Algebraic Structures in Double and Exceptional Field Theory Olaf - - PowerPoint PPT Presentation

Algebraic Structures in Double and Exceptional Field Theory

Olaf Hohm

‚ O.H, Zwiebach, 1701.08824 ‚ O.H., Kupriyanov, L¨

ust, Traube, 1709.10004

‚ O.H., Samtleben, 1707.06693 & to appear ‚ O.H., work in progress

Centro Atomico Bariloche, Argentina, January 2018

1

Road Map

‚ Duality covariant formulation in 1) gauged supergravity

(‘embedding tensor formalim’) and 2) double/exceptional field theory requires redundant or unphysical objects ñ ‘higher equivalences’

‚ analogous features in algebraic topology and homotopy theory, where

‘8-algebras’ allow one “to live with slightly false algebraic identities in a new world where they become effectively true.” [D. Sullivan]

‚ Features of physical theories usually taken for granted

[ e.g.: “continuous symmetries ” Lie algebras” ] hold only ‘up to homotopy’, which quite likely provides deep pointers for (so far) elusive underlying mathematical structure of DFT/ExFT

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Overview

‚ Strongly Homotopy (sh) or 8-Algebras ‚ Field Theories and L8 Algebras Ñ weakly constrained DFT? ‚ Leibniz (or Loday) Algebras and their Chern-Simons Gauge Theory ‚ Topological Phase of E8p8q ExFT as Leibniz Chern-Simons Theory ‚ General Remarks and Outlook

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Strongly Homotopy Lie or L8 Algebras

An L8 algebras is a graded vector space

[Zwiebach (1993), Lada & Stasheff (1993)]

X “

à

nPZ

Xn , equipped with multilinear and graded antisymmetric brackets or maps x1, . . . , xn ÞÑ ℓnpx1, . . . , xnq P Xn´2`ř

i |xi| ,

satisfying, for each n “ 1, 2, 3, . . ., the generalized Jacobi identities

ÿ

i`j“n`1

p´1qipj´1q ÿ

σ

p´1qσǫpσ; xq ℓj `

ℓipxσp1q, . . . , xσpiqq, xσpi`1q, . . . , xσpnq

˘ “ 0

with the sum over all permutations of n objects with partially ordered arguments (‘unshuffles’), σp1q ď ¨ ¨ ¨ ď σpiq, σpi ` 1q ď ¨ ¨ ¨ ď σpnq, and Koszul sign ǫpσ; xq , determined for any graded algebra with xixj “ p´1qxixj xjxi by x1 ¨ ¨ ¨ xk “ ǫpσ; xq xσp1q ¨ ¨ ¨ xσpkq

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Explicit L8-relations

For n “ 1 we learn that ℓ1 ” Q is nil-potent: ℓ1pℓ1pxqq “ 0 For n “ 2 we learn that ℓ1 is a derivation of ℓ2 ” r¨, ¨s: ℓ1pℓ2px1, x2qq “ ℓ2pℓ1px1q, x2q ` p´1qx1ℓ2px1, ℓ1px2qq For n “ 3 we learn that ℓ2 ” r¨, ¨s satisfies Jacobi only ‘up to homotopy’ 0 “ ℓ2pℓ2px1, x2q, x3q ` 2 terms

` ℓ1pℓ3px1, x2, x3qq ` ℓ3pℓ1px1q, x2, x3q ` 2 terms

For n “ 4 we learn that ℓ2ℓ3 ` ℓ3ℓ2 is zero ‘up to homotopy’, i.e., up to the the failure of ℓ1 to act as a derivation on ℓ4 plus infinitely more relations

5

Constructing L8 Algebras

Given a bilinear antisymmetric 2-bracket r¨, ¨s, is there L8 with ℓ2 “ r¨, ¨s ? Yes: extend the space V by a second copy V ˚ X1 “ V ˚

