Quantization of 2-plectic Manifolds

Transcription

1 School of Mathematical and Computer Sciences Heriot Watt University, Edinburgh Based on: Bayrischzell Workshop, Josh DeBellis, CS and Richard Szabo arxiv: (JMP), arxiv: (JHEP) CS and Richard Szabo, in preparation

2 Two-plectic Manifolds Multisymplectic manifolds are a natural generalization of symplectic manifolds. Symplectic manifolds Manifold M with closed 2-form ω such that ι v ω = 0 v = 0. Poisson structure Phase spaces in Hamiltonian dynamics. Starting point for quantization. p-plectic manifolds Manifold M with closed p + 1-form ω such that ι v ω = 0 v = 0. 1-plectic: symplectic, 2-plectic: 3-form ω (Often) Nambu-Poisson structure multiphase spaces in Nambu mechanics. Starting point for higher quantization (?) Why should we be interested in such manifolds? Why should we quantize them?

3 D1-D3-Branes and the Nahm Equation D1-branes ending on D3-branes can be described by the Nahm equation. D1-branes ending on D3-branes: dim D1 D3 A Monopole appears. X i u(n): transverse fluctuations Nahm equation: (s = x 6 ) d ds Xi + ε ijk [X j, X k ] = 0 Note SO(3)-invariance. Solution: X i = r(s)g i with r(s) = 1 s, Gi = ε ijk [G j, G k ] Nahm, Diaconescu, Tsimpis

4 D1-D3-Branes and the Nahm Equation The D1-branes end on the D3-branes by forming a fuzzy funnel. dim D1 D3 Solution: X i = r(s)g i r(s) = 1 s, Gi = ε ijk [G j, G k ] Matches profile from SUGRA analysis The D1-branes form a fuzzy funnel: G i form irrep of su(2): coordinates on fuzzy sphere S 2 F D1-worldvolume polarizes: 2d 4d Myers

5 Lifting D1-D3-Branes to M2-M5-Branes The lift to M-theory is performed by a T-duality and an M-theory lift IIB D1 D3 T-dualize along x 5 : IIA D2 D4 Interpret x 4 as M-theory direction: M M2 M5

6 The Basu-Harvey lift of the Nahm Equation M2-branes ending on M5-branes yield a Nahm equation with a cubic term. M M2 M5 A Self-Dual String appears. Substitute SO(3)-inv. Nahm eqn. d ds Xi + ε ijk [X j, X k ] = 0 by the SO(4)-invariant equation d ds Xµ + ε µνρσ [X ν, X ρ, X σ ] = 0 Solution: X µ = r(s)g µ with Basu, Harvey, hep-th/ r(s) = 1 s, G µ = ε µνρσ [G ν, G ρ, G σ ]

7 The Basu-Harvey lift of the Nahm Equation Also M2-branes end on M5-branes by forming a fuzzy funnel. M M2 M5 Solution: X µ = r(s)g µ r(s) = 1 s, G µ = ε µνρσ [G ν, G ρ, G σ ] Matches profile from SUGRA analysis The M2-branes form a fuzzy funnel: G µ form a rep of so(4): coordinates on fuzzy sphere S 3 F M2-worldvolume polarizes: 3d 6d 3-form structure appears quantization of S 3 required.

8 Further Motivation There are more appearances of 2-plectic manifolds in string-/m-theory. M5-brane perspective: Turning on 3-form background, C = θdx 0 dx 1 dx 2 + θ dx 0 dx 1 dx 2, the self-dual strings move in a space with [x 0, x 1, x 2 ] = θ and [x 3, x 4, x 5 ] = θ. Chu, Smith, Baez et al.: The phase space of the bosonic string is believed to be 2-plectic Using T-duality, Lüst, , conjectured that closed strings on T 3 in 3-form background lead to: [x i, x j ] ε ijk p k, [x i, p j ] δ i j, [p i, p j ] ε ijk p k. Note that here, the Jacobi identity is not satisfied.

9 Outline Quantization axioms for 2-plectic manifolds Quantized 2-plectic manifolds as vacua of 3-algebra models Another approach: Quantization via Groupoids

10 Axioms of Quantization Quantization is nontrivial and far from being fully understood. Classical level: states are points on a Poisson manifold M. observables are functions on M. Quantum level: states are rays in a complex Hilbert space H. observables are hermitian operators on H. Full Quantization A full quantization is a map ˆ : C (M) End (H ) satisfying 1 f ˆf is linear over C, f = f ˆf = ˆf. 2 the constant function f = 1 is mapped to the identity on H. 3 Correspondence principle: {f 1, f 2 } = g [ ˆf 1, ˆf 2 ] = ĝ. 4 The quantized coordinate functions act irreducibly on H. Problem: Groenewold-van Howe: no full quant. for T R n or S 2 (T 2 OK)

11 Loopholes to the obstructions to full quantizations There are three possible weakenings to the set of axioms for quantization. Three approaches to weaken the axioms of a full quantization: Drop irreducibility condition Quantize a subset of C (M) Correspondence principle applies only to O( ) The first two yield prequantization and geometric quantization. The last approach leads eventually to deformation quantization. My favorite method for this talk: Berezin quantization (or physicists fuzzy geometry), a hybrid of geometric and deformation quantization.

