[-1cm] membrane sigma-models, generalized geometry, and...

Membrane sigma-models, generalizedgeometry, and Nambu-Poisson structures

Peter Schupp

Jacobs University Bremen

NTU Taipei – July 2013

Outline

I introductory overview

I strings in non-geometric backgrounds

I membrane model and quantization

I effective actions

Motivation

Planck scale quantum geometry

Heuristic argument: quantum + gravity

“The gravitational field generated by the concentration of energyrequired to localize an event in spacetime should not be so strong as tohide the event itself to a distant observer.”

→ fundamental length scale, spacetime uncertainty

∆x ≥√

~G

c3≈ 1.6× 10−33cm

⇒ need to generalize usual notion of smooth geometry

Noncommutative geometry: model quantum geometry of spacetime

Noncommutative Gauge Theory

Particle physics on noncommutative spaces

I noncommutative electrodynamics based on Seiberg-Witten maps

S =

∫d4x

(−1

4Tr(Fµν ? Fµν) + Ψ ? i D/ Ψ

)Fµν = ∂µAν − ∂νAµ − i [Aµ ?, Aν ]

Aµ[A, θ] = Aµ +1

4θξν Aν ,∂ξAµ + Fξµ+ . . .

I novel interactions, Lorentz violation, UV/IR mixing

I also: NC Standard Model, NC GUTs, NC bundles and gerbes

Noncommutative Gravity

Gravity on noncommutative spaces

I general relativity on noncommutative spacetime

I theoretical laboratory for physics beyond QFT/GR

problem: fundamental length incompatible with spacetime symmetries

⇒ The symmetry (Hopf algebra) must be twisted, e.g.:

F = exp

(− i

2θab Va ⊗ Vb

), [Va,Vb] = 0

∆?(f ) = F∆(f )F−1 f ? g = F(f ⊗ g)

twisted tensor calculus, deformed Einstein equations

Exact NC black hole solution with rotational symmetry

star product (twist): [xi ?, xj ] = 2iλεijkxk ,

V ?W = VW +∞∑n=1

B(n,ρ

λ)Ln

ξ+V Ln

ξ−W

metric in isotropic coordinates

ds2 = −(

1− a

ρ

)dt2 +

r 2

ρ2(dx2 + dy 2 + dz2)

with r = (ρ+ a/4)2/ρ, a = 2M, ρ2 = x2 + y 2 + z2

⇒ quantized, quasi 2-dimensional “onion”-spacetime:

ρ = 2jλ = nλ; n = 0, 1, 2, . . .

PS, Solodukhin (2008)

Macroscopic and microscopic non-commutativity

Noncommutativity in electrodynamics and string theory

I electron in constant magnetic field ~B = Bez :

L =m

2~x2 − e~x · ~A with Ai = −B

2εijx

j

limm→0L = e

B

2x iεijx

j ⇒ [x i , x j ] =2i

eBεij

I bosonic open strings ending on D-brane

SΣ =1

4πα′

∫Σ

(gij∂ax i∂ax j − 2πiα′Bijε

ab∂ax i∂bx j)

in low energy limit gij ∼ (α′)2 → 0:

S∂Σ = − i

2

∫∂Σ

Bijxi x j ⇒ [x i , x j ] =

(i

B

)ij

C-S Chu, P-M Ho (1998)

Strings and Noncommutative Geometry

Microscopic non-commutativity: string theory

I dynamics of open strings ending on D-branes:electrodynamics (gauge theory)

I in closed string background 2-form B-field →non-commutative electrodynamics

string endpoints become non-commutative (in α′ → 0 limit) with starproduct ? depending on background fields:

f ? g = f · g +i

2

∑θij ∂i f · ∂jg −

~2

4

∑θijθkl ∂i∂k f · ∂j∂lg

− ~2

6

(∑θij∂jθ

kl ∂i∂k f · ∂lg − ∂k f · ∂i∂lg)

+ . . .

closed-open string relations:

1

g + B=

1

G +Φ+ θ for Φ = −B : θ = B−1

open string noncommutative gauge theory

I 2+1 points on boundary of disk: ordering

I 2-tensor (Θ, B-field) ⇒ 2-bracket

I quantization: noncommutative ?-product

closed string “nonassociative gravity”

I 3+1 points on sphere: orientation

I 3-tensor (R, H-flux) ⇒ 3-bracket

I quantization: nonassociative ?-product

Effective Actions

Open string effective action

SDBI =

∫dnx

1

gsdet

12 (g + B + F ) =

∫dnx

1

gsdet

12 (G +Φ + F )

commutative ↔ non-commutative duality

Expand to second order, ignore (cosmological) constants ⇒

SDBI =

∫dnx| − g | 12

4gsg ijgkl(B+F )ik(B+F )jl (Maxwell/Yang-Mills)

SNCDBI =

∫dnx|θ|− 1

2

4gsgij gklX i , X kX j , X l (Matrix Model)

Covariant coordinates: X i = x i + Ai

Commutative ↔ non-commutative duality fixes form of action

Massless bosonic modes

I open strings: Aµ, φi → gauge and scalar fields

I closed strings: gµν , Bµν , Φ → background geometry, gravity

Closed string effective actionWeyl invariance (at 1 loop) requires vanishing beta functions:

βµν(g) = βµν(B) = β(Φ) = 0⇓

equations of motion for gµν , Bµν , Φ

⇑closed string effective action∫

dDx | − g | 12(

R − 1

12e−Φ/3HµνλHµνλ − 1

6∂µΦ∂µΦ + . . .

)Noncommutative version of this?

Flux compactification

Compactification relates string theory to 3+1 dimensional observablyphenomenology and cosmology. Fluxes stabilize moduli and can lead togeneralized geometric structures; patching by string symmetries.

Non-geometric flux backgroundsT-dualizing a 3-torus with 3-form H-flux gives rise to geometric and

non-geometric fluxes HabcTa−→ f a

bcTb−→ Qab

cTc−→ Rabc

Hull (2005), Shelton, Taylor, Wecht (2005)

Q-flux: T-duality transitions between local trivializations → T-foldsR-flux: metric and B-field not even locally defined; non-geometric strings

→ non-commutative non-associative structuresLust (2010), Blumenhagen, Plauschinn (2010)

Non-geometric Backgrounds

Geometrized non-geometry: membrane sigma modelQ-space: T-fold with twisted bc at σ′1 ∈ S1 ⇒ branch cut

R-space: AKSZ membrane sigma model, not a string theory:membrane geometrizes the non-geometric background

branch surface can be interpreted as open string world sheet

σ′ 1

σ1

σ1

σ2

σ2

C Σ3

I

Σ2

I

dim. reduction Stokes

AKSZ sigma-models

AKSZ construction: action functionals in BV formalism of sigma modelQFT’s for symplectic Lie n-algebroids E

Alexandrov, Kontsevich, Schwarz, Zaboronsky (1995/97)

Poisson sigma model2-dimensional topological field theory, E = T ∗M

S(1)AKSZ =

∫Σ2

(ξi ∧ dX i +

1

2Θij(X ) ξi ∧ ξj

),

with Θ = 12 Θij(x) ∂i ∧ ∂j , ξ = (ξi ) ∈ Ω1(Σ2,X

∗T ∗M)

perturbative expansion ⇒ Kontsevich formality maps

(valid on-shell ([Θ,Θ]S = 0) as well as off-shell, e.g. twisted Poisson)

AKSZ sigma-models

Courant sigma modelTFT with 3-dimensional membrane world volume Σ3

S(2)AKSZ =

∫Σ3

(φi ∧ dX i +

1

2hIJ α

I ∧ dαJ − PIi (X )φi ∧ αI

+1

6TIJK (X )αI ∧ αJ ∧ αK

)with embeddings X : Σ3 → M, 1-form α, aux. 2-form φ, fibre metric h,anchor matrix P, 3-form T .

