On the Scalar Curvature for the Noncommutative Four Torus

# On the Scalar Curvature for the Noncommutative Four Torus

###### Abstract.

The scalar curvature for the noncommutative four torus , where its flat geometry is conformally perturbed by a Weyl factor, is computed by making the use of a noncommutative residue that involves integration over the 3-sphere. This method is more convenient since it does not require the rearrangement lemma and it is advantageous as it explains the simplicity of the final functions of one and two variables, which describe the curvature with the help of a modular automorphism. In particular, it readily allows to write the function of two variables as the sum of a finite difference and a finite product of the one variable function. The curvature formula is simplified for dilatons of the form , where is a real parameter and is an arbitrary projection, and it is observed that, in contrast to the two dimensional case studied by A. Connes and H. Moscovici, unbounded functions of the parameter appear in the final formula. An explicit formula for the gradient of the analog of the Einstein-Hilbert action is also calculated.

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## 1. Introduction

The computation of scalar curvature [9, 17] for noncommutative two tori was stimulated by the seminal work [10] of A. Connes and P. Tretkoff on the Gauss-Bonnet theorem for these -algebras, and its extension in [14] to general translation-invariant conformal structures. Flat geometries of [2, 3], whose conformal classes are represented by positive Hochschild cocycles [4], are conformally perturbed by means of a positive invertible element , where is a dilaton [10]. Local geometric invariants, such as scalar curvature, can then be computed by considering small time asymptotic expansions, which depend on the action of the algebra on a Hilbert space and the distribution at infinity of the eigenvalues of a relevant geometric operator, namely the Laplacian of the conformally perturbed metric.

Following these works, the local differential geometry of noncommutative tori equipped with curved metrics has received considerable attention in recent years [1, 16, 11, 15, 12]. See also [29, 13]. It should be mentioned that conformal geometry in the noncommutative setting is intimately related to twisted spectral triples and we refer to [8, 10, 9, 27, 26, 21] for detailed discussions. Also it is closely related to the spectral action computations in the presence of a dilaton [5, 6]. For noncommutative four tori , the scalar curvature is computed in [15] and it is shown that flat metrics are the critical points of the analog of the Einstein-Hilbert action. Also noncommutative residues for noncommutative tori were studied in [18, 25, 15] (see also [30]). We refer to [31, 22] and [24] for detailed discussions on noncommutative residues for classical manifolds.

A crucial tool for the local computations on noncommutative tori has been Connes’ pseudodifferential calculus, developed for -dynamical systems in [2], which can be employed to work in the heat kernel scheme of elliptic differential operators and index theory (cf. [20]). An obstruction in these calculations, which is a purely noncommutative feature, is the appearance of integrals of functions over the positive real line that are -algebra valued. This is overcome by the rearrangement lemma [10, 9] (cf. [1, 17, 23]), which uses the modular automorphism of the state implementing the conformal perturbation and delicate Fourier analysis to reorder the integrands and computes the integrals explicitly. The integrals are then expressed as somewhat complicated functions of the modular automorphism acting on relevant elements of the -algebra. This lemma has been generalized in [23], and the work in [12] is an instance where the generalization is used.

A striking fact about the final formulas for the curvature of noncommutative tori is their simplicity and their fruitful properties such as being entire. Considering the numerous functions from the rearrangement lemma that get involved in hundreds of terms in the computations, the final simplicity indicates an enormous amount of cancellations, which are carried out by computer assistance. One of the aims of this paper is to explain this simplicity by computing the scalar curvature for without using the rearrangement lemma. We then study the curvature formula for the dilatons that are associated with projections in . The gradient of the Einstein-Hilbert action is also calculated, which prepares the ground for studying its associated flow in future works.

This article is organized as follows. In §2 we recall the formalism and notions used in [15] concerning the conformally perturbed Laplacian on . In §3 we use a noncommutative residue that involves integrations on the 3-sphere to compute the scalar curvature. This method is quite convenient as it does not require any help from the rearrangement lemma and it is advantageous as it explains the simplicity of the final formula (3). Also it readily allows to write the function of two variables in (3) as the sum of a finite difference and a finite product of the one variable function. In §4 the curvature formula is simplified for dilatons of the form , where is a real parameter and is an arbitrary projection. It is observed that, in contrast to the two dimensional case studied in [9], unbounded functions of the parameter appear in the final formula. In §5, we compute an explicit formula for the gradient of the analog of the Einstein-Hilbert action in terms of finite differences (cf. [9, 23]) of a one variable function that describes this action [15].

