Quantization of the NC φ 3 model in 2 and 4. and the Kontsevich model. Harold Steinacker. University of Vienna

Quantization of the NC φ

model in 2 and 4 dimensions

and the Kontsevich model

Harold Steinacker

University of Vienna

Saclay, march 2006

Outline:

• Noncommutative Quantum Field Theory UV/ IR mixing and renormalization the Grosse-Wulkenhaar term

Matrix model formulation of the φ³ model

• The Kontsevich model review of relevant facts

• Quantization, renormalization and “solution” of the NC φ³ model 2 dimensions

4 dimensions

• Outlook

Noncommutative Quantum Field Theory

Motivation and relevance

1. Standard model of high-energy physics, gravity based on spacetime-continuum

idealization, seems unplausible;

⇒ quantized space? Heisenberg 1938

2. Gravity & Quantum Mechanics ⇒ space should have structure at l_P 3. String theory: strings ending on D-branes

D-branes in B–field background ⇒ strings induce NC field theory (NCFT) on D-brane

⇒ D-branes = NC space (independent of l_p!)

Non-commutative geometry, field theory

• Manifold M → NC algebra A of functions on M (with rep. on H)

“pointless geometry” (von Neumann 1955) NC (differential) geometry (Connes)

simplest example Rⁿ_θ:

[ˆx_i, ˆx_j] = iθ_ij (cp. Quantum Mechanics, phase space)

usually ∃ derivatives ∂_i, integral = trace, some symmetries

• Field theory on NC space:

C(M) → A → L(H)

φ(x) → ˆφ(ˆx) → φ

e.g. plane waves e^ikx → e^{ik ˆx}, spherical harmonics (fuzzy sphere), ...

Formulation of field theory is possible, many examples

Example: the quantum plane R

“coordinate-functions” ˆx_i, i = 1, 2 satisfy CCR [ˆx_i, ˆx_j] = iθ_ij, θ_ij ... a.s. tensor, “background-field”

generate algebra A_θ ∼= Heisenberg-algebra ( (ˆx₁, ˆx₂) ↔ (x, iθ_dx^d )) representation on Hilbert space H ∼= L²(R) as in Quantum Mechanics Scalar field: φ = φ(ˆx) ∈ A_θ resp. φ ∈ L(H) ... lin. operator, matrix

e.g. localized wave-packets: coherent states φ_~a = |~aih~a|

differential calculus

∂ˆ_iφ = −i[˜x_i, φ] for x˜_i := θ_ij⁻¹xˆ_j

note: a priori, NC does not imply existence of UV - cutoff, but UV/IR relation (cp. Quantum mechanics: squeezed states!)

NC scalar field theory

consider some NC space, algebra A (e.g. R^d_θ, T_θ², S_N² , CP_N² , ...) use representation of algebra A on Hilbert space H

Field φ(x) φ ∈ L(H) ... Hermitian operator on H trace replaces integral

Example:

S = T r  1

2∂_iφ∂_iφ + 1

2m²φ² + g 4φ⁴

 can write e.g. φ(x) = R dk φ_k : e^ikx : etc.,

Quantization

formally defined by (Euclidean) path integral hφ_k1 · · · φ_kli = R [DΦ]e^−S φ_k1 · · · φ_kl

R [DΦ]e^−S , [DΦ] = Y

dφ_k

⇒Wick’s theorem, however distinction planar ↔ nonplanar diagrams propagator: as usual, hφ_kφ_k⁰i = δ_kk⁰ _k2+m¹ ₂

one-loop planar and non-planar self-energy diagrams:

Γ⁽²⁾_P := g R _d^d_k

(2π)^d

1

k²+m² ∼ gΛ^d−2, Γ⁽²⁾_{N P}(p) := g R _d^d_k

(2π)^d

e^ikθp

k²+m² ∼ g 

1

1/Λ²+p²θ²

^d−2₂ Γ⁽²⁾_{N P}(p) is finite as long as p 6= 0, but IR singularity as p → 0

... UV/IR mixing (Minwalla, Van Raamsdonk, Seiberg) central feature of NC field theories,

serious obstacle to perturbative renormalization!

nontrivial relation UV ↔ IR

momentum dependence of effektive action Γ⁽²⁾(p)

⇒ modes p → 0 are suppressed (UV/IR)

spontaneous symmetry breaking, phase transition hpi 6= 0 ... “striped” phase (Gubser, Sondhi)

verified numerically (Ambjorn, Catterall; Martin; Bietenholz, Hofheinz, Nishimura)

one way to overcome this problem:

The Grosse-Wulkenhaar term

add “confining” potential to action, consider S =

Z  1

2∂_iφ∂_iφ+Ω² x˜_iφ˜x_iφ + 1

2m²φ² + g 4φ⁴



suppresses IR !

