Non-Conformal Geometry on Noncommutative Two Tori

Non-conformal geometry on noncommutative two tori

Dissertation

Presented in Partial Fulﬁllment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

Chao Xu, B.A.

Graduate Program in Department of Mathematics

The Ohio State University

2019

Dissertation Committee:

Henri Moscovici, Advisor

Ovidiu Costin

Michael Davis

David Penneys c Copyright by

Chao Xu

2019 Abstract

On the spectral triple of a noncommutative manifold (A,H,D), despite the absence of underlying space of points, one can still consider its scalar curvature in terms of spectral information of the Dirac operator D, for example using short-time asymptotic expansion of the heat kernel e−tD2 .

In the recent decade, the conformal theory on a noncommutative two tori was ﬁrstly started by Connes and Tretkoﬀ(nee Cohen), and later greatly developed by Connes,

Moscovici and many others. Noncommutative conformal geometry on a noncommutative torus Aθ is the study of quantized Gaussian curvature under noncommutative conformal change of “metric” by a positive operator-valued Weyl factor k = eh, h∗ = h.

In this dissertation, by using Lesch and Moscovici’s extension of Connes’ pseudo-diﬀerential calculus to the Heisenberg modules, we will calculate the scalar curvature of a non- conformal change of metric by means of two commuting positive operator-valued factors k1, k2.

The ﬁrst part of this paper, inspired by work by L.Dabrowski and S.Andrzej, contains extension of the rearrangement lemma that was systematized by M.Lesch, to non-conformal

P 2 2 operators, by which we mean the elliptic operators with principal symbol j kj ξj with distinct k1, ..., km.

By adapting the technique used by Y.Liu, we interpret the result of rearrangement as generalized hyper-geometric functions on Grassmannians, generalizing the conformal

ii results of Y.Liu , namely when k1 = k2.

Second part of this paper consists of calculation of scalar curvature density associated

m to a non-conformal Laplacian operator ∆k1,...,km on a m-torus AΘ .

Third part is calculation of index density of a non-conformal Dirac operator Dk1,k2;E(g,θ) on the Heisenberg module E(g, θ).

In appendix A, we will justify our terminology “non-conformal”. We show that such a non-conformal Dirac operator on Heisenberg module amounts to a conformal change

∗ on the endomorphism algebra End o (T (A )) of the cotangent bundle together with a Aθ θ change of complex structure.

In appendix B, we put the elementary but crucial lemma of Gaussian averages, which will be used in the proof of extended rearrangement lemma.

In appendix C, we list the deﬁnitions of hyper-geometric functions and their generalization, and propositions that will be needed in the statement of the rearrangement lemma.

iii Acknowledgments

I would like to show my most sincere gratitude to my advisor Henri Moscovici for his generosity to share his insightful ideas with me and his great support to my academic career through out these years.

I also thank my colleague and good friend Yang Liu for guiding me in pseudo-diﬀerential calculus, and the other meaningful discussion we had during his stay in OSU, and during several other noncommutative geometry conferences. It is his method of handling the

“rearrangement integral” diﬀerently that enable me to extend rearrangement lemma to this non-conformal case.

I would like to thank David Penneys for the noncommutative geometry seminars, for the lessons on Von Neumann algebra and subfactor theory. Also I thank Tao Yang, Joseph

Migler for helpful discussion about various topic on noncommutative geometry.

In addition, my academic life would not have been possible without all the uncondi- tional love and support from my wife, my parents, my grandparents.

iv Vita

2012 ...... B.A. Zhejiang University

2013-present ...... Graduate Teaching Associate, The Ohio State University.

Fields of Study

Major Field: Department of Mathematics

v Table of Contents

Page

Abstract ...... ii

Acknowledgments ...... iv

Vita...... v

1. Introduction ...... 1

1.1 What are the objects in noncommutative geometry ...... 1 1.2 Noncommutative conformal geometry on two tori ...... 4 1.3 Metric change beyond conformal ...... 5 1.4 Noncommutative scalar curvature, heat expansion, Zeta functions . . . .8 1.5 Heat kernel, symbolic calculus and rearrangement lemma ...... 11 1.5.1 Resolvent expansion ...... 12 1.5.2 Trace formula ...... 12

2. Notation and preliminaries ...... 14

2.1 noncommutative tori Aθ ...... 14 2 2.1.1 Heisenberg modules E(g, θ) over a two-torus Tθ ...... 15 2.2 Multiindex ...... 18 2.3 Functional calculus ...... 18 2.3.1 A⊗n+1 as an alternating multiplier functional on A⊗n ...... 20 2.3.2 Fubini condition, substitution lemma ...... 23 2.4 Gaussian average ...... 25 2.5 Partitions ...... 26 2.6 Hypergeometric integral and hypergeometric functions from rearrangement 27 2.6.1 Hypergeometric integral ...... 28 2.6.2 Hypergeometric functions on Grassmannian ...... 28

vi 3. Main technical lemma ...... 31

3.1 Rearrangement lemma ...... 33 3.2 Proof of rearrangement lemma ...... 36

4. Main result ...... 42

4.1 Application on untwisted module over m-torus ...... 42 4.1.1 Pseudodiﬀerential calculus on torus ...... 42 P 2 2 4.1.2 Asymmetric Laplacian P = j kj δj ...... 44 4.1.3 Density R2(P )...... 51 4.1.4 Gauss-Bonnet function relations ...... 56 4.2 Application on twisted module over 2-torus ...... 57 4.2.1 Symbol calculus on Heisenberg module ...... 58 γ 4.2.2 Index density R2 ...... 61

Appendices 71

2 A. non-conformal metric from conformal change on End(T (Tθ)) ...... 71 A.1 Conformal twist ...... 71 A.2 Complex diﬀerential forms Ω(p,q)(A)...... 74 A.3 Construction of asymmetric Dirac ...... 76 A.3.1 motivation from Riemannian geometry ...... 76 A.3.2 twisted Ω0 ...... 77 A.3.3 twisted Ω1 ...... 78 A.3.4 twisted Ω(0,1) and twisted Dolbeault operator ...... 78 A.4 Twisted Dolbeault operator on E(g, θ)...... 80

B. Gaussian integral ...... 82

C. Generalized hypergeometric function ...... 87

Bibliography ...... 92

vii Chapter 1: Introduction

1.1 What are the objects in noncommutative geometry

In noncommutative geometry, cf.[7], one studies geometric spaces (topological, diﬀer- entiable, measurable, etc.) in terms of coordinates, in the form of algebra of operators.

For example, by a noncommutative topological space one means a C∗ algebra A.

In the commutative case A = C(X) for some topological space X by Gelfand-Naimark

Theorem.

By a noncommutative measure space one means a Von Neumann algebra B. In the commutative case B = L∞(X, µ) for some σ-ﬁnite measure space (X, µ).

A spectral triple (A,H,D) is, by deﬁnition, collection of information containing

∗-algebra A , a Hilbert space H , self-adjoint unbounded operator D ∈ L(H) , and represenatation of A → B(H), such that [D, A] ⊂ B(H). This will be a very important object in noncommutative geometry, because many of the geometric property will be characterized in terms of spectral property of D and [D, A].

1 The Riemannian metric g on a manifold M, if spinc structure V → E → M exist, can be captured by the corresponding spectral triple (C∞(M),L2(M,E),D) as

distg(p, q) = sup |f(p) − f(q)|, [D, f] = df f,|df|≤1

In [8, 9], Connes proved that a commutative spectral triple will recover the spinc structure if the following axioms, ﬁrstly listed in [10], are satisﬁed.

1. D has compact resolvent, namely (D−λ)−1 is a compact operator on H and therefore

this self-adjoint operator D has eigenvalues λn → ∞, n → ∞.

1 1 m 2. The eigenvalues of D are of growth rate m , namely λn = O(n ) as n → ∞. Here m is the dimension of the underlying manifold.

3. D should be an “order one” operator, in the sense that [[D, a], JbJ] = 0, for any

a, b ∈ A, assuming the existence of a C-skewlinear operator J : A → A such that JAJ −1 being an Ao action.

4. δ = [|D|, ·] should deﬁnes a derivation on A, and both A and [D, A] are δ-smooth,

l namely A, [D, A] ⊂ ∩lDom(δ )

∞ l 5. The smooth part of H, H = ∩lDom(D ) should be ﬁnitely generated projective

Hilbert A-module, with A-valued inner product (, ) , and

−m < ξ, η >H= T rω((ξ, η)|D| )

where T rω on the right hand side is the Dixmier trace associated to taking limit

along ultra ﬁlter ω ∈ L∞(R)∗, which detect logarithm growth in partial sum:

PN 1 λj(A) T rω(A) = lim ω log(N)

2 6. Moreover, there exists an orientation class [c] in Hochschild Homology Hn(A), cf.[7,

8, 9, 10, 38], such that if (A, H,D) is an odd spectral triple, π(c) = 1; if (A, H,D)

is an even spectral triple, there should be a grading map γ : H → H, with

γ2 = 1, γ∗ = γ, γD = Dγ, π(c) = γ

Example 1.1.1. A natural noncommutative example of spectral triple is from a noncom-

2 mutative torus Aθ, see Def. 2.1, together with Dq = δ1 + qδ2, I(q) 6= 0 and H = L (A, τ) being the Hilbert space closure w.r.t. the tracial state τ on A by the GNS construc-

n1 n2 n1 n2 tion. Note that δj(U1 U2 ) = 2πinjU1 U2 . It is not hard to verify that (Aθ, H,Dq) is a spectral triple.

Example 1.1.2 (even spectral triple of Dolbeault operator on Aθ, cf.[6]). A natural even spectral triple can also be constructed from a noncommutative torus. As the cotangent bundle on a two torus T2 is trivializable, one can identify (0, 1)-forms

(0,1) 2 ∼ ∞ 2 Ω (T ) = C (T )

by dividing by dz = dx − qdy. With this intuition, Hilbert space H = H0 ⊕ H0 thought of as L2-functions and L2 antiholomorphic one forms on the torus, with

Ö è D+ D = D−

− + − ∗ where D = i(δ1 + qδ2), and D = (D ) = i(δ1 +qδ ¯ 2) is the adjoint w.r.t. inner product on H.

3 1.2 Noncommutative conformal geometry on two tori

In the classical case, conformal classes of Hermitian metric on a Riemann surface Σ has a bijective correspondence with complex structures on the Σ. Ifg ˜ = e2φg is the conformally changed Hermitian metric from g tog ˜, then the change of Hodge star operators

2−2p p satisﬁes ∗g˜ = e ∗g on space of p-forms Ω (Σ). Therefore the inner product on one forms

R ¯ 2φ remains unchanged, while inner product on zero forms is (f1, f2)g˜ = Σ f1f2e dvol(g). Motivated by the above observation, in [6], the conformal change of metric on a non-

2 commutative two torus Tθ is to perturb the inner product on the space of functions A

∗ ∗ ∗ 2 by a state φ on A, namely from (a, b) = τ(a b) to (a, b)φ = φ(a b) = τ(a bk ) for some

2 positive element k ∈ A. Let Hφ = L (A, φ) be the Hilbert space of functions w.r.t. the state φ, while keeping the same the inner product on (0, 1)-forms. The linear isomorphism

W : H0 → Hφ is an isomorphism of Hilbert spaces and right-A-Hilbert modules. In [6],

Connes and Moscovici studied the conformal change on the torus Aθ by studying the Ö Ö èè ∂¯∗φ (0,1) 1 spectral triple Aθ,Hφ ⊕ H ,D = , which is pulled back by W as ∂¯

Ö è Ö è ¯∗ (0,1) Rk∂ ik(δ1 + iδ2) (Aθ,H0 ⊕ H ,Dk),Dk = = ¯ ∂Rk i(δ1 − iδ2)k

D− φ (0,1) H0 H

∂¯ W

Hφ

¯ The conformal Dolbeault operator can be identiﬁed as ∂k = (δ1 + iδ2)Rk, where δ1, δ2

2 are standard derivation on Tθ, and Rk is right multiplication by k.

2 Later in [36], Moscovici and Lesch extend to vector bundles over Tθ. They studied the

1 ∗φ note that the adjoint is twisted, δj 6= −δj

4 2 Dolbeault operator twisted by any Heisenberg module E(g, θ), cf. Def. 2.1.1, over Tθ. ¯ The twisted Dolbeault operator can be identiﬁed as ∂E (∇1 + i∇2)Rk, where ∇1, ∇2 is the unique compatible connection on vector bundle E(g, θ).

¯∗ ¯ ¯∗ ¯ The Dolbeault Laplacian ∂ ∂, twisted or not, will always have principal symbol σ2(∂ ∂) =

2 2 2 k (ξ1 + ξ2 ). Y. Liu in [47] computed the density functional R2(Q) of Q = k∆k for ar- bitrary dimension, and rewrote the density functional into summation of hypergeometric modular functions.

1.3 Metric change beyond conformal

Despite the fact that the two torus is one of the most well understood noncommutative manifolds, it is not clear in general which unbounded operator deserves to be called a metric. This paper is an attempt to go beyond conformal metric.

In [17], the scalar curvature associated to a non-conformal type operator with principal

2 2 2 symbol ξ1 + k ξ2 was computed, and it was ﬁrst pointed out without proof in [17] how to possiby extend the rearrangement lemma in [35] to ﬁt more examples.

Formally speaking, an obvious extension would be to consider diﬀerential operators P

P 2 2 with principal symbol σ2(P ) = kj ξj . Through out this paper, we refer to such operator

P as non-conformal Laplacians type operator, if k1, ..., km are mutually commuting positive elements in the endomorphism algebra(Aθ in the untwisted case, and Aθ0 in the twisted case by an Heisenberg module E(g, θ)).

5 2 In the case of the Adk-twisted spectral triple of conformal Dirac operator

Ö è Ö è ¯∗ (0,1) Rk∂ ik(δ1 + iδ2) (Aθ,H0 ⊕ H ,Dk),Dk = = ¯ ∂Rk i(δ1 − iδ2)k

we will also generalize the conformal Dirac Dk to non-conformal Dirac operators

Ö è D+ D = D−

− + if the principal symbol of D is σ1(D ) = k1ξ1 + ik2ξ2, with k1k2 = k2k1.

Note that in such case the triple (A, H,D) is not spectral triple, not even a twisted spectral triple! But the method of Gilkey( cf. [33]) still applies to calculation of scalar curvature density , deﬁned in 1.4.1, associated to D2.

Example 1.3.1 (Connes’s example involving Peierls Operator). In [7], the Dirac operator

D on S(R) is deﬁned as

Df(s) = sf(s) − (T f 0)(s) 2πs with (T f)(s) = f(s + 1) + f(s − 1) + 2cos( )f(s) + λf(s) θ

Here D is a non-conformal operator when Re(λ) is large enough. In fact, S(R) is a Heisenberg module (deﬁned in §2.1.1) E(J, θ) with

Ö è 1 J = −1

2Given σ ∈ Aut(A), a σ-twisted triple (A, H,D) does not satisfy [D, A] ⊂ B(H), instead only the σ-commutators [D, a]σ = Da − σ(a)D are bounded for all a ∈ A.

