Quantization of non-geometric flux backgrounds Dionysios Mylonas - - PowerPoint PPT Presentation

Quantization of non-geometric flux backgrounds

Dionysios Mylonas

School of Mathematical and Computer Science Heriot-Watt university, Edinburgh

EMPG November 14, 2012 Based on: D.M, P. Schupp and R. Szabo, JHEP 1209 (2012) 012, [arXiv:1207.0926]

Background and motivation σ-models Deformation quantization Convolution product quantization

Outline

The past: Background and motivation. From a membrane σ-model to string theory on the boundary. Deformation quantization a l´ a Kontsevich. Convolution product quantization. Recap. The future: ?

Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation σ-models Deformation quantization Convolution product quantization

This is the end...

You have... 36d 9hr and 25min left to the END OF THE WORLD!

Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation σ-models Deformation quantization Convolution product quantization

This is the end...

You have... 36d 9hr and 25min left to the END OF THE WORLD! “Two things are infinite: the universe and human stupidity; and I’m not sure about the the universe.”

Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation σ-models Deformation quantization Convolution product quantization Open and closed string NCity Toroidal backgrounds CFT analysis

Open and closed strings

Open string with constant B-field

• Xi(τ, σ), Xj(τ, σ′)
• σ=σ′=0,2π = i θij

where θ = −2π α′ (1 + F2)−1 F and F = B − F.

(Seiberg & Witten, Chu & Ho)

Commutator not well defined for closed strings. Jacobiator 3-bracket: [Xi, Xj, Xk] := lim

σi→σ

• Xi(τ, σ1), Xj(τ, σ2)
• , Xk(τ, σ3)
• +cyclic

For the linearized SU(2) WZW model with H = dB = 0 [Xi, Xj, Xk] ∼ Hijk, i.e. the target space is non-associative.

(Blumenhagen & Plauschinn, 1010.1263)

Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation σ-models Deformation quantization Convolution product quantization Open and closed string NCity Toroidal backgrounds CFT analysis

T-duality frames

Consider T3 with non-vanishing H-flux. Tx3 Tx2 Tx1 H-flux f-flux (nilmanifold) Q-flux (T-fold) R-flux (non-geometric)

• Xi(τ, σ), Xj(τ, σ′)
• = 0
• Xi(τ, σ), X∗

j (τ, σ′)

• = 0

Tx2

− →

• Xi(τ, σ), Xj(τ, σ′)
• = 0
• Xi(τ, σ), X∗

j (τ, σ′)

• = 0

where X∗

i = Xi,L − Xi,R ∈ M∗. Should use doubled geometry.

In this context the same type of nonassociativity was found and a NA target space algebra on R-space was proposed.

(L¨ ust, 1010.1361)

Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation σ-models Deformation quantization Convolution product quantization Open and closed string NCity Toroidal backgrounds CFT analysis

Boundary conditions

Closed str on M × M∗ X(τ, σ + 2π) = e 2π i θ X(τ, σ) X∗(τ + 2π, σ) = e 2π i θ X∗(τ, σ) where θ = −n H and n ∈ Z the dual momentum along the T-dualised direction. The situation resembles the open string case: NCFT on D-branes T → FT on intersecting D-branes Are there some kind of closed string “D-branes”? Are they dynamical solutions in doubled gravity? NA SW map?

(L¨ ust, 1010.1361)

Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation σ-models Deformation quantization Convolution product quantization Open and closed string NCity Toroidal backgrounds CFT analysis

Linearized CFT

Flat space with constantl H-flux [Xi, Xj, Xk] = i α θijk where θijk ∼ H. α = 0 for the H-flux background α = 1 after an odd number of T-duality transformations. 3-product conjecure for constant R-flux (f1⋄f2⋄f3)(x) := exp π2 2 θijk∂x1

i ∂x2 j ∂x3 k

• f1(x1)f2(x2)f3(x3)
• xi=x

(Blumenhagen et al, 1106.0316)

Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation σ-models Deformation quantization Convolution product quantization More background Model building Reduction Boundary conditions

Definitions

Poisson manifold Given a bivector field Θ = 1

2 Θij(x) ∂i ∧ ∂j on a smooth manifold

M a skew-symmetric bracket {−, −}Θ can be defined. This is a Poisson structure if the Schouten-Nijehuis bracket [Θ, Θ]S is zero. A quasi-Poisson structure doesn’t satisfy the Jacobi identity. Lie algebroid A Lie algebroid is a vector bundle E → M endowed with a Lie bracket [−, −]E on smooth sections of E and an anchor map ρ : E → TM. The tangent map to ρ is a Lie algebra homomorphism. Courant algebroid: E is further equipped with a metric −, − and a Jacobiator.

Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation σ-models Deformation quantization Convolution product quantization More background Model building Reduction Boundary conditions

Poisson σ-model

A Poisson manifold M (= symplectic Lie 1-algebroid with the canonical symplectic structure on T ∗M). A 2d string worldsheet Σ2. A differential form on Σ2 is given by the embedding X = (Xi) : Σ2 → M and an auxilliary 1-form field on Σ2 ξ = (ξi) ∈ Ω1(Σ2, X∗T ∗M), i ∈ {1, . . . , d}. Action S(1) =

• Σ2
• ξi ∧ dXi + 1

2 Θij(X) ξi ∧ ξj

• This is a topological field theory on C∞(TΣ2, T ∗M).

(Cattaneo & Felder, math.QA/0102108)

Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation σ-models Deformation quantization Convolution product quantization More background Model building Reduction Boundary conditions

Courant σ-model

Courant algebroid (= symplectic Lie 2-algebroid). A 3d membrane worldvolume Σ3. α = (αI) ∈ Ω1(Σ3, X∗E) and φ = (φi) ∈ Ω2(Σ3, X∗T ∗M). Choose a local basis of sections {ψI} of E → M s..t. the fibre metric hIJ := ψI, ψJ is constant, I ∈ {1, . . . , 2d}.

• Def. the anchor matrix ρ(ψI) = PIi(x) ∂i, and the 3-form

TIJK(x) := [ψI, ψJ, ψK]E. Action S(2) =

• Σ3
• φi ∧ dXi

+ 1 2 hIJ αI ∧ dαJ − PIi(X) φi ∧ αI + + 1 6 TIJK(X) αI ∧ αJ ∧ αK

Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation σ-models Deformation quantization Convolution product quantization More background Model building Reduction Boundary conditions

H-space

Standard Courant algebroid C = TM ⊕ T ∗M twisted by H = 1

6 Hijk(x) dxi ∧ dxj ∧ dxk.

Structure maps:

• Antisymmetrized H-twisted Courant-Dorfman bracket.
• The usual pairing between TM and T ∗M and ρ=trivial.

Assume TM ∼ = M × Rd to keep only H-flux. Write (αI) := (α1, . . . , αd, ξ1, . . . , ξd) and integrate out φi. H-twisted Poisson σ-model

• S (1)

=

• Σ2
• ξi ∧ dXi + 1

2 Θij(X) ξi ∧ ξj

• +
• Σ3

1 6 Hijk(X) dXi ∧ dXj ∧ dXk

Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation σ-models Deformation quantization Convolution product quantization More background Model building Reduction Boundary conditions

H-space

Standard Courant algebroid C = TM ⊕ T ∗M twisted by H = 1

6 Hijk(x) dxi ∧ dxj ∧ dxk.

Structure maps:

• Antisymmetrized H-twisted Courant-Dorfman bracket.
• The usual pairing between TM and T ∗M and ρ=trivial.

Assume TM ∼ = M × Rd to keep only H-flux. Write (αI) := (α1, . . . , αd, ξ1, . . . , ξd) and integrate out φi. H-twisted Poisson σ-model

• S (1)

=

• Σ2
• ξi ∧ dXi + 1

2 Θij(X) ξi ∧ ξj

• +
• Σ3

1 6 Hijk(X) dXi ∧ dXj ∧ dXk

Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation σ-models Deformation quantization Convolution product quantization More background Model building Reduction Boundary conditions

H-space

Standard Courant algebroid C = TM ⊕ T ∗M twisted by H = 1

6 Hijk(x) dxi ∧ dxj ∧ dxk.

Structure maps:

• Antisymmetrized H-twisted Courant-Dorfman bracket.
• The usual pairing between TM and T ∗M and ρ=trivial.