Observation: there is a duality x ↔ p at Ω = 1 (Langmann-Szabo ) Result (Grosse - Wulkenhaar): perturbatively renormalizable for Ω 6= 0

in 2 and 4 dimensions (RG techniques) technically difficult, uses matrix formulation:

scalar field ≡ hermitian matrix φ_ij = hi|φ|ji

Matrix model formulation:

scalar field φ ∈ L(H) recall: ∂_iφ = −i[˜x_i, φ] ⇒

S = Z

−(˜x_iφ˜x_iφ − ˜x_ix˜_iφφ) + Ω²x˜_iφ˜x_iφ + µ²

2 φ² + i˜λ 3! φ³ simplifies for Ω = 1 to

S = R (˜ x

x ˜

+

)φ

+

φ

= T r



2

J φ

+

φ



.

where

J = 2(2πθ)²(X

i

˜

x_ix˜_i + µ²

2 ) ... harmonic oscillator ! choose basis of eigenstates:

in d = 2 : J |ni = 4π (n + ¹₂ + ^µ₂^2θ)|ni, n ∈ {0, 1, 2, ...}

d = 4 : J |n₁, n₂i = 8π²θ (n₁ + n₂ + 1 + ^µ₂^2θ)|n₁, n₂i, n_i ∈ {0, 1, 2, ...}.

The regularized (Euclidean) NC φ

model for Ω = 1

regularization (cutoff): H = C^N such that

d = 2 : J |ni = 4π (n + ¹₂ + ^µ₂^2θ)|ni, n ∈ {0, 1, 2, ...N }

d = 4 : J |n₁, n₂i = 8π²θ (n₁ + n₂ + 1 + ^µ₂^2θ)|n₁, n₂i, n_i ∈ {0, 1, 2, ...N }.

introduce counterterms R Aφ + ¹₂δµ²φ² (+ one more in d = 4, later), can eliminate either linear or quadratic term:

S = T r

− _2iλ¹ M²φ +˜ ^iλ_3! φ˜³

= T r

1

2M X² + ^iλ_3! X³ − _3(iλ)¹ ₂M³ using shift

φ = φ +˜ 1

iλJ = X + 1 iλM where

M = p

J² + 2(iλ)A

= Kontsevich model !

Quantization

Z(M ) = Z

D ˜φ exp(−T r

− 1

2iλM²φ +˜ iλ 3!

φ˜³

) = e^{F (M )} Kontsevich model, for fixed given (diagonal) matrix M as above Correlators or “n-point functions”

hφ_i1j1...φ_injni = 1 Z

Z

Dφ exp(−S) φ_i1j1....φ_injn (recall: φ_ij ∼ hi|φ|ji ... evaluation of field, cp. ∼ hx|φ|yi)

Renormalization condition (as for free case λ = 0):

hφ₀₀φ₀₀i = 1 2π

1

µ²_Rθ + 1, hφ₀₀i = 0 Nontrivial task: show that all n-point functions have a

well-defined limit N → ∞ (with nontrivial dependence on indices i, j).

Computation of correlators

obtained simply by taking derivatives of F (M ):

hφ_iki = h ˜φ_iki − ^J_iλ^ik

= −^J_iλ^ik + _Z¹ 2iλ_∂(M^∂2)_ik R D ˜φ exp(−T r

− _2iλ¹ M²φ +˜ ^iλ_3! φ˜³ )

= −^J_iλ^ik + 2iλ_∂(M^∂2)_ik F (M ) etc. (this is particular for the φ³ model!)

