquad \omega_{p} \mapsto \omega_p \;.
Proof

(skip proof)

By def. , as a polynomial algebra the CE-algebra of 𝔏𝔩(KU/BU(1))\mathfrak{L}\mathfrak{l}(\mathrm{KU}/BU(1)) is obtained from the CE-algebra of 𝔩(KU/BU(1))\mathfrak{l}(\mathrm{KU}/BU(1)) by adding a shifted copy of each generator. We denote by ω 2p1\omega_{2p-1} the shifted copy of ω 2p\omega_{2p} and by c˜ 2-\tilde{c}_2 the shifted copy of h 3h_3. The differential is then defined by

dω 2p+2=h 3ω 2p,dω 2p+1=h 3ω 2p1+c˜ 2ω 2p,dh 3=0,dc˜ 2=0. d\omega_{2p+2}=h_3 \wedge \omega_{2p}\;, \qquad d\omega_{2p+1}=h_3 \wedge \omega_{2p-1} + \tilde{c}_2\wedge \omega_{2p}\;, \qquad dh_3=0,\qquad d\tilde{c}_2=0\;.

Next, again by def. , the CE-algebra of 𝔏𝔩(KU/BU(1))/\mathfrak{L}\mathfrak{l}(\mathrm{KU}/BU(1))/\mathbb{R} is obtained by adding a further degree 2 generator c 2c_2 and defining the differential as

dω 2p+2=h 3ω 2p+c 2ω 2p+1, dω 2p+1=h 3ω 2p1+c˜ 2ω 2p, dc 2=0,dc˜ 2=0, dh 3=c 2c˜ 2. \array{ d\omega_{2p+2}= h_3 \wedge \omega_{2p} + c_2\wedge \omega_{2p+1}\;, & \qquad d\omega_{2p+1}= h_3 \wedge \omega_{2p-1} + \tilde{c}_2\wedge \omega_{2p}\;, \\ dc_2=0\;, \qquad d\tilde{c}_2=0\;, & \qquad \quad dh_3= - c_2\wedge \tilde{c}_2\;. }

The proof for 𝔏𝔩(ΣKU/BU(1))/\mathfrak{L}\mathfrak{l}(\Sigma\mathrm{KU}/BU(1))/\mathbb{R} is completely analogous.

\,

Hence postcomposition with ϕ T\phi_T

sends the doubly dimensionally reduced IIA/B D-brane cocycles

to some other super L L_\infty-cocycles.

The following theorem says that

indeed it takes them into each other:

\,

Theorem

(Fiorenza-Sati-Schreiber 16b, theorem 5.3)

The following diagram commutes

In particular this means that

  1. (π 9 IIA/IIB) *(μ F1 IIA/IIB)=c 2 IIB/IIA(\pi^{IIA/IIB}_9)_\ast \left( \mu^{IIA/IIB}_{F1} \right) = - c_2^{IIB/IIA}.

  2. (π 9 IIA) *e 9 IIA()| 8+1:C IIAC IIB(\pi_9^{IIA})_\ast - e_9^{IIA} \wedge (-)\vert_{8+1} \;\colon\; C^{IIA} \mapsto C^{IIB}.

\,

The first of these two conditions is

the super L L_\infty-version of the axiom for

topological T-duality due to Bouwknegt-Evslin-Mathai 04.

\,

\,

To understand the second condition

pass to the correspondence super L L_\infty-algebra.

\,

Proposition

(Fiorenza-Sati-Schreiber 16b, prop. 6.2)

We have a diagram of super L L_\infty-algebras of the following form

such that on the classifying 3-cocycles

ν\nu is given by the Poincaré form

𝒫:=e 9 IIAe 9 IIB \mathcal{P} := e_9^{\mathrm{IIA}} \wedge e_9^{\mathrm{IIB}}

as

p B *(μ F1 IIB)p A *(μ F1 IIA)=d𝒫. p_B^\ast (\mu_{{}_{F1}}^{\mathrm{IIB}} )- p_A^\ast (\mu_{{}_{F1}}^{\mathrm{IIA}}) = d \mathcal{P} \,.

