exceptional geometry and string compactifications Henning Samtleben - - PowerPoint PPT Presentation

ENS de Lyon meets SISSA Lyon 12/2017 Henning Samtleben

exceptional geometry and string compactifications

Theoretical physics group (équipe 4) at Laboratoire de physique, ENS de Lyon

condensed matter theory

statistical physics mathematical physics

head of group: Jean-Michel Maillet

Marco Marciani Takashi Kameyama Baptiste Pezelier Valentin Raban Benjamin Roussel Jérôme Thibaut Lavi Kumar Upreti

Theoretical physics group (équipe 4)

Clément Cabart Christophe Goeller Savish Goomanee Callum Gray Yannick Herfray Sylvain Lacroix Thibaud Louvet Raphaël Menu faculty postdocs PhD students

condensed matter theory

statistical physics mathematical physics

Angel Alastuey Jeremie Bouttier David Carpentier Pascal Degiovanni François Delduc Pierre Delplace Andrey Fedorenko Krzysztof Gawedzki Peter Holdsworth Karol Kozlowski Etera Livine Marc Magro Jean Michel Maillet Giuliano Niccoli Edmond Orignac Tommaso Roscilde Henning Samtleben Lucile Savary

Theoretical physics group (équipe 4) at Laboratoire de physique, ENS de Lyon condensed matter theory

topological matter: topological insulators, topological interacting phases, Dirac phases,

hydrodynamics, topological superconductivity, dynamical systems

relativistic phases in condensed matter:

graphene, Weyl/Dirac semimetals, quantum transport, effects of disorder

strong correlations in boson and fermion systems:

Tomonaga-Luttinger liquids, Mott transition

mesoscopic physics:

quantum nanoelectronics, electron quantum optics, decoherence, quantum technologies

quantum magnetism: frustrated systems, spin ladders,

magnetic monopole quasi-particles, Coulomb and topological phase transitions

non-equilibrium quantum many-body systems:

quantum quenches, correlation spreading, many-body localization

quantum correlations in many-body systems: entanglement and beyond Bose-Einstein condensation: long-range effects

statistical physics

+ related activity in (all) other groups macroscopic fluctuation theory disordered systems: functional renormalization group, random field systems,

elastic manifolds, depinning

critical Casimir forces, magnetic, fluid and quantum systems emergent electrodynamics: lattice gauge theories for spin systems solvable lattice models and their connections with enumerative/algebraic combinatorics quantum plasmas: path integrals, recombination, ionic criticality

Theoretical physics group (équipe 4) at Laboratoire de physique, ENS de Lyon

mathematical physics

integrable systems: quantum separation of variables and correlation functions,

quantum critical models, integrable probability

conformal field theory: nonequilibrium CFT AdS5 x S5 string theory: integrable deformations of (string) sigma models supersymmetric field theories: 6D SCFT

, M5 branes, higher gauge theories

supergravity: supersymmetry on curved space, duality symmetries quantum gravity: TQFTs, discrete path integrals, holography & entanglement,

random maps and 2DQG

asymptotic analysis: multiple integrals, Riemann–Hilbert problems

Theoretical physics group (équipe 4) at Laboratoire de physique, ENS de Lyon

ENS de Lyon meets SISSA Lyon 12/2017 Henning Samtleben

based on work with

• O. Hohm (MIT), C. Pope (Texas A&M),
• A. Baguet, M. Magro (ENSL)

exceptional geometry and string compactifications

A) Kaluza-Klein theory & Riemannian geometry

• utline

Henning Samtleben ENS Lyon

string compactifications integrability and modified IIB supergravity

gravity and extended geometry applications

D) String theory & generalized geometry E) M theory & exceptional geometry

1919 : extra dimensions in Einstein’s general relativity: D = 4 + 1 1970’s, then 2000 — : D = 10 + 10 1980’s, then 2010 — : D = 11 + ??