6 It is a Aθ-A 1 Morita equivalence bimodule with left A 1 action θ θ

2πi s V1f(s) = e θ f(s),

V2f(s) = f(s − 1)

According to the standard derivation deﬁned in [36]

θ sf(s) = ∇ f 2πi 2 0 ∗ ∗ 0 T f = (V1 + V1 + V2 + V2 + λ) · f := k∇1f θ Df(s) = (k∇ − i ∇ )f 1 2π 2

∗ ∗ Once λ is outside of σ(Spec(V1 + V1 + V2 + V2 )) ⊂ R, k will be invertible.

The above example is important in many aspects, for example as it is related to the

∗ ∗ Peierls Operators T = V1 + V + V2 + V + λ ∈ A 1 , which is related to the Quantum 1 2 θ Hall effect, cf. [7]. The operator T was intensively studied on its spectral properties (such as spectral gap) by quantum physicists, e.g. in [4, 5] . This operator D satisfies our definition as a non-conformal Dirac operator. it is included in my earlier calculation for the special case when k1 = 1, cf. [44].

In Appendix A, we will justify this terminology ”non-conformal”. Following the exact logic of how one obtain the (twisted) spectral spectral triple of conformal Dirac operator in

[6], we will follow the phenomenon in commutative Riemann surface that a non-conformal change of metric is changing the complex structure too.

Intuitively, we start with a change of metric locally from g = dx2 + dy2 tog ˜ =

2 2 2 2 k1dx + k2dy , and keep tract of the change of ortho-normal frames, change of inner product in both Ω0 and Ω1. Moreover, we introduce a new change of complex structure,

7 therefore also altering the space of anti-holomorphic one forms Ω(0,1). Putting all eﬀects together, we ends up getting our new partial-bar operator being

−1 » −1 −1 » k1k2 (δ1k1 − iδ2k2 ) k1k2

−1 This also recovers the conformal case k (δ1 − iδ2) if we set k1 = k2.

1.4 Noncommutative scalar curvature, heat expansion, Zeta functions

The standard method in geometry for scalar curvature, from Riemannian metric g to

1 ∞ Levi-Civita connection ∇, to scalar curvature function Scal = − 2 Rijij ∈ C (M), does not work in the noncommutative world!

Instead, with the Laplacian operator ∆ being our initial information, one can compute scalar curvature through spectrum info of ∆, for example through heat kernel expansion method.

2 Theorem 1.4.1 (Gilkey,[33]). Let dim(M) = m, ∆p = (d + δ) be the Laplacian on p-forms. For any elliptic operator P of order 2, the heat expansion is

Z −tP X j−m X j−m T r(e ) ∼t→0 t 2 kj(x, P )dvol(x) = t 2 aj(P ) j M j

then for the Laplace-Beltrami operator ∆0 on functions,

−Rijij Z k2(x, ∆0) = , a2(∆0) = k2(x, P )dvol(x) 24π M

8 Lemma 1.4.2. If ∆ is an elliptic diﬀerential operator of order n (deﬁned in Def. 4.1.1)

2 −t∆ on a m-torus H0 = L (AΘ) , the heat trace TrH0 (ae ), a ∈ Aθ has short-time asymptotic expansion

∞ X j−m −t∆ n TrH0 (ae ) ∼t&0 aj(a, ∆)t j=0

Similarly if ∆E is an elliptic diﬀerential operator of order n (deﬁned in Def. 4.2.1)

−t∆E on the Heisenberg module E(g, θ) over a two torus Aθ, the heat trace TrE (ae ), a ∈ Aθ0 has short-time asymptotic expansion

∞ X j−2 −t∆E n TrHE (ae ) ∼t&0 aj(a, ∆E )t j=0

Sketch of proof. The proof is identical to the one in [36], namely on one hand by the

Cauchy integral formula

1 Z 1 Z e−t∆ = e−tλ(λ − ∆)−1dλ = e−tλ(∆ − λ)−2dλ 2πi C 2πi C

On the other hand, the trace formula on H0 and HE , [6, 36], guarantee that the trace of

(∆ − λ)−2 is equivalent to R σ((∆ − λ)−2)dξ asymptotically as λ → ∞.

Finally, the subtle estimation of the remainder of the asymptotic resolvent approximation

−1 X σ(∆ − λ) ∼ bj, ord(bj) = −n − j, j in [33] §1.7, guarantee a correspondence of short-time heat expansion in t and resolvent expansion in (ξ, λ).

For the j-th term bj(ξ, λ) of order −n − j, the corresponding short-time heat coeﬃcient

9 j−n 3 is t n , as can be shown using homogeneity of bj.

Z Z −tλ e bj(ξ, λ)dλdξ Rm C Z Z −λ λ 1 = e bj(ξ, ) dλdξ Rm tC t t Z Z −λ − −n−j 1 1 = e t n bj(t n ξ, λ) dλdξ Rm tC t Z Z −λ − −n−j −1 − m = e t n bj(ξ, λ)dλt n dξ Rm tC Z Z j−m −λ =t n e bj(ξ, λ)dλdξ Rm tC Z Z j−m −λ =t n e bj(ξ, λ)dλdξ Rm C

Deﬁnition 1.4.1. The total scalar curvature associated to ∆ is the second heat coeﬃcient a2(·, ∆) in the short-time heat expansion in Lem. 2.3.0.1.

The scalar curvature functional R2(∆) is the unique element in Aθ (or Aθ for R2(∆E )), such that

τ(aR2(∆)) = a2(a, ∆), ∀a ∈ Aθ

As pointed out in [37], R2 is the Radon–Nikodym derivative of a2(·, ∆) over τ(·).

The scalar curvature can also be deﬁned by zeta function of ∆.

Deﬁnition 1.4.2 (zeta function of ∆). Let ∆−1 denote the zero extension of the inverse of ∆ on Ker(∆)⊥. The zeta function

−s ζa(s, ∆) := Tr(a∆ )

3note that λ is of order n

10 This zeta function is connected to the short-time heat expansion in Lem.2.3.0.1 through

Mellin transform, since

1 Z ∞ λ−s = ts−1e−tλdt, Re(s), Re(λ) > 0, Γ(s) 0 and thus

Z ∞ 1 s−1 −t∆ ζa(s, ∆) = t Tr(ae (1 − Pker∆))dt. (1.4.0.1) Γ(s) 0

P αn The Mellin transform converts short-time asymptotics f(t) ∼t&0 ant to a meromor- phic function Mf(s) with exact poles at s = −αn of residue an. Therefore Γ(s)ζa(s, ∆) has poles at s = 1, 0, −1, ... with residues a0(a, ∆), a2(a, ∆) − Tr(aPKer∆), a4(a, ∆), ...

Corollary 1.4.3. a2(a, ∆) = ζa(0, ∆).

1.5 Heat kernel, symbolic calculus and rearrangement lemma

In Lem. 2.3.0.1 we already brieﬂy showed that the short-time heat expansion for an elliptic operator ∆ of order 2 has the form

∞ −t∆ X (j−2)/2 Tr(ae ) ∼t→0+ aj(a, P )t j=0

In this section we will ﬁll in more detail to the sketch proof of Lem. 2.3.0.1, in order to compute the scalar curvature a2(a, ∆).

11 1.5.1 Resolvent expansion

The symbol of resolvent (∆ − λ)−1 has an asymptotic expansion

−1 X σ(∆ − λ) ∼ bj(ξ, λ) j≥0

∞ 2 c in a way such that bj(ξ, λ) ∈ C (R × R+, Aθ) is of order −2 − j as a λ-parameter

−1 Pn depending symbol, and the n-th remainder σ(∆ − λ) − bj is of order −3 − j.

∞ 2 c A parameter depending symbol q(ξ, λ) ∈ C (R × R+, Aθ) is of order r if

1 l I J l 2 r−|J|− 2 |δ ∂ξ ∂λq| ≤ CI,J,l(1 + |ξ| + |λ| ) .

1.5.2 Trace formula

In [36] Thm 6.2, it was shown that for any pseudo-diﬀerential operator P = Op(f(·, λ)) on the Heisenberg module E(g, θ), with λ-parameter depending symbol f(ξ, λ) ∈ C∞(R2 ×

c R+, Aθ0 ) in the above sense, the asymptotic behavior of Tr(P ) is fully captured by the “integral on cotangent bundle” just like in the classical case.

Proposition 1.5.1 (Trace of ΦDO, [36]). Let rank(E) = |cθ + d| denotes the rank of

E(g, θ), then

Z −∞ Tr(P ) = rank(E) τAθ0 f(ξ, λ)dξ + O(λ ). R2

Together with trace formula on a m-torus, we have:

Corollary 1.5.2. For an ΦDO on a m-torus AΘ

Z Z 1 −λ aj(a, ∆) = e τAθ (abj(ξ, λ))dλdξ. 2πi Rm C

12 For an ΦDO on a Heisenberg module E(g, θ) over a 2-torus Aθ

Z Z 1 −λ aj(a, ∆E ) = rank(E) e τAθ0 (abj(ξ, λ))dλdξ 2πi R2 C

Other than the constant multiple in the trace formula, the other diﬀerence between the untwisted case (over m-torus AΘ) and the twisted case (over the Heisenberg module

E(g, θ)) is the different rules of composition of ΦDOs. This will only effect the exact expression of b2. Let p2 + p1 + p0 be the symbol of ∆. Then bj can always be written in terms of b0, ..., bj−1 and derivations of coefficients in p2, p1, p0:

X α0 α1 αn I bj = b0 ρ1b0 ··· ρnb0 ξ

X α0 αn I = (b0 ⊗ · · · ⊗ b0 · ρ1 ⊗ · · · ⊗ ρn)ξ

X α I = (b0 · ρ)ξ where ρ is derivations of coeﬃcients in symbol of P .

P 2 2 2 P 2 For example, if P = j kj δj , which is an extension of ∆k2 = k ( j δj ) in [47], then ρ

2 2 could be δj(kl ), or δjδl(kr ) for any j, l, r = 1, ..., m. As is already used in [17], one can extend the rearrangement lemma summarized by Lesch in [35] as long as all k1, ..., km commutes. Using rearrangement lemma, cf. [6],[35],[17],[47],

α I one can integrate each summand b0 ξ over ξ and λ, as in Cor. 4.2.5, and end up with

P 2 2 −1 functions of adjoint actions. In our case, where b0 = ( j kj ξj − λ) , the resulting func-

tions are evaluated on yj = Adkj , adjoint actions by k1, ..., km.

13 Chapter 2: Notation and preliminaries

This notation section consists of three parts: noncommutative tori , multi-index ab- breviations and the smooth functional calculus .

2.1 noncommutative tori Aθ

We denote by Aθ the algebra of smooth functions on a noncommutative m torus, which is generated by unitaries U1, ..., Um such that

2πiθij UiUj = e UjUi

m for some skew-symmetric m × m matrix θ = (θij) ∈ Mn(R ).

m Let α denotes the action of T on the space of functions Aθ by shifting, namely

m m m α : T → Aut(Aθ), λ = (λ1, ..., λm) 7→ αλ, λ ∈ T ⊂ C

that is uniquely determined by αλ(Ui) = λiUi.

14 The induced inﬁnitesimal action of the Lie algebra deﬁnes the standard smooth structure on Aθ as the m standard derivations δ1, ..., δm, where

  0 if i 6= j δj(Ui) =  2πiUj if i = j

When we need to be speciﬁc about the dimension m of a torus, we may also use the

m notation Tθ .

2 2.1.1 Heisenberg modules E(g, θ) over a two-torus Tθ

Finitely generated projective modules of a noncommutative manifold A are called the noncommutative vector bundle over A, because of the famous theorem of Serra and

Swan. Let K ⊂ B(H) be the ideal of compact operators. The idempotents in A ⊗ K that correspond to those ﬁnitely generated projective modules on A generate the topological

K0 group K0(A).

The topological K-groups of a noncommutative torus Aθ were shown to be the same as in the commutative case, using the six term exact sequence of Pimsner and Voiculescu

2 2 [41]. In particular, when A = Tθ is of dimension two, K0(Aθ) is the abelian group Z generated by the trivial bundle [e] and the Powers-Rieﬀel projection [θ], cf.R1,R2,R3,R4.

Moreover, when θ ∈ R \ Q, the trace map

tr : K0(Aθ) → R

15 4 is an abelian group embedding that maps K0(Aθ) positively onto Z + θZ ⊂ R, such that tr([1]) = 1, tr([θ]) = θ.

For coprime integers c, d ∈ Z, such that cθ + d > 0 , and any pair of integers a, b ∈ Z Ö è a b such that g = ∈ SL(2, Z), there is an explicit construction of a right-Aθ mod- c d ule E(g, θ) that realizes [cθ + d] ∈ K0(Aθ).

Now let us recall the explicit deﬁnition of E(g, θ). Our convention of bimodule structure is from [36]. Ö è a b Deﬁnition 2.1.1 (Heisenberg module E(g, θ)). For g = ∈ SL(2, Z), θ0 = gθ = c d aθ+b cθ+d , E(g, θ) = S(R × Z/cZ) is the space of Schwartz functions on |c| copies of lines.

A Heisenberg module of Aθ is a left-Aθ0 right-Aθ bimodule E(g, θ), and the bimodule structure is deﬁned as follows:

Let Ui denote generator of Aθ, Vi denote generators of Aθ0 , for any f(t, α) ∈ S(R ×

Z/cZ)

cθ + d (fU )(t, α) = e2πi(t−αd/c)f(t, α)(fU )(t, α) = f(t − , α − 1) (2.1.1.1) 1 2 c 2πi( t − α ) 1 (V f)(t, α) = e cθ+d c f(t, α)(V f)(t, α) = f(t − , α − a) (2.1.1.2) 1 2 c

Moreover, the left-Aθ0 right-Aθ Hilbert bimodule structure is determined by the com-

5 6 2 patibility of A-inner product (·, ·)r , Aθ0 -inner product (·, ·)l , and the L inner product

4 a positive map preserves positive elements, K0(Aθ) carries ordering as a Grothendieck group, an element x ∈ K0(Aθ) is positive if and only if x can be represented by some idempotent 5 Aθ-inner product is the Hilbert module inner product on E(g, θ), namely (ξ1a1, ξ2a2)r = ∗ a1(ξ1, ξ2)ra2, ∀ξ1, ξ2 ∈ E(g, θ), ∀a1, a2 ∈ Aθ 6 ∗ similarly, (b1ξ1, b2ξ2)l = b1(ξ1, ξ2)lb2, ∀ξ1, ξ2 ∈ E(g, θ), ∀b1, b2 ∈ Aθ0

16 (·, ·).

7 Moreover, E(g, θ) is a Morita equivalence bimodule between Aθ and Ag.θ , so

op op End (E) = Aθ0 , EndA 0 (E) = A . Aθ θ θ

0 Let τ, τ be the normalized trace on Aθ and Aθ0 respectively. For f, g ∈ E(g, θ)

Z (f, g) := X f(t, α)g(t, α)dt (2.1.1.3) α 0 |cθ + d|τ (g, f)l = (f, g) = τ(f, g)r (2.1.1.4)

On E(g, θ), there exists a “standard connection” that is compatible with its bimodule structure!

