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For me this note [1] was part of the motivation for developing synthetic variational calculus in infinity-toposes with +Igor Khavkine [2]. Of course there are many other motivations for that, too, but the superalgbra extensions obtained from the Noether charges of the super p-branes are a particularly striking example of the higher Noether theorem. This is discussed a bit in the talk slides at [3].

[1] https://ncatlab.org/schreiber/show/Lie+n-algebras+of+BPS+charges

[2] https://ncatlab.org/schreiber/show/Synthetic+variational+calculus

[3] https://ncatlab.org/schreiber/show/Obstruction+theory+for+parameterized+higher+WZW+terms

[1] https://ncatlab.org/schreiber/show/Lie+n-algebras+of+BPS+charges

[2] https://ncatlab.org/schreiber/show/Synthetic+variational+calculus

[3] https://ncatlab.org/schreiber/show/Obstruction+theory+for+parameterized+higher+WZW+terms

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**Introduction to rational homotopy theory**[1] the video recording of my 7th lecture in the super-p-brane seminar [2] is now available here:

http://www.karlin.mff.cuni.cz/~soucek/zaznamy_prednasek/Urs%20Schreiber/7_seminar/index.html

[1] https://ncatlab.org/nlab/show/rational+homotopy+theory

[2] https://ncatlab.org/schreiber/show/From+the+Superpoint+to+T-Folds

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Yesterday in Prague +Igor Khavkine gave a beautiful exposition of

**synthetic PDE theory**. The video recording is now available here: http://www.karlin.mff.cuni.cz/~soucek/zaznamy_prednasek/SDG%20PDE/index.html Post has attachment

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I am happy that we finally have this out on the arXiv: https://arxiv.org/abs/1702.01774 See [1] for more. I will also speak about this stuff at

[1] https://ncatlab.org/schreiber/show/M-Theory+from+the+Superpoint

[2] https://stringmath2017.desy.de/

**String Math 2017**[2].[1] https://ncatlab.org/schreiber/show/M-Theory+from+the+Superpoint

[2] https://stringmath2017.desy.de/

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My understanding is that in 1992 Zwiebach not only discovered that the n-point functions turn the closed string's BRST complex into an L-infinity algebra [1], but he thereby really discovered the very concept of L-infinity algebra [2] in the first place. Stasheff, hearing him speak about this, realized that this was an evident variant of the A-infinity algebras that he had famously studied long before, and then babtized them "L-infinity algebras". But it is somewhat remarkable that mathematicians had not made that step themselves. On the other hand, in the dual dg-algebra guise they secretly did, namely in Sullivan's rational homotopy theory, viewed under its equivalence to the Quillen model. But again, in that dual guise in fact the physicists also did that around that time, namely in the supergravity literature ("FDA"s). This somewhat convoluted history of the concept of L-infinity algebras is discussed in a llittle more detail here: https://ncatlab.org/nlab/show/L-infinity-algebra#History .

More recently, the Bagger-Lambert model for the nonabelian solitonic M2-brane [3] seemed to feature "Filippov 3-algebras". While these may be viewed as certain L-infinity algebras with metric [4], their reception sometimes suggests that the memory of the fundamental role of L-infinity algebras in field theory in general, and in string theory in particular, may need refreshment [5]. Similarly, these days people are excited about the huge gauge symmetry in the "higher spin" tensionless limit of string field theory, and seem to forget that before taking the tensionless limit then the full theory exhibits a much richer symmetry still, namely Zwiebach's L-infinity symmetry of the closed string.

For all these reasons it is good to see Zwiebach come back to the topic of L-infinity algebras in a broader perspective:

Olaf Hohm, Barton Zwiebach

"L∞ Algebras and Field Theory"

https://arxiv.org/abs/1701.08824

[1] https://ncatlab.org/nlab/show/string+field+theory

[2] https://ncatlab.org/nlab/show/L-infinity-algebra

[3] https://ncatlab.org/nlab/show/BLG+model

[4] https://ncatlab.org/nlab/show/n-Lie+algebra#ReferencesAsMetricLienAlgebras

[5] https://ncatlab.org/nlab/print/L-infinity+algebras+in+physics

More recently, the Bagger-Lambert model for the nonabelian solitonic M2-brane [3] seemed to feature "Filippov 3-algebras". While these may be viewed as certain L-infinity algebras with metric [4], their reception sometimes suggests that the memory of the fundamental role of L-infinity algebras in field theory in general, and in string theory in particular, may need refreshment [5]. Similarly, these days people are excited about the huge gauge symmetry in the "higher spin" tensionless limit of string field theory, and seem to forget that before taking the tensionless limit then the full theory exhibits a much richer symmetry still, namely Zwiebach's L-infinity symmetry of the closed string.

