Matrix Models and Scalar Field Theories on Noncommutative Spaces

Transcription

1 Matrix Models and Scalar Field Theories on Noncommutative Spaces Based on: School of Mathematical and Computer Sciences Heriot Watt University, Edinburgh Young Researchers in Mathematical Physics, D. O Connor and CS, JHEP ) 066 CS, SIGMA ) 50 M. Ihl, C. Sachse and CS, JHEP ) 091

2 Outline 1 Partition functions in Matrix Models 2 The Fuzzy Sphere 3 Fuzzy Scalar Field Theory 4 Perturbative Expansion 5 Large N Limit & Saddle Point Method 6 Comparison to Known Results 7 Modification of the Model 8 Conclusions

3 Path integrals Path integrals or partition functions are difficult to compute. In QFT, we have to compute path integrals/partition functions: Z = Dϕ e i S[ϕ] Various problems with this: Wick rotation Have to introduce cut-off/regulator Perturbative expansions do not see topological effects Degeneracies for gauge equivalences However, there are examples where Z can be computed. Focus here: d = 0.

4 The Hermitian Matrix Model The One-)Hermitian Matrix Model can be solved in various ways. One-Hermitian Matrix Model Z = dµ D Φ) e tr r Φ2 +g Φ 4 ) Ingredients: Φ: field taking values in Hermitian N N matrix S[Φ] := tr r Φ 2 + g Φ 4) : action dµ D Φ) := i j drφ ij) i<j diφ ij): Dyson measure Large degeneracy: Invariance of tr Φ n ) under Φ ΩΦΩ 1 First approach: Perturbation theory Φ 2 -term gives a propagator 1 r Φ 4 -term a vertex with coupling g Double line Feynman diagrams. With Φ 3, used in dynamic triangulation of surfaces.)

5 The Hermitian Matrix Model The One-)Hermitian Matrix Model can be solved in various ways. Now: Exact solution for Z = dµ D Φ) e tr r Φ2 +g Φ 4 ) Use invariance: Φ = ΩΛΩ, Λ = diagλ 1,..., λ N ). N dµ D Φ) = dλ i λ j λ k ) 2 i=1 j>k }{{} Vandermonde determinant Absorbing the constant factor dµ H Ω), we obtain: Z = N i=1 dλ i e 2 i>j ln λ i λ j r i λ2 i +g i λ4 i dµ H Ω) }{{} Haar measure I) Coulomb-gas of repulsive eigenvalues II) Degeneracy gone. Continue with saddle point, orthogonal polynomials, etc.

6 Motivation for Studying Fuzzy Geometry Fuzzy spaces appear naturally in string theory. They also can serve as regulators of QFTs. Planck-Scale Structure of Spacetime Smooth structure of spacetime probably not to arbitrary scales. The most prominent modifications: SUSY and Noncommutativity. Fuzzy Geometry: NC on compact symplectic Riemannian spaces arise naturally in string theory Regularization of Field Theories Field theories on fuzzy spaces: finite-dimensional matrix models. QFTs are finite and path integrals well-defined. Advantages over lattice approach: Isometries preserved, no fermion doubling, analytical handle Numerical Simulations are easily done

7 The Fuzzy Sphere Idea: Truncate the spherical harmonics at a certain angular momentum. As usual, do not quantize space itself, but algebra of functions. Here: Spherical harmonics Y lm with l = 0,.., and m = l,..., l. Quantization: Truncate angular momentum l L Result: The space becomes fuzzy Multiplication will not close any more: Y l1...y l2... = Y l1 +l However: can deform product to star product [x i, x j ] iε ijk x k, where x i R 3 S 2 yields a closed, truncated algebra. Note: Truncated spherical harmonics can be arranged in matrix.

8 The Fuzzy Sphere The truncated coordinate ring is mapped to an L-particle Hilbert space. Recall the Hopf fibration S 2 = CP 1 0 U1) S 3 CP 1 0. Functions depending on Y lm, l L can be written in terms of homogeneous coordinates z α x i z α σ i αβ z β) on CP 1 as fx) = fz) = α i,β i f α 1...α L β 1...β L z α1...z αl z β1... z βl z 2L with α i, β i = 1, 2. Reduction: C 2 S 3 CP 1 ) Quantization as in flat case, z α, z β ) â α, â β ): ˆfx) = ˆfz) = αi,βi f α 1...α L β 1...βL â α 1...â α L 0 0 â β1...â βl In this way Berezin quantization ), also fuzzy X CP n...