standard Courant algebroid:C = TM ⊕ T ∗M with natural frame (%i , χ

i ), metric 〈%i , χj〉 = δij

H-space sigma-model

H-space sigma-modelrelevant for geometric flux compactifications: C = TM ⊕ T ∗M twistedby 3-form flux H = 1

6 Hijk(x)dx i ∧ dx j ∧ dxk

H-twisted Courant–Dorfman bracket[(Y1, α1) , (Y2, α2)

]H

:=([Y1,Y2]TM , LY1α2 − LY2α1

− 12 d(α2(Y1) − α1(Y2)

)+ H(Y1,Y2,−)

)metric: natural dual pairing⟨

(Y1, α1) , (Y2, α2)⟩

= α2(Y1) + α1(Y2)

anchor map: projection ρ : C → TMnon-trivial bracket and 3-bracket

[%i , %j ]H = Hijk χk , [%i , %j , %k ]H = Hijk

H-space sigma-model

H-space sigma-model action

S(2)WZ =

∫Σ3

(φi ∧dX i +αi ∧dξi −φi ∧αi +

1

6Hijk(X )αi ∧αj ∧αk

).

where (αI ) = (α1, . . . , α2d) ≡ (α1, . . . , αd , ξ1, . . . , ξd)

If Σ2 := ∂Σ3 6= ∅, we can add a boundary term ⇒boundary/bulk open topological membrane action

S(2)WZ = S

(2)WZ +

∫Σ2

1

2Θij(X ) ξi ∧ ξj .

(other boundary terms are possible, but will not be considered here)

H-space sigma-model

H-twisted Poisson sigma-modelIntegrating out the two-form fields φi yields the AKSZ action

S(1)AKSZ =

∫Σ2

(ξi ∧ dX i +

1

2Θij(X ) ξi ∧ ξj

)+

∫Σ3

1

6Hijk(X )dX i ∧ dX j ∧ dX k ,

which is the action of the H-twisted Poisson sigma-model with targetspace M. Consistency of the equations of motion require Θ to beH-twisted Poisson, i.e.

[Θ,Θ]S =∧3Θ](H) 6= 0

⇒ the Jacobi identity for the bracket is violated.

R-space sigma-model

From H to Q to RClosed strings in Q-space via two T-duality transformations on 3-torusT3; locally filtration of T2 over S1, globally not well-defined (T-fold).Closed string world sheet C = R× S1, coordinates (σ0, σ1), windingnumber p3, twisted boundary conditions at σ′1.

Closed string non-commutativity expressed via Poisson brackets:

x i , x jQ = Q ijk p k and x i , p jQ = 0 = p i , p jQ

Another T-duality transformation sends Q ijk 7→ R ijk , pk 7→ pk and the

Poisson brackets to the twisted Poisson structure

x i , x jΘ = R ijk pk , x i , pjΘ = δi j and pi , pjΘ = 0 .

Lust (2010,2012)

R-space sigma-model

The hidden open stringCFT computation: insert twist field at σ′1 ∈ S1 → generates branch cut

There are indications that the appropriate R-space theory is a membranesigma model, not a string theory:

I open strings do not decouple from gravity in R-space

I membrane theory geometrizes the non-geometric R-flux background

⇒ extend world sheet C to membrane world volume Σ3 = R× (S1 ×R);resulting branch surface can be interpreted as open string world sheet:

σ′ 1

σ1

σ1

σ2

σ2

C Σ3

I

Σ2

I

closed ↔ open string duality

R-space sigma-model

R-space sigma-modelGeneral Courant sigma-model with standard Courant algebroidC = TM ⊕ T ∗M, twisted by a trivector flux R = 1

6 R ijk(x) ∂i ∧ ∂j ∧ ∂k .

Roytenberg’s R-twisted Courant-Dorfman bracket[(Y1, α1) , (Y2, α2)

]R

:=([Y1,Y2]TM + R(α1, α2,−) ,

LY1α2 − LY2α1 − 12 d(α2(Y1) − α1(Y2)

))non-trivial bracket and 3-bracket

[χi , χj ]R = R ijk %k , [χi , χj , χk ]R = R ijk .