For all these reasons it is good to see Zwiebach come back to the topic of L-infinity algebras in a broader perspective:

Olaf Hohm, Barton Zwiebach

"L∞ Algebras and Field Theory"

https://arxiv.org/abs/1701.08824

[1] https://ncatlab.org/nlab/show/string+field+theory

[2] https://ncatlab.org/nlab/show/L-infinity-algebra

[3] https://ncatlab.org/nlab/show/BLG+model

[4] https://ncatlab.org/nlab/show/n-Lie+algebra#ReferencesAsMetricLienAlgebras

[5] https://ncatlab.org/nlab/print/L-infinity+algebras+in+physics

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On Feb 17 +Igor Khavkine will be speaking in the DG Seminar at Charles University Prague on

differential equations (PDEs) in synthetic differential geometry

(SDG), one that would seamlessly generalize the traditional theory to

a range of enhanced contexts, such as super-geometry, higher (stacky)

differential geometry, or even a combination of both. SDG greatly

enriches the notion of a smooth space, admitting both infinite

dimensional manifolds and manifolds with infinitesimal directions, in

the same way as ordinary manifolds. This freedom allows us to give a

precise definition of (a parametrized family of) formal solutions and

to directly define the category of PDEs to consist of morphisms that

preserve formal solutions, without invoking the Cartan distribution.

Within this framework, we prove that the prolongation of a PDE

coincides with its universal family of formal solutions and, inspired

by the work of Marvan, that our PDE category is equivalent to the

(Eilenberg-Moore) category of coalgebras over the jet functor comonad.

Through Marvan's original result, this shows that our PDE category is

equivalent to Vinogradov's, when restricted to ordinary manifolds.

This is joint work with Urs Schreiber [https://arxiv.org/abs/1701.06238].

**A synthetic approach to the formal theory of PDEs**(https://arxiv.org/abs/1701.06238)**Abstract:**We give an abstract formulation of the formal theory partialdifferential equations (PDEs) in synthetic differential geometry

(SDG), one that would seamlessly generalize the traditional theory to

a range of enhanced contexts, such as super-geometry, higher (stacky)

differential geometry, or even a combination of both. SDG greatly

enriches the notion of a smooth space, admitting both infinite

dimensional manifolds and manifolds with infinitesimal directions, in

the same way as ordinary manifolds. This freedom allows us to give a

precise definition of (a parametrized family of) formal solutions and

to directly define the category of PDEs to consist of morphisms that

preserve formal solutions, without invoking the Cartan distribution.

Within this framework, we prove that the prolongation of a PDE

coincides with its universal family of formal solutions and, inspired

by the work of Marvan, that our PDE category is equivalent to the

(Eilenberg-Moore) category of coalgebras over the jet functor comonad.

Through Marvan's original result, this shows that our PDE category is

equivalent to Vinogradov's, when restricted to ordinary manifolds.

This is joint work with Urs Schreiber [https://arxiv.org/abs/1701.06238].

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With +John Huerta we finally have a writeup of the proof of the emergence of

The “brane scan” classifies consistent Green–Schwarz strings and membranes in terms of the invariant cocycles on super-Minkowski spacetimes. The “brane bouquet” generalizes this by consecutively forming the invariant higher central extensions induced by these cocycles, which yields the complete brane content of string/M-theory, including the D-branes and the M5-brane, as well as the various duality relations between these. This raises the question whether the super-Minkowski spacetimes themselves arise as maximal invariant central extensions. Here we prove that they do. Starting from the simplest possible super-Minkowski spacetime, the superpoint, which has no Lorentz structure and no spinorial structure, we give a systematic process of consecutive maximal invariant central extensions, and show that it discovers the super-Minkowski spacetimes that contain superstrings, culminating in the 10- and 11-dimensional super-Minkowski spacetimes of string/M-theory and leading directly to the brane bouquet.

**"M-Theory from the Superpoint"**, see the pdf here: https://ncatlab.org/schreiber/print/M-Theory+from+the+SuperpointThe “brane scan” classifies consistent Green–Schwarz strings and membranes in terms of the invariant cocycles on super-Minkowski spacetimes. The “brane bouquet” generalizes this by consecutively forming the invariant higher central extensions induced by these cocycles, which yields the complete brane content of string/M-theory, including the D-branes and the M5-brane, as well as the various duality relations between these. This raises the question whether the super-Minkowski spacetimes themselves arise as maximal invariant central extensions. Here we prove that they do. Starting from the simplest possible super-Minkowski spacetime, the superpoint, which has no Lorentz structure and no spinorial structure, we give a systematic process of consecutive maximal invariant central extensions, and show that it discovers the super-Minkowski spacetimes that contain superstrings, culminating in the 10- and 11-dimensional super-Minkowski spacetimes of string/M-theory and leading directly to the brane bouquet.

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Lecture notes for today's session "Super p-branes" of the Lecture series

[1] https://ncatlab.org/schreiber/show/From+the+Superpoint+to+T-Folds

**From the Superpoint to Super T-Folds**[1] are now in section "5 Branes" here: https://ncatlab.org/nlab/print/geometry+of+physics+--+fundamental+super+p-branes#Branes[1] https://ncatlab.org/schreiber/show/From+the+Superpoint+to+T-Folds

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Hier ist eine deutsche Version des erweiterten Vortragsskiptes: https://ncatlab.org/nlab/print/Topologie

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This Wednesday in Bonn, I give the the introductory lecture for my course on point-set

**Topology**later this summer. The slides are here: https://ncatlab.org/nlab/print/topology#IntroductionWait while more posts are being loaded