9 Fuzzy Scalar Field Theory: Definition. Scalar field theory on the fuzzy sphere is a finite hermitian matrix model Quantized algebra of functions on the fuzzy sphere: span â α 1...â α L 0 0 â β1...â βl ) = MatL + 1) Laplacian and integration on the fuzzy sphere: L i = iε ijk x j k [L i, ], C 2, S2 da f 4πR2 N The action of real scalar field theory on the fuzzy sphere: S tr β[l i, Φ][L i, Φ] + r Φ 2 + g Φ 4) We define the partition function Z = dµ D Φ) e tr β[l i,φ][l i,φ]+r Φ 2 +g Φ 4 ) with the Dyson measure dµ D Φ) = i j drφ ij ) i<j diφ ij ) tr ˆf)

10 Fuzzy Scalar Field Theory vs. HMMs Fuzzy scalar field theory is significantly harder than matrix models usually considered. Compare Fuzzy Scalar Field Theory Z = dµ D Φ) e tr β[l i,φ][l i,φ]+r Φ 2 +g Φ 4 ) to One-Hermitian Matrix Model: Z = dµ D Φ) e tr r Φ2 +g Φ 4 ) Diagonalization trick Φ = ΩΛΩ does not work due to L i. State of the art: Hermitian matrix model with one external matrix Z = dµ D Φ) e tr V AΦ)+r Φ2 +g Φ 4 ) Solution: splitting Φ = ΩΛΩ, as well as character expansion exp tr V AΦ))) = ρ f ρ χ ρ AΦ) Itzykson and Di Francesco, Ann. Poincare ) 117

11 Fuzzy Scalar Field Theory: Simplifications Symmetry arguments yield simplifications of our model. Fuzzy scalar field theory: Z = dµ D Φ) e tr β[l i,φ][l i,φ]+r Φ 2 +g Φ 4 ) N = 2 : Z = dλ 1 dλ 2 λ 1 λ 2 ) 2 e βλ 1 λ 2 ) 2 +rλ 2 1 +λ2 2 )+gλ4 1 +λ4 2 )) Symmetries: 1. dµ D Φ) = dµ D ΩΦΩ ) dµ D Φ) e S = dµ D Φ) e S 0 S 0 = n s n tr Φ n ) + n,m s nm tr Φ n ) tr Φ m ) dµ D Φ) fφ) d N 2 Φ µ fφ µ τ µ ) S 0 = n s n tr Φ 2 ) ) n 3. [L i,1] = 0, λ λ tr [L i, Φ] 2 ) i>j λ i λ j ) 2m k ) nl

12 Perturbative expansion: Motivation A high temperature expansion yields a useful approximation. Idea: Treat the kinetic term e tr β[l i,φ][l i,φ]) perturbatively. Motivation: Similar to character expansion High temperature expansion successfully used on the lattice. Specific heat up to Oβ 8 ) for N = 2: Group theory allows everything else to be treated exactly.

13 Perturbative expansion: Principles The angular variables can be integrated out in the perturbative series. Z = dµ D Φ) e tr β[l i,φ][l i,φ]+r Φ 2 +g Φ 4 ) Introduce generators τ a of un) such that Φ = Φ a τ a. Kinetic term of the action: tr Φ[L i, [L i, Φ]]) = tr τ a [L i, [L i, τ b ]]) }{{} =:K ab tr Φ τ a ) tr Φ τ b ) Expand exponential: e β tr Φ[L i,[l i,φ]]) = 1 βk ab Φ a Φ b + β2 2 K abφ a Φ b ) 2 β3 6 K abφ a Φ b ) 3 + Oβ 4 ), Need to compute: k I k := dµ H Ω) K ai b i tr ΩΛΩ τ ai ) tr ΩΛΩ τ bi ) i=1

14 Perturbative expansion: Principles The angular variables can be integrated out in the perturbative series. Example: I 2 := dµ H Ω)K a1 b 1 K a2 b 2 tr ΩΛΩ τ a1 ) tr ΩΛΩ τ b1 ) tr ΩΛΩ τ a2 ) tr ΩΛΩ τ b2 ) Use tr A) tr B) = tr A B) and AB CD = A C)B D): I 2 = dµ H Ω) K a1 b 1 K a2 b 2 ) tr Ω Ω Ω Ω) 4 Λ) 4 Ω )τ a1 τ b1 τ a2 τ b2 ) Orthogonality relation: dµ H Ω) [ρω)] ij [ρ Ω)] kl = 1 dimρ) δ ilδ jk Decompose into irreducible representations of sun)!

15 Perturbative expansion: Results I Up to Oa 3 ), the perturbative expansion is easily doable. Explicit results: tr K) I 1 = N 2 1 tr tr K) Λ2 ) N 3 N tr Λ)2, 10 tr K) I 2 = I 3 = N 2 tr K) 1 + N 2) ) + tr K) N 2) tr K 2) N 36 + N N 2) 2 ) tr Λ4 )+ ) tr K) N 2) tr K 2) N N N 2) 2 ) tr Λ3 ) tr Λ) N 2) tr K) N 2 + N 4) tr K) N 2 + N 4) tr K 2) N N N 2) 2 ) tr Λ2 ) tr K) N 2) tr K) N 2) tr K 2)) N 36 + N N 2) 2 ) tr Λ2 ) tr Λ) tr K) N 2) tr K) N N N 2) 2 ) 6 + N 2) tr K 2) tr Λ)4 For I 3, 76 Young tableaux to be considered Mathematica. Notice terms tr Λ n ) m.