Assume TM ∼ = M × Rd to keep only H-flux. Write (αI) := (α1, . . . , αd, ξ1, . . . , ξd) and integrate out φi. H-twisted Poisson σ-model

• S (1)

=

• Σ2
• ξi ∧ dXi + 1

2 Θij(X) ξi ∧ ξj

• +
• Σ3

1 6 Hijk(X) dXi ∧ dXj ∧ dXk

Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation σ-models Deformation quantization Convolution product quantization More background Model building Reduction Boundary conditions

H-space

Standard Courant algebroid C = TM ⊕ T ∗M twisted by H = 1

6 Hijk(x) dxi ∧ dxj ∧ dxk.

Structure maps:

• Antisymmetrized H-twisted Courant-Dorfman bracket.
• The usual pairing between TM and T ∗M and ρ=trivial.

Assume TM ∼ = M × Rd to keep only H-flux. Write (αI) := (α1, . . . , αd, ξ1, . . . , ξd) and integrate out φi. H-twisted Poisson σ-model

• S (1)

=

• Σ2
• ξi ∧ dXi + 1

2 Θij(X) ξi ∧ ξj

• +
• Σ3

1 6 Hijk(X) dXi ∧ dXj ∧ dXk

Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation σ-models Deformation quantization Convolution product quantization More background Model building Reduction Boundary conditions

H-space

Standard Courant algebroid C = TM ⊕ T ∗M twisted by H = 1

6 Hijk(x) dxi ∧ dxj ∧ dxk.

Structure maps:

• Antisymmetrized H-twisted Courant-Dorfman bracket.
• The usual pairing between TM and T ∗M and ρ=trivial.

Assume TM ∼ = M × Rd to keep only H-flux. Write (αI) := (α1, . . . , αd, ξ1, . . . , ξd) and integrate out φi. H-twisted Poisson σ-model

• S (1)

=

• Σ2
• ξi ∧ dXi + 1

2 Θij(X) ξi ∧ ξj

• +
• Σ3

1 6 Hijk(X) dXi ∧ dXj ∧ dXk

Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation σ-models Deformation quantization Convolution product quantization More background Model building Reduction Boundary conditions

H-space

Standard Courant algebroid C = TM ⊕ T ∗M twisted by H = 1

6 Hijk(x) dxi ∧ dxj ∧ dxk.

Structure maps:

• Antisymmetrized H-twisted Courant-Dorfman bracket.
• The usual pairing between TM and T ∗M and ρ=trivial.

Assume TM ∼ = M × Rd to keep only H-flux. Write (αI) := (α1, . . . , αd, ξ1, . . . , ξd) and integrate out φi. H-twisted Poisson σ-model

• S (1)

=

• Σ2
• ξi ∧ dXi + 1

2 Θij(X) ξi ∧ ξj

• +
• Σ3

1 6 Hijk(X) dXi ∧ dXj ∧ dXk

Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation σ-models Deformation quantization Convolution product quantization More background Model building Reduction Boundary conditions

H-space

Standard Courant algebroid C = TM ⊕ T ∗M twisted by H = 1

6 Hijk(x) dxi ∧ dxj ∧ dxk.

Structure maps:

• Antisymmetrized H-twisted Courant-Dorfman bracket.
• The usual pairing between TM and T ∗M and ρ=trivial.

Assume TM ∼ = M × Rd to keep only H-flux. Write (αI) := (α1, . . . , αd, ξ1, . . . , ξd) and integrate out φi. H-twisted Poisson σ-model

• S (1)

=

• Σ2
• ξi ∧ dXi + 1

2 Θij(X) ξi ∧ ξj

• +
• Σ3

1 6 Hijk(X) dXi ∧ dXj ∧ dXk

Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation σ-models Deformation quantization Convolution product quantization More background Model building Reduction Boundary conditions

H-space

Standard Courant algebroid C = TM ⊕ T ∗M twisted by H = 1

6 Hijk(x) dxi ∧ dxj ∧ dxk.

Structure maps:

• Antisymmetrized H-twisted Courant-Dorfman bracket.
• The usual pairing between TM and T ∗M and ρ=trivial.