Einstein’s general relativity (1915)

Henning Samtleben ENS Lyon

Riemannian geometry

space-time metric gµν dynamics: Einstein-Hilbert action fundamental symmetry: space-time diffeomorphisms ξµ and possible matter couplings with straightforward generalisation to N space-time dimensions

Henning Samtleben ENS Lyon

metric, gauge potential, dilaton {gµν, Aµ, φ} S = Z dNx p |det g| ⇣ R − 1 2∂µφ ∂µφ − 1 4 eαφ FµνF µν⌘ after compactification of the extra dimension: T1

N-dimensional general relativity with matter:

Einstein—Maxwell—dilaton theory

A) Kaluza-Klein theory (1919)

S = Z dN+1x p |det G| R[G]

dynamics

GMN = ✓eaφgµν + eφAµAν eφAµ eφAµ eφ ◆

general relativity in N + 1 space-time dimensions

etc.

(N = 4)

A) Kaluza-Klein theory (1919)

Henning Samtleben ENS Lyon

metric, gauge potential, dilaton {gµν, Aµ, φ} S = Z dNx p |det g| ⇣ R − 1 2∂µφ ∂µφ − 1 4 eαφ FµνF µν⌘ after compactification of the extra dimension: T1

N-dimensional general relativity with matter:

fundamental symmetries: space-time diffeomorphisms , gauge transformations Λ

ξµ

“geometrization

• f gauge symmetry”

S = Z dN+1x p |det G| R[G]

dynamics

GMN = ✓eaφgµν + eφAµAν eφAµ eφAµ eφ ◆

Einstein—Maxwell—dilaton theory general relativity in N + 1 space-time dimensions

etc.

D) string theory & generalized geometry

Henning Samtleben ENS Lyon

reproducing gravitational and gauge interactions with the three-form flux fundamental symmetries: space-time diffeomorphisms , gauge transformations

ξµ Λµ

1970’s: string theory — theory of extended objects (in ten space-time dimensions) universal sector: Einstein—Kalb-Ramond—dilaton

D) string theory & generalized geometry

Henning Samtleben ENS Lyon

with the three-form flux fundamental symmetries: space-time diffeomorphisms , gauge transformations

ξµ Λµ

Kaluza-Klein question : can this structure be embedded in some “higher-dimensional geometry” ..? with combining into some “higher-dimensional diffeomorphism” ..? universal sector: Einstein—Kalb-Ramond—dilaton

“geometrization of gauge symmetry”

??

D) string theory & generalized geometry

Henning Samtleben ENS Lyon

Kaluza-Klein question : can this structure be embedded in some “higher-dimensional geometry” ..? with combining into some “higher-dimensional diffeomorphism” ..?

• W. Siegel (1993), C. Hull, B. Zwiebach, O. Hohm (2009), …

physics double field theory

• N. Hitchin, M. Gualtieri (2003), …

generalized geometry mathematics

GMN = ✓ gµν −gµρBρν Bµρgρν gµν − BµρgρσBσν ◆

universal sector: Einstein—Kalb-Ramond—dilaton

D) string theory & generalized geometry

Henning Samtleben ENS Lyon

generalized metric: 2D–dimensional “space” (with a section condition) unifying

— compatible with the SO(D,D) group structure — closure requires a section condition on the fields:

fields live on D–dimensional slices in the 2D–dimensional “space” generalized connections and curvature:

Dorfmann bracket on the generalized tangent bundle — vanishing of the generalized torsion tensor does not fully determine the connection — notion of a generalized Ricci tensor and Ricci scalar, no generalized Riemann tensor

generalized diffeomorphisms:

GMN = ✓ gµν −gµρBρν Bµρgρν gµν − BµρgρσBσν ◆

physics double field theory generalized geometry mathematics

D) string theory & generalized geometry

Henning Samtleben ENS Lyon

generalized Ricci scalar

unified “geometrical” action space for non-geometric compactifications SO(D,D) covariance of the equations: compact reduction formulas

generalized frame field patching coordinates and dual coordinates

D–dimensional slices in the D+D–dimensional “space”

momentum coordinates and dual winding coordinates T-duality covariant formulation