∂ ∇ f(t, α) = f(t, α) 1 ∂t c ∇ f(t, α) = 2πiµ(g, θ)tf(t, α), µ(g, θ) = 2 cθ + d

The connection is compatible with the diﬀerential structure on Aθ, and compatible up to a scalar on Ag.θ

0 ∇j(a · f · b) = δj(a) · f · b + a · ∇j(f) · b + a · f · δj(b), ∀a ∈ Ag.θ, b ∈ Aθ

It’s obvious that [∇1, ∇2] = 2πiµI.

7 aθ+b Here g.θ denotes the Mobius action g.θ = cθ+d

17 2.2 Multiindex

First throughout the paper we use I for an orderless multiindex with entries chosen from {1, ..., m}. Given a multiindex I = (I1, ..., In), higher order derivation on A such as

I Q δ means j δIj for δj ∈ Der(A). It will also be frequently used in our main result and

Q application to abbreviate kI = j kIj , if k1, ..., km ∈ A mutually commutes.

Also note that when it comes to ∇I will not be well-deﬁned, for the standard connection on the Heisenberg module (§2.1.1). When it comes to diﬀerential operators on the

Heisenberg module, we use ordered multiindex.

In most cases, we treat I = (I1, ..., In) as n discrete variables. But it will sometimes be convenient to think of I as a ﬁxed multiindex, for example in (B.0.0.4) , and read the

I ˆ ˆ multiplicity of each number. We will denote the power of ξ1, ..., ξm in ξ by I1, ..., Im , ˆ I Qm Ij namely ξ = j=1 ξj .

On the other hand, we will also come across ordered multiindex α = (α0, ..., αn) over

Z, and l = (l1, ..., lm) over Q similarly.

2.3 Functional calculus

The notation of smooth functional calculus in this paper when handling pseudo- diﬀerential calculus follows notation in [47] and [35].

Starting from an elliptic operator P of order d , one can obtain the heat coeﬃcient

−tP aj(a, P ) ∈ Aθ in the heat expansion of P , since the heat trace Tr(ae ) has short-time

18 asymptotic expansion

∞ −tP X (j−m)/2 Tr(ae ) ∼t→0+ aj(a, P )t (2.3.0.1) j=0

P α0 α1 αn −1 The heat coeﬃcient aj, is of the form b0 ρ1b0 ρ2...ρnb0 , where b0 = (σd(P ) − λ)

−1 is the principal symbol of the resolvent (P − λI) , and ρ1, ..., ρn are derivatives of the principal symbol of P .

The general process of the Connes-Tretkoﬀ-Moscovici Rearrangement lemma, is to rearrange the heat coeﬃcient aj, under a simple identity

ak = kk−1ak = kAd(k−1)(a)

After rearrangement, one obtains some operator-valued functions of adjoint actions that will not be visible when the algebra A is commutative, as then conjugation Adk will be trivial.

To take a closer look into such rearrangement, when n > 1, one have adjoint actions on each component, for example

kak2bk = k4Ad(k−3)(a)Ad(k−1)(b)

(j) Following the convention in [35], we would realize these adjoint actions Adk in the algebra A⊗n+1, which acts on A⊗n as an alternating multipliers functional.

19 2.3.1 A⊗n+1 as an alternating multiplier functional on A⊗n

For any nuclear Frechet algebra A, the functional calculus of these adjoint actions xj are realized within the C∗ algebra closure8 of A⊗n+1, which acts on A⊗n as a functional through alternating multiplication pairing

A⊗n+1 × A⊗n → A deﬁned by

(a0 ⊗ · · · an) · (b1 ⊗ bn) 7→ a0b1a1 ··· bnan (2.3.1.1)

This is always well-deﬁned on algebraic tensor products, but in general may not extend onto all C∗ algebra closures, cf.[35]. Fortunately, our C∗ algebra of interest, the algebra of

9 noncommutative m-torus Aθ, is nuclear . So is the Frechet subalgebra of smooth elements

⊗n Aθ. Therefore there will be no ambiguity in taking norm closure on A , which we denote by A⊗¯ n. As pointed out in [35], the multiplication operator does extend onto minimal

⊗(n+1) tensor product closure A⊗min A, so we could view (2.3.1.1) as a representation of A as A-valued functional on A⊗n:

A⊗(n+1) → L(A⊗¯ n, A).

2 For example, using the above representation, we can rewrite an expression b0ρ1b0ρ2b0

2 as (b0 ⊗ b0 ⊗ b0) · (ρ1 ⊗ ρ2).

8There is no ambiguity of C∗ algebra norm on A⊗n+1 when A is nuclear. Our C∗ algebra of interest, Aθ is nuclear. So is the Frechet subalgebra of smooth elements Aθ. 9A nuclear C∗ algebra has the same maximal and minimal tensor product norm on its tensor product algebra.

20 For j = 0, 1, ..., n, the left multiplication on the j + 1-st entry in A⊗n is

(j) a := (1, ··· , a, ··· , 1) : (c1, ··· , cn) 7→ c1 ··· cjacj+1 ··· cn j

For 1 ≤ j ≤ n, if a is invertible, the adjoint action by a on the j-th entry is

(j) −1 (j−1) (j) −1 Ada := (a ) a :(c1, ··· , cn) 7→ c1 ··· a cjacj+1 ··· cn

The adjoint action allows us to switch order of multiplication inside the noncommutative algebra A, for example if a, x ∈ A and a is invertible, then xa = aAda(x).

P n To further motivate our main result, Thm. 3.1.1, let f(a) = cna , cn ∈ C be either a polynomial or analytic function in a. Then with a similar idea to switch order of x and a using adjoint actions, we have

X n X n n xf(a) = cnxa = cna Ada (x) = f(aAda)(x)

If f0(a), ..., fn(a) are n + 1 polynomials or analytic function in a, then using functional

⊗n+1 calculus in A , for any ρ1, ..., ρn ∈ A, the alternating product of fj(a) and ρj becomes

⊗n a function of adjoint actions acting on ρ1...ρn ∈ A .

n (1) (1) (2) Y (j) f0(a)ρ1...ρnfn(a) = f0(a)f1(aAda )f2(aAda Ada )...fn(a Ada ) · (ρ1...ρn) (2.3.1.2) j=1

Equation (2.3.1.2) is the exact formula used in the rearrangement lemma for conformal

α0 αn 2 P 2 −1 operator, in which case one needs to rearrange b0 ρ1...ρnb0 , for b0 = (k ξ − λ) .

21 In our case, if k1, ..., km are m mutually commuting positive elements in A, and b0 =

P 2 2 −1 α0 αn ( kj ξj − λ) , it’s obvious that what is needed to rearrange b0 ρ1...ρnb0 should be a multivariable version of (2.3.1.2).

For 1 ≤ l ≤ m, 1 ≤ j ≤ n, we will still refer to the adjoint actions of each kj as the

Modular operators. We will abbreviate these frequently used operators Ad(l) as follows kj

y(l) := Ad(l) j kj

If f(k1, ..., km) is an polynomial or analytic function in k1, ..., km, then using the power series of f we have

xf(k1, ..., km) = f(k1y1, ..., kmym)(x)

If f0(k1, ..., km), ..., fn(k1, ..., km) are n + 1 polynomials or analytic functions in k1, ..., km, and ρ1, ..., ρn ∈ A, then

n Ñ j j é Y Y (l) Y (l) f0(k1, ..., km)ρ1...ρnfn(k1, ..., km) = f0(k1, ..., km) fj k1 y1 , ..., km ym · (ρ1...ρn) j=1 l=1 l=1 (2.3.1.3)

From equation (2.3.1.3) we can see that the adjoint action by kj in the 0-th and s-th

Qs l place, which is composition of adjacent adjoint actions l=1 yj, will also be frequently used. So we denote them by

y(1,2,··· ,l) := Ad(1)Ad(2) ··· Ad(l) :(b , ··· , b ) 7→ k−1b ··· b k b ··· b j kj kj kj 1 n j 1 l j l+1 n

The rearrangement lemma in our case, provide functions that are evaluated at non-

(1,2,··· ,l) 2 negative operators 1 − (yj ) , so we will denote

(l) (1,2,··· ,l) 2 zj := 1 − (yj )

22 If T1,T2, ··· ,Tn are n mutually commuting self-adjoint operators on a Hilbert space

∗ ∗ H, the unital C algebra C (I,T1,T2, ··· ,Tn) generated by T1, ..., Tn in B(H) is a commutative C∗ algebra. By Gelfand-Naimark, it is isomorphic to C(X) for some X ⊂

Qn n i=1 Spec(Ti) ⊂ R .

∗ For any f(x1, ··· , xn) ∈ C(X), its isomorphic image in C (I,T1,T2, ··· ,Tn) will be de-

∗ ∗ noted as f(T1, ··· ,Tn). The isomorphism C (X) → C (I,T1,T2, ··· ,Tn) is simply the algebra endormorphism extension of the substitution maps xj 7→ Tj, j = 1, ..., n. By spectral theory this isomorphism can be realized through the spectral integral.

The R-parametrized family of projections Ej(λ), λ ∈ R associated to each self adjoint

∗ operator Tj, is the projection of C (Tj) deﬁned by

Z ∗ ∗ g(λ)dEj(λ) = g(Tj), ∀g ∈ C (Spec(Tj)) = C (Tj) R

Ej(λ) deﬁnes a projection-valued measure on R. If by dE(λ1, ..., λn) = dE1(λ1) ··· dEn(λn) we mean the product measure on X ⊂ Rn, then the functional calculus substitution is given by

Z f(T1, ··· ,Tn) = f(λ1, ..., λn)dE(λ1, ..., λn) X

2.3.2 Fubini condition, substitution lemma

Let’s recall operator substitution lemma in the paper of M.Lesch ([35] Lemma 2.1). If fx(λ) is an integrable family of functions, T is an operator whose spectrum is contained in domain of all fx. The operator substitution lemma gives a Fubini type criteria to change R order of integrating in x and evaluating at λ = T . Namely if F (λ) = fx(λ)dx, this R substitution lemma tell us when will fx(T )dx be the same as F (T ).

23 The following lemma speciﬁes the Fubini type condition needed to change order of integral with evaluation. Moreover, the substitution lemma will make sense of an operator valued substitution ξ˜ = kξ in certain integral.

Lemma 2.3.1 (operator substitution lemma). Let R1, ··· ,Rn be mutually commuting

Qn operators in Banach algebra B, and X = j=1 Spec(Rj), Ω ⊂ R. R For a function f :Ω × X → R, let F (µ) = Ω f(ω, µ)dω.

1. If f is a function that satisﬁes Fubini condition

Z sup |f(ω, µ1, ··· , µn)|dω < ∞ (2.3.2.1) Ω µ∈X

Then Z f(ω, R1, ··· ,Rn)dω = F (R1, ··· ,Rn) Ω

2. Under the same condition as above, if moreover there is another positive operator

R0 that also commutes with all Rj. And Ω = [0, ∞)(or R) . Then

Z −1 f(ωR0,R1, ··· ,Rn)dω = R0 F (R1, ··· ,Rn) Ω

Proof of operator substitution lemma. The ﬁrst half of lemma 2.3.1 is directly from Lesch’s

[35], while we paraphrase the second half to ﬁt in our situation. So we will prove the second half.

Let dEj(µj) be the spectral projection measure of Rj, then

Z f(ωR0,R1, ··· ,Rn)dω Ω Z = f(ωµ0, µ1, ··· , µn)dE0(µ0) ··· dEn(µn)dω Ω×Q Spec(Rj )

24 The Fubini theorem allows switching the order of the integral freely. After applying a substitutionω ˜ = µ0ω, Ω = R+ or R remains unchanged under rescaling, and

Z Z −1 f(ωR0,R1, ··· ,Rn)dω = µ0 f(ω, µ1, ··· , µn)dE0(µ0) ··· dEn(µn)dω Ω Ω×Q Spec(Rj ) −1 =R0 F (R1, ··· ,Rn)

In this paper, we will frequently be using this lemma in the proof of the main theorem(Theorem 3.1.1). For the ﬂuency of the proof, we will not explicitly quote this lemma every single time. The reader should keep in mind that every time we deal with an op-

R erator valued integral x∈X f(A, x)dx for some function f(a, x) on spec(A) × X, we will need the Fubini condition Z |f(A, x)|dx < ∞. X

2.4 Gaussian average

In [47], the key treatment of the rearrangement integral

Z Z −λ α0 αn e b0 ρ1...ρnb0 dλdξ Rm C is in the integral over ξ ∈ Rn. Through the Mellin Transform, the rearrangement integral reduces to integral of the form

Z ÅZ ã e−(Hξ,Hξ)ξI dξ dµ Q Rm

25 where Q is some n − 1 simplex , dµ is some measure over Q, H is a positive self-adjoint linear map on Rn and I is some multiindex. The inner integral is the Gaussian average of ξI with respect to symmetric bilinear form H2.

Given a positively deﬁnite bilinear form H2 on Rm, the Gaussian average of a function f(ξ) is Z e−(Hξ,Hξ)f(ξ)dξ Rm

In this paper, we only need Gaussian average of monomials ξI , with respect to some positively deﬁnite m by m matrix H. Such integral is denoted as:

Z G(H,I) := e−(Hξ,Hξ)ξI dξ (2.4.0.1) Rm

Detail computation of G(H,I) can be found in section B.

2.5 Partitions

Computation in section B involves summation over all possible splitting of a given index set I into pairs. We will recall the set theoretic concept here, and simplify the notation for our convenience.

A partition of a set S is a set of subsets {Φλ|λ ∈ Λ}, such that each element s ∈ S belongs to one and only one subset Φλ. For example, a partition of {1, 2, 3} could be {{1, 3}, {2}}.

Deﬁnition 2.5.1. Fix a set I of even cardinality, say |I| = 2k.A 2-partition of I is a partition in which every chosen subset consists of two elements. Let P2(I) be the set of all 2-partitions of I.

For example, P2({a, b, c, d}) = {{{a, b}, {c, d}}, {{a, c}, {b, d}}, {{a, d}, {b, c}}}.

26 To simplify the notation, we will denote the above 2-partition by

P2({a, b, c, d}) = {(a, b)(c, d), (a, c)(b, d), (a, d)(b, c)} instead.

n Given a symmetric bilinear form A on R , ﬁxing an orthornormal basis {e1 ··· , en}

n 2k on R , and let Ai,j = A(ei ⊗ ej), for any even multiindex set I ∈ {1, 2, ··· , n} , we can

⊗k evaluate A at a 2-partition α ∈ P2(I) as the α1α2...α2k-entry of the 2k-tensor A .

⊗k Aα := (A , α) = Aα1,α2 ··· Aα2k−1,α2k . (2.5.0.1)

2.6 Hypergeometric integral and hypergeometric functions from rearrangement

We will show that the rearrangement integral

Z Z −λ α0 αn e b0 ρ1...ρnb0 dλdξ Rm C reduces to a hypergeometric integral10 (of the Gaussian average function).

Z G(H,I)dµ Q

Firstly, we should be more precise with what we mean by ”hypergeometric intergral”.