R-space sigma-model

R-space sigma-model action

S(2)R =

∫Σ3

(φi ∧

(dX i − αi

)+ αi ∧ dξi +

1

6R ijk(X ) ξi ∧ ξj ∧ ξk

)+

1

2

∫Σ2

g ij(X ) ξi ∧ ∗ξj ,

where we have added a non-topological term involving g ij , to ensureconsistency of R ijk 6= 0.

Integrating out the 2-form field φ yields:

S(2)R =

∫Σ2

ξi∧dX i +

∫Σ3

1

6R ijk(X ) ξi∧ξj∧ξk +

∫Σ2

1

2g ij(X ) ξi∧∗ξj .

assume now constant R ijk and g ij and consider e.o.m. for X . . .

R-space sigma-model

⇒ ξi = dPi and the action reduces to a pure boundary action:

S(2)R =

∫Σ2

(dPi ∧dX i +

1

2R ijk Pi dPj ∧dPk

)+

∫Σ2

1

2g ij dPi ∧∗dPj ,

which can be rewritten as

S(2)R =

∫Σ2

−1

2Θ−1

IJ (X )dX I ∧ dX J +

∫Σ2

1

2gIJ dX I ∧ ∗dX J ,

with

Θ−1 =(Θ−1

IJ

)=

(0 −δi jδi j R ijk pk

),

(gIJ

)=

(0 00 g ij

)and X = (X I ) = (X 1, . . . ,X 2d) := (X 1, . . . ,X d ,P1, . . . ,Pd).

⇒ effective target space = phase space

The “closed string metric” gIJ acts only on momentum space.

R-space sigma-model

Linearized actionGeneralized Poisson sigma-model

S(2)R =

∫Σ2

(ηI ∧ dX I +

1

2ΘIJ(X ) ηI ∧ ηJ

)+

∫Σ2

1

2G IJ ηI ∧ ∗ηJ ,

with auxiliary fields ηI and

Θ =(ΘIJ)

=

(R ijk pk δi j−δi j 0

),

(G IJ)

=

(g ij 00 0

)obeying the usual closed-open string relations, w.r.t. Θ−1 and g .

In phase-space component form:

S(2)R =

∫Σ2

(ηi∧dX i+πi∧dPi+

1

2R ijk Pk ηi∧ηj+ηi∧πi

)+

∫Σ2

1

2g ij ηi∧∗ηj ,

with (ηI ) = (η1, . . . , η2d) ≡ (η1, . . . , ηd , π1, . . . , πd).

R-space sigma-model

Non-commutative, non-associative phase spaceΘ is an H-twisted Poisson bi-vector: [Θ,Θ]S =

∧3Θ](H), where

H = 16 R ijk dpi ∧ dpj ∧ dpk = dB , and B = 1

6 R ijk pk dpi ∧ dpj .

Twisted Poisson brackets

x i , x jΘ = R ijk pk , x i , pjΘ = δi j and pi , pjΘ = 0 .

Corresponding Jacobiator:

x i , x j , xkΘ = R ijk ,

where x I , xJ , xKΘ := [Θ,Θ]S(x I , xJ , xK ) = ΠIJK and

(ΠIJK

)=

1

3

(ΘKL ∂LΘIJ + ΘIL ∂LΘJK + ΘJL ∂LΘKI

)=

(R ijk 0

0 0

).

Macroscopic non-associativity

Non-associativity in electrodynamics with magnetic sourcesA charged particle (charge e, mass m ≡ 1) experiences a magnetic field~B (with sources) only via the Lorentz force ~v = e~v × ~B.The Hamiltonian H = 1

2~v 2 is purely kinetic.

