[PPT] - Exceptional geometry for affine and other groups Axel Kleinschmidt PowerPoint Presentation

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Exceptional geometry for affine and other groups

Axel Kleinschmidt (Albert Einstein Institute, Potsdam) Exceptional Field Theory, Strings and Holography Texas A&M, 23 April 2018 Joint work with Guillaume Bossard, Martin Cederwall, Jakob Palmkvist and Henning Samtleben

[arXiv:1708.08936, Phys. Rev.

D96 (2017) 106022]

Exceptional geometry for affine and other groups – p.1

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Motivation

In theories of gravity and matter Geometry Matter diffeomorphisms gauge transformations

δξgµν = 2∂(µξν) δλAµ = ∂µλ

(xµ space-time coordinates)

Exceptional geometry for affine and other groups – p.2

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Motivation

In theories of gravity and matter Geometry Matter diffeomorphisms gauge transformations

δξgµν = 2∂(µξν) δλAµ = ∂µλ

(xµ space-time coordinates)

Two sides of the same coin (waffle)?

Exceptional geometry for affine and other groups – p.2

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Motivation

In theories of gravity and matter Geometry Matter diffeomorphisms gauge transformations

δξgµν = 2∂(µξν) δλAµ = ∂µλ

(xµ space-time coordinates)

Two sides of the same coin (waffle)? Idea: Find common structure for both

⇒ Generalised/exceptional geometry

Exceptional geometry for affine and other groups – p.2

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Example: Generalised geometry

[Gualtieri, Hitchin 2004]

Metric gµν, diffeomorphisms with vector ξµ Two-form Bµν, gauge parameter co-vector λµ Structure: TM ⊕ T ∗M over space-time M with coords. xµ

Exceptional geometry for affine and other groups – p.3

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Example: Generalised geometry

[Gualtieri, Hitchin 2004]

Metric gµν, diffeomorphisms with vector ξµ Two-form Bµν, gauge parameter co-vector λµ Structure: TM ⊕ T ∗M over space-time M with coords. xµ Generalised Lie derivative (a.k.a. Dorfman derivative) w.r.t. joint parameter ΛM = (ξµ, λµ)

LΛV M = ΛN∂NV M +

∂MΛN − ∂NΛM

V N

Raise/lower with ηMN = ( 0 1

1 0 )

⇒ O(d, d) structure

Derivatives ∂M = (∂µ, ˜

∂µ) and set ˜ ∂µ = 0

Algebra closes: [LΛ1, LΛ2] = L[Λ1,Λ2]C

Exceptional geometry for affine and other groups – p.3

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Example: Generalised geometry (II)

Considering generalised metric

HMN =

gµν

−gµρBρν Bµρgρν gµν − BµρgρσBσν

∈ O(d, d)

gives standard transformations from LΛHMN.

Exceptional geometry for affine and other groups – p.4

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Example: Generalised geometry (II)

Considering generalised metric

HMN =

gµν

−gµρBρν Bµρgρν gµν − BµρgρσBσν

∈ O(d, d)

gives standard transformations from LΛHMN. In closed string (field) theory (and Kaluza–Klein theory)

gµν couples to momentum modes Bµν couples to winding modes

T-duality mixes these. Natural to introduce also new coordinates ˜

xµ for string fields [Siegel; Hull, Zwiebach]

Exceptional geometry for affine and other groups – p.4

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Example: Double geometry

If fields and parameters depend on XM = (xµ, ˜

xµ)

dimension of base manifold M is doubled non-trivial derivative ˜

∂µ. XM fundamental of O(d, d)

can still use same formula for generalised Lie derivative But: Closure now only with (strong) section constraint

ηMN∂MA(X) ∂NB(X) = 0 ⇔ ηMN∂M ⊗ ∂N = 0

for any A(X), B(X). Constraint is singlet of O(d, d)