## 2. Preliminaries

The noncommutative four torus is the universal -algebra generated by four unitaries , which satisfy the commutation relations

 UkUj=e2πiθkjUjUk,

where is an antisymmetric matrix. For simplicity elements of the form shall be denote by for any 4-tuple of integers .

There is a natural action of on this -algebra, which is defined by

 αs(Uℓ)=eis⋅ℓUℓ,s∈R4,ℓ∈Z4,

and is extended to a 4-parameter family of -algebra automorphisms . The infinitesimal generators of this action, denoted by , are defined on the smooth subalgebra

 C∞(T4Θ)={a∈C(T4Θ);the mapR4∋s↦αs(a)%issmooth},

which is a dense subalgebra of and can alternatively be defined as the space of elements of the form with rapidly decaying complex coefficients These derivations are determined by the relations and , if .

One can consider a complex structure on (cf. [15]) by introducing the analog of the Dolbeault operators

 ∂1=δ1−iδ3,∂2=δ2−iδ4,¯∂1=δ1+iδ3,¯∂2=δ2+iδ4,

and by setting

 ∂=∂1⊕∂2,¯∂=¯∂1⊕¯∂2,

which are maps from to

There is a canonical positive faithful trace , which is defined on the smooth algebra by

 φ0(∑ℓ∈Z4aℓUℓ)=a0.

Following the method introduced in [10], is viewed as the volume form, and conformal perturbation of the metric is implemented in [15] by choosing a dilaton and by considering the linear functional given by

 φ(a)=φ0(ae−2h),a∈C(T4Θ).

This is a KMS state and we consider the associated 1-parameter group of inner automorphisms given by

 σt(a)=eithae−ith,a∈C(T4Θ),

and will use the following operators substantially

 Δ(a)=σi(a)=e−haeh,∇(a)=logΔ(a)=[−h,a],a∈C(T4Θ).

Denoting the inner product associated with the state by

 (a,b)φ=φ(b∗a),a,b∈C(T4Θ),

the Hilbert space completion of with respect to this inner product is denoted by , and the analog of the de Rham differential is defined in [15] by

 d=∂⊕¯∂:Hφ→H(1,0)φ⊕H(0,1)φ.

The Hilbert spaces and are respectively the completions of the analogs of -forms and -forms, namely the spaces and , with the appropriate inner product related to the conformal factor.

The Laplacian is then computed and shown to be anti-unitarily equivalent to the operator

 △φ=eh¯∂1e−h∂1eh+eh∂1e−h¯∂1eh+eh¯∂2e−h∂2eh+eh∂2e−h¯∂2eh.

## 3. Scalar Curvature for T4Θ and its Functional Relations

The scalar curvature of the conformally perturbed metric on is the unique element such that

 ress=1Trace(a△−sφ)=φ0(aR),∀a∈C∞(T4Θ).

Since the linear functional

 ∫−P=ress=0Trace(P△−sφ)

defines a trace on the algebra of pseudodifferential operators [7, 19], it follows from the uniqueness of traces on the algebra of pseudodifferential operators [18, 15, 25] that it coincides with the noncommutative residue defined in [15]. Therefore there exists a constant such that for any

 ∫−P=c∫S3φ0(ρ−4(ξ))dΩ,

where is the homogeneous term of order in the expansion of the symbol of , and is the invariant measure on the sphere . Therefore, in order to compute the curvature, we can write

 ress=1Trace(a△−sφ)=ress=0Trace(a△−s−1φ)=∫−a△−1φ=cφ0(∫S3ab2(ξ)dΩ),

where is the homogeneous term of order in the asymptotic expansion of the symbol of the parametrix of . Hence

 R=c∫S3b2(ξ)dΩ.

We compute by applying Connes’ pseudodifferential calculus [2] to the symbol of , which is the sum of the homogeneous components

 a2(ξ)=eh4∑i=1ξ2i,a1(ξ)=4∑i=1δi(eh)ξi,a0(ξ)=4∑i=1(δ2i(eh)−δi(eh)e−hδi(eh)).

That is, we solve the following equation explicitly up to

 (b0+b1+b2+⋯)∘(a0+a1+a2)∼1,

which in general yields

 b0=a−12=(eh4∑i=1ξ2i)−1,bn=−∑2+j+|ℓ|−k=n,0≤j1).

Here, for any , denotes and denotes . Note that the composition rule for pseudodifferential symbols [2],

 ρ∘ρ′=∑ℓ∈Z4≥01ℓ!∂ℓρ(ξ)δℓ(ρ′(ξ)),

is used in the derivation of the above recursive formula for .