10There are many diﬀerent hypergeometric functions, each with multiple integral representations. In this paper, the notion of hypergeometric integral comes from generalization of an integral representation 1 R α0−1 α1−1 of the Gauss hypergeometric function 2F1, see §C, of the form 0 (1 − t) t f(t)dt

27 2.6.1 Hypergeometric integral

By hypergeometric integral , we mean an integral over some standard simplex Qn and

n with very speciﬁc measure µ as follows. Let Qn be the n dimensional simplex in R , namely

n Qn := {(s1, ··· , sn) ∈ R |s1, ··· , sn ≥ 0, s1 + ··· + sn ≤ 1} (2.6.1.1)

Let

1 α0−1 α1−1 αn−1 dµα = Qn (1 − s1 − · · · − sn) s1 ··· sn ds1...dsn (2.6.1.2) j=0 Γ(αj)

11 be a measure on the simplex Qn that depends on α.

2.6.2 Hypergeometric functions on Grassmannian

As the rearrangement integral reduces to hypergeometric integral of the Gaussian average functions G(H,I), we can step by step evaluate this hypergeometric integral

Z G(H,I)dµα(s) Qn

Firstly let us ﬁx the notation for the bilinear form H in the Gaussian average G(H,I).

(l) It turns out that H = H(Z, s) is a function on Z = (zj )1≤j≤m,1≤l≤n ∈ Mm×n(R),and s = (s1, ..., sn) ∈ Qn, valued in m × m positively deﬁnite symmetric matrix. Therefore

11 Note that the volume of measure µα is 1 Z 1 α0−1 α1−1 αn−1 µα(1) = Qn (1 − s1 − · · · − sn) s1 ··· sn ds1...dsn = j=0 Γ(αj) Qn Γ(α0 + ... + αn)

28 R Q G(H,I)dµ will be a function on Z ∈ Mm×n(R).

Deﬁnition 2.6.1. By H we denote the m × m diagonal matrix such that

(l) H = H(Z; s) = H((zj ); (s1, ..., sn)) = diag (H11, ..., Hmm)

q Pn (l) where the diagonal entries Hjj = 1 − l=1 zj sl

In §B we recalled the classical result that Gaussian average G(H,I) taking some kind

−2 of average of H over P2(I), as stated in Prop. B.0.4, that

1 m −1 X −2 G(H,I) = π 2 det(H ) H 2|I|/2 σ σ∈P2(I)

The ﬁrst function we need to deﬁne, is the result after applying Prop. B.0.4.

Definition 2.6.2. Given α = (α0, ...αn), multiindex I, if Qn is n − 1-simplex as defined in (2.6.1.1), dµα is the measure associated to α as defined in (2.6.1.2), and H is an m × m ˜ diagonal matrix defined as in (2.6.1), then Fα;I (Z) is defined as the hypergeometric integral

−1 P −2 of det(H ) σ∈P2(I) Hσ . In other word

Z ˜ (1) (n) −1 X −2 Fα;I (z1 , ..., zm ) := det(H ) Hσ dµα(s) (2.6.2.1) Qn σ∈P2(I)

−2 By deﬁnition of H as in Def. 2.6.1, and the deﬁnition of Hσ as in (2.5.0.1), the

−1 −2 integrand det(H )Hσ is product of diﬀerent powers of Hjj.

2l (l1,...,lm) Qm j Therefore we deﬁne the second function, Fα0,...,αn (Z), as hypergeometric integral of j=1 Hjj .

2 Pn (l) The power is chosen so that −lj is the power of Hjj = (1 − l=1 zj sl).

29 q Pn (l) Deﬁnition 2.6.3. Given α = (α0, ..., αn), l = (l1, ..., lm), if Hjj = 1 − l=1 zj sl is the

l j-th diagonal entry of an m × m diagonal matrix H as in Deﬁnition 2.6.2, then Fα(Z) is deﬁned by the following hypergeometric intergral:

Z m Z m n l (1) (n) Y −2lj Y X (l) −lj Fα(z1 , ..., zm ) := Hjj dµ(α)(s) = (1 − zj sl) dµ(α)(s) (2.6.2.2) Qn j=1 Qn j=1 l=1

l Remark. The exact relation between the second function Fα(Z) and the ﬁrst function ˜ Fα;I (Z) will be proved in Cor. B.0.5.

l In Prop. C.0.2 that Fα(Z) is the integral representation of the hypergeometric function on Grassmannian.

So we use the common deﬁnition of hypergeometric functions on Grassmannian as the third function Fn×m(α1, ..., αn, l1, ..., lm; c) . It is, the analytic function (multivalued if α, l not all integer) with a speciﬁc series expansion near the origin.

Q (α ; P M ) Q (β ; P M ) X j j l jl l l j jl M11 Mmn Fm×n(α1, ..., αm, β1, ..., βn; c;(zjl)) = P Q z11 ...zmn (c; Mjl) Mjl! M11,...,Mjl,...,Mmn≥0 j,l j,l Γ(a + n) (a; n) := is the Pochhammer symbol. Γ(a)

According to Prop. C.0.2, Fn×m(αα,ll) is ,up to a constant multiple, the second function

(l1,...,lm) ˜ l Fα0,...,αn , with α0 = c − α1 − ... − αn. Therefore, it will suﬃce to just use Fα;I (Z) and Fα. We will leave interested reader to section C for more detail and references.

30 Chapter 3: Main technical lemma

Our main purpose for this section, will be to state and prove a rearrangement lemma, that extend the conformal rearrangement lemma as in [35],[6], to handle elliptic operator

m Pm 2 2 P on a m-torus A = Tθ , with asymmetric principal symbol p2 = j=1 kj ξj , for some m-tuple of positive elements kj ∈ that mutually commute with each other. And we will adopt method of in [47], to reveal the identity of the resulting functions as generalized hypergeometric functions on Grassmannian.

Remark. Note that in the case m = 2, we also allow the operator to be deﬁned on any

P γ ﬁnitely generated right-A module E,cf. [36], in which case the operator P = |γ≤2| aγ∇ is deﬁned as endormorphism valued derivations, namely aγ ∈ B = EndA(E), and ∇1, ∇2 are connections on E that are compatible with the B−A-bimodule structure, cf.[36]. Then we will need k1, k2 ∈ B as well.

As is already brieﬂy explained in section 1.5, through out the paper we will stick to the following notation:

P is a elliptic diﬀerential operator of order 2 on Aθ (or E(g, θ)), with principal symbol

X 2 2 p2 = kj ξj , kj ∈ Aθ( or Aθ0 ) j

31 The symbol of P is

σ(P ) = p2 + p1 + p0 while symbol of P − λI is denoted as

σ(P − λI) = p2(λ) + p1 + p0 = (p2 − λ) + p1 + p0

Symbol of Resolvent of P has formal asymptotic expansion

∞ −1 X σ(P − λ) ∼ bj (3.0.0.1) j=0

1 with bj being homogeneous in ξ and |λ| 2 , of order −2 − j.

2 Although on H0 = L (Aθ, τ) and E(g, θ), the composition of symbols of pseudo- diﬀerential operators follows diﬀerent rules, namely (4.1.1.1) v.s. (4.2.1.3). It is straight forward to prove by induction that in both cases bj has the same pattern, that it is a sum

⊗α α0 α1 αn of an alternating products (b0 · ρ) = b0 ρ1b0 ...ρnb0 , with ρj being derivations of the coeﬃcients of ξ’s in σ(P ) = p2 + p1 + p0.

Lemma 3.0.1. With notation in section 1.5, the bj component of the Resolvent is of the

P ⊗α I form bj = (b0 · ρ)ξ where

α = (α0, ..., αn),ρρ = (ρ1, ..., ρn) and

−2(α0 + ... + αn) + |I| = −2 − j, n ≤ j + 1

Moreover, ρj is derivations of k1, ..., km and (p1)1, ..., (p1)m, p0.

32 By 4.2.5, computation of the density Rj(P ) therefore breaks down to computation of the following quantaty:

Deﬁnition 3.0.1. Given α = (α0, ..., αn) ∈ N, and multiindex I such that

|I| = −2 − j + 2(α0 + ... + αn)

⊗α I integral of the corresponding summand b0 ξ in ξ and λ is denoted as T (α; I)

1 Z Z −λ α0 αn I T(α0, ..., αn; I) := e b0 ⊗ · · · ⊗ b0 ξ dλdξ (3.0.0.2) 2πi Rm C

2 Example 3.0.1. One of the possible summand of b2 of some elliptic operator P is b0 ⊗ b0ξiξj,the corresponding function is T (2, 1; {i, j}).

Remark. The above discussion works

3.1 Rearrangement lemma

As a generalization of the result in [47], in which conformal rearrangement lemma was shown to produce special functions in the Lauricella Type D hypergeometric function family12, our nonconformal rearrangement lemma produces a slightly larger family of special functions, the generalized hypergeometric function on Grassmannian 13.

˜ l As explained in section 2.6, the three functions Fα;I (Z), Fα(Z), or Fn×m(αα,ll; Z) are related. T (α; I) can be expressed as either one of them.

Theorem 3.1.1 (Asymmetric rearrangement Lemma). Given α0, ..., αn ≥ 1 ,and multi-

(l) (1) (n) index I of length |I|. Let Z = (zj ) ∈ Mm×n(R) be the matrix of mn variables z1 , ..., zm

(l) (l) (1,...,l) 2 , and Z = (zj ),zj = 1 − (yj ) as deﬁned in section 2.3.

12deﬁned in section C 13see section C,Deﬁnition C.0.4

33 Let k(0) = k(0) ··· k(0) as in section 2.2, and let Qm k(0) = k(0) ··· k(0) be the volume factor I i1 i2k j=1 j 1 m under our change of metric.

ˆ ˆ ˆ I Lastly,using notation in section 2.2, let I = (I1, ..., Im) be the m-tuple such that ξ = ˆ Qm Ij j=1 ξj

1. In terms of Gaussian average, without evaluating G(H,I) that is deﬁned in 2.4.0.1,

1 Z T(α0, ..., αn; I) = (0) (0) G(H,I)dµ(α)(s) (3.1.0.1) Qm Q j=1 kj kI n

ˆ 2. If Ij is odd for some j, then

T(α0, ..., αn; I) = 0

ˆ ˆ 3. If I1 = 2a1, ..., Im = 2am, |I| = 2k, then

T(α0, ..., αn; I) m/2 π ˜ = Fα ,··· ,α ;I (Z) (3.1.0.2) k Qm (0) (0) 0 n 2 j=1 kj ·kI ˆ m/2 1+I1 1+Iˆm π (2a1)!...(2am)! ( 2 ,..., 2 ) = Fα0,··· ,αn (Z) (3.1.0.3) 2k Qm (0) (0) (a )!...(a )! 2 j=1 kj ·kI 1 m m/2 ˆ ˆ π (2a1)!...(2am)! 1 + I1 1 + Im = Fn×m(αα, , ..., ; Z) 2k Qm (0) (0) (a )!...(a )!Γ(α + ... + α ) 2 2 2 j=1 kj ·kI 1 m 0 n (3.1.0.4)

˜ l where Fα;I (Z) is defined in Definition 2.6.2, , Fα(Z) is defined in Definition 2.6.3,

Fn×m(αα,ll; Z) is deﬁned in Deﬁnition C.0.4.

Before proving Theorem 3.1.1, let’s look at some examples.

34 Example 3.1.1. If m = 2, b2 term is the noncommutative scalar density. Summand of

⊗(α0,α1) I ⊗(α0,α1,α2) I b2 is either b0 ξ or b0 ξ , such that the homogeneity condition reads |I| =

−4 + 2|α|.

Let 1ij be Kroneker delta.

1. If n = 1, α = (2, 1), and I = {j, j}, then

π ˜ T (2, 1; {j, j}) = F2,1;{j,j}(Z) (0) (0) (0) 2 2k1 k2 (kj )

π (2)! (1/2,1/2)+ej π (1/2,1/2)+ej T (2, 1; {j, j}) = F2,1 (Z) = F2,1 (Z) (0) (0) (0) 2 (1)! (0) (0) (0) 2 4k1 k2 (kj ) 2k1 k2 (kj ) π T (2, 1; {j, j}) = F1×2(2, 1, 1/2 + 11j, 1/2 + 12j; Z) (0) (0) (0) 2 2k1 k2 (kj ) Γ(2 + 1) π (4) = FD (2, 1, 1/2 + 11j, 1/2 + 12j; z1, z2) (0) (0) (0) 2 4k1 k2 (kj )

2. If n = 2, α = (3, 1, 1), and I = {1, 1, 1, 1, 2, 2}, then

π ˜ T (3, 1, 1; {1, 1, 1, 1, 2, 2}) = F3,1,1;{1,1,1,1,2,2}(Z) (0) (0) (0) 4 (0) 2 8k1 k2 (k1 ) (k2 )

π (4)!(2)! (1/2,1/2)+2e1+e2 T (3, 1, 1; {1, 1, 1, 1, 2, 2}) = F3,1,1 (Z) (0) (0) (0) 4 (0) 2 (2)!(1)! 64k1 k2 (k1 ) (k2 ) 3π (5/2,3/2) = F3,1,1 (Z) (0) (0) (0) 4 (0) 2 8k1 k2 (k1 ) (k2 ) 3π T (3, 1, 1; {1, 1, 1, 1, 2, 2}) = F1×2(3, 1, 1, 5/2, 3/2; Z) (0) (0) (0) 4 (0) 2 8k1 k2 (k1 ) (k2 ) Γ(3 + 1 + 1) π (5) = FD (3, 1, 1, 5/2, 3/2; z1, z2) (0) (0) (0) 4 (0) 2 64k1 k2 (k1 ) (k2 )

35 3.2 Proof of rearrangement lemma

The proof is adapted from idea in [47] to apply Mellin Transform in a way that results in hypergeometric type integrals. The essential diﬀerence between such method and the routine symbolic calculus rearrangement lemma, is the way integral in ξ is evaluated. In earlier literature, unless the operator has good enough angular symmetry14, integral in ξ means direct evaluation using integration by part m times. The complexity of computation increases as dimension m of the manifold increases. In [47] and this paper, on the contrary, integral in ξ is packed into the operation of taking Gaussian average G(H,I), which we can better understand using Combinatorics15.

Proof of Theorem 3.1.1. By Prop. B.0.5 and Prop. C.0.2, equation ?? will imply equa- ˜ tions ?? and ??. By definiton of Fα;I as in Definition 2.6.2, equation 3.1.0.1 is equivalent with equation ??. Therefore it suffices to prove equation 3.1.0.1.

Let’s break down the proof of equation ?? into several steps.

First step is to switch b0 with ρj as in (2.3.1.3).