Exceptional geometry for affine and other groups – p.5

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Example: Double geometry

If fields and parameters depend on XM = (xµ, ˜

xµ)

dimension of base manifold M is doubled non-trivial derivative ˜

∂µ. XM fundamental of O(d, d)

can still use same formula for generalised Lie derivative But: Closure now only with (strong) section constraint

ηMN∂MA(X) ∂NB(X) = 0 ⇔ ηMN∂M ⊗ ∂N = 0

for any A(X), B(X). Constraint is singlet of O(d, d) Field theories constructed on doubled space with constraint double field theory (DFT)

[Aldazabal, Andriot, Berman, Betz, Blumenhagen, Deser, Geissb¨ uhler, Hassler, Hohm, Hull, Larfors, Kwak, L¨ ust, Marques, Musaev, Nunez, Park, Patalong, Penas, Perry, Plauschinn, Rudolph, Zwiebach,...]

Exceptional geometry for affine and other groups – p.5

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Exceptional geometry

String theory also has U-duality ⊃ T-duality. Type II on T d

Ed+1 ⊃ O(d, d)

Exceptional geometry and exceptional field theory (ExFT):

⇒ Redo DFT analysis for U-duality groups

t t t t t

Ed+1

Exceptional geometry for affine and other groups – p.6

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Exceptional geometry

String theory also has U-duality ⊃ T-duality. Type II on T d

Ed+1 ⊃ O(d, d)

Exceptional geometry and exceptional field theory (ExFT):

⇒ Redo DFT analysis for U-duality groups

t t t t t

Ed+1 n En

coord rep. section rep.

5 SO(5, 5) 16 10 6 E6 27 27 7 E7 56 1 ⊕ 133 8 E8 248 1 ⊕ 248 ⊕ 3875

momentum and brane winding

Exceptional geometry for affine and other groups – p.6

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Exceptional geometry

String theory also has U-duality ⊃ T-duality. Type II on T d

Ed+1 ⊃ O(d, d)

Exceptional geometry and exceptional field theory (ExFT):

⇒ Redo DFT analysis for U-duality groups

t t t t t

Ed+1 n En

coord rep. section rep.

5 SO(5, 5) 16 10 6 E6 27 27 7 E7 56 1 ⊕ 133 8 E8 248 1 ⊕ 248 ⊕ 3875

momentum and brane winding

Closure of generalised Lie derivative and ExFT for n ≤ 8

[Berman, Blair, Cederwall, Ciceri, Coimbra, Godazgar2, Guarino, Hohm, Hull, Inverso, AK, Malek, Musaev, Nicolai, Palmkvist, Park, Perry, Rosabal, Samtleben, Strickland-Constable, Suh, Thompson, Waldram,...]

Exceptional geometry for affine and other groups – p.6

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Why is this interesting?

En covariant formulations for different duality frames

‘geometric’ origin of gauged supergravities via generalised Scherk–Schwarz reduction [Baguet, du

Bosque, Hassler, Hohm, Inverso, L¨ ust, Malek, Samtleben]

Uplift formula for lower-dimensional solutions [Godazgar2,

Guarino, Kr¨ uger, Nicolai, Pilch, Varela]

Non-geometric solutions and orbits of exact solutions

[Bakhmatov, Berman, Hassler, Jensen, AK, L¨ ust, Rudolph, Musaev]

Construction of M-theory effective action [Bossard, AK]

Exceptional geometry for affine and other groups – p.7

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Why is this interesting?

En covariant formulations for different duality frames

‘geometric’ origin of gauged supergravities via generalised Scherk–Schwarz reduction [Baguet, du

Bosque, Hassler, Hohm, Inverso, L¨ ust, Malek, Samtleben]

Uplift formula for lower-dimensional solutions [Godazgar2,

Guarino, Kr¨ uger, Nicolai, Pilch, Varela]

Non-geometric solutions and orbits of exact solutions

[Bakhmatov, Berman, Hassler, Jensen, AK, L¨ ust, Rudolph, Musaev]

Construction of M-theory effective action [Bossard, AK] More speculatively on the horizon Fundamental symmetries of M-theory, e.g.