Assume TM ∼ = M × Rd to keep only H-flux. Write (αI) := (α1, . . . , αd, ξ1, . . . , ξd) and integrate out φi. H-twisted Poisson σ-model

• S (1)

=

• Σ2
• ξi ∧ dXi + 1

2 Θij(X) ξi ∧ ξj

• +
• Σ3

1 6 Hijk(X) dXi ∧ dXj ∧ dXk

Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation σ-models Deformation quantization Convolution product quantization More background Model building Reduction Boundary conditions

R-space

R-twisted C-D bracket; all other structure maps don’t change. Non-topological quasi-Poisson σ-model, constant R-flux S(2)

R =

• Σ2
• ηI ∧ dXI + 1

2 ΘIJ(X) ηI ∧ ηJ

• +
• Σ2

1 2 GIJ ηI ∧ ∗ηJ Θ =

• ΘIJ

= Rijk pk δij −δij

• and
• GIJ

= gij

• [Θ, Θ]S = 3Θ♯(H) , where H = 1

6 Rijk dpi ∧ dpj ∧ dpk.

The twisting is provided by a U(1)-gerbe in momentum space with 2-connection B = 1

6 Rijk pk dpi ∧ dpj.

{xI, xJ}Θ = ΘIJ(x) (antisym.) reproduces L¨ ust’s NA algebra.

Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation σ-models Deformation quantization Convolution product quantization More background Model building Reduction Boundary conditions

R-space

R-twisted C-D bracket; all other structure maps don’t change. Non-topological quasi-Poisson σ-model, constant R-flux S(2)

R =

• Σ2
• ηI ∧ dXI + 1

2 ΘIJ(X) ηI ∧ ηJ

• +
• Σ2

1 2 GIJ ηI ∧ ∗ηJ Θ =

• ΘIJ

= Rijk pk δij −δij

• and
• GIJ

= gij

• [Θ, Θ]S = 3Θ♯(H) , where H = 1

6 Rijk dpi ∧ dpj ∧ dpk.

The twisting is provided by a U(1)-gerbe in momentum space with 2-connection B = 1

6 Rijk pk dpi ∧ dpj.

{xI, xJ}Θ = ΘIJ(x) (antisym.) reproduces L¨ ust’s NA algebra.

Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation σ-models Deformation quantization Convolution product quantization More background Model building Reduction Boundary conditions

R-space

R-twisted C-D bracket; all other structure maps don’t change. Non-topological quasi-Poisson σ-model, constant R-flux S(2)

R =

• Σ2
• ηI ∧ dXI + 1

2 ΘIJ(X) ηI ∧ ηJ

• +
• Σ2

1 2 GIJ ηI ∧ ∗ηJ Θ =

• ΘIJ

= Rijk pk δij −δij

• and
• GIJ

= gij

• [Θ, Θ]S = 3Θ♯(H) , where H = 1

6 Rijk dpi ∧ dpj ∧ dpk.

The twisting is provided by a U(1)-gerbe in momentum space with 2-connection B = 1

6 Rijk pk dpi ∧ dpj.

{xI, xJ}Θ = ΘIJ(x) (antisym.) reproduces L¨ ust’s NA algebra.

Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation σ-models Deformation quantization Convolution product quantization More background Model building Reduction Boundary conditions

R-space

R-twisted C-D bracket; all other structure maps don’t change. Non-topological quasi-Poisson σ-model, constant R-flux S(2)

R =

• Σ2
• ηI ∧ dXI + 1

2 ΘIJ(X) ηI ∧ ηJ

• +
• Σ2

1 2 GIJ ηI ∧ ∗ηJ Θ =

• ΘIJ

= Rijk pk δij −δij

• and
• GIJ

= gij

• [Θ, Θ]S = 3Θ♯(H) , where H = 1

6 Rijk dpi ∧ dpj ∧ dpk.

The twisting is provided by a U(1)-gerbe in momentum space with 2-connection B = 1

6 Rijk pk dpi ∧ dpj.

{xI, xJ}Θ = ΘIJ(x) (antisym.) reproduces L¨ ust’s NA algebra.

Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation σ-models Deformation quantization Convolution product quantization More background Model building Reduction Boundary conditions

R-space

R-twisted C-D bracket; all other structure maps don’t change. Non-topological quasi-Poisson σ-model, constant R-flux S(2)

R =

• Σ2
• ηI ∧ dXI + 1

2 ΘIJ(X) ηI ∧ ηJ

• +
• Σ2

1 2 GIJ ηI ∧ ∗ηJ Θ =

• ΘIJ

= Rijk pk δij −δij

• and
• GIJ

= gij

• [Θ, Θ]S = 3Θ♯(H) , where H = 1

6 Rijk dpi ∧ dpj ∧ dpk.

The twisting is provided by a U(1)-gerbe in momentum space with 2-connection B = 1

6 Rijk pk dpi ∧ dpj.

{xI, xJ}Θ = ΘIJ(x) (antisym.) reproduces L¨ ust’s NA algebra.

Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation σ-models Deformation quantization Convolution product quantization More background Model building Reduction Boundary conditions

R-space

R-twisted C-D bracket; all other structure maps don’t change. Non-topological quasi-Poisson σ-model, constant R-flux S(2)

R =

• Σ2
• ηI ∧ dXI + 1

2 ΘIJ(X) ηI ∧ ηJ

• +
• Σ2

1 2 GIJ ηI ∧ ∗ηJ Θ =

• ΘIJ

= Rijk pk δij −δij

• and
• GIJ

= gij

• [Θ, Θ]S = 3Θ♯(H) , where H = 1

6 Rijk dpi ∧ dpj ∧ dpk.

The twisting is provided by a U(1)-gerbe in momentum space with 2-connection B = 1

6 Rijk pk dpi ∧ dpj.

{xI, xJ}Θ = ΘIJ(x) (antisym.) reproduces L¨ ust’s NA algebra.

Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation σ-models Deformation quantization Convolution product quantization More background Model building Reduction Boundary conditions

Boundary conditions

Consider M = T2 × S1. Elliptic monodromies induce twisted BCs on closed str. Canonical quantization results to NC fibres: {xi, xj}Q = Qijk ˜ p k and {xi, ˜ p j}Q = 0 = {˜ p i, ˜ p j}Q, with constant Q-flux. Winding number dependance. Closed str. on an orbifold → Open str. on the covering space. In CFT, monodromy=twist field at some pt ⇒ branch pt. Stokes’ theorem: Σ3 → Σ2 ⇒ branch cut I.

Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation σ-models Deformation quantization Convolution product quantization Motivation Kontsevich’s formalism Constant R-flux Seiberg-Witten maps

Motivation

Choose BCs on I appropriate for an open str. twisted Poisson σ-model. Take the topological limit g ≪ R. The propagator is

• XI(w) ηJ(z)
• = i

2π δIJ dzφh(z, w) ,

where dz := dz ∂

∂z + dz ∂ ∂z and φh(z, w) is the harmonic

angle. The Feynman diagram expansion reproduces Kontsevich’s graphical expansion for global deformation quantization of a twisted Poisson structure.

(Cattaneo & Felder, math.QA/9902090, hep-th/0111028)

Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation σ-models Deformation quantization Convolution product quantization Motivation Kontsevich’s formalism Constant R-flux Seiberg-Witten maps

Star product

Formality map = a sequence of L∞ morphisms Un. f ⋆ g :=

∞

• n=0

( i )n n! Un(Θ, . . . , Θ)(f, g) Jacobiator quantization, i.e. Π = [Θ, Θ]S. [f, g, h]⋆ :=

∞

• n=0

( i )n n! Un+1(Π, Θ, . . . , Θ)(f, g, h) ≡ Φ(Π)(f, g, h)

Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation σ-models Deformation quantization Convolution product quantization Motivation Kontsevich’s formalism Constant R-flux Seiberg-Witten maps

Derivation properties

∃ a formality condition s.t. Un is an L∞ morphism. Associator. [f, g, h]⋆ = 2 i

• (f ⋆ g) ⋆ h − f ⋆ (g ⋆ h)
• Derivation properties.

i Φ(dΘΠ) (f, g, h) = f ⋆ [g, h, k]⋆ − [f ⋆ g, h, k]⋆+ +[f, g ⋆ h, k]⋆ − [f, g, h ⋆ k]⋆ + [f, g, h]⋆ ⋆ k where dΘΠ := [Π, Θ]S. l.h.s. is equal to zero for constant R-flux. “Pentagon identity”.

Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation σ-models Deformation quantization Convolution product quantization Motivation Kontsevich’s formalism Constant R-flux Seiberg-Witten maps

Constant R-flux

Nonassociative star product. f ⋆ g = µ2

• exp

i

2 ΘIJ ∂I ⊗ ∂J

• (f ⊗ g)
• Associator.

[f, g, h]⋆ = 4 i

• ˜

⋆

• sinh

2

4 Rijk ∂i⊗∂j⊗∂k

• (f ⊗g⊗h)

˜ p→p

The associator defines a quantization of the Nambu-Poisson structure defined by Π [f, g, h]⋆ = 6 i {f, g, h}Θ + O

• 2

. Leibnitz rule = ⇒ pentagon identity at the quantum level.

Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation σ-models Deformation quantization Convolution product quantization Motivation Kontsevich’s formalism Constant R-flux Seiberg-Witten maps

Seiberg-Witten maps (generalities)

YM

1 gauge field aµ, parameter λ 2 δλaµ = ∂µλ

NCYM

1

ˆ Aµ and ˆ Λ

2 δˆ

Λ ˆ

Aµ = ∂µˆ Λ + i [ˆ Λ , ˆ Aµ]⋆ ˆ A(a) and ˆ Λ(λ, a) are the Seiberg-Witten maps.

(Seiberg-Witten)

Covariantizing map. D : xµ − → ˆ xµ = xµ + θµν ˆ Aν(x) Gauge transformations. θ → θ′ = θ (1 + f θ)−1 , then D(f ⋆θ′ g) = Df ⋆θ Dg

(Jur˘ co, Schupp & Wess)

Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation σ-models Deformation quantization Convolution product quantization Motivation Kontsevich’s formalism Constant R-flux Seiberg-Witten maps

Seiberg-Witten maps (on gerbes)

On gerbes:

H = dBα on each patch, Fαβ := Bβ − Bα = daαβ on overlaps, λαβγ on triple overlaps.

Θ can be locally untwisted by Bα to: Θα = Θ (1 − Bα Θ)−1. Θα (Poisson) and ⋆α (associative) are related by covariantizing maps computed from aαβ.

(Jur˘ co, Schupp & Wess)

Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation σ-models Deformation quantization Convolution product quantization Motivation Kontsevich’s formalism Constant R-flux Seiberg-Witten maps

Seiberg-Witten maps (on phase space)

Replace patch index α by a constant momentum vector ˜ p. Θ˜

p =

Rijk ˜ pk δij −δij

• and

B˜

p =

Rijk (pk − ˜ pk)

• 1-connection is given by:

a˜

p,˜ p ′ = Rijk pi (˜

pk − ˜ p ′

k) dpj

1 For H fixed the D’s are constructed from the gauge potential:

A = AI(x) dxI = ai(x, p) dxi + ˜ ai(x, p) dpi .

2 Relationship between ⋆0 and ⋆˜

p.

f ⋆ g =

• D˜

p(f ⋆ g)

• ˜

p→p =

• D˜

pf ⋆0 D˜ pg

• ˜

p→p

SW map can be computed in closed form.

Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation σ-models Deformation quantization Convolution product quantization Lie-2 algebras/Lie-2 groups R-space and Q-space Lie 2-algebras Integrating 2-groups Convolution product

Definitions

Objective Find a suitable higher analogue of Weyl quantization. A 2-vector space is a linear category V = (V0, V1) of

1 a vector space of objects V0 2 a vector space of morphisms V1 3 a source and a target maps s, t : V1 ⇒ V0 4 an inclusion map ✶ : V0 → V1, v → ✶v.

Lie 2-algebras A Lie 2-algebra is a 2-vector space V together with an antisymmetric bilinear bracket [−, −]V : V × V → V and an antisymmetric trilinear Jacobiator isomorphism on objects satisfying a higher Jacobi identity.

Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation σ-models Deformation quantization Convolution product quantization Lie-2 algebras/Lie-2 groups R-space and Q-space Lie 2-algebras Integrating 2-groups Convolution product

Definitions

Objective Find a suitable higher analogue of Weyl quantization. A 2-vector space is a linear category V = (V0, V1) of

1 a vector space of objects V0 2 a vector space of morphisms V1 3 a source and a target maps s, t : V1 ⇒ V0 4 an inclusion map ✶ : V0 → V1, v → ✶v.

Lie 2-algebras A Lie 2-algebra is a 2-vector space V together with an antisymmetric bilinear bracket [−, −]V : V × V → V and an antisymmetric trilinear Jacobiator isomorphism on objects satisfying a higher Jacobi identity.

Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation σ-models Deformation quantization Convolution product quantization Lie-2 algebras/Lie-2 groups R-space and Q-space Lie 2-algebras Integrating 2-groups Convolution product

More definitions

L∞- algebras An L∞-algebra is a graded vector space V with a collection of totally (graded) antisymmetric n-brackets [−, . . . , −] : n V → V

• f degree n − 2 satisfying higher or homotopy Jacobi identities.