GMN = ✓ gµν −gµρBρν Bµρgρν gµν − BµρgρσBσν ◆

[Hohm, Lust, Zwiebach]

physics double field theory generalized geometry mathematics

E) M theory & exceptional geometry

Henning Samtleben ENS Lyon

theory of strings and branes (in eleven space-time dimensions) unifying the various string theories fundamental symmetries: space-time diffeomorphisms , gauge transformations

ξµ

low energy theory: D = 11 supergravity can this structure be embedded in some “higher-dimensional geometry” ..? with combining into some “higher-dimensional diffeomorphism” ..?

[Hull, Berman, Perry, Waldram, Coimbra, Strickland-Constable, Thompson, West, Godazgar, Cederwall, Aldazabal, Grana, Marques, Rosabal, Hohm, H.S., …]

exceptional geometry / exceptional field theory

with the four-form flux

E) M theory & exceptional geometry

Henning Samtleben ENS Lyon

generalized metric: E7: (4+56)—dimensional “space” with a section condition exceptional geometry / exceptional field theory unifying

— compatible with the ED(D) group structure — closure requires a section condition on the fields:

fields live on (4+7) / (4+6)–dimensional slices

extended Dorfmann bracket

• n the extended tangent bundle

generalized diffeomorphisms: inequivalent slices describe IIA and IIB supergravity, respectively

IIA IIB c.f. embedding of

AD subgroups

… combines with dual magnetic fields …

E) M theory & exceptional geometry

Henning Samtleben ENS Lyon

generalized metric: E7: (4+56)—dimensional “space” with a section condition exceptional geometry / exceptional field theory generalized diffeomorphisms: generalized torsion, connection, curvature (analogue of Ricci tensor) unique action invariant under generalized diffeomorphisms: E7(7) covariant exceptional field theory

¼ Z L ^ R þ 1 24 gDMMNDMMN

• þ 24

1 4 MMNF MF N

þ e1Ltop VðM; eÞ:

D = 4+56 with section condition upon solving the section condition: reformulation of the original theories IIA and IIB supergravity accommodated in the same framework four-dimensional field theory with fields in infinite-dimensional representations, and infinite-dimensional gauge structure

Henning Samtleben ENS Lyon

manifestly duality covariant formulation of maximal supergravity

D=4 maximal sugra

global E7(7)

reduction of these theories to D=4 dimensions yields maximal supergravity with (hidden) global E7(7) symmetry

T7 T6

D=11 sugra

IIB sugra

E7(7) covariant exceptional field theory

¼ Z L ^ R þ 1 24 gDMMNDMMN

• þ

1 4 MMNF MF N

þ e1Ltop VðM; eÞ:

D = 4+56 with section condition

GL(7) solution to section condition GL(6) solution to section condition

[Cremmer, Julia 1979]

E7(7) : exceptional field theory — applications

Henning Samtleben ENS Lyon

manifestly duality covariant formulation of maximal supergravity

D=4 maximal sugra

global E7(7)

reduction of these theories to D=4 dimensions yields maximal supergravity with (hidden) global E7(7) symmetry

T7 T6

D=11 sugra

IIB sugra

E7(7) covariant exceptional field theory

¼ Z L ^ R þ 1 24 gDMMNDMMN

• þ

1 4 MMNF MF N

þ e1Ltop VðM; eÞ:

D = 4+56 with section condition

GL(7) solution to section condition GL(6) solution to section condition

exceptional field theory explains the symmetry enhancement

E7(7) : exceptional field theory — applications

Henning Samtleben ENS Lyon

manifestly duality covariant formulation of maximal supergravity

D=4 maximal sugra

gauge group SO(8)

also allows a compact description of non-trivial reductions

D=11 sugra

E7(7) covariant exceptional field theory

¼ Z L ^ R þ 1 24 gDMMNDMMN

• þ

1 4 MMNF MF N

þ e1Ltop VðM; eÞ:

D = 4+56 with section condition

GL(7) solution to section condition

S7 x AdS4

E7(7) : exceptional field theory — applications

• þ

1 4 MMNF MF N

þ e1Ltop VðM; eÞ:

¼ Z L ^ R þ 1 24 gDMMNDMMN

Henning Samtleben ENS Lyon

manifestly duality covariant formulation of maximal supergravity

D=4 maximal sugra

gauge group SO(8)

S7 x AdS4

captured by a twisted torus (Scherk-Schwarz) reduction of ExFT

[Kaloper, Myers, Dabholkar, Hull, Reid-Edwards, Dall'Agata, Prezas, HS, Trigiante, Hohm, Kwak, Aldazabal, Baron, Nunez, Marques, Geissbuhler, Grana, Berman, Musaev, Thompson, Rosabal, Lee, Strickland-Constable, Waldram, Dibitetto, Roest, Malek, Blumenhagen, Hassler, Lust, Cho, Fernández-Melgarejo, Jeon, Park, Guarino, Varela, Inverso, Ciceri, …, …]

D=11 sugra

E7(7) covariant exceptional field theory D = 4+56 with section condition

GL(7) solution to section condition

also allows a compact description of non-trivial reductions

E7(7) : exceptional field theory — applications

Henning Samtleben ENS Lyon

D=11 sugra

• þ

1 4 MMNF MF N

þ e1Ltop VðM; eÞ:

¼ Z L ^ R þ 1 24 gDMMNDMMN

MMN(x, Y ) = UM

K(Y ) MKL(x) UN L(Y )

=3

AµM(x, Y ) = ρ−1(Y ) (U −1)KM(Y ) AµK(x)

=3

Bµν α(x, Y ) = ρ−2(Y ) Uαβ(Y ) Bµν β(x)

D=4 maximal sugra

gauge group SO(8)

reduction via generalized Scherk-Schwarz ansatz in ExFT

in terms of an E7(7)—valued twist matrix and scale factor UM N(Y )

ρ(Y )

system of consistency equations generalized parallelizability no general classification of its solutions (Lie algebras vs Leibniz algebras)

=3

• (U −1)M

P (U −1)N L ∂P UL K 912 !

= ρ XMN

K

E7(7) covariant exceptional field theory

GL(7) solution to section condition

E7(7) : exceptional field theory — applications

Henning Samtleben ENS Lyon

twist matrix associated to SO(8) structure constants

=3

U ∈ SL(8)

=3

U = δij a(y2) yi b(y2) yi c(y2) d(y2)

• D=11 sugra

MMN(x, Y ) = UM

K(Y ) MKL(x) UN L(Y )

=3

AµM(x, Y ) = ρ−1(Y ) (U −1)KM(Y ) AµK(x)

=3

Bµν α(x, Y ) = ρ−2(Y ) Uαβ(Y ) Bµν β(x)

D=4 maximal sugra

gauge group SO(8)

=3

• Y M

− →

• Y AB, YAB
• −

→

• Y i8, Y ij, YAB
• =3

yi ≡ Y i8 coordinates twist matrix 56 28 + 28 7 + 21 + 28

=3

y2 ≡ yiyi

• þ

1 4 MMNF MF N

þ e1Ltop VðM; eÞ:

¼ Z L ^ R þ 1 24 gDMMNDMMN

E7(7) covariant exceptional field theory

GL(7) solution to section condition

E7(7) : exceptional field theory — applications

Henning Samtleben ENS Lyon

encodes complicated reduction formulas for the original fields

D=11 sugra

MMN(x, Y ) = UM

K(Y ) MKL(x) UN L(Y )

=3

AµM(x, Y ) = ρ−1(Y ) (U −1)KM(Y ) AµK(x)

=3

Bµν α(x, Y ) = ρ−2(Y ) Uαβ(Y ) Bµν β(x)