E10 [Julia; Damour, Henneaux, Nicolai] E11 [West]

New variables for quantisation?

Exceptional geometry for affine and other groups – p.7

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Aim of this talk

Exceptional geometry for affine and other groups – p.8

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Aim of this talk

Extend construction of exceptional geometry to E9 discuss infinite-dimensional affine group E9 identify correct coordinate representation identify appropriate section condition check closure of algebra elementary check of generalised Scherk–Schwarz ansatz vs. gauged supergravity in D = 2

Exceptional geometry for affine and other groups – p.8

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Aim of this talk

Extend construction of exceptional geometry to E9 discuss infinite-dimensional affine group E9 identify correct coordinate representation identify appropriate section condition check closure of algebra elementary check of generalised Scherk–Schwarz ansatz vs. gauged supergravity in D = 2 Along the way: Find different view on existing constructions!

Exceptional geometry for affine and other groups – p.8

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What is E9?

✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇

1 2 3 4 5 6 7 8

E9 = extension of E8 loop group.

Current algebra formulation: e8-valued Laurent series in z (horizontal) e8 generators: T A (A = 1, . . . , 248)

T A, T B

= fABCT C , T A|T B = ηAB

Current modes T A

m = T A ⊗ zm (m ∈ Z), T A

= T A

T A

m, T B n

= fABCT C

m+n + ηABm δm+n,0K

K is the central extension:

K, T A

m

= 0

Exceptional geometry for affine and other groups – p.9

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What is E9? (II)

So far: T A

m, K. Also derivation d

d, T A

m

= −mT A

m ,

[d, K] = 0 ⇒ Kac–Moody algebra e9 = span

T A

m, K, d

Exceptional geometry for affine and other groups – p.10

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What is E9? (II)

So far: T A

m, K. Also derivation d

d, T A

m

= −mT A

m ,

[d, K] = 0 ⇒ Kac–Moody algebra e9 = span

T A

m, K, d

Irreducible highest representations labelled by e8 irrep r(λ),

level k and weight h. Use Fock space notation: ‘Ground states’ |v ∈ r(λ)

T A

n |v = 0

for n > 0,

K|v = k|v, d|v = h|v

The T A

0 act as e8 rotations in r(λ). Excited states are

combinations of

|V =

T Ai

−ni|v

(irrep=remove sing. vectors)

Exceptional geometry for affine and other groups – p.10

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E9 representations

From e8 label λ and level k ⇒ e9 label Λ Denote e9 irrep by R(Λ)h. Often h = 0 or irrelevant here.

Exceptional geometry for affine and other groups – p.11

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E9 representations

From e8 label λ and level k ⇒ e9 label Λ Denote e9 irrep by R(Λ)h. Often h = 0 or irrelevant here. Any e9 irrep can be written as sequence of e8 irreps. E.g. start with with trivial e8 irrep r(0) at k = 1

⇒ basic/fundamental representation R(Λ0)h R(Λ0)h = 1h ⊕ 248h+1 ⊕ (1 ⊕ 248 ⊕ 3875)h+2 ⊕ . . .

‘ground state’ ‘first excited level’ ‘second excited level’

Exceptional geometry for affine and other groups – p.11

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E9 representations

From e8 label λ and level k ⇒ e9 label Λ Denote e9 irrep by R(Λ)h. Often h = 0 or irrelevant here. Any e9 irrep can be written as sequence of e8 irreps. E.g. start with with trivial e8 irrep r(0) at k = 1

⇒ basic/fundamental representation R(Λ0)h R(Λ0)h = 1h ⊕ 248h+1 ⊕ (1 ⊕ 248 ⊕ 3875)h+2 ⊕ . . .