After canonical quantization:

[r i , r j ] = 0 , [r i , v j ] = iδij [v i , v j ] = ieεijkBk

⇒ translations are generated by U(~a) = exp(i~a · ~v)

U(~a1)U(~a2) = e−ieΦ12 U(~a1+~a2) , Φ12 = flux through triangle (~a1,~a2)

(non)associativity:

[U(~a1)U(~a2)]U(~a3) = e−ieΦ123 U(~a1)[U(~a2)U(~a3)] ,

where Φ123 is the flux through the tetrahedron (~a1,~a2,~a3).

infinitesimally:

[v 1, [v 2, v 3]] + [v 2, [v 3, v 1]] + [v 3, [v 1, v 2]] = e∇ · ~B

~v can be realized as a linear operator −i∇− e~A only in the associativeregime, i.e.:

I for ∇ · B = 0: no sources, no flux, associativity;

I for ∇ · B 6= 0: non-associativity, unless Φ123/(2π) is an integer.

In the latter case ∇ · ~B must consist of delta functions, so that the totalflux does not change continuously when the ~a’s change ⇒ monopolesfurthermore, the Dirac quantization condition must be satisfied

Jackiw (1985)

Quantization

Path integral quantizationMapping the open string endpoints to finite values and imposing naturalboundary conditions, we are let to the following schematic functionalintegrals that reproduce Kontsevich’s graphical expansion of globaldeformation quantization. For multivector fields Xr of degree kr :

Un(X1, . . . ,Xn)(f1, . . . , fm)(x) =

∫e

i~ S

(2)R SX1 · · · SXn Ox(f1, . . . , fm) ,

where m = 2− 2n +∑

r kr , SXr = i~∫

Σ2

1kr ! X

I1...Irr (X ) ηI1 · · · ηIr , and

Ox(f1, . . . , fm) =

∫X (∞)=x

[f1

(X (q1)

)· · · fm

(X (qm)

)](m−2)

,

with 1 = q1 > q2 > · · · > qm = 0 and ∞ distinct points on the boundaryof the disk ∂Σ2; the path integrals are weighted with the full gauge-fixedaction and the integrations taken over all fields including ghosts.

Cattaneo, Felder (2000)

Quantization

Kontsevich formality mapsUn maps n multivector fields to a differential operator

Un(X1, . . . ,Xn) =∑Γ∈Gn

wΓ DΓ(X1, . . . ,Xn) ,

where the sum is over all possible diagrams with weight

wΓ =1

(2π)2n+m−2

∫Hn

n∧i=1

(dφhe1

i∧ · · · ∧ dφh

ekii

).

The star product and the 3-bracket are given by

f ? g =∞∑n=0

( i ~)n

n!Un(Θ, . . . ,Θ)(f , g) =: Φ(Θ)(f , g) ,

[f , g , h]? =∞∑n=0

( i ~)n

n!Un+1(Π,Θ, . . . ,Θ)(f , g , h) =: Φ(Π)(f , g , h) .

Quantization

Relevant diagrams involve the bivector Θ = 12 ΘIJ∂I ∧ ∂J . . .

θ1

ψ1p1

θ1

ψ1

θ2

ψ2

p1

p2

Quantization

. . . and the trivector Π = 16 ΠIJK∂I ∧ ∂J ∧ ∂K = dΘΘ = [Θ,Θ]S :

θ

ψ

φ

p1

For constant Π all other diagrams factorize and their weights can beexpressed in terms of these three diagrams (up to permutations).

Quantization

Formality conditionThe Un define L∞-morphisms and satisfy

d·Un(X1, . . . ,Xn) +1

2

∑ItJ=(1,...,n)I,J 6=∅

εX (I,J )[U|I|(XI) , U|J |(XJ )

]G

=∑i<j

(−1)αij Un−1

([Xi ,Xj ]S,X1, . . . , Xi , . . . , Xj , . . . ,Xn

),

relating Schouten brackets to Gerstenhaber brackets.Kontsevich (1997)

This implies in particular

d?Φ(Θ) = i ~Φ(dΘΘ) ,

which explicitly quantifies the lack of associativity of the star product:

(f ? g) ? h − f ? (g ? h) = ~2 i [f , g , h]? = ~

2 i Φ(Π)(f , g , h) .