12 Berezin Quantization of CP 1 S 2 The fuzzy sphere is the Berezin quantization of CP 1. Hilbert space H is the space of global holomorphic sections of a certain line bundle: H = H 0 (M, L). For M = CP 1 : L := O(k). H k = span(zα1...z αk ) = span(â α 1...â α k 0 ) Coherent states For any z M: coherent st. z H. Here: z = 1 k! ( z αâ α) k 0. Quantization Quantization is the inverse map on the image Σ = σ(c (M)) of ( ) f(z) = σ( ˆf) z z = tr z z ˆf, Bridge: P := z z z z

13 Axioms of Generalized Quantization We propose a generalization of the quantization axioms to p-plectic manifolds. Problem is notoriously difficult, and many people tried to extend geometric quantization. Berezin quantization should be easier. Keep: a complex Hilbert space H and End (H ) as observables. Generalized quantization axioms A full quantization is a map ˆ : Σ End (H ), Σ C (M) satisfying 1 f ˆf is linear over C, f = f ˆf = ˆf. 2 the constant function f = 1 is mapped to the identity on H. 3 Correspondence principle: lim 0 i σ( [ ˆf 1,..., ˆf n ] ) {f 1,..., f n } = 0 L 2 If M is a Poisson manifold, this holds for Berezin quantization.

14 Quantization of R 3 The quantized Nambu-Heisenberg algebra corresponds to the space R 3 λ. We start from R 3 with ω = ε ijk dx i dx j dx k. We find {f, g, h} = ε ijk x i f x j g x k h. What is the geometry of [ˆx, ŷ, ẑ] = i 1? One possible interpretation as R 3 λ : Take a fuzzy sphere with Hilbert space H 0 (CP 1, O(k)). Define: [ˆx 1, ˆx 2, ˆx 3 ] = i,j,k Radius of this fuzzy sphere: R F,k = ε ijk ˆx iˆx j ˆx k = i 6R3 k k 3 k 6. Now discretely foliate R 3 by fuzzy spheres. R 3 λ. 1 H k

15 Brief Review: The IKKT model Noncommutative geometries arise as stable solutions of the IKKT matrix model. Fully dimensionally reduce maximally SUSY Yang-Mills theory: S = tr ([X I, X J ][X I, X J ]+µ I (X I ) 2 +C IJK X I [X J, X K ]+fermions) where X I u(n), I = 0,..., 9. Stable classical solutions: Moyal spaces µ i = C ijk = 0 [X i, X j ] θ ij 1 Fuzzy sphere C 123 = 1 [X i, X j ] ε ijk X k NC Hpp-waves C 123 = 1, µ 1 = µ 2 = µ [X 1, X 2 ] θ 12 1 [X 3, X i ] θ ij X j First two: BPS. The third: Nappi-Witten algebra. Hpp-waves: 4d Cahen-Wallach symm. space in Brinkman coords.: ds 2 4 = 2α β dx + dx + γ 2 dz 2 1 ( 4 β2 γ 2 z 2 b ) (dx + ) 2 Identification: X 1 + ix 2 z, X 3 J, X 4 1.

16 The Dimensionally Reduced BLG Model The theory corresponding to SYM is the BLG model, which we dimensionally reduce. Basu-Harvey Bagger, Lambert and Gustavsson developed a Chern-Simons matter theory for M2-branes. Reduced form: S = 1 2 ( Aµ X I, A µ X I) + i 2 ( Ψ, Γ µ A µ Ψ ) I=1 µ 2 1,I ( X I, X I) + i 2 µ ( 2 Ψ, Γ3456 Ψ ) + C IJKL ( [X I, X J, X K ], X L) + i ( 4 Ψ, ΓIJ [X I, X J, Ψ] ) 1 ( 12 [X I, X J, X K ], [X I, X J, X K ] ) + 1 (( 6 ɛµνλ A µ, [A ν, A λ ] )) + 1 (( [Aµ 4γ 2, A ν ], [A µ, A ν ] )). Matter fields X I, Ψ in vector space forming an orth. rep. of the gauge algebra in which A µ lives. ( 3-Lie algebra ) This model should take over the role of the IKKT model!

17 Solutions in the 3-Lie Algebra Model Stable solutions of the IKKT model have a 3-Lie algebra analogue. Stable classical solutions: R 3 λ µ I = C IJKL = 0 [X i, X j, X k ] ε ijk 1 Fuzzy S 3 C 1234 = 1 [X i, X j, X k ] ε ijkl X l NC Hpp-waves C 1234 = 1, µ 1 = µ 2 = µ [X 1, X 2, X 3 ] θ [X 4, X i, X j ] θ ijk X k First two solutions again BPS. The third: 5d Hpp-wave backgrnd.: ds 2 4 = 2α β dx + dx + γ 2 dz β2 ( γ 2 z 2 b ) (dx + ) 2 Identification: X 1 + ix 2 z, X 3 z 3, X 4 J, X 5 1.