A 2-term L∞-algebra is one with underlying graded vector space V = V0 ⊕ V1; it has vanishing n-brackets for n > 3. 2-term L∞-algebras are the same things as Lie 2-algebras.

(Baez & Crans, math.QA/0307263)

Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation σ-models Deformation quantization Convolution product quantization Lie-2 algebras/Lie-2 groups R-space and Q-space Lie 2-algebras Integrating 2-groups Convolution product

Even more definitions

A tensor or monoidal category C = (C0, C1) has an exterior product ⊗ : C × C → C an identity object ✶ ∈ C0 unity isomorphisms ✶X := ✶ ⊗ X ∼ = X ∼ = X ⊗ ✶ in C1 , ∀ X ∈ C0 associator isomorphisms PX,Y,Z : (X ⊗ Y ) ⊗ Z

≈

→ X ⊗ (Y ⊗ Z) , ∀ X, Y, Z ∈ C0. For higher associators to be consistent, ✶X and P must satisfy:

1 the pentagon identities (5 bracketings of 4 objects) 2 the triangle identities (associator is compatible with the unity)

Braiding C is braided if ∃ BX,Y : X ⊗ Y

≈

− − → Y ⊗ X , ∀ X, Y ∈ C0. B are called commutativity relations.

Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation σ-models Deformation quantization Convolution product quantization Lie-2 algebras/Lie-2 groups R-space and Q-space Lie 2-algebras Integrating 2-groups Convolution product

Lie-2 groups

A 2-group is a monoidal category in which every object and morphism has an inverse. A Lie 2-group is a pair G = (G0, G1) of objects in the category of smooth manifolds and smooth maps with

1 source and target maps s, t : G1 ⇒ G0 2 vertical multiplication ◦ : G1 × G1 → G1 3 horizontal multiplication functor ⊗ : G × G → G 4 identity object 1 5 inversion functor (−)−1 : G → G 6 associator, left and right units as before 7 units and counits g ⊗ g−1 ∼

= 1 ∼ = g−1 ⊗ g

Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation σ-models Deformation quantization Convolution product quantization Lie-2 algebras/Lie-2 groups R-space and Q-space Lie 2-algebras Integrating 2-groups Convolution product

R-space and Q-space Lie 2-algebras

Let V ∼ = R2d and define [−, −]R : V ∧ V → V by the R-space commutation relations. This is a pre-Lie algebra. V0 = V1 = V and let d := [−] : V1 → V0 be the identity map. This is a 2-term L∞-algebra = the R-space Lie 2-algebra V . V can not be directly integrated to a Lie 2-group. Take the Q-space algebra g. 2-term L∞-algebra: ˜ V1 = R

˜ d

− − → ˜ V0 = g

• with 3-cocycle:

j(ˆ xi, ˆ xj, ˆ xk) = Rijk . This is the Q-space Lie 2-algebra ˜ V .

Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation σ-models Deformation quantization Convolution product quantization Lie-2 algebras/Lie-2 groups R-space and Q-space Lie 2-algebras Integrating 2-groups Convolution product

Integrating 2-groups

(g, R, j) exponentiates (G, U(1), ϕ), where G=d-dim Heisenberg and ϕ=associator.

To exponentiate [j] ∈ H3(g, R) to a compact element [ϕ] ∈ H3(G, U(1)), we need to restrict the space of 3-cocycles to a lattice Λ ∼ = Zd. Λ injects into Γ= cocompact lattice in G. G/Γ is a Heisenberg nilmanifold. Λ is equipped with an inner product and a dual pairing Σ.

ˆ ˜ pi → ˆ pi with braiding : Bg,h =

• g h , β(g, h)
• .

This braided monoidal category G is the Lie 2-group that integrates the Lie 2-algebra V .

Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation σ-models Deformation quantization Convolution product quantization Lie-2 algebras/Lie-2 groups R-space and Q-space Lie 2-algebras Integrating 2-groups Convolution product

Integrating 2-groups

(g, R, j) exponentiates (G, U(1), ϕ), where G=d-dim Heisenberg and ϕ=associator.

To exponentiate [j] ∈ H3(g, R) to a compact element [ϕ] ∈ H3(G, U(1)), we need to restrict the space of 3-cocycles to a lattice Λ ∼ = Zd. Λ injects into Γ= cocompact lattice in G. G/Γ is a Heisenberg nilmanifold. Λ is equipped with an inner product and a dual pairing Σ.