‘ground state’ ‘first excited level’ ‘second excited level’

|V = V 0|0h + V 1

AT A −1|0h + V 2 ABT A −1T B −1|0h + . . .

Can be summarised by q-series: qh

1 + 248q + 4124q2 + . . .

Related to CFT bosons on E8 torus and modular j-function.

Exceptional geometry for affine and other groups – p.11

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E9 representations (II)

Sugawara construction in the enveloping algebra

L(k)

m =

1 2(k + g∨)

n∈Z

ηAB : T A

n T B m−n :

(g∨ = 30 dual Coxeter number, k level of e9 irrep.) Satisfy Virasoro algebra (ck = k dim e8

k+g∨ )

L(k)

m , L(k) n

= (m − n)L(k)

m+n + ck

12m(m2 − 1)δm+n,0

L(k)

m , T A n

= −nT A

m+n

(semi-direct product)

Standard relation to OPEs of conserved currents and energy-momentum tensor Tµν ∼ JµJν

Exceptional geometry for affine and other groups – p.12

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E9 representations (III)

Will also require tensor products of e9 irreps. Goddard-Kent-Olive coset construction for R(Λ) ⊗ R(Λ′) at levels k and k′:

Lcos

n

≡ L(k)

n

⊗ 1 1 + 1 1 ⊗ L(k′)

n

− L(k+k′)

n

Then

Lcos

n , T A m

= 0
n R(Λ) ⊗ R(Λ′)

⇒ e9 reps multiplied by Virasoro characters (=q-series)!!

Exceptional geometry for affine and other groups – p.13

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E9 representations (III)

Will also require tensor products of e9 irreps. Goddard-Kent-Olive coset construction for R(Λ) ⊗ R(Λ′) at levels k and k′:

Lcos

n

≡ L(k)

n

⊗ 1 1 + 1 1 ⊗ L(k′)

n

− L(k+k′)

n

Then

Lcos

n , T A m

= 0
n R(Λ) ⊗ R(Λ′)

⇒ e9 reps multiplied by Virasoro characters (=q-series)!!

E.g. for basic representation

R(Λ0)0 ⊗ R(Λ0)0 = Vir3

1,1R(2Λ0)0 ⊕ Vir3 2,1R(Λ7)−3/2 ⊕ Vir3 2,2R(Λ1)−15/16

Coefficients are in minimal Virasoro q-series (Ising model).

Exceptional geometry for affine and other groups – p.13

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E9 geometry: coordinates

By extrapolation of En coordinates, E9 exceptional geometry requires coordinates XM → |X

|X ∈ R(Λ0)

(h = 0 for simplicity)

Reasonable from E8 decomposition

|X =

X0 + ηABXA

1 T A −1 + . . .

|0

KK circle 3 → 2 E8 ExFT coords

Corresponding derivative ∂M are bra vectors ∂| ∈ R(Λ0)

Exceptional geometry for affine and other groups – p.14

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E9 geometry: section constraint

Constraint on ∂1| ⊗ ∂2| ∈ R(Λ0) ⊗ R(Λ0) Writing out the coset Virasoro characters:

R(Λ0) ⊗ R(Λ0) = (1 + q2 + O(q3))R(2Λ0) ⊕ (q2 + O(q3))R(Λ7) ⊕ (q + O(q2))R(Λ1)

Exceptional geometry for affine and other groups – p.15

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E9 geometry: section constraint

Constraint on ∂1| ⊗ ∂2| ∈ R(Λ0) ⊗ R(Λ0) Writing out the coset Virasoro characters:

R(Λ0) ⊗ R(Λ0) = (1 + q2 + O(q3))R(2Λ0) ⊕ (q2 + O(q3))R(Λ7) ⊕ (q + O(q2))R(Λ1)

For any En, the section constraint always leaves only the very leading terms in the symmetric part and the antisymmetric parts of the product: 1 · R(2Λ0) + q · R(Λ1)