Computing and restricting it to by the substitutions

 ξ1=cos(ψ),ξ2=cos(θ)sin(ψ), ξ3=sin(θ)cos(ϕ)sin(ψ),ξ4=sin(θ)sin(ϕ)sin(ψ),

with we perform its integral over the sphere and find that

 ∫S3b2(ξ)dΩ = ∫2π0∫π0∫π0b2(ξ)sin(θ)sin2(ψ)dψdθdϕ
 (1) = 4∑i=1((−2π2)b0δiδi(eh)b0+(2π2)b0δi(eh)1ehδi(eh)b0+5π22b0δi(eh)b0δi(eh)b0 +(3π2)b0ehb0δiδi(eh)b0+(−8π2)b0ehb0δi(eh)b0δi(eh)b0 +(−2π2)b0ehb0ehb0δiδi(eh)b0+(−π2)b0δi(eh)b0ehb0δi(eh)b0 +(2π2)b0ehb0δi(eh)b0ehb0δi(eh)b0+(4π2)b0ehb0ehb0δi(eh)b0δi(eh)b0) =π24∑i=1(−e−hδ2i(eh)e−h+32e−hδi(eh)e−hδi(eh)e−h).

The fact that, over , reduces to is crucial in the last equation, which leads to such a simple final formula.

We then use the following identities [10, 9, 17] to write the expression (1) in terms of and :

 e−hδi(eh)=g1(Δ)(δi(h)),e−hδ2i(eh)=g1(Δ)(δ2i(h))+2g2(Δ,Δ)(δi(h)δi(h)),

where

 (2) g1(u)=u−1logu,g2(u,v)=u(v−1)log(u)−(u−1)log(v)log(u)log(v)(log(u)+log(v)).

This yields

 (3) ∫S3b2(ξ)dΩ = 1cR = π24∑i=1(−e−hΔ−1g1(Δ)(δi(h))−2e−hΔ−1(g2(Δ,Δ)(δi(h)2)) +32e−hΔ−1(g1(Δ)(δi(h))g1(Δ)(δi(h)))) =

where

 k(s)=−e−sg1(es)=e−s−1s,
 (4) H(s,t) = −2e−s−tg2(es,et)+32e−s−tg1(es)g1(et) = e−s−t((es−1)(3et+1)t−(es+3)s(et−1))2st(s+t).

This formula matches with the one obtained in [15] (up to the multiplicative factor ).

###### Theorem 3.1.

Let

 ~k(s)=esk(s),~H(s,t)=es+tH(s,t),

where and are the functions in the final formula for the scalar curvature. We have

 (5) ~H(s,t)=2~k(s+t)−~k(s)t+32~k(s)~k(t).
###### Proof.

It follows from (4) and the following relation between the functions introduced in (2):

 g2(u,v) = ∫10susg1(vs)ds=1log(v)(uv−1log(uv)−u−1log(u)) = 1log(v)(g1(uv)−g1(u)).

## 4. Projections and the Scalar Curvature

Similar to the illustration in [9] of the scalar curvature of for dilatons associated with projections, we consider dilatons of the form , where and is an arbitrary projection, and simplify the expression (3) for these cases. We shall also study the behaviour of the functions of the parameter that appear in the final formula.

###### Proposition 4.1.

Let be a projection. For the dilaton , , the formula for the scalar curvature reduces to

 R=e−sp(f1(s)△(p)+f2(s)△(p)p+f3(s)p△(p)+f4(s)p△(p)p),

where and

 f1(s)=14(−2sinh(s)+cosh(s)−1),f2(s)=12sinh2(s2), f3(s)=−s+sinh(s)2−cosh(s)−14,f4(s)=s−sinh(s).
###### Proof.

Our method is quite similar to the one used in [9]. That is, we first use the identity

 △(p)=p△(p)p+p△(p)(1−p)+(1−p)△(p)p+(1−p)△(p)(1−p),

to decompose to the sum of eigenvectors of with eigenvalues . Therefore

 k(∇)(△(h)) = sk(∇)(△(p)) = sk(0)(p△(p)p+(1−p)△(p)(1−p)) +sk(−s)(p△(p)(1−p))+sk(s)((1−p)△(p)p) = −s(p△(p)p+(1−p)△(p)(1−p)) +(1−es)(p△(p)(1−p))+(e−s−1)((1−p)△(p)p).

Then, using the identity , one can see that

 H(∇,∇)(δi(h)δi(h)) =s22((H(s,−s)+H(−s,s))+(H(s,−s)−H(−s,s))(1−2p))(δi(p)δi(p)) =s22(2(cosh(s)−1)s2+4(s−sinh(s))s2(1−2p))(δi(p)δi(p)) =((cosh(s)−1)+2(s−sinh(s))(1−2p))(δi(p)δi(p)) =(2s−2sinh(s)+cosh(s)−1−4(s−sinh(s))p)(δi(p)δi(p)).

