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Emergence170208

superstring Lie 2-algebra [33], to be denoted string: d\p 1 10 ⋆ ↔ string R 9,1|16 extended super Minkowski super Lie 2-algebra super Minkowski super Lie algebra and **the** supermembrane Lie 3-algebra, denoted m2brane: d\p 2 11 ⋆ ↔ m2brane R 10,1|32 extended super Minkowski super Lie 3-algebra super Minkowski super Lie algebra An exposition of **the**se structures as objects in higher Lie **the**ory appears in Huerta’s **the**sis [33]. In **the**ir dual incarnation as “FDA”s **the**se algebras are **the** extended super-Minkowski spacetimes considered by Chryssomalakos et al. [15]. We follow **the**ir idea, and call extensions of super-Minkowski spacetime to super Lie n-algebras such as string and m2brane extended super-Minkowski spacetimes, noticing that in fact **the**se are “higher extended” in **the** sense of extensions of super Lie n-algebras. Now that each entry in **the** old brane scan is identified with a higher super Lie algebra in this way, something remarkable happens: **the**re appear new cocycles on **the**se extended super-Minkowski spacetimes, cocycles which do not show up on plain super-Minkowski spacetime itself. (In homotopy **the**ory this is in fact a familiar phenomenon, it is **the** hallmark of **the** construction of **the** ‘Whitehead tower’ of a topological space.) And indeed, in turns out that **the** new invariant cocycles thus found do correspond to **the** branes that were missing **from** **the** old brane scan [25]: On **the** super Lie 3-algebra m2brane **the**re appears an invariant 7-cocycle, which corresponds to **the** M5-brane, on **the** super Lie 2-algebra string IIA **the**re appears a sequence of (p + 2)-cocycles for p ∈ {0, 2, 4, 6, 8, 10}, corresponding to **the** type IIA D-branes, and on **the** superstring Lie 2-algebra string IIB **the**re appears a sequence of (p + 2)-cocycles for p ∈ {1, 3, 5, 7, 9}, corresponding to **the** type IIB D-branes. Under **the** identification of super Lie n-algebras with formal duals of “FDA”s, **the** algebra behind this statement is in fact an old result: For **the** M5-brane and **the** type IIA D-branes this is due to Chryssomalakos et al. [15], while for **the** type IIB D-branes this is due to Sakaguchi [43, section 2]. In fact **the** 7-cocycle on **the** supermembrane Lie 3-algebra that, according to Bandos et al. [8], corresponds to **the** M5-brane was already discovered in **the** 1982 by D’Auria and Fré [2, equations (3.27) and (3.28)]. If we again denote each of **the**se fur**the**r cocycles by **the** super Lie n-algebra extension which it classifies according to [25, prop. 3.5] [23, **the**orem 3.1.13], and name **the**se extensions by **the** super p-brane species whose WZW-term is given by **the** cocycle, **the**n we obtain **the** following diagram in **the** category of super L ∞ -algebras: 4

m5brane m2brane d5brane d3brane d1brane d0brane d2brane d4brane d7brane R 10,1|32 d6brane string het string IIA d8brane d9brane string IIB R 9,1|16+16 R 9,1|16 R 9,1|16+16 Hence in **the** context of higher super Lie algebra, **the** “old brane scan” is completed to a tree of consecutive invariant higher central extensions emanating out of **the** super-Minkowski spacetimes, with one leaf for each brane species in string/M-**the**ory and with one edge whenever one brane species may end on ano**the**r [25, section 3]. This was called **the** “brane bouquet” in [25, def. 3.9 and section 4.5]. Interestingly, a fair bit of **the** story of string/M-**the**ory is encoded in this purely super Lien-algebraic ma**the**matical structure. This includes in particular **the** pertinent dualities: **the** KKreduction between M-**the**ory and type IIA **the**ory, **the** HW-reduction between M-**the**ory and heterotic string **the**ory, **the** T-duality between type IIA and type IIB, **the** S-duality of type IIB, and **the** relation between type IIB and F-**the**ory. All of **the**se are reflected as equivalences of super Lie n-algebras obtained **from** **the** brane bouquet, this is shown in [26, 27, 34]. The diagram of super L ∞ -algebras that reflects **the**se L ∞ -equivalences looks like a canditate to fill Polchinski’s famous cartoon picture of M-**the**ory [42, figure 1] [49, figure 4] with ma**the**matical life: D0brane D2brane D4brane D6brane D8brane KK string IIA HW m5brane m2brane d=11 N=32 ns5brane d=10 d=10 N=16 N=16 string het string IIB d=10 d=10 N=16+16 R d−1,1|N d=10 d=10 N=16+16 N=16+16 N=16+16 (p, q)string IIB Dstring T (p, q)1brane (p, q)3brane (p, q)5brane (p, q)7brane (p, q)9brane S 5

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