18 Further Features of the 3-Algebra Matrix Model The 3-algebra model has more interesting features. Recall: IKKT expanded around sol.: NC YM theory on sol. Here: 3-algebra model exp. around sol.: expected part +... Close connection to the cubic osp(1 32)-cubic matrix model: S CSM = str ( M 3),

19 The Groupoid Approach to Quantization The elements of geometric quantization can be written in a groupoid language. Groupoids: Objects + composable, invertible arrows between them. Why groupoids? Quantization of the dual of a Lie algebra: Twisted convolution algebra of integrating group Poisson struct. Lie algebroid Conv. alg. on Lie groupoid Construction avoids Hilbert spaces, useful for 2-plectic case Procedure (Eli Hawkins, math/ , Weinstein, Renault,...) 1 Integrating groupoid s, t : Σ M, ω, ω = 0, t Poisson 2 Construct a prequantization of Σ with data (L, ) 3 Endow Σ with a groupoid polarization 4 Construct a twist element 5 Obtain twisted polarized convolution algebra of Σ.

20 Example: Groupoid quantization of R 2 The Moyal plane is conveniently reproduced in groupoid language. Starting point: M = R 2, Poisson structure π ij, i, j = 1, 2. 1 Lie groupoid: Σ = M M, coords. (x i, y i ), ω = dx i dy i s(x i, y i ) = (x i πij y j ) and t(x i, y i ) = (x i 1 2 πij y j ) Note: t is indeed a Poisson map: {t f, t g} ω = t {f, g} π x i + π ij (y j + y j) x i + π ij (y j y j) x i π ij (y j + y j) From this: pr 1, pr 2 and m. ω = p 1 ω m ω + p 2 ω = 0 2 Prequantization: L trivial line bundle over Σ, F = i2πω 3 Polarization: Induced by symplectic prepotential θ = x i dy i 4 Twist element: θ = σ0 1 dσ 0 = d( 1 2 πij y i y j ) 5 Twisted polarized convolution algebra: Moyal product on M Hawkins

21 Towards a Groupoid Quantization of R 3 There are two groupoid approaches quantizing R 3 : 2-groupoids and loop spaces. How do we extend this to R 3? 1 Strict approach: Look at 2-Groupoids (2-Hilbert spaces) etc. (work in progress, cf. Freed, Baez, Rogers...) 2 Trick: Map the 2-plectic forms back to symplectic ones. Consider the following double fibration: Transgression LM S 1 ev pr M LM T : Ω k+1 (M) Ω k (LM), T = pr! ev (T ω) x (v 1 (τ),..., v k (τ)) := dτ ω(v 1 (τ),..., v k (τ), ẋ(τ)) S 1 Previous application: Lift ADHMN construction to M-theory CS, Papageorgakis&CS, Palmer&CS

22 Towards a Groupoid Quantization of R 3 The symplectic manifold LR 3. Explicitly, this works as follows: We start from R 3 with 2-plectic form ϖ = ε ijk dx i dx j dx k. Transgression yields a symplectic form on loop space LR 3 : ω = dτ This gives a Poisson bracket dσ ε ijk ẋ k (τ)δ(τ σ) δx i (τ) δx j (σ) {x i (τ), x j (σ)} := ε ijk ẋ k (τ)δ(τ σ) Corresponding commutator relation previously derived in M-theory. Kawamoto&Sasakura Bergshoeff&Berman&van der Schaar&Sundell

23 Groupoid Quantization of R 3 The groupoid approach can be extended to higher structures using loop space. Starting point: M = R 3, Poisson structure π ijk ẋ k, i, j = 1, 2. 1 Lie groupoid: Σ = LM LM, coords. (x i, y i ), ω = dx i y i s(x i, y i ) = (x i πijk y j ẋ k ) and t(x i, y i ) = (x i 1 2 πijk y j ẋ k ) Note: t is indeed a Poisson map: {t f, t g} ω = t {f, g} π x i +π ij (y j +y j)ẋ k x i +π ijk (y j y j)ẋ k x i π ijk (y j +y j)ẋ k From this: pr 1, pr 2 and m. ω = p 1 ω m ω + p 2 ω = 0 2 Prequantization: L trivial line bundle over Σ, F = i2πω 3 Polarization: Induced by symplectic prepotential θ = x i dy i 4 Twist element: θ = σ0 1 dσ 0 = d( 1 2 πijk y i y jẋk) 5 Twisted polar. conv. alg.: [x i (τ), x j (σ)] = δ(τ σ)π ijk ẋ k (τ)

24 Conclusions Summary and Outlook. Summary: Naive approach to quantizing 2-plectic manifolds works OK IKKT-like model can be written down, has expected features. Groupoids offer a more general approach to quantization Future directions: Extend quantization of R 3 to 2-groupoid. Extend groupoid constructions to other spaces (S 3 ). Unify picture: Higher Poisson structures? Courant algebroids? Rewrite BLG model using the new function algebras.

25 School of Mathematical and Computer Sciences Heriot Watt University, Edinburgh Bayrischzell Workshop,