⇒ only need to show: Z(M ) = e^{F (M )} depends smoothly on M , well-defined limit N → ∞.

however: nontrivial, requires renormalization.

first: perform perturbative computations to get better feeling

Perturbative computations:

rewrite action S = T r

1

4(J φ² + φ²J ) + ^iλ_3! φ³ − Aφ

= T r(¹₂φⁱ_j (G_R)^j;l_i;k φ_l^k + ^iλ_3! φ³ − Aφ + ¹₄(δJ φ² + φ²δJ )) finite (renormalized) kinetic term (G_R)^j;l_i;k = ¹₂δ_lⁱδ_j^k(J_i^R + J_j^R)

propagator:

∆^i;k_j;l = hφⁱ_jφ^k_l i = δ_lⁱδ_j^k 2

J_i^R + J_j^R = δ_lⁱδ_j^k 1/(4π²θ)

i + j + (µ²_Rθ + 2), where n = n in 2D, n = n₁ + n₂ in 4D.

J^R|n₁, n₂i = 8π²θ(n + 1 + ^µ2R₂^θ)|n₁, n₂i (4d), J^R|ni = 4π(n + ^1+µ

2 Rθ

2 )|ni, (2d)

δJ ∼ δµ² = (µ² − µ²_R) ... part of the counter-term

1-point function

one-loop contribution to the 1-point function:

hφ_iii = 1

J_i^RA_i − iλ 2

1 J_i^R

X

k

2

J_i^R + J_k^R + O(λ²) ... divergent, unless canceled by counterterm A.

2D: A ∼ (iλ) log(N )

4D: A_i ∼ (iλ)N log(N )1 + (iλ) log(N )J_i.

⇒ need further counterterm: either A = a1 + cJ or equivalently φ → φ + c, c ∼ iλ log(N ).

can be absorbed by redefinition of Kontsevich-model

M =

qJ˜² + 2(iλ)a − (iλ)²c² J˜ = J + (iλ)c.

2-point function

hφ_klφ_lki for l 6= k: has one-loop planar contribution

gives

hφ_klφ_lki = _JR²

k +J_l^R − 2(iλ)_(J^hφ_R^kk+φlli

k +J_l^R)²

+_(JR ⁴

k +J_l^R)²

P

j

(iλ)² J_k^R+J_j^R

1

J_l^R+J_j^R − ^δJl+δJ₂ ^k

+ O(λ⁴) implies mass renormalization in 4D

δµ² ∼ (iλ)²

256 π⁶θ⁴ log N (no mass renormalization in 2D)

hφ_llφ_kki for l 6= k: vanishes at tree level, one-loop nonplanar contribution

which gives

hφ_llφ_kki = hφ_kkihφ_lli + 1 4

(iλ)² J_k^R J_l^R

 2

J_k^R + J_l^R

² is finite.

Goal: nonperturbative proof of renormalizability:

need renormalized Kontsevich model Z(M ),

with eigenvalues m_i appropriately rescaled as N → ∞.

The Kontsevich model

defined by

Z^Kont(M ) = e^{FKont(M )} = _N¹ R dX expn

T r 

−^{M X}₂ ² + i^X₆³o

= _N¹ R d ˜φ expn

T r

1

2iλM²φ −˜ _3!ⁱ φ˜³ − ¹₃M³o where N = R dX expn

−T r 

M X² 2

o .

depends only on eigenvalues m_i of M ... hermitian N × N matrix introduced by Kontsevich 1991

suitable variables:

t_r = −(2r − 1)!! X

i

m^−(2r+1)_i ,

Remarkable fact (Kontsevich): F^Kont(M ) = F^Kont(t_r) is generating function of intersection numbers (topological characteristics) of moduli spaces of punctured Riemann surfaces (⇒rational coefficients!)

more precisely: (Kontsevich, Itzykson-Zuber) consider perturbative expansion

Z^{Kont (N )} = Z

dµ_M(X) exp( i

6T rX³) = X

k≥0

Z_k^{Kont (N )}(M )

Z_k^{Kont (N )}(M ) = (−1)^k (2k)!

Z

dµ_M(X) T rX³ 6

^2k . then

• Z_k^{Kont (N )}(M ) is polynomial in the t_r of degree 3k, for deg t_r = 2r + 1.

• Z_k^{Kont (N )}(t_r) is independent of N for N ≥ 3k, depends only on t_r, 2r + 1 ≤ 3k.

⇒ Z^Kont(t_r) = P

k≥0 Z_k^Kont(t_r) well-defined for N → ∞ as asymptotic expansion.

Further facts: (Kontsevich)

• Z^Kont(t_r) is a τ -function for the Korteweg-de Vries equation i.e. u = _∂t^∂22

0

ln Z^Kont satisfies

∂u

∂t₁ = ∂

∂t₀

 1 12

∂²u

∂t²₀ + 1 2u²



, ∂

∂t_n u = ∂

∂t₀ R_n+1.