\,

This now is the super L L_\infty-version of the

refined formulation of topological T-duality

due to Bunke-Schick 04:

\,

Proposition

(Fiorenza-Sati-Schreiber 16b, prop. 6.4)

The integral transform of pull-push through this correspondence is an isomorphism

(π 9 IIA) *ν *(π 9 IIB) *:H μ F1 IIA( 9,1|16+16¯,𝔩(KU))H μ F1 IIB( 9,1|16+16,𝔩(ΣKU)) (\pi_9^{\mathrm{IIA}})_\ast \circ \nu^\ast \circ (\pi_9^{\mathrm{IIB}})^\ast \;:\; H_{\mu_{F1}^{\mathrm{IIA}}} \left( \mathbb{R}^{9,1\vert \mathbf{16} + \overline{\mathbf{16}}}, \mathfrak{l}(\mathrm{KU}) \right) \longrightarrow H_{\mu_{F1}^{\mathrm{IIB}}} \left( \mathbb{R}^{9,1\vert \mathbf{16} + {\mathbf{16}}}, \mathfrak{l}(\Sigma \mathrm{KU}) \right)

Moreover, it identifies the type IIA D-brane cocycles with those of type IIB (from def. ), as in Theorem :

exp(f 2 IIB)C IIB =(π 9 IIA) *ν *(π 9 IIB) *(exp(f 2 IIA)C IIA) =(π 9 IIA) *(exp(𝒫)(π 9 IIB) *(exp(f 2 IIA)C IIA)), \begin{aligned} \exp(-f_2^{\mathrm{IIB}}) \wedge C^{\mathrm{IIB}} & =\; (\pi_9^{\mathrm{IIA}})_\ast \circ \nu^\ast \circ (\pi_9^{\mathrm{IIB}})^\ast \left( \exp(-f_2^{\mathrm{IIA}}) \wedge C^{\mathrm{IIA}} \right) \\ & =\; (\pi_9^{\mathrm{IIA}})_\ast \left(\, \exp(\mathcal{P}) \wedge (\pi_9^{\mathrm{IIB}})^\ast \left( \, \exp(-f^{\mathrm{IIA}}_2) \wedge C^{\mathrm{IIA}} \, \right) \,\right) \end{aligned} \,,
Proof

One checks that

(π 9 IIA) *(exp(𝒫)(π 9 IIB) *())=(π 9 IIA) *()e 9 IIA()| 8+1. (\pi_9^{\mathrm{IIA}})_\ast \left( \exp(\mathcal{P}) \wedge (\pi_9^{\mathrm{IIB}})^\ast \left( - \right) \right) = (\pi_9^{\mathrm{IIA}})_\ast(-) - e_9^{IIA} \wedge (- )\vert_{8+1} \,.

With this the statement follows from theorem , item 2.

\,

\,

Prop. is the super L L_\infty-algebraic analog of Bunke-Schick 04, theorem 3.13.

In fact on CE-algebras this is the Buscher rules for RR-fields

also known as Hori’s formula (Hori 99).

\,

To see why the above correspondence

follows from the double dimensional reduction isomorphism ϕ T\phi_T

it is useful to break up the doubly dimensionally reduced cocycles (μ F1,Dp IIA/B) \mathcal{L}( \, \mu_{F1,Dp}^{IIA/B} \, )

into the part for the fiber bundle and the part for the K-cocycles:

\,

Definition

The delooped T-duality Lie 2-algebra B𝒯 1B \mathcal{T}_1

is the homotopy fiber of the cup square of the first Chern class

B𝒯 1BBAc 1c˜ 1AB 3 B \mathcal{T}_1 \overset{}{\longrightarrow} B \mathbb{R} \oplus B \mathbb{R} \overset{\phantom{A}c_1 \cup \tilde c_1 \phantom{A}}{\longrightarrow} B^3 \mathbb{R}

hence by prop.

CE(B𝒯 1)={dc 2=0,dc˜ 2=0 dh 3=c 2c˜ 2}. CE( B \mathcal{T}_1 ) \;=\; \left\{ \array{ d c_2 = 0, \;\; d \tilde c_2 = 0 \\ d h_3 = - c_2 \wedge \tilde c_2 } \right\} \,.

\,

Proposition

(FSS 16b, prop. 7.3)

The dimensionally reduced twisted K-theory coefficients from prop.

sit in a homotopy fiber sequence of the form

𝔩(KUΣKU) 𝔩((KU/BU(1))/S 1) B𝒯 1 \array{ \mathfrak{l}(\, KU \oplus \Sigma KU \,) &\longrightarrow& \mathfrak{l}(\, \mathcal{L}( KU/BU(1))/S^1 \,) \\ && \downarrow \\ && B \mathcal{T}_1 }

exhibiting (via NSS 12) an ∞-action

of the T-duality Lie 2-algebra from def.

on 𝔩(KUΣKU)\mathfrak{l}(\, KU \oplus \Sigma KU \,).