Quantization

The formality condition implies derivation properties:

I For a function h, the Hamiltonian vector field dΘh = −, h ismapped to the inner derivation d?h = i

~ [ h ,−]? = i~Φ(dΘh), where

h = Φ(h) ≡∑∞

n=0( i ~)n

n! Un+1(h,Θ, . . . ,Θ).

I A Poisson structure preserving vector field X (dΘX = 0) is mapped

to a differential operator X =∑∞

n=0( i ~)n

n! Un+1(X ,Θ, . . . ,Θ)satisfying X (f ? g) = X (f ) ? g + f ? X (g).

I The formality condition d?Φ(Π) = i~Φ(dΘΠ) and higher derivationproperties encode quantum analogs of the derivation property andfundamental identity for a Nambu-Poisson structure.

I In particular, in the present case, where dΘΠ = 0:

[f ?g , h, k]?−[f , g ?h, k]?+[f , g , h?k]? = f ?[g , h, k]?+[f , g , h]??k .

Quantization

Explicit formulas

I Dynamical non-associative star product: f ? g ≡ f ?p g , with

f ?p g = ·[e

i ~2 R ijk pk ∂i⊗∂j e

i ~2

(∂i⊗∂ i−∂ i⊗∂i

)(f ⊗ g)

]I Replacing the dynamical variable p with a constant p we obtain an

associative Moyal-Weyl type star product ? := ?p.

I Triple products and 3-bracket:

(f ? g) ? h =[?(

exp(~2

4 R ijk ∂i ⊗ ∂j ⊗ ∂k)(f ⊗ g ⊗ h)

) ]p→p

[f , g , h]? =4 i

~

[?(

sinh(~2

4 R ijk ∂i ⊗ ∂j ⊗ ∂k)(f ⊗ g ⊗ h)

) ]p→p

I Trace property:∫

[f , g , h]? = 0

Seiberg-Witten maps

Seiberg-Witten mapThe map from ordinary to NC gauge theory is related to the equivalencemap D of star products ?, ?′ and is a quantum analog of Moser’s lemma.

Let F = dA and ρ the flow generated by the vector field AΘ = Θ(A,−):

B :

ρMoser

Θ

ρ

quantization // ?

D

B + F : Θ′quantization // ?′

where Θ′ = Θ(1 + ~F Θ)−1 and D(f ?′ g) = Df ?Dg .

The noncommutative gauge field A is obtained from Dx =: x + A,such that ordinary gauge transform of A ⇒ NC gauge transform of A.→ explicit expression for the SW map for arbitrary Θ(x)→ can be globalized (and extended to gerbes)

Jurco, PS, Wess (2000-2002)

Seiberg-Witten maps

Twisted Poisson structure, NC gerbesPoisson structure twisted by closed 3-form H: [Θ,Θ]S =

∧3 Θ]H

For covering by contractible open patches labeled by α, β, γ, . . .:

H|α = dBα , (Bβ − Bα)|α∩β = Fαβ = dAαβ

Θ can be locally untwisted by Bα: Θα := Θ(1− ~BαΘ)−1.

quantization of Θ → nonassociative ?quantization of Θα, Θβ → associative ?α, ?β related by Dαβ

for more details: Aschieri, Bakovic, Jurco, PS (2010)

SW maps for R-twisted Poisson structurestrivial gerbe → replace patch label α by the (constant) vector p:

Θ =

(~R ijk pk δi j−δi j 0

)Θp =

(~R ijk pk δi j−δi j 0

)Bp =

(0 00 R ijk (pk − pk)

)Θ: twisted Poisson Θp: Poisson H = dBp = 1

2 R ijkdpidpjdpk

Seiberg-Witten maps

Gauge potential: A = AIdx I = ai (x , p)dx i + ai (x , p)dpi

Maps between associative ? and ?′ are generated byApp′ = R ijkpi (pk − p′k)dpj with Fpp′ = R ijk(pk − p′k)dpidpj .

Special case p = 0: canonical Moyal-Weyl star product ?0.