P 2 (1,...,l) 2 2 P 2 2 Lemma 3.2.1. If we denote ηl = kj (yj ) ξj for l = 1, ..., n, and η0 = kj ξj . Then

n 1 α0 αn Y b ρ1 ··· ρnb = · (ρ1...ρn) (3.2.0.1) 0 0 αl j=0 (ηj − λ)

P 2 P 2 2 −1 −1 Proof. Let p(ξ1, ..., ξm, λ) = ( ξj − λ),then b0 = ( kj ξj − λ) = p(k1ξ1, ..., kmξm, λ) . Using notation as in §2.3 that y(l) = (Ad(j−1))−1Ad(j), and y(1,...,l) = Ql y(s), then we j kj kj j s=1 j

14 2 P 2 for example if p2 = k ( j ξj ), then one can use polar coordinate to greatly simplify integral in ξ 15as explained in section B

36 have

−1 ρb0 = p(k1y1ξ1, ..., kmymξm, λ) (x) n α0 αn α0 Y (1,...,l) (1,...,l) −αl b0 ρ1 ··· ρnb0 = b0 p(k1y1 ξ1, ..., kmym ξm, λ) · (ρ1...ρn) l=1

Second step is to choose a special contour C = iR and apply Mellin transform.

This change of contour can be done because

n −λ −α0 −αn −λ Y (j) −αj λ 7→ e (p2 − λ) ⊗ · · · ⊗ (p2 − λ) = e (p2 − λ) j=0

⊗¯ n+1 is a A -valued holomorphic function on C\(0, ∞), and p2 = η0 is strictly positive when |ξ|= 6 0.

16 Lemma 3.2.2. When |ξ|= 6 0, the contour C can be chosen to be iR. Using ηj deﬁned in lemma 3.2.1, then

Z −λ α0 αn e b0 ⊗ · · · ⊗ b0 dλ C Z ∞ n 1 =i e−ix Y dx ∈ A⊗¯ n+1 αj −∞ j=0 (ηj − ix)

By Mellin transform, for j = 0, 1, ··· , n

1 Z −αj αj −1 −sj (ηj −ix) (ηj − ix) = sj e dsj Γ(αj) [0,∞)

16 R ∞ −ix Qn 1 Fubini condition for e αj dx is satisﬁed because −∞ j=0 (ηj −ix)

n Y 1 1 1 | α | < min( P , min ( P )) (η − ix) j αj 0≤l≤n αj j=0 j |x| |ηl|

1 P is in L (R(x)) when αj > 1.

37 Therefore

Z −λ α0 αn e b0 ⊗ · · · ⊗ b0 dλ C α −1 Z ∞ Z n n 0 αn−1 (P sj −1)ix−P sj ηj s0 ...sn =i e j=0 j=0 dsn...ds0dx −∞ [0,∞)n+1 Γ(α0)...Γ(αn)

(1,...,l) Pn P 2 2 2 − j=1 sj ηj Note that ηl = kj (yj ) ξj are positive elements when |ξ| 6= 0, therefore e is Schwartz functions in s. As a result the integral over [0, ∞)n+1 is absolutely converging, the order of integral in s0, ..., sn can also be freely changed.

R ∞ R ∞ (s0+c)ix Note that any integral of the form −∞ 0 e g(s0)ds0dx can be evaluated by considering the zero extension G(s0) of g(s0) onto (−∞, ∞). By Fourier theory, it is simply 2πG(−c). Therefore

Lemma 3.2.3. After choosing contour C = iR, and applying Mellin Transform, the contour integral becomes

1 Z −λ α0 αn e b0 ⊗ · · · ⊗ b0 dλ 2πi C α −1 n αl−1 Z Pn Pn (1 − P s ) 0 Q s −(1− sl)η0− slηl l l l=1 l = e l=1 l=1 dsn...ds1 Qn Γ(α0)...Γ(αn) Z Pn −(η0− (η0−ηl)sl) = e l=1 dµ(α)(s) Qn

Proof. By Fourier theory, for any Schwartz function g ∈ S([, ∞)),any c ∈ R,

  g(−c) c < 0  1 Z ∞ Z ∞  (s0+c)ix g(0) e g(s0)ds0dx = G(−c) = 2 c = 0 2π −∞ 0    0 c > 0

38 n Therefore if we denote step functions on A ⊂ [0, ∞) by 1A, then

1 Z ∞ Z ∞ (s0+s1+...+sn−1)ix e g(s0)ds0dx = g(1 − s1 − ... − sn)1Qn(s1,...,sn) 2π −∞ 0

α −1 Pn n Qn j − j=0 sj ηj up to a zero measure set s1+...+sn = 1 in [0, ∞) . Finally, let g(s0) = j=0 sj e .

At this point the rearrangement integral

Z Z 1 −λ α I e b0 ξ dλdξ Rn 2πi C is reduced to

Z Z Pn −(η0− (η0−ηl)sl) I e l=1 ξ dµ(α)(s)dξ n R Qn

P 2 2 P 2 (1,...,l) 2 2 Note that η0 = kj ξj , and ηl = kj (yj ) ξj . The third step is to apply the ˆ substitution lemma in §2.3.2 to substitute each kjξj by ξj.

ˆ Lemma 3.2.4. By the substitution lemma in [35], under substitution ξj = kjξj for j =

1, ..., m, we have the following changes

X ˆ2 X (1,...,l) 2 ˆ2 η0 = ξj , ηj = (yj ) ξj 1 1 ξI = ξˆI , dξ = dξˆ (0) Qm (0) kI j=1 kj

As a result

(1,...,l) (l) −(η −Pn (η −η )s ) − Pm [1−Pn (1−(y )2)s ]ξˆ2 − Pm [1−Pn z s ]ξˆ2 −(Hξ,Hˆ ξˆ) e 0 l=1 0 l l = e j=1 l=1 j l j = e j=1 l=1 j l j = e

39 And the rearrangement integral becomes

Z Z 1 −λ α I e b0 ξ dλdξ Rn 2πi C Z Z 1 −(Hξ,Hξ) I = (0) (0) e ξ dµ(α)dξ (3.2.0.2) Qm n Q j=1 kj kI R n where H = H(Z, s) is the diagonal m by m matrix as in deﬁnition 2.6.1.

The last step is to show that H is in fact positively deﬁnite. In fact, in order to prove equation (??), it suﬃces to check the positivity of H. It will follow that e−(Hξ,Hξ) is Schwartz function in ξ, so the order of integral in dµ(s) and dξ can be freely switched.

And then the rearrangement integral will reduces to

Z Z Z 1 −(Hξ,Hξ) I 1 (0) (0) e ξ dξdµ(α)(s) = (0) (0) G(H,I)dµ(α)(s) Qm Q n Qm Q j=1 kj kI n R j=1 kj kI n

n+1 Lemma 3.2.5. If s = (s1, ··· , sn) ∈ Qn, t = (t0, ··· , tn) ∈ R+ , then

F (t, s) := t0 − (t0 − t1)s1 − (t0 − t2)s2 − · · · − (t0 − tn)sn > 0 (3.2.0.3)

Proof. Note that Qn is subset of the cone generated by the vertices ej = (0, ··· , 1, ··· , 0) ∈

n 17 R , and therefore every point in Qn is a conical combination of the vertices ej, j = 1, ..., n.

By the linearity of F (t, s) in s, it suﬃces to check that F (t, s) at the vertices s = ej. And obviously

F (t, ej) = t0 − (t0 − tj) = tj > 0

17 P conical combination of a set of vectors v1, ..., vn is linear combination cjvj with coeﬃcient cj ≥ 0

40 Corollary 3.2.6. H(Z, s) is positively deﬁnite.

(1,...,l) 2 Proof. For j = 1, ..., m, let’s evaluate function F (t, s) at t0 = 1 and tl = (yj ) , then

n X (1,...,l) 2 F (t, s) = 1 − (1 − (yj ) )sl = Hjj l=1

By functional calculus, positivity of F (t, s) as a scalar value function and positivity of

(1,...,l) 2 (yj ) implies Hjj > 0.

Therefore H = diag(H11, ..., Hmm) > 0

This ﬁnishes the proof of equation (??) and the whole theorem 3.1.1.

41 Chapter 4: Main result

4.1 Application on untwisted module over m-torus

An immediate application will be to compute the local expression of heat expansion coeﬃcient functionals18, on an m dimensional torus A, for any elliptic operator P on

P 2 2 the A with principal symbol of P is σ2(P ) = j kj ξj , and k1, ..., km ∈ A are mutually commuting positive elements.

4.1.1 Pseudodiﬀerential calculus on torus

Let’s recall some basic facts about symbolic calculus on a torus Aθ.

2 2 On the Hilbert space of L functions L (Aθ, τ), obtained through GMS construction corresponding to the unique trace τ, let the algebra of smooth functions Aθ acts by right multiplication.

18 Associated to any a ∈ A and an elliptic operator P , the j-th coeﬃcient Vj(a, P ) in the heat expansion (2.3.0.1) as a functional on A, has its local expression Rj(P ) ∈ A, such that

Vj(a, P ) = τ(aRj(P )), ∀a ∈ A

42 Definition 4.1.1 (Differential operators on Aθ). On Aθ, a differential operator P of order n is of the form

X I P = aI δ , aI ∈ Aθ |I|≤n

m Symbol space of Pseudo-diﬀerential operators on Aθ is denoted as S(R , A), which consists of smooth maps from Rm to A of ﬁnite growth rates in ξ ∈ Rm as |ξ| → ∞.

m 2 Given f(ξ) ∈ S(R , A), the corresponding pseudo-diﬀerential operator Pf ∈ L(L (A, τ)) is an (unbounded) linear operator , such that

Z ˇ 2 Pf a = f(s)αs(a)ds, ∀a ∈ L (A, τ)

ˇ 1 R −is·ξ m where f(s) = (2π)m e f(ξ)dξ is simply the Fourier inverse on R .

P n m P n m Note that if f = n,m fnmξ1 ξ2 is, then Pf = n,m fnmδ1 δ2 is a diﬀerential operator on

Aθ. And reversely, if P is an pseudo-diﬀerential operator, let σ(P ) denotes the symbol of

P .

Let A, B be pseudo-diﬀerential operator on Aθ, cf. [36], the symbol of composition

AB formally can be expanded in the following way.

1 σ(A) ∗ σ(B) := σ(AB) = X (Djσ(A)) · (∇jσ(B)) (4.1.1.1) j=0 j!

Where D is derivation along ﬁber of cotangent bundle, namely derivation in ξ, while

∇ is horizontal derivation on torus, namely δ, and

j j X (D A) · (∇ B) = (Dn1 ··· Dnj A)(∇n1 · · · ∇nj B). 1≤n1,··· ,nj ≤m

Let p(λ) denotes the symbol of the elliptic diﬀerential operator P − λ. p(λ) = p2(λ) + p1 + p0, p2(λ) = p2 − λ.

43 −1 Let b = b0 + b1 + ··· be the symbol of resolvent (P − λI) , bj(ξ, λ) is of order −2 − j in

2 −2−j the sense that bj(cξ, c λ) = c bj(ξ, λ) for any c > 0.

P j j From the formal expansion 1 = b ∗ p(λ) = j(D b) · (∇ p(λ)), one can solve for b0, b1, b2, ....

−1 b0 = p2(λ)

b1 = −[b0p1 + (Db0) · (∇p2(λ))]b0 1 b = −[b p + b p + (Db ) · (∇p ) + (Db ) · (∇p (λ)) + (D2b ) · (∇2p (λ))]b 2 0 0 1 1 0 1 1 2 2 0 2 0

P 2 2 4.1.2 Asymmetric Laplacian P = j kj δj

P 2 In [47], the operator ∆k = k j δj associated to a positive element k ∈ Aθ is concep- tualized as the conformal metric, as it can be viewed as noncommutative version of the

P 2 conformal metric gk = kg0 = k j dxj . Whether or not such operator is the correct candidate for being the ”noncommutative metric”, it deﬁnitely captures some very fundamental information of the corresponding Lapla- cian operator. In [6], after conformal change of the trace τ into state φk(a) = τ(ak), the Ö è 0 k(δ + iδ ) 1 2 (0,1) Dirac operator Dk = , acting on Hilbert space H0 ⊕H of (δ1 − iδ2)k 0 Ö è ∆+ 0 2 k functions and anti-holomorphic one forms. The Laplacian operator Dk = − 0 ∆k 2 P 2 + − will have principle symbol k j ξj for both ∆k and ∆k .

A natural extension to ∆k, also the easiest possible application of our modiﬁed rearrangement lemma, will be to consider the asymmetric Laplacian operator P =

P 2 2 j kj δj . The term asymmetric is adopted from [17], in which rearrangement lemma was

44 2 2 2 ﬁrst extended to handle elliptic operator on two-torus that has principal symbol ξ1 +k ξ2 .

Density R2(P )

j−2 Through Pseudo-differential calculus, the heat coefficient Vj(a, P ) as coefficient of t 2 in heat expansion (2.3.0.1), can be written in terms of bj, as the term of homogeneous order −2 − j in (3.0.0.1). In fact, it turns out the functional Vj( ,P ) on A has local expression Rj(P ) ∈ A, such that

Vj(a, P ) = τ(aRj(P )), ∀a ∈ A 1 Z Z Rj(P ) = bj(ξ, λ)dλdξ 2πi Rm C

19 where C is some contour with positive orientation that bounds R+ in its interior.

Pm 2 2 Our ﬁrst application is to compute the R2(P ) for P = j=1 kj δj on an m torus A, such that all k1, ..., km ∈ A are positive (self-adjoint therefore too) elements. In [47], the

R2(P ) density is computed for the case when k1 = k2, and written in terms of Hypergeo- metric functions. We should verify our result in this special case.

−tP Note that when m = 2, Vm(a, P ) is the constant term in heat expansion of e , namely total scalar curvature, and R2(P ) is the local expression, the corresponding ”diﬀerential form”.

Recall that the resulting functions from rearrangement lemma use yj = Adkj , Adjoint

⊗n actions by k1, ..., km, as variables. More precise, the function acting on A will have mn

19 Once it is veriﬁed that bj is holomorphic in λ ∈ C\R+ , then the C can be any such curve.

45 (l) (l) variables yj with j = 1, ..., m and l = 1, ..., n, where yj acts on the j-th component in

A⊗n.

Theorem 4.1.1.

m/2 2 π X kj (1,1) 2 2 R2(P ) = K (Y)(δj(k ) · δj(k )) Qm (0) k2k2 j,v,u v u j=1 kj j,u,v v u πm/2 k2 + X j K(2)(y)(δ2(k2)) Qm (0) k2 j,u j u j=1 kj j,u u

where

(y(1))2 (1,1) 1 v ˜ ˜ j ˜ 1. Kj,v,u(Y) = [(− 2 − 1j )F2,1,1;{u,u,v,v} + F3,1,1;{j,j,v,v,u,u} + 2 F2,2,1;{j,j,v,v,u,u}](Z)

(1) (l) 2 With Z = (..., 1 − (yj ··· yj ) , ...)1≤j≤m,1≤l≤2

(2) ˜ 1 ˜ 2. Kj,u (y) = [−F3,1;{j,j,u,u} + 2 F2,1;{u,u}](z)

(1) 2 With z = (..., 1 − (yj ) , ...)1≤j≤m

Because the component of degree 1 and 0 in p = σ(P ) vanishes, the symbolic calculus is greatly simpliﬁed.