\,

Proposition

(FSS 16b, prop. 7.5)

The homotopy fiber of the cyclified IIA/B cocycles (theorem )

projected to the T-duality Lie 2-algebra B𝒯 1B \mathcal{T}_1 (def. )

are the super L L_\infty-algeberas p A * 9,1|16+16¯^p_{A}^\ast \widehat{ \mathbb{R}^{9,1 \vert \mathbf{16} + \overline{\mathbf{16}}} } and p B *( 9,1|16+16^)p_B^\ast( \widehat{ \mathbb{R}^{9,1\vert \mathbf{16} + \mathbf{16}} } ) from prop.

hence the Bunke-Schick 04-type correspondence

follows from theorem

via \infty-functoriality of homotopy fibers:

\,

Prop. implies that we may study T-duality

in terms of principal 2-bundles for the T-duality 2-group

over 9d super-manifolds locally modeled on 8,1|16+16\mathbb{R}^{8,1\vert \mathbf{16} + \mathbf{16}}.

This is the supergeometric refinement

of the 2-bundle perspective on T-folds due to (Nikolaus 14).

\,

\,

Integral lift

The above super L L_\infty-algebra

is to be the curvature/Chern character data

of twisted differential cohomology

on supermanifolds.

\,

Consider the (∞,1)-topos over the site of all super-formal manifolds

HSh (SuperFormalMfd). \mathbf{H} \coloneqq Sh_\infty( SuperFormalMfd ) \,.

Hence XHX \in \mathbf{H} is a “super formal smooth ∞-groupoid”.

\,

Proposition

(Schreiber, dcct)

This (∞,1)-topos H\mathbf{H} carries a progression of adjoint modalities

as shown on the right

where

  1. each item denotes an idempotent (∞,1)-(co-)monad;

  2. specifically

    1. DiscΓ\flat \coloneqq Disc \circ \Gamma is the comonad induced from the terminal geometric morphism;

    2. =loc 0|1A{{= loc_{\mathbb{R}^{0\vert 1}}} \atop {\phantom{A}}}

      is the A1-localization at the superpoint;

  3. each LRL \dashv R denotes a pair of adjoint (∞,1)-functors;

  4. each F>FF' \gt F means that (F(X)X)(F(X)X)(F(X) \simeq X) \Rightarrow (F'(X) \simeq X).

\,

Moreover, with H\mathbf{H} also its

tangent (∞,1)-toposTHT \mathbf{H} of parameterized spectrum objects in H\mathbf{H}

inherits the same kind of progression.

\,

In conclusion

this progression of adjoint modalities

is the abstract reflection that inside H\mathbf{H}

there is a good theory of

  1. differential cohomology\;\; \int \dashv \flat \dashv \sharp

  2. differential geometry\;\; Et\Re \dashv \Im \dashv E\!t

  3. supergeometry\;\; loc 0|1\rightrightarrows \dashv \rightsquigarrow \dashv loc_{\mathbb{R}^{0\vert 1}}

\,

Regarding the first point:

The ()(\int \dashv \flat) fracture squares

dRE^ AAAA dRE^ dRE^ E^ dRE^ E^ AAA E^. \array{ && \int_{dR} {\hat E} && \stackrel{\phantom{AAAA}}{\longrightarrow} && \flat_{dR}{\hat E} \\ & \nearrow & & \searrow & & \nearrow && \searrow \\ \int_{dR} \flat {\hat E} && && {\hat E} && && \int \flat_{dR} \hat E \\ & \searrow & & \nearrow & & \searrow && \nearrow \\ && \flat {\hat E} && \underset{\phantom{AAA}}{\longrightarrow} && \int \hat E } \,.

exhibit each E^T *H\hat E \in T_\ast \mathbf{H} as a differential cohomology theory (Bunke-Nikolaus-Völkl 13)

connectionformsontrivialbundles deRhamdifferential curvatureforms curvature deRhamtheorem flatdifferentialforms geometricbundleswithconnection rationalizedbundle topol.class Cherncharacter geometricbundleswithflatconnection comparisonmap shapeofbundle \array{ && {{connection\;forms}\atop{on\;trivial\;bundles}} && \stackrel{de\;Rham\;differential}{\longrightarrow} && {{curvature}\atop{forms}} \\ & \nearrow & & \searrow & & \nearrow_{\mathrlap{curvature}} && \searrow^{\mathrlap{de\;Rham\;theorem}} \\ {{flat}\atop{differential\;forms}} && && {{geometric\;bundles}\atop{with\;connection}} && && {{rationalized}\atop{bundle}} \\ & \searrow & & \nearrow & & \searrow^{\mathrlap{topol.\;class}} && \nearrow_{\mathrlap{Chern\;character}} \\ && {{geometric\;bundles}\atop{with\;flat\;connection}} && \underset{comparison \;map}{\longrightarrow} && {{shape}\atop{of\;bundle}} }

(the differential cohomology hexagon).