Generalization of SW maps for non-associative structuresA construction directly based on twisted Θ is spoiled by [Θ,Θ]S -terms.These can be avoided in the present case by choosing ai (x , p) = 0!

I general coordinate transformations generated byΘ(A,−) = ai (x , p)∂i

I Nambu-Poisson maps: choose A = R(a2,−) for any 2-form a2;→ higher “Nambu-Poisson” gauge theory.

I map from associative to nonassociative: Dp generated byAp = 1

2 R ijkpi pkdpj can be explicitly computed and satisfies

f ? g =[Dpf ?0 Dpg

]p→p

Nambu-Poisson and Membranes

Remarks on Nambu-Poisson structures

I The trivector Π = 16 R ijk∂i ∧ ∂j ∧ ∂k is an example of a

Nambu-Poisson tensor.

I Nambu mechanics: multi-Hamiltonian dynamics with generalizedPoisson brackets; e.g. Euler’s equations for the spinning top :

d

dtLi = Li ,

~L2

2,T with f , g , h ∝ εijk ∂i f ∂jg ∂kh

I more generally:

f , h1, . . . , hp = Πi j1...jp (x) ∂i f ∂j1 h1 · · · ∂jphp

f0, · · · , fp, h1, · · · , hp = f0, h1, · · · , hp, f1, · · · , fp+ . . .

. . .+ f0, . . . , fp−1, fp, h1, · · · , hpI Our construction can be used to quantize these objects.


Nambu-Dirac-Born-Infeld action(B Jurco & PS 2012)

commutative ↔ non-commutative duality implies

Sp-DBI =

∫dnx

1

gmdet

p2(p+1) [g ] det

12(p+1)

[g + (C + F )g−1(C + F )T

]=

∫dnx

1

Gmdet

p2(p+1)

[G]

det1

2(p+1)[G +(Φ+F ) G−1

(Φ+F )T]

This action interpolates between early proposals based on supersymmetryand more recent work featuring higher geometric structures.


Expansion of actionignore a cosmological constant term and let F = C + F

Sp-DBI =

∫dnx

1

2(p + 1)gmdet

12 (g) tr

[g−1F g−1FT

]+ . . .

the coupling constant gm is dimensionless for:

I strings on D3 with 2-form field strength (Maxwell/Yang-Mills)

I 2-branes on 5-brane with 3-form field strength ( M2-M5 system)

I p-branes on 2(p + 1)-brane with p + 1 form field strength

consider p = 2, p′ = 5 and expand further (k = Fkli Fjkl):

det16 (1 + k) =

√1 +

1

3trk − 1

6trk2 +

1

36(tr k)2 + . . . .

⇒ exact match with κ-symmetry computation ofCederwall, Nilsson, Sundell, “An Action for the superfive-brane” (1998)


From higher gauge theory to matrix model. . .dual NC model in background independent gauge ΠCT = −1

expanding to lowest order (ignore a non-cosmological constant) ⇒semi-classical/infinite-dimensional version of a matrix model∫

dp+1x1

|Π|1

p+1

1

2(p + 1)gm

· gi0j0 · · · gip jpX j0 , . . . , X jpX i0 , . . . , X ip

quantize:

1

2(p + 1)gmTr(

gi0j0 · · · gip jp

[X j0 , . . . , X jp

] [X i0 , . . . , X ip

])

Conclusion

Summary

I Membrane sigma model with R-flux: Target space doubling arisesnaturally in this model and geometrizes the flux.

I Perturbative quantization of the model leads to a nonassociativedynamical star product. Its Jacobiator provides the quantization ofthe 3-bracket.

I The construction can be used to quantize Nambu-Poisson structures.

I Duality of commutative vs. non-commutative description fixeseffective action for D/M-branes (↔ generalized geometry)⇒ (higher) DBI action and gauge theory with qg/string corrections

Outlook

I A similar equivalence should fix the effective action for closed stringsincluding all corrections ⇒ qg/string corrections to theEinstein-Hilbert action (plus Kalb-Ramond field and dilaton).

Thanks for listening!

[-1cm] membrane sigma-models, generalized geometry, and...

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