−1 X 2 2 −1 b0 = (p − λ) = ( kj ξj − λ) j b1 = −[(Db0) · (∇q2)]b0 1 1 b = −[(Db ) · (∇q ) + (D2b ) · (∇2q )]b = (D((Db ) · (∇p)b )) · (∇q )b − (D2b ) · (∇2q )b 2 1 2 2 0 2 0 0 0 2 0 2 0 2 0

By Leibniz Rule, b2 is sum of the following 4 terms, over i, j = 1, ··· , m

a)( D(Db0)i)j(∇p)ib0(∇p)jb0

b)( Db0)i(D(∇p)i)jb0(∇p)jb0

46 c)( Db0)i(∇p)i(Db0)j(∇p)jb0

1 2 2 d) − 2 (D b0)ij(∇ q2)ijb0

As we will need to treat multiple summations, we will omit the summation notation

P if there is no ambiguity. For example ,instead of i,j aibj, we just put aibj. On the other hand, to avoid confusion between derivation δj and the Kronecker delta, we will denote the later one by 1.

(D(Db0)i)j(∇p)ib0(∇p)jb0

2 2 2 3 (D b0)ij(∇p2)ib0(∇p2)jb0 = −(b0Dij(p2) − 2b0Di(p2)Dj(p2))∇i(p2)b0∇j(p2)b0

2 (Dp2)i = 2ki ξi,

2 2 i (D p2)ij = 2kj 1j,

X 2 2 (∇p2)i = δi(kl )ξl , l

2 (D b0)ij(∇p2)ib0(∇p2)jb0

2 2 j 3 2 2 2 2 2 2 = − (2b0kj δi − 8b0ki kj ξiξj)δi(kv)ξv b0δj(ku)ξub0

2 2 2 2 2 2 3 2 2 2 2 2 2 = − 2b0kj δj(kv)b0δj(ku)b0ξv ξu + 8b0ki kj δi(kv)b0δj(ku)b0ξiξjξv ξu

2 2 2 2 2 2 3 4 2 2 2 2 2 = − 2b0kj δj(kv)b0δj(ku)b0ξv ξu + 8b0kj δj(kv)b0δj(ku)b0ξj ξv ξu + odd part

47 I I The last equation uses the fact that G(ξ ) = 0 if ξ has odd for some ξj.

1 R R By rearrangement lemma, m (D(Db ) ) (∇p) b (∇p) b dλdξ is, up to overall 2πi R C 0 i j i 0 j 0

m/2 −1 factor π (k1 ··· km)

2 1 ˜ 4 1 ˜ 2 2 = (−2kj 2 2 2 F2,1,1;{u,u,v,v}(Z) + 8kj 3 2 2 2 F3,1,1;{j,j,v,v,u,u}(Z))(δj(kv) · δj(ku)) 2 kvku 2 kj kvku k2 j 1 ˜ ˜ 2 2 = 2 2 (− F2,1,1;{u,u,v,v} + F3,1,1;{j,j,v,v,u,u})(Z))(δj(kv) · δj(ku)) (4.1.2.1) kvku 2

(Db0)i(D(∇p)i)jb0(∇p)jb0

2 (Db0)iD((∇p2)i)jb0(∇p2)jb0 = −b0(Dp2)iD((∇p2)i)jb0(∇p2)jb0

2 (Dp2)i = 2ki ξi,

2 D((∇p2)i)j = 2δi(kj )ξj,

X 2 2 (∇p2)j = δj(ks )ξs s

(Db0)iD((∇p2)i)jb0(∇p2)jb0

2 2 2 2 2 = − 4b0ki δi(kj )b0δj(ks )b0ξiξjξs

2 2 2 2 2 2 = − 4b0kj δj(kj )b0δj(ks )b0ξj ξs + odd part

1 R R By rearrangement lemma, m (Db ) (D(∇p) ) b (∇p) b dλdξ is, up to overall 2πi R C 0 i i j 0 j 0

m/2 −1 factor π (k1 ··· km)

48 2 1 ˜ 2 2 = −4kj 2 2 2 F2,1,1;{j,j,s,s}(Z)(δj(kj ) · (δj(ks )) 2 kj ks 1 ˜ 2 2 = − 2 F2,1,1;{j,j,s,s}(Z)(δj(kj ) · (δj(ks )) ks k2 j v ˜ 2 2 = − 2 2 1j F2,1,1;{j,j,u,u}(Z)(δj(kv) · (δj(ku)) (4.1.2.2) kvku

(Db0)i(∇p)i(Db0)j(∇p)jb0

2 (Db0)i(∇p2)i(Db0)j(∇p2)jb0 = b0(Dp2)i(∇p2)ib0(Dp2)jb0(∇p2)jb0

2 (Dp2)i = 2ki ξi,

2 (Dp2)j = 2kj ξj

X 2 2 (∇p2)i = δi(kv)ξv , v X 2 2 (∇p2)j = δj(ku)ξu u

Therefore

(Db0)i(∇p2)i(Db0)j(∇p2)jb0

2 2 2 2 2 2 2 2 = 4b0ki δi(kv)b0kj δj(ku)b0ξiξv ξjξu

4 2 2 2 2 2 2 2 2 = 4kj b0yj (δj(kv))b0δj(ku)b0ξj ξv ξu + odd part

49 1 R R By rearrangement lemma, m (Db ) (∇p) (Db ) (∇p) b dλdξ is, up to overall 2πi R C 0 i i 0 j j 0

m/2 −1 factor π (k1 ··· km)

4 (1) 2 1 ˜ 2 2 =4kj (yj ) 3 2 2 2 F2,2,1;{j,j,v,v,u,u}(Z)(δj(kv) · (δj(ku)) 2 kj kvku k2 (y(1))2 j j ˜ 2 2 = 2 2 F2,2,1;{j,j,v,v,u,u}(Z)(δj(kv) · (δj(ku)) (4.1.2.3) kvku 2

1 2 2 − 2 (D b0)ij(∇ p2)ijb0

1 2 2 3 2 1 2 2 2 − 2 (D b0)ij(∇ q2)ijb0 = −b0(Dp2)i(Dp2)j(∇ p2)ijb0 + 2 b0(D p2)ij(∇ p2)ijb0

2 (Dp2)i = 2ki ξi

2 (Dp2)j = 2kj ξj

2 2 i (D p2)ij = 2kj 1j

2 2 2 (∇ p2)ij = δiδj(ks )ξs

Therefore

1 − (D2b ) (∇2q ) b 2 0 ij 2 ij 0 3 2 2 2 2 2 2 i 2 2 = − 4b0ki kj δiδj(ks )b0ξiξjξs + b0kj 1jδiδj(ks )b0ξs

3 4 2 2 2 2 2 2 2 = − 4b0kj δjδj(ks )b0ξj ξs + b0kj δjδj(ks )b0ξs + odd part

50 1 R R 1 2 2 By rearrangement lemma, m − (D b ) (∇ p ) b dλdξ is, up to overall factor 2πi R C 2 0 ij 2 ij 0

m/2 −1 π (k1 ··· km)

4 1 ˜ 2 2 2 1 ˜ 2 2 = −4kj 2 2 2 F3,1;{j,j,s,s}(δj (ks )) + kj 2 F2,1;{s,s}(δj (ks )) 2 kj ks 2ks k2 j ˜ 1 ˜ 2 2 = 2 [−F3,1;{j,j,s,s} + F2,1;{s,s}](δj (ks )) (4.1.2.4) ks 2

4.1.3 Density R2(P )

1 R R Adding up (4.1.2.1)-(4.1.2.4), we will get the density functional R (P ) = m b dλdξ, 2 2πi R C 2

Theorem 4.1.2. [Main result 1]

m/2 2 π X kj (1,1) 2 2 R2(P ) = K (Y)(δj(k ) · δj(k )) Qm (0) k2k2 j,v,u v u j=1 kj j,u,v v u πm/2 k2 + X j K(2)(y)(δ2(k2)) Qm (0) k2 j,u j u j=1 kj j,u u

where

(y(1))2 (1,1) 1 v ˜ ˜ j ˜ 1. Kj,v,u(Y) = [(− 2 − 1j )F2,1,1;{u,u,v,v} + F3,1,1;{j,j,v,v,u,u} + 2 F2,2,1;{j,j,v,v,u,u}](Z)

(1) (l) 2 With Z = (..., 1 − (yj ··· yj ) , ...)1≤j≤m,1≤l≤2

(2) ˜ 1 ˜ 2. Kj,u (y) = [−F3,1;{j,j,u,u} + 2 F2,1;{u,u}](z)

(1) 2 With z = (..., 1 − (yj ) , ...)1≤j≤m

51 m/2 −1 Proof. Add up (4.1.2.1)-(4.1.2.4), the density is ,up to overall factor π (k1 ··· km) =

πm/2 Qm (0) , j=1 kj

k2 j 1 ˜ ˜ 2 2 = 2 2 (− F2,1,1;{u,u,v,v} + F3,1,1;{j,j,v,v,u,u})(Z))(δj(kv) · δj(ku)) kvku 2 k2 j v ˜ 2 2 − 2 2 1j F2,1,1;{j,j,u,u}(Z)(δj(kv) · (δj(ku)) kvku k2 (y(1))2 j j ˜ 2 2 + 2 2 F2,2,1;{j,j,v,v,u,u}(Z)(δj(kv) · (δj(ku)) kvku 2 k2 j ˜ 1 ˜ 2 2 + 2 [−F3,1;{j,j,s,s} + F2,1;{s,s}](δj (ks )) ks 2 k2 (y(1))2 j 1 v ˜ ˜ j ˜ 2 2 = 2 2 [(− − 1j )F2,1,1;{u,u,v,v} + F3,1,1;{j,j,v,v,u,u} + F2,2,1;{j,j,v,v,u,u}](δj(kv) · δj(ku)) kvku 2 2 k2 j ˜ 1 ˜ 2 2 + 2 [−F3,1;{j,j,s,s} + F2,1;{s,s}](δj (ks )) ks 2

The result can be then expressed in terms of generalized hypergeometric function

(l) Fα . The result will be much less compact and convenient, however, as the combinato-

F˜α;I rial constant C = ˆ ˆ from equation (B.0.0.4) varies from one case to another. I (1/2+I1,...,1/2+Im) Fα

To handle R2(P ), all we need is the following

52 Lemma 4.1.3. For Theorem 4.1.2, the result only involves multiindex I with |I| ≤ 6.

All possible CI are listed as below:

2! C = = 1 (4.1.3.1) {u,u} 21   4!  222! = 3, if u = v C{u,u,v,v} = (4.1.3.2)  2!2!  22 = 1, if u 6= v   6!  233! = 15, if j = u = v   4!2! C{j,j,u,u,v,v} = 3, either j = u 6= v, or j = v 6= u, or u = v 6= j (4.1.3.3)  232!    2!2!2!  23 = 1, if j, u, v distinct

πm/2 Proposition 4.1.4. R2(P ) is, upto overall factor Qm (0) , j=1 kj

2 (1) 2 X kj 1 (1/2,...)+e +e (1/2,...)+e +e +e (yj ) (1/2,...)+e +e +e = [(− )F u v + F j u v + F j u v ](δ (k2) · δ (k2 )) k2k2 2 2,1,1 3,1,1 2 2,2,1 j v j u j,u,v distinct v u 2 (1) 2 X kj 3 (1/2,...)+e +e (1/2,...)+e +e +e 3(yj ) (1/2,...)+e +e +e + [(− )F u v + 3F j u v + F j u v ](δ (k2) · δ (k2 )) k2k2 2 2,1,1 3,1,1 2 2,2,1 j v j u j=v6=u v u 2 (1) 2 X kj 1 (1/2,...)+e +e (1/2,...)+e +e +e 3(yj ) (1/2,...)+e +e +e + [(− )F u v + 3F j u v + F j u v ](δ (k2) · δ (k2 )) k2k2 2 2,1,1 3,1,1 2 2,2,1 j v j u j=u6=v v u 2 (1) 2 X kj 3 (1/2,...)+e +e (1/2,...)+e +e +e 3(yj ) (1/2,...)+e +e +e + [(− )F u v + 3F j u v + F j u v ](δ (k2) · δ (k2 )) k2k2 2 2,1,1 3,1,1 2 2,2,1 j v j u j6=v=u v u 2 (1) 2 X kj 9 (1/2,...)+e +e (1/2,...)+e +e +e 15(yj ) (1/2,...)+e +e +e + [(− )F u v + 15F j u v + F j u v ](δ (k2) · δ (k2 )) k2k2 2 2,1,1 3,1,1 2 2,2,1 j v j u j=v=u v u 2 X kj (1/2,...)+e +e 1 (1/2,...)+e + [−F j s + F s ](δ2(k2)) k2 3,1 2 2,1 j s j6=s s 2 X kj (1/2,...)+e +e 1 (1/2,...)+e + [−3F j s + F s ](δ2(k2)) k2 3,1 2 2,1 j s j=s s

Proof. apply equation(B.0.0.4), lemma(4.1.3), to result in prop(4.1.2)

2 P As the asymmetric Laplacian P will reduces to ∆k2 = k ( ) when k1 = k2 = ··· = km.

Prop (4.1.2) can recover density R2(∆k) in [47].

53 Comparison with conformal case

In the conformal case, all kjs coincide. To avoid confusion with the multivariable

(l) (l) (l−1) −1 (l) (l) notation y = (··· , yj , ··· )j,l, let’s denote Adk = (k ) k by yˆ .

It’s immediate that any function of modular actions by k1, ··· , km will be reduced to a function of modular actions by k:

N ∗ (l) Lemma 4.1.5. If k1 = k2 = ··· = km, then for any f ∈ C (yj ), 1≤j≤m 1≤l≤n

(l) f(··· , yj , ··· )1≤j≤m,1≤l≤n = f| (l) (l) yj =yˆ ,∀j,l

(l) The generalized hypergeometric functions Fα (Z) will be reduced as follows:

Proposition 4.1.6. If k1 = ··· , = km, then

(l1,··· ,lm) (l1+···+lm) ˆ Fα0,··· ,αn (Z) = Fα0,··· ,αn (Z) where l (l) (l) Y (r) 2 Z = (··· , zj , ··· )1≤j≤m, zj = 1 − (yj ) 1≤l≤n r=1 while l ˆ (l) (l) Y (r) 2 Z = (··· , zˆ , ··· )1≤l≤n, zˆ = 1 − (yˆ ) r=1

Proof.