Using the identity we sum the above expressions and find that the formula (3), for the dilaton , reduces to

 12(−2sinh(s)+cosh(s)−1)△(p)+(−s+sinh(s)−cosh(s)2+12)p△(p) +sinh2(s2)△(p)p+2(s−sinh(s))p△(p)p,

up to multiplication from left by .

In contrast to the two dimensional case (cf. [9]), the functions of the variable that appear in the statement of Proposition 4.1 are not bounded as they tend to as . The graphs of these functions are given below and some relations between these functions are investigated. First we graph .

Graph of the function .

Among these functions, is the only one that is bounded below, whereas the other functions are neither bounded above nor bounded below. In fact is a non-negative even function.

Graph of the function .

The function , similar to , does not satisfy any symmetry properties.

Graph of the function .

The last function is obviously an odd function whose graph is given here:

Graph of the function .

It is interesting to observe that these functions, which describe the scalar curvature for the dilaton , where is an arbitrary projection, satisfy the following relations:

 f1(s)+f1(−s)=f2(s)+f2(−s)=−(f3(s)+f3(−s))=−sinh2(s2),
 f3(s)−f3(−s)=−12f4(s)=sinh(s)−s.

## 5. Gradient of the Einstein-Hilbert Action

Denoting the Einstein-Hilbert action associated with the dilaton by , we compute its gradient, namely an explicit formula for an element that represents the derivative at of , where are selfadjoint smooth elements. The final formula for the gradient is expressed in terms of finite differences of the function , obtained in the proof of Theorem 5.3 of [15]. We recall from the proof of this theorem that

 Ω(h)=φ0(R)=4∑i=1φ0(e−hT(∇)(δi(h))δi(h)),

where

 T(s)=−2s+es−e−s(2s+3)+24s2.

The fact that this function is non-negative played a crucial role in identifying the extrema of the Einstein-Hilbert action in [15].

###### Theorem 5.1.

For any selfadjoint , we have

where

 ω1(s)=−2sinh(s)+sinh(2s)−cosh(2s)+14s2,
 ω2(s,t)=

###### Proof.

We have

 Ω(h+εa) =4∑i=1φ0(e−h−εaT(∇h+εa)(δi(h+εa))δi(h+εa)) =4∑i=1(φ0(e−h−εaT(∇h+εa)(δi(h))δi(h))+εφ0(e−h−εaT(∇h+εa)(δi(a))δi(h)) +εφ0(e−h−εaT(∇h+εa)(δi(h))δi(a))+ε2φ0(e−h−εaT(∇h+εa)(δi(a))δi(a))).

Therefore

 ddε⏐ε=0Ω(h+εa)

Using the following lemmas we obtain the explicit formula,

where

 ω1(s)=−T(s)−T(−s)e−s,
 ω2(s,t) = E(s,t)+L(s,t)−P(s,t)−Q(s,t) = e−s−t−1s+tT(s)+e−s−t(T(−t)−T(s)s+t+T(t)−T(−s)s+tet) −(T(s+t)e−s−1s+T(t)−T(s+t)se−s+T(s+t)−T(s)t) −(T(−s−t)e−s−te−s−1s+T(t)−T(s+t)se−s+T(s+t)−T(s)t).

Then one can find the above explicit functions in the statement of the theorem by direct computer assisted computations. ∎

For simplicity in the notation, in the following lemmas, can be taken to be any of the canonical derivations introduced in §2. The proofs follow closely the techniques given in [9] for the computation of the gradient of linear functionals similar to (see also [23]).

###### Lemma 5.1.

We have

 φ0((ddε⏐ε=0e−h−εa)G(∇)(x)x)=φ0(ae−hE(∇,∇)(xx)),

where

 E(s,t)=e−s−t−1s+tG(s).
###### Proof.

Using

 ddε∣∣ε=0e−h−εa=1−e∇∇(a)e−h,

we have

 φ0((ddε⏐ε=0e−h−εa)G(∇)(x)x) = φ0(1−e∇∇(a)e−hG(∇)(x)x) = φ0(ae−he−∇−1∇(G(∇)(x)x)) = φ0(ae−hE(∇,∇)(xx)).

###### Lemma 5.2.

For any , we have

 φ0(e−hddε⏐ε=0G(∇h+εa)(x)x)=φ0(ae−hL(∇,∇)(xx)),

where

 L(s,t)=e−s−t(G(−t)−G(s)s+t+G(t)−G(−s)s+tet).
###### Proof.

Writing and using the following identity [9]