Exceptional geometry for affine and other groups – p.15

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E9 geometry: section constraint

Constraint on ∂1| ⊗ ∂2| ∈ R(Λ0) ⊗ R(Λ0) Writing out the coset Virasoro characters:

R(Λ0) ⊗ R(Λ0) = (1 + q2 + O(q3))R(2Λ0) ⊕ (q2 + O(q3))R(Λ7) ⊕ (q + O(q2))R(Λ1)

For any En, the section constraint always leaves only the very leading terms in the symmetric part and the antisymmetric parts of the product: 1 · R(2Λ0) + q · R(Λ1) Express via coset Virasoro Cn = 32Lcos

n

as

∂1| ⊗ ∂2| (C0 − 1 + σ) = 0

permutes the two derivatives

Right reduction in E8 expansion

Exceptional geometry for affine and other groups – p.15

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E9 geometry: section constraint (II)

∂1| ⊗ ∂2| (C0 − 1 + σ) = 0

can be supplemented by

∂1| ⊗ ∂2|C−n = 0 (n > 0) (∂1| ⊗ ∂2| + ∂2| ⊗ ∂1|)C1 = 0

due to structure of tensor product. Together: full E9 section constraint!

Exceptional geometry for affine and other groups – p.16

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E9 geometry: section constraint (II)

∂1| ⊗ ∂2| (C0 − 1 + σ) = 0

can be supplemented by

∂1| ⊗ ∂2|C−n = 0 (n > 0) (∂1| ⊗ ∂2| + ∂2| ⊗ ∂1|)C1 = 0

due to structure of tensor product. Together: full E9 section constraint! Now construct geometry valid modulo these constraints.

Exceptional geometry for affine and other groups – p.16

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E9 generalised Lie derivative

Generalised Lie derivative in index notation for En (n ≤ 7)

[Berman, Cederwall, AK, Thompson]

LξV M = ξN∂NV M + ZMN PQ∂NξP V Q ZMN PQ is e9-adjoint-valued in N P and M Q and has known

relation to section constraint. For E9 in index-free notation

Lξ|V = ∂V |ξ|V + ∂ξ|(C0 − 1)|ξ ⊗ |V

Exceptional geometry for affine and other groups – p.17

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E9 generalised Lie derivative

Generalised Lie derivative in index notation for En (n ≤ 7)

[Berman, Cederwall, AK, Thompson]

LξV M = ξN∂NV M + ZMN PQ∂NξP V Q ZMN PQ is e9-adjoint-valued in N P and M Q and has known

relation to section constraint. For E9 in index-free notation

Lξ|V = ∂V |ξ|V + ∂ξ|(C0 − 1)|ξ ⊗ |V

Closing the algebra with this fails. Already for n = 8 need additional constrained parameter Σ [Hohm, Samtleben;

Cederwall, Rosabal]

Exceptional geometry for affine and other groups – p.17

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E9 generalised Lie derivative (II)

Determine E9 representation of constrained Σ from tensor hierarchy. Here: ΣN M with M index satisfying the same section constraints as ∂M ≡ ∂| Write as |ΣπΣ| (not a direct product in general!).

Exceptional geometry for affine and other groups – p.18

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E9 generalised Lie derivative (II)

Determine E9 representation of constrained Σ from tensor hierarchy. Here: ΣN M with M index satisfying the same section constraints as ∂M ≡ ∂| Write as |ΣπΣ| (not a direct product in general!). To preserve q-grading need to make ansatz

Lξ,Σ|V = ∂V |ξ|V + ∂ξ|(C0 − 1)|ξ ⊗ |V + πΣ|C−1|Σ ⊗ |V

for grading

Exceptional geometry for affine and other groups – p.18

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E9 generalised Lie derivative (II)

Determine E9 representation of constrained Σ from tensor hierarchy. Here: ΣN M with M index satisfying the same section constraints as ∂M ≡ ∂| Write as |ΣπΣ| (not a direct product in general!). To preserve q-grading need to make ansatz