• Virasoro constraints: L_mZ = 0, m ≥ −1

for suitable operators L_m (differential operators in the t_r)

• genus expansion: (generally for matrix models...) ln Z^Kont = F^Kont = X

g≥0

F_g^Kont

by drawing the Feynman diagrams on a Riemann surface.

... allows to obtain explicit solution:

genus 0: (Makeenko and Semenoff, ... )

F₀^Kont = ¹₃ P

i m³_i − ¹₃ P

i(m²_i − 2u₀)^3/2 − u₀ P

i(m²_i − 2u₀)^1/2 +^u₆³⁰ − ¹₂ P

i,k lnn_(m2

i−2u₀)^1/2+(m²_k−2u₀)^1/2 m_i+m_k

o higher genus: (Itzykson-Zuber)

F₁^Kont = ₂₄¹ ln _1−I¹

1, F₂^Kont = ₅₇₆₀¹ h

5_(1−I^I4

1)³ + 29_(1−I^I3I2

1)⁴ + 28_(1−I^I23

1)⁵

i , etc. where

I_k = −(2k − 1)!!P

i

1

(m²_i−2u₀)^{k+ 12} , u₀ = −P

i

√ 1

m²_i−2u₀ = I₀.

generally: all F_g^Kont with g ≥ 2 are given by finite sums of polynomials in I_k/(1 − I₁)^2k+13 .

explicitly (just for illustration ...) (Itzykson-Zuber)

F₀^Kont = ^t_3!³⁰ + t₁^t_3!³⁰ + 

t₂ ^t_4!⁴⁰ + 2^t_2!^{21 t}_3!³⁰

+ 

t₃ ^t_5!⁵⁰ + 3t₁t₂^t_4!⁰⁴ + 6^t_3!^{30 t}_3!³¹ +h

t₄ ^t_6!⁶⁰ + 

6^t_2!²² + 4t₁t₃ _t5 0

5! + 24^t_3!^{30 t}_4!⁴¹ + 12t₂^t_2!^{21 t}_4!⁴⁰i + . . .

24F₁^Kont = t₁ + _t2 1

2! + t₀t₂

+ 

2^t_3!³¹ + t₃ ^t_2!⁰² + 2t₀t₁t₂ +

6^t_4!⁴¹ + t₄ ^t_3!⁰³ + 4^t_2!^{20 t}_2!²² + 6t₀t₂^t_2!²¹ + 3t₁t₃ ^t_2!²⁰ +

24^t_5!⁵¹ + t₅^t_4!⁴⁰ + 24t₀t₂^t_3!³¹ + (4t₁t₄ + 7t₂t₃) ^t_3!³⁰ + 16t₁^t_2!^{20 t}_2!²² + 12t₃^t_2!^{20 t}_2!²¹ + . . .

Application to the NC φ

model

set m_i =

qJ˜_i² + const ≈







i, d = 2

(i₁ + i₂), d = 4 (degenerate) as given by the φ³ model with harmonic oscillator potential note:

• t₀, (t₁) and u₀ = −P

i

√ 1

m²_i−2u₀ and correlation functions are divergent a priori

• only the combination q

m²_i − 2u₀ = λ^−2/3

qJ˜_k² + 2b

enters, where b = (iλ)a − λ^4/3 u₀ + λ²c²/2 (a, c ... bare parameters).

will show: ˜J and b are finite after renormalization,

for suitable a = a_N, c = c_N, µ² = µ²_N (divergent!) rendering correlation functions well-defined in the limit N → ∞.

Renormalization and finiteness

1-point function at genus 0:

hφ_kki_g=0 = 2iλ ^∂

∂ ˜J_k² F₀( ˜J²) − ^J_iλ^k

= _iλ¹ (

qJ˜_k² + 2b − ˜J_k) + c + P

j

√ (iλ)

J˜_k²+2b

qJ˜_j²+2b+( ˜J_j²+2b)

= finite!