ℓ1

Ý Ñ X0 “ V

ℓ1pv˚q “ v , ℓ3pu, v, wq “ ´Jacpu, v, wq˚ But: ‘trivial’ because v „ w iff v ´ w “ ℓ1p¨q, no further extension Non-trivial if Jacobiator lives in proper subspace or, more generally, in image of linear map D : U Ñ V : Jacp¨, ¨, ¨q “ Dfp¨, ¨, ¨q. Theorem: X2 – KerpDq

ℓ1“ι

Ý Ý Ý Ñ X1 “ U

ℓ1“D

Ý Ý Ý Ý Ñ X0

carries L8 structure, provided rImpDq, V s Ă ImpDq, with ℓ3p¨, ¨, ¨q “ ´fp¨, ¨, ¨q and generally non-trivial ℓ4p¨, ¨, ¨, ¨q

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Field Theories & Weakly Constrained DFT

Dictionary L8 algebra Ð

Ñ field theory: ¨ ¨ ¨ Ý Ñ X1

ℓ1

Ý Ñ X0

ℓ1

Ý Ñ X´1

ℓ1

Ý Ñ X´2 Ý Ñ ¨ ¨ ¨

χ ξ Ψ EOM Gauge transformations and field equations: δξΨ “ ℓ1pξq ` ℓ2pξ, Ψq ´ 1

2 ℓ3pξ, Ψ, Ψq ` ¨ ¨ ¨

0 “ ℓ1pΨq ´ 1

2 ℓ2pΨ, Ψq ´ 1 3! ℓ3pΨ, Ψ, Ψq ` ¨ ¨ ¨

gauge algebra closes ‘up to homotopy’: trivial parameters ξ “ ℓ1pχq Example: Courant algebroid/gauge structure of DFT, with ℓ2 “ r¨, ¨sc, defines L8 algebra with ℓ4 “ 0

[Roytenberg & Weinstein (1998)]

Ñ generalization to weakly constrained? Indeed, in general L8 non-trivial

ℓ2pχ1, χ2q “ xDχ1, Dχ2y p “ BMχ1 BMχ2 “ 0 q

Ñ still very non-trivial (non-local projected product needed)

[A. Sen (2016)] 7

Leibniz Algebras and their Chern-Simons Theory

Leibniz (or Loday) algebra: vector space with product ˝, satisfying x ˝ py ˝ zq “ px ˝ yq ˝ z ` y ˝ px ˝ zq If ˝ antisymmetric ñ Lie algebra Defines symmetry variations: δxy “ Lxy ” x ˝ y that close:

rLx, Lysz ” LxpLyzq ´ LypLxzq “ x ˝ py ˝ zq ´ y ˝ px ˝ zq “ px ˝ yq ˝ z “ Lx˝yz

(Anti-)symmetrizing in x, y:

rLx, Lysz “ Lrx,ysz ,

Ltx,yuz “ 0 Thus, t, u defines ‘trivial vector’. Jacobiator is trivial:

ÿ

antisym

3rrx1, x2s, x3s ´ tx1 ˝ x2, x3u “ 0 ‘Trivial space’ forms ideal of bracket: r¨, t , us “ t¨, ¨u. Thus: Theorem: Any Leibniz algebra defines L8 algebra with ℓ2 “ r¨, ¨s

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Leibniz-valued Gauge Fields and Chern-Simons Action

Leibniz-valued one-form with gauge transformations δλAµ “ Dµλ ” Bµλ ´ Aµ ˝ λ This closes up to ‘higher gauge transformations’ (c.f. trivial parameters). Generalized Chern-Simons action SCS ”

ż

d3x ǫµνρ @ Aµ , BνAρ ´ 1

3Aν ˝ Aρ

D

is gauge invariant provided the inner product x , y is invariant and

xx, t¨, ¨uy “ 0 @x ñ situation in 3D gauged SUGRA in embedding tensor formalism

[de Wit, Nicolai & Samtleben (2001–2002)]