\,

In particular if

the non-differential twisted generalized cohomology theory EE^E \coloneqq \int \hat E is prescribed

then a differential refinement E^\hat E is obtained by

setting:

Ω flat(,𝔩(E)) E^ (pb) E E, \array{ && \Omega_{flat}(-, \mathfrak{l}(E)) \\ & \nearrow && \searrow \\ \hat E && (pb) && E_{\mathbb{R}} \\ & \searrow && \nearrow \\ && E } \,,

where Ω(,𝔩(E))\Omega(-,\mathfrak{l}(E)) denotes the sheaf of flat super L-∞ algebra valued differential forms.

\,

The super L L_\infty-cocycles discussed above are examples

of such flat super L-∞ algebra valued differential forms,

namely they are precisely the left invariant differential forms.

\,

Hence the above super L-∞ algebra is

the image under the Chern character of interesting

super twisted differential cohomology theory.

\,

Need to globalize this super-tangent space wise!

\,

To that end, consider the second stage of adjoint modalities \Re \dashv \Im

\,

Definition

Given X,YHX,Y\in \mathbf{H} then a morphism f:XYf \;\colon\; X\longrightarrow Y is a local diffeomorphism if its naturality square of the infinitesimal shape modality \Im

X X f f Y Y \array{ X &\longrightarrow& \Im X \\ \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{\Im f}} \\ Y &\longrightarrow& \Im Y }

is a homotopy pullback square.

Definition

Given XHX \in \mathbf{H}, its infinitesimal disk bundle T infXXT_{inf} X\to X is the pullback of the unit of the infinitesimal shape modality \Im along itself

T infX X X X. \array{ T_{inf} X &\stackrel{}{\longrightarrow}& X \\ \downarrow && \downarrow \\ X &\longrightarrow& \Im X } \,.
Definition

Let VGrp(H)V \in Grp(\mathbf{H}) be an ∞-group.

A VV-manifold is an XHX \in \mathbf{H} such that there exists a VV-atlas, namely a correspondence of the form

U et et V X \array{ && U \\ & {}^{\mathllap{et}}\swarrow && \searrow^{\mathrlap{et}} \\ V && && X }

with both morphisms being local diffeomorphisms, def. , and the right one in addition being an epimorphism, hence an atlas.

Proposition

(Khavkine-Schreiber 17, Wellen 17)

For XX a VV-manifold, def. , then its infinitesimal disk bundle T infXXT_{inf} X \to X, def. , is associated to a GL(V)GL(V)-principal Fr(X)XFr(X) \to X – to be called the frame bundle, modulated by a map to be called τ X\tau_X, producing homotopy pullbacks of the form

T infX V/GL(V) X τ X BGL(V)AAAAAFr(X) * X τ X BGL(V). \array{ T_{inf} X &\longrightarrow& V/GL(V) \\ \downarrow && \downarrow \\ X &\stackrel{\tau_X}{\longrightarrow}& \mathbf{B} GL(V) } \;\;\phantom{AAAAA}\; \array{ Fr(X) &\longrightarrow& \ast \\ \downarrow && \downarrow \\ X &\stackrel{\tau_X}{\longrightarrow}& \mathbf{B} GL(V) } \,.

\,

This provides all the ingredients

to incarnate super topological T-duality globally

in terms of twisted super-differential cocycles on super-stacks

that tangent complex-wise restrict to the above super L L_\infty-cocycle models.

\,

To be done.

\,

References

Detailed pointers to the literature are contained in the above text.

Here we only list the references to the original work reported on here.

\,

Our discussion of L-infinity algebra cohomology is due to

The observation of the brane bouquet in super L L_\infty-algebra and the general construction of higher WZW terms from higher L L_\infty-cocycles is due to

The homotopy-descent of the M5-brane cocycle and of the type IIA D-brane cocycles is due to

The derivation of supersymmetric topological T-duality, rationally, and of the higher super Cartan geometry for super T-folds is due to

The derivation of the process of higher invariant extensions that leads from the superpoint to 11-dimensional supergravity:

General discussion of twisted cohomology is in

The rational homotopy theory of parameterized spectra is discussed in

The synthetic differential Cartan geometry is discussed in

A textbook account of some of the story is in


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Last revised on September 6, 2018 at 11:52:21. See the history of this page for a list of all contributions to it.