ˆ ˜ pi → ˆ pi with braiding : Bg,h =

• g h , β(g, h)
• .

This braided monoidal category G is the Lie 2-group that integrates the Lie 2-algebra V .

Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation σ-models Deformation quantization Convolution product quantization Lie-2 algebras/Lie-2 groups R-space and Q-space Lie 2-algebras Integrating 2-groups Convolution product

Integrating 2-groups

(g, R, j) exponentiates (G, U(1), ϕ), where G=d-dim Heisenberg and ϕ=associator.

To exponentiate [j] ∈ H3(g, R) to a compact element [ϕ] ∈ H3(G, U(1)), we need to restrict the space of 3-cocycles to a lattice Λ ∼ = Zd. Λ injects into Γ= cocompact lattice in G. G/Γ is a Heisenberg nilmanifold. Λ is equipped with an inner product and a dual pairing Σ.

ˆ ˜ pi → ˆ pi with braiding : Bg,h =

• g h , β(g, h)
• .

This braided monoidal category G is the Lie 2-group that integrates the Lie 2-algebra V .

Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation σ-models Deformation quantization Convolution product quantization Lie-2 algebras/Lie-2 groups R-space and Q-space Lie 2-algebras Integrating 2-groups Convolution product

Explicit construction

Exponentiation of the Lie 2-algebra generators. ˆ Za = exp

• 2π i (Σ−1)ia ˆ

xi ˆ Pξ = exp

• i ξi ˆ

pi

• ,

where a = 1, . . . , d and ξ = (ξi) ∈ Rd. Commutation relations. ˆ Za ⊗ ˆ Zb = ˆ Pξab

R ⊗ ˆ

Zb ⊗ ˆ Za ˆ Za ⊗ ˆ Pξ = e 2π i (Σ−1)ia ξi ˆ Pξ ⊗ ˆ Za ˆ Pξ ⊗ ˆ Pξ′ = ˆ Pξ′ ⊗ ˆ Pξ where

• ξab

R

i = −4π2 (Σ−1)ja Rijk (Σ−1)kb .

Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation σ-models Deformation quantization Convolution product quantization Lie-2 algebras/Lie-2 groups R-space and Q-space Lie 2-algebras Integrating 2-groups Convolution product

Explicit construction

Associativity relation. ˆ Za ⊗ ˆ Zb ⊗ ˆ Zc = e −2π i Rabc ˆ Za ⊗ ˆ Zb ⊗ ˆ Zc , where Rabc = 2π2 Rijk (Σ−1)ia (Σ−1)jb (Σ−1)kc. The tricharacter ϕ(m, n, q) = e −2π i Rabc ma nb qc , where ma ∈ Λ∗ ∼ = Zd, obeys the required pentagonal cocycle condition. Higher associativity relations in G were computed. ⇒ Quantization of the fundamental identity for Nambu-Poisson structures is encoded in G .

Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation σ-models Deformation quantization Convolution product quantization Lie-2 algebras/Lie-2 groups R-space and Q-space Lie 2-algebras Integrating 2-groups Convolution product

Convolution star product

Consider C∞(T ∗M) on T ∗M = Td × (Rd)∗. Embed it as an algebra object A of G . Categorify the Weyl map: defined it as the linear isomorphism W

• e i kI xI

= ˆ W(m, ξ) := exp

• i kI ˆ

xI , where (kI) = (k1, . . . , kd, ξ1, . . . , ξd) with ki = 2π (Σ−1)ia ma and extended by linearity (Fourier expansion). Star product definition W (f ⊛ g) := W (f) ⊗ W (g) The NA ⊛-product satisfies the associativity relation of the category; i.e. (A , ⊛) really is an object of G .

Dionysios Mylonas Quantization of non-geometric flux backgrounds

Background and motivation σ-models Deformation quantization Convolution product quantization Lie-2 algebras/Lie-2 groups R-space and Q-space Lie 2-algebras Integrating 2-groups Convolution product

Summary

Courant σ-model − → twisted Poisson σ-model that offers a geometric interpretation of the R-flux background. Deformation quantization gave us the nonassociative star product as well as quantization of Nambu-Poisson brackets. SW maps from NA to associative star products were computed. Using the appropriate categorical formalism the Weyl quantization map has been categorified and the petinent NA convolution product was computed. The two products coincide giving us a categorical version of Kathotia’s theorem.

Dionysios Mylonas Quantization of non-geometric flux backgrounds