16 Perturbative expansion: Results II Up to Oa 3 ), the perturbative expansion is easily doable. The terms tr Λ n ) m recombine to a multitrace matrix model: dµ H Ω) e tr β[l i,φ][l i,φ]) = dµ H Ω) e m,n αm,na,n) tr Φm ) n and the action gets replaced: tr β[l i, Φ][L i, Φ] + r Φ 2 + g Φ 4) tr r Φ 2 + g Φ 4) + α m,n a, N) tr Φ m ) n m,n Many consistency checks available, e.g. comparison with N = 2. Diagonalization trick Φ = ΩΛΩ now works again!

17 Saddle point approximation The saddle point approximation gives a rought picture of the actual result. To solve: take the large N-limit and perform saddle point approx. Rewrite: λ i λ i N ) = λx), 0 < x < 1, N i=0 N 1 0 dx Rescale: β = N θ β β, r = N θr r, g = N θg g, λx) = N θ λ λx) Partition function: Z = Dλ exp N 2 S) With moments c n := dx λ n x), we have the action: β S =βn 3 c 2 c 2 1) β 2 N 4 ) c c 2 β 3 4N 5 ) 2c c 1 c 2 + c 3 + βnrc 2 + βngc 4 N 2 dx dy log λx) λy)

18 Analysis of the Phase Diagram Contrary to the hermitian matrix model, there are three phases. Recall: Eigenvalues form one-dimensional Coulomb gas. Eigenvalue distribution affects potential. Phases: symmetric single cut symmetric double cut asymmetric single cut Can compute the classical eigenvalue distribution for each case. For example: symmetric single cut [ δ, δ] up to Oβ 3 ): u λ) = 4 r β + 12π β ) ) ) 2 c g + π β 2 2 δ g + π β 2 2 λ2 δ 2 λ 2

19 Analysis of the Phase Diagram Contrary to the hermitian matrix model, there are three phases. Phases from existence bounds and choosing lowest free energy: Analytic results Numerical results Flores, O Connor, Martin, hep-th/ , Panero, hep-th/

20 Comparison with the Matrix Model Phase Diagram The matrix model phase diagram suggests two phases. Fuzzy Scalar Field Theory Hermitian Matrix Model

21 Comparison with φ 4 -Theory on R 2 The lattice) model has two different phases. How well does our model approximate φ 4 -theory on R 2? Z = Dφ e d 2 x 1 2 φ)2 +bφ 2 +cφ 4 c, b = const. Fuzzy Scalar Field Theory Scalar Field Theory proof of existence: [Glimm, Jaffe, Spencer, 1974/1975] exact shape numerically: [Loinaz, Willey, hep-lat/ ], confirmed by [Lee, hep-th/ ]

22 Known Results: φ 4 -Theory on R 2 θ The fuzzy model corresponds actually to regularized NC φ 4 theory. In φ 4 -theory on R 2 θ : New phase predicted [Gubser, Sondhi, hep-th/ ], analytically confirmed [Chen, Wu, hep-th/ ] and numerically confirmed [Ambjorn, Catterall, hep-th/ ]. Indications for the new phase also found by regularization φ 4 -theory with fuzzy spaces by [Steinacker, hep-th/ ]. Removal of new phase would be an indicator of successful regularization of commutative φ 4 -theory. We are regularizing φ 4 on the Moyal plane R 2 θ.

23 Modification of the model The proposed modification of the model moves the triple point in the right direction. Modify the action, assuming momentum-dependent wave function regularization Z L C 2 ) 1 + κc 2 : S = tr βφc 2 + κc 2 C 2 )Φ + r Φ 2 + g Φ 4) Dolan, O Connor, Presnajder This implies the following modification in our analysis K ab Ǩab := K ab + κk ac K cb and ˇβ = β κn 2 1)) Rescaling of κ to keep highest order term yields ˇβ = β κ). The triple point moves off to infinity for increasing κ.

24 Conclusion Summary and Outlook. We discussed the following: Matrix models and how to solve them The Fuzzy/Berezin quantized sphere Scalar field theories on fuzzy spaces How to solve them using matrix model techniques Obtained rigurous definition of certain functional integrals Can treat similarly: Scalar QFT on fuzzy CP n Scalar QFT on fuzzy CP n R matrix quantum mechanics) Future directions: Study regularization properties Deformed integrable hierarchies in these models? Relation to c > 1 string theories?

25 Matrix Models and Scalar Field Theories on Noncommutative Spaces School of Mathematical and Computer Sciences Heriot Watt University, Edinburgh Young Researchers in Mathematical Physics,