D=4 maximal sugra

gauge group SO(8)

• þ

1 4 MMNF MF N

þ e1Ltop VðM; eÞ:

¼ Z L ^ R þ 1 24 gDMMNDMMN

E7(7) covariant exceptional field theory

GL(7) solution to section condition

E7(7) : exceptional field theory — applications

ds2 = ∆−2/3(x, y) gµν(x) dxµdxν + Gmn(x, y)

• dym + K[ab]

m(y)Aab µ (x)dxµ

dyn + K[cd]

n(y)Acd ν (x)dxν

Gmn(x, y) = ∆2/3(x, y) K[ab]

m(y)K[cd] n(y) M ab,cd(x) ¼ ffiffiffi 4 Cmμνρ ¼ − 1 32 K½abm

• 2

ffiffiffiffiffi ffi jgj p εμνρστMab;NFστN þ ffiffiffi 2 p εabcdefΩcdef

μνρ

• − 1

4 ffiffiffi 2 p K½abkK½cdlZ½efmklðA½μabAνcdAρefÞ; Cμνρσ ¼ − 1 16 YaYb ffiffiffiffiffi ffi jgj p εμνρστDτMbc;NMNca þ 2 ffiffiffi 2 p εcdefgbF½μνcdAρefAσga þ 1 4

• ffiffiffi

2 p K½abkK½cdlK½efnZ½ghkln − YhYjεabcegjηdf

• A½μabAνcdAρefAσgh þ ΛμνρσðxÞ:

Cklmn ¼ ~ Cklmn þ 1 16 ~ ωklmnpΔ4=3mαβ ~ Gpq∂qðΔ−4=3mαβÞ; Cμkmn ¼ ffiffiffi 2 p 4 Z½abkmnAμab; Cμνmn ¼ ffiffiffi 2 p 4 K½abkZ½cdkmnA½μabAνcd;

• ffiffiffiffiffi

ffi p ffiffiffi p

• ffiffiffi

p

dictionary

Henning Samtleben ENS Lyon

encodes complicated reduction formulas for the original fields

D=11 sugra

MMN(x, Y ) = UM

K(Y ) MKL(x) UN L(Y )

=3

AµM(x, Y ) = ρ−1(Y ) (U −1)KM(Y ) AµK(x)

=3

Bµν α(x, Y ) = ρ−2(Y ) Uαβ(Y ) Bµν β(x)

D=4 maximal sugra

gauge group SO(8)

• þ

1 4 MMNF MF N

þ e1Ltop VðM; eÞ:

¼ Z L ^ R þ 1 24 gDMMNDMMN

E7(7) covariant exceptional field theory

GL(7) solution to section condition

E7(7) : exceptional field theory — applications

solves many old problems on consistent truncations IIB supergravity on AdS5 x S5 (holography)

heterotic string on group manifolds (string vacua)

ds2 = ∆−2/3(x, y) gµν(x) dxµdxν + Gmn(x, y) dym + K[ab]

Gmn(x, y) = ∆2/3(x, y) K[ab]

• ½
• ½
• Cmμνρ ¼ − 1

• 2

• − 1

• ffiffiffi

• A½μabAνcdAρefAσgh þ ΛμνρσðxÞ:

• ffiffiffiffiffi

• 1

Henning Samtleben ENS Lyon

encodes complicated reduction formulas for the original fields

D=11 sugra

MMN(x, Y ) = UM

K(Y ) MKL(x) UN L(Y )

=3

AµM(x, Y ) = ρ−1(Y ) (U −1)KM(Y ) AµK(x)

=3

Bµν α(x, Y ) = ρ−2(Y ) Uαβ(Y ) Bµν β(x)

D=4 maximal sugra

gauge group SO(8)

• þ

1 4 MMNF MF N

þ e1Ltop VðM; eÞ:

¼ Z L ^ R þ 1 24 gDMMNDMMN

E7(7) covariant exceptional field theory

GL(7) solution to section condition

E7(7) : exceptional field theory — applications

solves many old problems on consistent truncations IIB supergravity on AdS5 x S5 (holography)

heterotic string on group manifolds (string vacua)

ds2 = ∆−2/3(x, y) gµν(x) dxµdxν + Gmn(x, y) dym + K[ab]

Gmn(x, y) = ∆2/3(x, y) K[ab]

• ½
• ½
• Cmμνρ ¼ − 1

• 2

• − 1

• ffiffiffi

• A½μabAνcdAρefAσgh þ ΛμνρσðxÞ:

• ffiffiffiffiffi

• 1

truncate the equations of D=11 supergravity to a lower-dimensional theory, such that any solution to the truncated equations defines a solution of D=11 supergravity typically: commuting Killing vectors (tori) around curved backgrounds: non-trivial! (in general: impossible)

Henning Samtleben ENS Lyon

D=4 maximal sugra

gauge group SO(8)

• þ

1 4 MMNF MF N

þ e1Ltop VðM; eÞ:

¼ Z L ^ R þ 1 24 gDMMNDMMN

E7(7) covariant exceptional field theory

E7(7) : exceptional field theory — applications

built from Killing vectors on SO(p,q)/SO(p–1,q)

similar: twist matrix associated to SO(p,q) and CSO(p,q,r) structure constants

=3

U ∈ SL(8)

Henning Samtleben ENS Lyon

D=4 maximal sugra

gauge group CSO(p,q,r)

D=4 maximal sugra

gauge group SO(p,q)

built from Killing vectors on SO(p,q)/SO(p–1,q)

D=4 maximal sugra

gauge group SO(8)

similar: twist matrix associated to SO(p,q) and CSO(p,q,r) structure constants

=3

U ∈ SL(8)

• þ

1 4 MMNF MF N

þ e1Ltop VðM; eÞ:

¼ Z L ^ R þ 1 24 gDMMNDMMN

E7(7) covariant exceptional field theory

E7(7) : exceptional field theory — applications

Henning Samtleben ENS Lyon

D=4 maximal sugra

gauge group CSO(p,q,r)

D=4 maximal sugra

gauge group SO(p,q)

D=4 maximal sugra

gauge group SO(8)

similar: twist matrix associated to SO(p,q) and CSO(p,q,r) structure constants

=3

U ∈ SL(8)

dictionary

D=11 sugra

background: (warped) hyperboloids new compactifications and solutions spectra and moduli spaces

• þ

1 4 MMNF MF N

þ e1Ltop VðM; eÞ:

¼ Z L ^ R þ 1 24 gDMMNDMMN

E7(7) covariant exceptional field theory

E7(7) : exceptional field theory — applications

and the warpe

Henning Samtleben ENS Lyon

conclusion

geometrization of gauge interactions

generalized / exceptional geometry

unique theory with generalized diffeomorphism invariance in all coordinates (modulo section condition) upon an explicit solution of the section condition the theory reproduces full D=11 and full D=10 IIB supergravity

exceptional field theory

new tools for supersymmetric backgrounds consistent truncations moduli spaces non-geometric compactifications reproduces the modified supergravities arising in the study of integrable deformations of string sigma models

[with M. Magro]

Henning Samtleben ENS Lyon

• utlook — what’s next ?

hints towards a more fundamental formulation infinite-dimensional extensions inclusion of massive string modes underlying symmetries of string and M-theory further applications systematics of compactifications (integration of Leibniz algebras) higher order corrections (counterterms of N=8 supergravity)

candidat for a finite theory of quantum gravity [Bern,Dixon,Roiban, …]

loop calculations

duality invariant graviton amplitudes [Bossard,Kleinschmidt]

• ther exceptional groups

E6 : (5 + 27)—dimensional space with section condition E7 : (4 + 56)— … E8 : (3 + 248)— …

E9 : (2 + )— … [Bossard,Cederwall,Kleinschmidt,Palmkvist,HS] …

[Damour,Henneaux,Nicolai] [West]