Lξ,Σ|V = ∂V |ξ|V + ∂ξ|(C0 − 1)|ξ ⊗ |V + πΣ|C−1|Σ ⊗ |V

for grading

We show that this gauge algebra closes (modulo the section constraint) ← main result Proof depends on Virasoro action on product spaces Reduces as expected to E8

Exceptional geometry for affine and other groups – p.18

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General lessons

The operator in the section constraint can be written out explicitly as

C0 + σ − 1 = −ηαβTα ⊗ Tβ + σ − 1

Here: Tα generators of E9 and ηαβ inverse invariant form. Almost the same form is valid for any (Kac–Moody) group and any coordinate representation!

Exceptional geometry for affine and other groups – p.19

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General lessons

The operator in the section constraint can be written out explicitly as

C0 + σ − 1 = −ηαβTα ⊗ Tβ + σ − 1

Here: Tα generators of E9 and ηαβ inverse invariant form. Almost the same form is valid for any (Kac–Moody) group and any coordinate representation! Let g be a (simply-laced) Kac–Moody algebra and R(Λ) be the coordinate representation. The weak section constraint is associated with the symmetric product

R(Λ) ⊗s R(Λ) = R(2Λ) ⊕ . . .

and eliminates all dots.

Exceptional geometry for affine and other groups – p.19

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General lessons (II)

R(Λ) ⊗s R(Λ) = R(2Λ) ⊕ . . . R(2Λ) is distinguished by its Casimir eigenvalue. Casimir C2 = 1 2ηαβ : TαTβ :=

α>0

E−αEα + 1 2(H, H) + (ρ, H)

Weyl vector

in a highest weight representation: C2(R(Λ)) = 1

2(Λ, Λ + 2ρ)

In particular

C2(R(2Λ)) = 2C2(R(Λ)) + (Λ, Λ)

A vector |p ∈ R(Λ) satisfies the weak section constraint

|p ⊗ |p ∈ R(2Λ) iff (C2(R(2Λ)) − 2C2(R(Λ)) − (Λ, Λ))|p ⊗ |p = 0

Exceptional geometry for affine and other groups – p.20

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General lessons (III)

0 = (C2(R(2Λ)) − 2C2(R(Λ)) − (Λ, Λ))|p ⊗ |p =

ηαβTα ⊗ Tβ − (Λ, Λ)
|p ⊗ |p

Condition for minimal orbit in R(Λ) Equivalent to 1/2-BPS condition in momentum space Can show strong section constraint to be

0 =

−ηαβTα ⊗ Tβ + (Λ, Λ) − 1 + σ
|p ⊗ |q

Reduces to E9 expression since there (Λ0, Λ0) = 0.

Exceptional geometry for affine and other groups – p.21

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General lessons (III)

0 = (C2(R(2Λ)) − 2C2(R(Λ)) − (Λ, Λ))|p ⊗ |p =

ηαβTα ⊗ Tβ − (Λ, Λ)
|p ⊗ |p

Condition for minimal orbit in R(Λ) Equivalent to 1/2-BPS condition in momentum space Can show strong section constraint to be

0 =

−ηαβTα ⊗ Tβ + (Λ, Λ) − 1 + σ
|p ⊗ |q

Reduces to E9 expression since there (Λ0, Λ0) = 0. Generalised Lie derivative

Lξ|V = ∂V |ξ|V − ηαβ∂ξ|Tα|ξTβ|V + ((Λ, Λ) − 1)∂ξ|ξ|V

Exceptional geometry for affine and other groups – p.21

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Example I: SL(d)

Consider sl(d) in fundamental (Λ, Λ) = d−1

d

Traceless generators Kmn have inverse bilinear form

(Knm, Kqp)−1 = δm

q δp n − 1

dδm

n δp q

and fundamental action Kmn · vp = δp

nvm − 1 dδm n vp

Exceptional geometry for affine and other groups – p.22

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Example I: SL(d)