⇒ c =







0 , d = 2

−_(8π^iλ_2θ)2 ln(N ) + c⁰, d = 4

, and ˜J , b finite

b determined by renormalization condition hφ₀₀i = 0, solution b = O(λ).

in 4D: J = J + (iλ)c finite˜ ⇒ mass renormalization µ² = (iλ)²

256π⁶θ⁴ ln(N ) + µ²_R complete agreement with 1-loop result

counterterm a: (recall R aφ ) determined through implicit equation

b

iλ − a + (iλ)²c²/2 = −P

i

√(iλ)

J˜_i²+2b

⇒ a =







(iλ)

4π ln N + (f inite), d = 2

(iλ)

8π²θN ln N − _(8π^λ2²θ)⁴ ln(N )² + (f inite), d = 4 renormalized 1-point function in 4D:

hφ

i

=

(

q J ˜

+ 2b − ˜ J

) + c

+ (iλ)f

(k),

finite and well-defined as N → ∞

For certain point in moduli space (b = c⁰ = 0):

hφ_kki_g=0 = (iλ)X

j

1 J˜_j

 1

J˜_k + ˜J_j − 1 J˜₀ + ˜J_j



(1) coincides with one-loop result

Finiteness of general n-point function for any genus g

1) diagonal case:

hφ_i1i1...φ_inini_c ∼ ∂

∂m²_i

1i₁

... ∂

∂m²_i

ni_n

F (M ) easy to show using explicit form of F_g (polynomial in ^Ik

(1−I₁)^2k+13

for g ≥ 2):

above renormalization a = a_N, c = c_N, µ² = µ²_N

guarantees that all derivatives of F_g(M ) w.r.t. m²_i are finite and well-defined as N → ∞, as long as I₁ = −λ² P

i

1

( ˜J_i²+2b)^3/2 < 1 2) can be extended to general n-point functions

hφ_i1j1....φ_injni_c ∼ ∂

∂m²_i

1j₁

... ∂

∂m²_i

nj_n

F (M )

also well-defined for each genus g, using F (M²) = F (U⁻¹diag(m²_i)U ).

(slight complication due to degeneracy in 4D, but no essential problem)

Theorem: The (connected) genus g contribution to any given n-point function hφ_i1j1....φ_injni_c is finite and has a well-defined limit N → ∞ for all g, provided the couplings are renormalized as above

all correlation functions can be computed in principle for any genus g 2D:

a = (iλ)

4π ln N (+f inite) 4D:

c = −_(8π^iλ2θ)² ln(N ) (+f inite) µ² = _256π^(iλ)6²θ⁴ ln(N ) (+f inite)

a = _8π^(iλ)2θN ln N + _(8π^(iλ)2θ)⁴ ln(N )² (+f inite) note: model originally defined for imaginary coupling R iλφ³

can analytically continue to (sufficiently small) real coupling R λφ³ all correlation functions real

Singularity (phase transition, instability) for real coupling λ simultaneously for all genera g ≥ 1:

I₁ = −(iλ)² X

i

1

( ˜J_i² + 2b)^3/2 = 1 2D:

1 + µ²θ

λ ≈ ±0.0646989 4D:

µ²_Rθ + 2 = λ²

2(8π²θ)³ .

Example: 2-point function at genus 0 (in 4D) using identity D ˜φ_klφ˜_lk

E

= _m2^2iλ

k−m²_l D ˜φ_kk − ˜φ_ll E find

hφ

φ

i

= 2

√

√

coincides (!) with 1-loop for special point (b = c⁰ = 0) in moduli space.

(b=c⁰=0)

= 2

J˜_k + ˜J_l − (iλ)² 2 J˜_k + ˜J_l

X

j

1

J˜_j( ˜J_k + ˜J_j)( ˜J_l + ˜J_j). similarly

hφ_llφ_kki − hφ_kkihφ_lli = (iλ)²√ ¹

J˜_k²+2b

√ 1

J˜_l²+2b



√ 1

J˜_k²+2b+

√J˜_l²+2b

²

in agreement with perturbative computations (recall b = b(λ)).

Remarks and outlook:

• nontrivial interacting “solvable” NC 4D field theory

• renormalization determined by genus 0 contribution alone, (cp. general NCFT; IR divergences are suppressed here)

expect to hold generally for (scalar) NCFT with oscillator potential genus 0 is accessible more generally (matrix model techniques)!

• genus 0 contribution coincides exactly with 1-loop for special point in moduli space (not for higher genera) (??)

• is P

g convergent ?

• generalizations:

– D=6 ? (no longer super-renormalizable)

– extend to Ω 6= 1 (i.e. remove the oscillator potential) certainly doable “perturbatively” ... ?

NCFT are accessible through new analytical (matrix) methods !