(l1,··· ,lm) Fα0,··· ,αn (Z) 1 Z m n Y X (l) −lj =Qn (1 − zj sl) dµ(α)(s) l=0 Γ(αl) Qn−1 j=1 l=1 n kj =k,∀j 1 Z Pm X (l) − j=1 lj = Qn (1 − zˆ sl) dµ(α)(s) l=0 Γ(αl) Qn−1 l=1

(l1+···+lm) ˆ =Fα0,··· ,αn (Z)

54 Now proposition (4.1.6) can be applied to reduce R2(P ) in Prop. (4.1.4) to R2(∆k2 )

2 Pm 2 Proposition 4.1.7. Let ∆k2 = k j=1 δj , the density R2(∆k2 ) is , up to overall factor

πm/2 km

m + 2 m + 2 (m + 4)(y(1))2 [(− )F(2+m/2) + (m + 4)F(3+m/2) + F(3+m/2)](|∇k2|2) k2 2 2,1,1 3,1,1 2 2,2,1 m + [−(m + 2)F(2+m/2) + F(1+m/2)](|∆k2|) 3,1 2 2,1 where

2 2 X 2 2 |∇k | := (δj(k ) · δj(k )) j

2 X 2 2 |∆k | := δj (k ) j

πm/2 Proof. by Prop. (4.1.4) and Prop. (4.1.6), R2(∆k2 ) is , up to overall factor km

(1) 2 X 1 1 (2+m/2) (3+m/2) (y ) (3+m/2) (m − 1)(m − 2) [(− )F + F + F ](δ (k2) · δ (k2)) k2 2 2,1,1 3,1,1 2 2,2,1 j j j (1) 2 X 1 3 (2+m/2) (3+m/2) 3(y ) (3+m/2) + (m − 1) [(− )F + 3F + F ](δ (k2) · δ (k2)) k2 2 2,1,1 3,1,1 2 2,2,1 j j j (1) 2 X 1 1 (2+m/2) (3+m/2) 3(y ) (3+m/2) + (m − 1) [(− )F + 3F + F ](δ (k2) · δ (k2)) k2 2 2,1,1 3,1,1 2 2,2,1 j j j (1) 2 X 1 3 (2+m/2) (3+m/2) 3(y ) (3+m/2) + (m − 1) [(− )F + 3F + F ](δ (k2) · δ (k2)) k2 2 2,1,1 3,1,1 2 2,2,1 j j j (1) 2 X 1 9 (2+m/2) (3+m/2) 15(y ) (3+m/2) + [(− )F + 15F + F ](δ (k2) · δ (k2)) k2 2 2,1,1 3,1,1 2 2,2,1 j j j X (2+m/2) 1 (1+m/2) + (m − 1) [−F + F ](δ2(k2)) 3,1 2 2,1 j j X (2+m/2) 1 (1+m/2) + [−3F + F ](δ2(k2)) 3,1 2 2,1 j j 2 (1) 2 X 1 (m + 2) (2+m/2) (3+m/2) (m + 2)(m + 4)(y ) (3+m/2) = [(− )F + (m + 2)(m + 4)F + F ](δ (k2) · δ (k2)) k2 2 2,1,1 3,1,1 2 2,2,1 j j j X (2+m/2) m (1+m/2) + [−(m + 2)F + F ](δ2(k2)) 3,1 2 2,1 j j

55 Remark (result comparison to [47]). In [47], the two types of hypergeometric function

0 Ka,b(z),Ha,b,c(z, z ) relates to our general hypergeometric functions F as follows:

(a+b−2+m/2) (1) Ka,b(z; m) = Γ(a + b − 2 + m/2)Fa,b (z )|z(1)=z and

0 (a+b+c−2+m/2) (1) (2) Ha,b,c(z, z ; m) = Γ(a + b + c − 2 + m/2)Fa,b,c (z , z )|z(1)=z,z(2)=z0

If we replace our F in Prop. (4.1.7) by these two families of functions Ha,b,c and Ka,b, the density R2(∆k2 ) is,

(m + 2)πm/2 −1 1 (y(1))2 [ H + H + H ](|∇k2|2) km+2 Γ(1 + m/2) 2,1,1 Γ(2 + m/2) 3,1,1 Γ(2 + m/2) 2,2,1 πm/2 −1 1 + [ K + K ](|∆k2|) km Γ(1 + m/2) 3,1 Γ(m/2) 2,1

This recovers the density R2(∆k2 ) in [47]

4.1.4 Gauss-Bonnet function relations

In [16], associated to what we call in this paper the Dirac operator Dk = k(δ1 + iδ2)

20 ∗ under a conformal twist by Weyl factor k ∈ Aθ, the Laplacian operator ∆φ = DkDk = k∆k , was shown to satisfy

τ(R2(∆k)) = 0

20 ¯¯∗φ ¯ choice of notation ∆φ comes from the fact that this operator can be obtained as ∂∂ , where ∂ = δ1 + iδ2 is the standard partial bar operator on A, while the adjoint ∗φ is associated to a non-tracial state φ that comes from the conformal change of metric: φ(a) = τ(ak−2) , that change the Hilbert space inner product on space of functions H0(Aθ)

56 which we call the Gauss-Bonnet property, as it corresponds to identity of the classical

Gauss-Bonnet Theorem: Z KdV = 2πχ(M) M

2 on torus M = T . In fact, the second heat coeﬃcient a2(f, D) = τ(fR2(D)) in heat expansion

−tD X j−2 Tr(fe ) ∼t→0 a2j(f, D)t 2 j≥0 is total scalar curvature when f = 1, and Euler characteristic of a torus T 2 is 0.

Later in [6] a conceptual explanation was given to this Gauss-Bonnet property. It’s

sh sh s s shown that under variation ∆(s) = e ∆e = k ∆k , the heat coeﬃcient aj(1, ∆(s)) satisﬁes d j − 2 a (1, ∆(s)) = a (h, ∆(s)) ds j 2 j

And therefore the noncommutative scalar curvature a2(1, ∆k) is invariant under deforma- tion on k, and equal to 0 as in the case when k = 1.

In [47] the scalar curvature a2(f, ∆k) and its density functional R2(∆k) associated

2 to operator ∆k = k ∆ was computed in terms of Hypergeometric functions. As ∆k =

−1 k∆φk , the scalar curvature a2(1, ∆k) bounds to vanish as well. It turns out this all reduce to certain relations satisﬁed by Hypergeometric functions.

A natural question to ask, is whether the density functional R2(∆k1,k2 ) for our asym-

P 2 2 metric Laplacian ∆k1,k2 whose symbol is j kj ξj still satisﬁes Gauss-Bonnet property.

4.2 Application on twisted module over 2-torus

± In [6], modular scalar curvature density R2(∆k ) noncommutative two tori Aθ for conformal Dirac operator

57 Ö è D+ k + − + ∗ Dk = , with Dk = ik(δ1 + iδ2),Dk = (Dk ) = i(δ1 − iδ2)k and − Dk Ö è Ö è D+D− ∆+ 2 k k k Dk = = . − + − Dk Dk ∆k Then one could extend the computation onto Heisenberg module E(g, θ) over Aθ, cf. [36], for Ö è D+ k;E + − + ∗ Dk;E = , with Dk;E = ik(∇1 + i∇2),Dk;E = (Dk;E ) = i(∇1 − i∇2)k. − Dk;E Our next example is the asymmetric version of Dk;E on E(g, θ), namely Ö è D+ k1,k2;E + − Dk1,k2;E = , with Dk ,k ;E = i(k1∇1 + i∇2),Dk ,k ;E = i(∇1k1 − − 1 2 1 2 Dk1,k2;E i∇2k2).

4.2.1 Symbol calculus on Heisenberg module

0 aθ+b 0 With Aθ treated as the base manifold of E(g, θ), we treat Aθ , θ = g.θ = cθ+d as the endomorphism ring.

Definition 4.2.1 (Differential operators on E(g, θ)). An differential operator P on E(g, θ)

P is of the form P = I aI ∇I , with summation over ordered multi-index I, and coeﬃcients aI ∈ Aθ0 .

Deﬁnition 4.2.2 (Pseudo-diﬀerential operators on E(g, θ)). The space of symbol of

2 2 Pseudo-diﬀerential operators, S(R , Aθ0 ), consists of all Aθ0 -valued functions f : R → Aθ0 with at most polynomial growth in ξ ∈ R2 as |ξ| → ∞.

m 2 More precisely, we say f(ξ) is of order m, denoted as f ∈ S (R , Aθ0 ), if

1. for any multi-index I,J, there exist constant CIJ such that

I J m−|J| |δ (∂ξ )f| ≤ CIJ (1 + |ξ|)

58 f(λξ) 2. and the limit lim|ξ|→∞ λm exist, which is then the principal symbol of f.

Associated to a symbol f, the corresponding Pseudo-diﬀerential operator is

Z ˇ Op(f) := f(x)Wxdx (4.2.1.1) R2

ˇ 2 where f is inverse-Fourier transform of f, and x 7→ Wx ∈ B(H(g, θ)) is an R action on the Hilbert space completion of E(g, θ)21,twisted by a cocycle e : R2 × R2 → T ⊂ C.

Explicitly, for any u(t, α) ∈ E(g, θ) = S(R × Z/cZ), the W(x1,x2) action is deﬁned by Heisenberg action,

πiµ(x1x2+2tx2) (W(x1,x2)u)(t, α) = e u(t + x1) (4.2.1.2)

Example 4.2.1. Op(ξj) = i∇j, j = 1, 2, namely σ(∇j) = −iξj

Proof.

ZZ ix·ξ Op(ξj) = e ξjWxdξdx ZZ 1 ix·ξ d = − e (Wx)dξdx i dxj d = i |x=0 (Wx) = i∇j dxj

The algebra structure on space of symbols is governed by composition of Pseudo- diﬀerential operators. We will rephrase the composition formula as in [36] so as to have a more consistent form with the one we used for Pseudo-diﬀerential calculus on functions, namely (4.1.1.1).

21 completion under its Aθ − Aθ0 -Morita equivalence bimodule inner product, see Appendix

59 Proposition 4.2.1. If p, q are symbol of two (pseudo)diﬀerential operator on E(g, θ), then the symbol of composition p ∗ q = σ(Op(p) ◦ Op(q)) satisﬁes the following identity.

¯ δj j Let ∂xi denote partial derivatives in ξ, let δj = i( cθ+d + (−1) πµ∂j+1) be a mixture of

22 derivation on Aθ0 and partial derivatives in ξ. Then

1 p ∗ q(ξ) = Σ ∂γp(ξ)δ¯γq(ξ) (4.2.1.3) γ γ! ξ

23 ¯ 1 j To put this in contraction notation , Let Dξj = ∂ξj , and Dj = i( cθ+d δj +(−1) πµ∂j+1). Then

1 p ∗ q(ξ) = Dnp(ξ) · D¯ nq(ξ) (4.2.1.4) n! ξ

m 2 m0 2 Proof. As shown in [36], for p ∈ S (R , Aθ0 ), q ∈ S (R , Aθ0 ), the composition of f and g is, up to smoothing24,

−|I| X i I I (∂ξ p)(ξ)∂y |y=0 (α−y(q(ξ + By))) I I!

Ö è −πµ with B = πµ 2 Let’s recall that the R -automorphsm αy on both Aθ and Aθ0 is the one induced by conjugation of Wy, and is determined by its actions on generators U1,U2 ∈ Aθ and

22 ∂3 := ∂1 23 2 For symbolic calculus on L (Aθ, τ), the composition formula

1 X p ∗ q = ∂γ pδγ q γ can be rewritten as X 1 p ∗ q = Dnp · Dnq n! ξ x n n n P I I where Dξj = ∂j,Dxj = δj and Dξ p · Dx q = |I|=n ∂ pδ q 24 −∞ 2 namely up to any perturbation in S (R , Aθ0 )

60 V1,V2 ∈ Aθ0 , as follows:

2πi yj 2πiyj αy(Vj) = e cθ+d Vj, αy(Uj) = e Uj, j = 1, 2

2πi n1 n2 cθ+d <−(y1,y2),(n1,n2)> n1 n2 So α−y(V1 V2 ) = (e V1 V2 ) for all n1, n2 ∈ Z.

J |J| 0J 0 1 n1 n2 2πinj n1 n2 And ∂y |y=0 (α−y(·)) = (−1) δ , with δj = cθ+d δj : V1 V2 7→ cθ+d V1 V2 .

c J J |J| J2 J On the other hand ∂y |y=0 q(ξ+By) = ∂y |y=0 q(ξ1−πµy2, ξ2+πµy1) = (πµ) (−1) ∂ξ q(ξ),

c with J = (J1,J2) and J = (J2,J1).

It easily follows that

! 0 I 0 0c −|I| I |I| X J1 0 J |J | J i ∂y |y=0 (α−y(q(ξ + By))) = i (−1) (δ ) (πµ) ∂ξ J+J0=I J

|I| 0 I1 0 I2 ¯I = i (δ1 − πµ∂2) (δ2 + πµ∂1) q(ξ) = δ q(ξ)

Example 4.2.2.

2 2 σ(∇j ) = −ξj

σ(∇1∇2) = −ξ1ξ2 + iπµ

σ(∇2∇1) = −ξ1ξ2 − iπµ

γ 4.2.2 Index density R2

Our second application of Theorem 3.1.1 is to compute the index density functional

γ b2 on Heisenberg module E(g, θ) associated to the asymmetric Dirac operator Dk1,k2;E = Ö è D+ k1,k2;E + − , with Dk ,k ;E = i(k1∇1 + i∇2),Dk ,k ;E = i(∇1k1 − i∇2k2). − 1 2 1 2 Dk1,k2;E For any elliptic diﬀerential operator P , thus Fredholm, the index of P can be computed as follows

61 Lemma 4.2.2. Ind(P ) = tr(e−tP ∗P ) − tr(e−tP P ∗ ), ∀t > 0

Proof. There is 1-1 correspondence between eigenspaces of nonzeroes eigenvalues of PP ∗ and P ∗P : if P ∗P v = λv , then PP ∗(P v) = λ(P v).

∗ As a result, let λ1, ... ∈ R denotes all non-zero eigenvalues of P P , then

∗ ∗ X X tr(e−tP P ) − tr(e−tP P ) = (dim(ker(P )) + e−tλj ) − (dim(ker(P ∗)) + e−tλj ) = Ind(P )

Therefore, in the asymptotic expansion of tr(e−tP P ∗ ), tr(e−tP ∗P ) as t → 0, they diﬀer by a constant Ind(P ).

Ö è D+ Let’s recall that the heat expansion of ∆± = D±D∓ associated to D = D− on noncommutative torus itself, or on Heisenberg module over the torus, will always be of the form

± X j−2 −t∆ 2 ± Tr(fe ) ∼t→0 t τ(fRj ) j up to a constant multiple25.

Deﬁnition 4.2.3 (Index density ). For any j, we will denotes by upper-script γ taking

γ + − the graded sum, for example Rj := Rj − Rj .

γ + − By Index density of D, we are refering to when j = 2, R2 := R2 − R2 . From the

γ + above discussion we know that τ(R2 ) = Ind(D ).

± ± From now on, we will abbreviate Dk1,k2;E as D .