Consider sl(d) in fundamental (Λ, Λ) = d−1

d

Traceless generators Kmn have inverse bilinear form

(Knm, Kqp)−1 = δm

q δp n − 1

dδm

n δp q

and fundamental action Kmn · vp = δp

nvm − 1 dδm n vp

(Dual) section constraint vanishes identically

(−δm

q δp n + 1

dδm

n δp q)δa nvm δb qwp +

d − 1

vawb + vbwa = 0

Exceptional geometry for affine and other groups – p.22

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Example I: SL(d)

Consider sl(d) in fundamental (Λ, Λ) = d−1

d

Traceless generators Kmn have inverse bilinear form

(Knm, Kqp)−1 = δm

q δp n − 1

dδm

n δp q

and fundamental action Kmn · vp = δp

nvm − 1 dδm n vp

(Dual) section constraint vanishes identically

(−δm

q δp n + 1

dδm

n δp q)δa nvm δb qwp +

d − 1

vawb + vbwa = 0

Generalised Lie derivative

Lξvm = ξn∂nvm − (δa

qδp b − 1

dδa

b δp q)∂bξaδm q vp − 1

d∂nξnvm = ξn∂nvm − ∂nξmvn

Ordinary Riemann geometry

Exceptional geometry for affine and other groups – p.22

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Example II: O(d, d)

Consider o(d, d) in fundamental (Λ, Λ) = 1 Antisymmetric generators JMN have inverse form

(JMN, JPQ)−1 = ηNPηMQ − ηNQηMP

and fundamental action JMN · V Q = 2ηQ[NV M]

Exceptional geometry for affine and other groups – p.23

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Example II: O(d, d)

Consider o(d, d) in fundamental (Λ, Λ) = 1 Antisymmetric generators JMN have inverse form

(JMN, JPQ)−1 = ηNPηMQ − ηNQηMP

and fundamental action JMN · V Q = 2ηQ[NV M] (Dual) section constraint becomes

(−ηNPηMQ + ηNQηMP)ηANV MηBQW P + W AV B = ηABηPQV PW Q = 0

DFT section constraint

Exceptional geometry for affine and other groups – p.23

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Example II: O(d, d)

Consider o(d, d) in fundamental (Λ, Λ) = 1 Antisymmetric generators JMN have inverse form

(JMN, JPQ)−1 = ηNPηMQ − ηNQηMP

and fundamental action JMN · V Q = 2ηQ[NV M] (Dual) section constraint becomes

(−ηNPηMQ + ηNQηMP)ηANV MηBQW P + W AV B = ηABηPQV PW Q = 0

DFT section constraint Generalised Lie derivative

LξV M = ξN∂NV M + (−ηBPηAQ + ηBQηAP )∂CηCBξAηMQV P = ξN∂NV M − ∂PξNV N + ∂MξNV N

DFT geometry

Exceptional geometry for affine and other groups – p.23

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Generalised Scherk–Schwarz reductions

Exceptional geometry for affine and other groups – p.24

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Generalised Scherk–Schwarz reductions

DFT and ExFT give rise to gauged supergravity via GSS.

[Aldazabal, Baguet, Baron, Berman, Geissb¨ uhler, Gra˜ na, Hassler, Hohm, L¨ ust, Marques, Musaev, Rosabal, Samtleben, Thompson, Waldram,...]

⇒ section constraint becomes embedding tensor constraint

Here: Gauged supergravity in D = 2. Not fully developed.

[Samtleben, Weidner] have an embedding tensor θ in basic of

E9 with C−1(θ ⊗ θ) = 0

Exceptional geometry for affine and other groups – p.24

SLIDE 52

Generalised Scherk–Schwarz reductions

DFT and ExFT give rise to gauged supergravity via GSS.