ñ any Leibniz algebra with x , y as above defines Chern-Simons theory ñ general dimensions: tensor hierarchy (& corresponding L8 algebra)

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‘Unbroken Phase’ of E8p8q ExFT

Fields: eµa, AµM, BµM, MMN, coordinates: pxµ, Y Mq, M “ 1, . . . , 248 Action: S “

ż

d3x d248Y e

´ p

R ` 1 240DµMMNDµMMN ´ V pMq ` LtoppA, Bq

¯

Consider subsector that truncates MMN, say by setting: MMN “ δMN : E8p8q Ñ SOp16q However, we want unbroken phase, so we set (illegally): MMN “ 0 Perhaps justification in suitably reformulated/enlarged theory? c.f. unconstrained ‘doubled ’metric Ñ α1 corrections

[O.H., Siegel & Zwiebach (2013)]

first-order formulation with degenerate frame field?

[E. Witten (1988)] 10

Big Chern-Simons Theory

Leibniz algebra unifies 3D Poincar´ e and (doubled) generalized diffeos: Ξ “

`

ξa , λa ; ΛM , ΣM

˘

Ξ1 ˝ Ξ2 “

`

ξa

12 , λ12a ; ΛM 12 , Σ12M

˘

where [RM ” fMNK BNΛK ` ΣM, constraint: ΣM b BM “ 0, etc.] λ12a “ ǫabc λb

1 λc 2 ` 2 Λr1 NBNλ2sa

Σ12M “ L1Σ2M ` ΛN

2 BMR1N ´ 2 α ξa

r1BMλ2sa

etc. For algebra element A ”

`

ea, ωa; AM, BM

˘

inner product given by

xA, Ay “ ż

d248Y

´

2 α eaωa ` 2 AMBM ´ fKMNAMBKAN¯ CS action for Aµ precisely top. ExFT action! Generalization of: pure 3D (super-)gravity ” Chern-Simons theory

[Achucarro & Townsend (1986), Witten (1988)] 11

Consistent Kaluza-Klein to half-maximal D “ 3 SUGRA

Duality: Opd ` 1, d ` 1q, coordinates Y M ” Y rMNs, M, N fundamental ‘Doubled vector’ Υ ” pΛMN, ΣMNq satisfies Leibniz algebra w.r.t. Υ1 ˝ Υ2

” ´

LΥ1ΛMN

2

, LΥ1Σ2MN ` 1 4ΛKL

2

BMNKpΥ1qKL ¯

Generalized Scherk-Schwarz in terms of ‘doubled’ twist matrix: U ¯

M ¯ N ”

`

ρ´1UK

r ¯

MUL ¯ Ns , ´1

4 ρ´1pBKLUP ¯

MqUP ¯ N

˘

reads for the gauge vectors: Aµpx, Y q “ U ¯

M ¯ NpY q Aµ ¯ M ¯ Npxq

and is consistent provided U ¯

M ¯ N ˝ U ¯ K¯ L “ ´X ¯ M ¯ N, ¯ K¯ L ¯ P ¯ Q U ¯ P ¯ Q

where X is the constant embedding tensor.

ñ consistency of D “ 6, N “ p1, 1q & p2, 0q SUGRA on AdS3 ˆ S3

12

Outlook & Remarks

‚ algebraic structures beyond Lie arise naturally in string/M-theory ‚ tensor hierarchy of gauged SUGRA & ExFT suggests 8-algebra,

difficult/unnatural in terms of Lie algebra

‚ unifying algebraic structure of M-theory? Ñ Hermann and Martin’s talks

affine E9p9q works analogously to E8p8q

Ñ Guillaume’s talk

ñ Lie algebra theory may be the “slightly wrong” framework ‚ novel products in HSZ theory [O.H., Siegel & Zwiebach (2013)] Ñ interpretation as 8-algebra? Ñ more general story for (chiral) CFTs? Ñ Ralph’s talk ‚ global structure of doubled (extended) spaces? [O.H. & Zwiebach (2012)]

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