[Aldazabal, Baguet, Baron, Berman, Geissb¨ uhler, Gra˜ na, Hassler, Hohm, L¨ ust, Marques, Musaev, Rosabal, Samtleben, Thompson, Waldram,...]

⇒ section constraint becomes embedding tensor constraint

Here: Gauged supergravity in D = 2. Not fully developed.

[Samtleben, Weidner] have an embedding tensor θ in basic of

E9 with C−1(θ ⊗ θ) = 0

Ansatz involves loop group and part of Virasoro:

e−2σ(x,X)M(x, X) = UT

loop(X)e−2σ(x)M(x)Uloop(X)

ext. 2D coords
int. E9 coords

2D conf. factor twist matrix E9 scalars

Similar ansatz for KK scalar ρ

Exceptional geometry for affine and other groups – p.24

SLIDE 53

Generalised Scherk–Schwarz (II)

Ansatz for vector and gauge parameters

|V = U−T |V |ξ = U−T |ξ |ΣπΣ| = . . .

Underlined (flat) depend only on external constrained nature of Σ from current U−T∂MU in ansatz

Exceptional geometry for affine and other groups – p.25

SLIDE 54

Generalised Scherk–Schwarz (II)

Ansatz for vector and gauge parameters

|V = U−T |V |ξ = U−T |ξ |ΣπΣ| = . . .

Underlined (flat) depend only on external constrained nature of Σ from current U−T∂MU in ansatz Plugging all this into generalised Lie derivative gives

δξ|V = UT Lξ,Σ|V = θ|C−1|ξ ⊗ |V + ϑ|C0|ξ ⊗ |V

standard gaugings trombone gaugings

Constant θ and ϑ given by components of U

Exceptional geometry for affine and other groups – p.25

SLIDE 55

Generalised Scherk–Schwarz (III)

Gauge algebra consistent

[δξ1, δξ2] = δξ12

if

θ| ⊗ θ|C−1 + ϑ| ⊗ θ| (C0 + σ − 1) = 0 ϑ| ⊗ ϑ|C0 + θ| ⊗ ϑ|C−1 = 0

Recovers constraints from [Samtleben, Weidner]! Especially ‘shifted’ coset Virasoro for ordinary gaugings.

Exceptional geometry for affine and other groups – p.26

SLIDE 56

Summary and outlook

Construction of E9 exceptional geometry, including extra constrained parameters Section constraint through coset Virasoro operators Recovers beginnings of D = 2 gauged SUGRA General section constraint and generalised Lie der. w/o constrained parameters. General closure sometimes from tensor hierarchy algebra [Cederwall, Palmkvist]

Exceptional geometry for affine and other groups – p.27

SLIDE 57

Summary and outlook

Construction of E9 exceptional geometry, including extra constrained parameters Section constraint through coset Virasoro operators Recovers beginnings of D = 2 gauged SUGRA General section constraint and generalised Lie der. w/o constrained parameters. General closure sometimes from tensor hierarchy algebra [Cederwall, Palmkvist] Prospects Construction of E9 ExFT [see Franz’s talk] Inclusion of fermions (hard) Construction of solutions. Relation to exotic branes

[Bergshoeff, Berman, Musaev, Riccioni, Rudolph,...]

Exceptional geometry for affine and other groups – p.27

SLIDE 58

Summary and outlook

Construction of E9 exceptional geometry, including extra constrained parameters Section constraint through coset Virasoro operators Recovers beginnings of D = 2 gauged SUGRA General section constraint and generalised Lie der. w/o constrained parameters. General closure sometimes from tensor hierarchy algebra [Cederwall, Palmkvist] Prospects Construction of E9 ExFT [see Franz’s talk] Inclusion of fermions (hard) Construction of solutions. Relation to exotic branes

[Bergshoeff, Berman, Musaev, Riccioni, Rudolph,...]

Thank you for your attention!

Exceptional geometry for affine and other groups – p.27