25c.f. Thm 4.2.5

62 Lemma 4.2.3.

+ σ(D ) = k1ξ1 + ik2ξ2 = p

− 0 0 σ(D ) = k1ξ1 − ik2ξ2 + i(δ1(k1) − iδ2(k2)) =p ¯ + β where

X j−1 p(ξ) = k1ξ1 + ik2ξ2 = i kjξj (4.2.2.1) j

X 1−j p¯(ξ) = k1ξ1 − ik2ξ2 = i kjξj (4.2.2.2) j

0 0 X 2−j 0 β = i(δ1(k1) − iδ2(k2)) = i δj(kj) (4.2.2.3) j

Lemma 4.2.4. Let

± ± ± ± σ(∆ ) = P2 + P1 + P0

± with Pj being of order j in ξ. Then

+ − 2 X 2 2 P2 = P2 = |p| = kj ξj (4.2.2.4) j

+ X j+1−l 0 l+1−j 0 P1 = [i kjδl(kl) + i klδl(kj)]ξj (4.2.2.5) j,l

− X 1+j−l 0 0 P1 = i [δl(kl)kj + klδl(kj)]ξj (4.2.2.6) j,l

γ X j+1−l 0 l−j 0 P1 = i [kj(1 − yj)δl(kl) + ((−1) − 1)klδl(kj)]ξj (4.2.2.7) j,l

γ X j−l+2 0 0 P0 = i kjδj(δl(kl)) − 4πµk1k2 (4.2.2.8) j,l

63 Proof.

+ X ¯0 σ(∆ ) = p ∗ (¯p + β) = pp¯ + pβ + ∂jpδj(¯p + β) j

X 0 j = pp¯ + pβ + i ∂jp(δj(¯p + β) + (−1) πµ∂j+1p¯) j

2 X 0 X 0 j = |p| + pβ + i ∂jpδjp¯+ i ∂jp(δj(β) + (−1) πµ∂j+1p¯) |{z} j j P2 | {z } | {z } + + P1 P0

− X ¯0 σ(∆ ) = (¯p + β) ∗ p =pp ¯ + βp + ∂j(¯p + β)δjp j

X 0 j =pp ¯ + βp + i ∂jp¯(δj + (−1) πµ∂j+1)p j

2 X 0 X j = |p| + βp + i ∂jpδ¯ jp + πµi (−1) ∂jp∂¯ j+1p |{z} j j P2 | {z } | {z } − − P1 P0

64 Therefore

+ X j−1+2−l 0 X 1+j−1+1−l 0 P1 = i kjδl(kl)ξj + i kjδj(kl)ξl j,l j,l

X j+1−l 0 l+1−j 0 = [i kjδl(kl) + i klδl(kj)]ξj j,l

− X 2−j+l−1 0 X 1+1−j+l−1 0 P1 = i δj(kj)klξl + i kjδj(kl)ξl j,l j,l

X 1−j+l 0 0 = i [δj(kj)kl + kjδj(kl)]ξl j,l

X 1−j+l 0 = i [δj(kjkl)]ξl j,l

γ X j+1−l 0 l+1−j 0 X 1+j−l 0 0 P1 = [i kjδl(kl) + i klδl(kj)]ξj − i [δl(kl)kj + klδl(kj)]ξj j,l j,l

X j+1−l 0 l−j 0 0 0 = i [kjδl(kl) + (−1) klδl(kj) − δl(kl)kj − klδl(kj)]ξj j,l

X j+1−l 0 l−j 0 = i [kj(1 − yj)δl(kl) + ((−1) − 1)klδl(kj)]ξj j,l

γ X 0 X j P0 = i ∂jpδj(β) + 2πµi (−1) ∂jp∂j+1p¯ j j

X 1+j−1+2−l 0 0 = i kjδj(δl(kl)) − 4πµk1k2 j,l

X j−l+2 0 0 = i kjδj(δl(kl)) − 4πµk1k2 j,l

According to the trace formula in [36], for any λ-parameter dependent26 symbol

m 2 fλ(ξ) ∈ S (R , Aθ0 ) of Pseudo-diﬀerential operators on the Heisenberg module E(g, θ),

26 In a more delicate way than just a λ-family of operator, λ is treated as of of order 2, while ξ1, ξ2 are 2 of order 1. A more precise statement should be f(ξ, λ) ∈ S(R × R+, Aθ0 ), and f is of order m if

0 00 1 0 00 I I I 2 m−|I |−|I | 0 00 |δ ∂ξ ∂λ f(ξ, λ)| ≤ CI,I0,I00 (1 + |ξ| + |λ| ) , ∀I,I ,I

65 the trace of the corresponding Pseudo-diﬀerential operator Op(fλ) is related to the Aθ0 -

27 trace of fλ , just like in the classical case , as λ → ∞ asymptotically.

m 2 Theorem 4.2.5 (trace formula [36]). A symbol f(ξ, λ) ∈ S (R × R+, Aθ0 ) of operator on E(g, θ) is trace-class if m < −n = −2. In this case,

Z |cθ + d| −∞ Tr(Op(f)(·, λ)) = 2 τ(f(ξ, λ))dξ + O(λ ), λ → ∞ (4.2.2.9) 4π R2

As a corollary, the heat trace T r(e−t∆) can be computed through Cauchy Integral

1 Z −1 Z Tr(e−t∆) = Tr( e−tλ(λ − ∆)−1dλ) = Tr( e−tλ(λ − ∆)−2dλ) 2πi 2πit in which integral in λ is along any contour around R+ in clockwise orientation.

−2 −2 P∞ ˜ On the one hand, as (λ − ∆) is trace-class, by Thm 4.2.5, if σ(λ − ∆) ∼ j=0 bj(ξ, λ)

−2 ˜ −4−2j 2 is the expansion of (λ−∆) with bj ∈ S (R ×R+, Aθ0 ) , then heat coeﬃcient relates ˜ to bj as follows −|cθ + d| ZZ a (a, ∆) = τ(a˜b )dξdλ j 2πi(4π2) j

On the other hand, it’s much more convenient to compute only the resolvent (∆−λ)−1.

−1 P∞ −2−2j 2 If σ(∆ − λ) ∼ j=0 bj, in which bj ∈ S (R × R+, Aθ0 ), then note that since

−2 d −1 R −tλ 1 R −tλ˜ (λ − ∆) = dλ (∆ − λ) , through Integral by part, e bjdλ = t e bjdλ. Therefore

27For a classical Pseudo-diﬀerential operator Op(f) on a vector bundle V → E → M, trace can be computed as Z Z Z T r(Op(f(x, ξ))) = Kf (x, x)dx = trEnd(V )f(x, ξ)dξdx 2 M M R R i(x−y)·ξ in which Kf (x, y): e f(x, ξ)dξ is kernel of Op(f(x, ξ)).

66 |cθ + d| ZZ a (a, ∆) = τ(ab )dξdλ j 2πi(4π2) j

Corollary 4.2.6. Associated to operator Dk1,k2;E(g,θ), the heat expansion densities

ZZ ± |cθ + d| ± R = b (ξ, λ)dξdλ ∈ A 0 . j 2πi(4π2) j θ

And more speciﬁcally, the index density

ZZ γ |cθ + d| γ R = b (ξ, λ)dξdλ ∈ A 0 2 2πi(4π2) 2 θ

Index of D+ can be computed with the Index density bγ = b+ − b− through: k1,k2;E(g,θ) 2 2 2

|cθ + d| ZZ Ind(D−) = τ(Rγ) = τ(bγ(ξ, λ))dξdλ (4.2.2.10) 2 8π3i 2

The index density is calculated through standard process, by applying the slightly modiﬁed composition formula (4.2.1.3) on Heisenberg module.

−1 P 2 2 −1 Lemma 4.2.7. Let b = b0 = P2 = ( j kj ξj − λ) , then

± ± X 0 b1 = −bP1 b − i ∂jbδjP2b (4.2.2.11) j

± ± ± ± X 0 ± X ± 0 b2 = −bP0 b − b1 P1 b − i ∂jbδjP1 b − i ∂jb1 δjP2b j j

X j 1 2 02 − i ∂jb(−1) πµ∂j+1P2b − D (b) · δ P2b (4.2.2.12) j 2 and

γ γ b1 = −bP1 b (4.2.2.13)

γ γ + + − − X 0 γ X 0 γ X γ 0 b2 = −bP0 b + (bP1 bP1 b − bP1 bP1 b) + i ∂jbδjP2bP1 b − i ∂jbδjP1 b − i ∂jb1 δjP2b j j j (4.2.2.14)

67 Apply Thm 3.1.1, it follows that

Theorem 4.2.8 (Main result 2).

γ |cθ + d| (1/2,1/2) γ R2 = [ − F1,1 (Z)(P0 ) 4πk1k2 1 l(j) (1) (2) + + 1 l(j) (1) (2) − − + 2 F1,1,1(Z , Z )((P1 )j ⊗ (P1 )j) − 2 F1,1,1(Z , Z )((P1 )j ⊗ (P1 )j) 2kj 2kj

1 l(j,l) (1) (2) 0 2 γ − i 2 Cj,lF2,1,1 (Z , Z )((δj(kl )) ⊗ (P1 )j) 2kl 1 l(j) (1) (2) γ 0 2 − i 2 F1,1,1(Z , Z )((P1 )j ⊗ (δj(kj ))) kj

1 (1,2) 2 l(j,l) (1) (2) γ 0 2 + i 2 Cj,l(yj ) F1,1,2 (Z , Z )((P1 )j ⊗ (δj(kl ))) 2kl l(j) 0 γ + iF2,1 (Z)(δj(P1 )j)]

where the Hyper geometric functions are as in deﬁnition 2.6.3, evaluated at Z = (z1, z2),

2 (l) (l) (l) (l) (1,...,l) 2 zj = 1 − yj or Z = (z1 , z2 ),zj = 1 − (yj ) .  1, if j 6= l  ± The constant Cj,l = C{j,j,l,l} = = 1 + 21j,l. And (P1 )j is coeﬃcient of ξj  3, if j = l ± ± P ± in P1 , namely it’s deﬁned by P1 = j(P1 )jξj.

γ ± ± Proof. It’s rather direct application of the rearrangement lemma on terms like, bP0 b, bP1 bP1 b.

P P 2 For example bP1bP1b = j1, j2b(P1)j1 b(P1)j2 bξj1 ξj2 is equivalent to j b(P1)jb(P1)jbξj in integral on ξ ∈ R2.

P 0 γ P 0 γ For terms like i j ∂jbδjP2bP1 b, i j ∂jbδjP1 b, the result follows after putting

2 2 0 0 X 0 2 2 ∂jb = −b∂jP2b = −2kj b ξj, δjb = −bδjP2b = − bδ j(kl )bξl l

68 P γ 0 28 For the term −i j ∂jb1 δjP2b, we’ve applied integral by part on ∂j , and

Z Z X γ 0 X γ 0 −i ∂jb1 δjP2bdξ = i b1 ∂j(δjP2b)dξ j j Z X γ 0 2 0 2 2 = i b1 [δj(kj )bξj + δj(kl )∂j(b)ξl ]dξ j Z X γ 0 2 0 2 2 2 2 = i b1 [2δj(kj )bξj − 2δj(kl )b kj ξjξl ]dξ j Z X γ 0 2 0 2 2 2 2 = −2i bP1 b[δj(kj )bξj − δj(kl )b kj ξjξl ]dξ j Z X γ 0 2 0 2 2 2 2 2 = −2i b(P1 )jb[δj(kj )b − δj(kl )b kj ξl ]ξj dξ j

γ γ + One of the key feature of R2 is that τ(R2 ) = Ind(D ). As proved in [36], index of

γ Dk,k;E = |c|. The can be veriﬁed on R2 .

γ Proposition 4.2.9. τ(R2 ) = |c|

There are a few useful properties about HGF that will be used.

n+1 2 Lemma 4.2.10. Let α = (α0, ..., αn) ∈ Z , l = (l1, l2) ∈ R used in deﬁnition 2.6.3 of Ö è Ö è zl 1 − Ql (y(j))2 l 1 n l 1 j=1 1 n+1 n HGF Fα(Z , ..., Z ), where Z = = . Let ι : Z → Z be l Ql (j) 2 z2 1 − j=1(y2 ) as deﬁned in [47], ιαα = (α0 + αn, α1, ..., αn−1).

l l F | n n = F α z1 =z2 =0 ιαα

28 0 It is crucial to keep in mind that IBP on δj is not allow, because we are computing the exact Index γ density R2 :

69 2.

l 1 n τ(kI Fα(Z , ..., Z )(ρ1 ⊗ ... ⊗ ρn))

l 1 n =τ(k F (Z , ..., Z )| n n (ρ ⊗ ... ⊗ ρ )) I α z1 =z2 =0 1 n

l 1 n−1 =τ(kI Fιαα(Z , ..., Z )(ρ1 ⊗ ... ⊗ ρn−1)ρn)

l 1 Fα0 = Γ(α0)

Ö è z1 Lemma 4.2.11. For matrix Z = , we will use entry-wise multiplication notation: z2

Ö è Z z1 = z1−1 Z − 1 z2 z2−1

1. 2 l Z 2l l Y 2lj l Fα0,α1 ( ) = y Fα1,α0 (Z) = yj Fα1,α0 (Z) Z − 1 j=1

Z τ(k F l (Z)(a)b) = τ(k aF l ( )(b)) I α0,α1 I α0,α1 Z − 1 Z =τ(k yI F l ( )(b)a) = τ(k yI y2lF l (Z)(b)a) I α0,α1 Z − 1 I α1,α0

70 Appendix A: non-conformal metric from conformal change on 2 End(T (Tθ))

A.1 Conformal twist

If one change the metric on the base manifold Aθ, or similarly on the endomorphism bundle Aθ0 , the Hilbert bimodule structure on E(g, θ) changes correspondingly, as explained in [36]. We will interpret such change of Hermitian metric on E(g, θ) as a change of the left side Aθ0 -module structure by an modular action.

Firstly let’s view a noncommutative torus A with conformally changed metric as a

A−A bimodule with new multiplication from one side. Recall the deﬁnition of the Hilbert

2 29 space Hφ = L (A, φ) ,used in [6],[36] etc., for any state φ on A, as the Hilbert space

30 ∗ corresponding to φ under GNS construction , with inner product ([a], [b])φ = φ(a b).

Now lets specify the state to be a conformal state, namely φ(a) = τ(ak−2), ∀a ∈ A. Then the map

W : L2(A, τ) → L2(A, φ),W (a) = [ak] is isomorphism of Hilbert space.

The isomorphism W pushes forward the standard bimodule structure31 on L2(A, τ) onto

29a state is a positive, normalized linear functional

30 ∗ completion of A under the inner product < a, b >φ= φ(a b) 31a · [x] · c = [axc], [x] ∈ L2(A, τ), a, b ∈ A

71 L2(A, φ) as

a · [x] · b = [axk−1bk], [x] ∈ L2(A, φ), a, b ∈ A

op 2 32 As an Aθ ⊗ A module, L (A, τ) admits an unitary representation , because

(a · [x] · b, [y]) = τ(b∗x∗a∗y) = τ(x∗a∗yb∗) = ([x], a∗ · [y] · b∗) for any [x], [y] ∈ L2(A, τ), for any a, b ∈ A.

op 2 Therefore, the pushed forwarded Aθ ⊗ A module on L (A, φ) is also unitary. Namely

(a · [x] · b, [y]) =φ(k∗b∗(k−1)∗x∗a∗y) = τ(x∗a∗y(k−1)∗b∗(k−1)∗)

= φ(x∗a∗yk−1b∗k) = ([x], a∗ · [y] · [b∗]) for any [x], [y] ∈ L2(A, φ), for any a, b ∈ A.

Moreover, the Hilbert bimodule structure on L2(A, τ) 33 will be pushed forward onto

L2(A, φ) as

−1 ∗ −1 ([a], [b])r = k a bk

−2 ∗ ([a], [b])l = ak b

([a], [b]) = φ(a∗b) = τ(a∗bk−2)

It’s straight forward to verify that τ(([b], [a])l) = ([a], [b]) = τ(([a], [b])r).

Tensoring (to the right) to any right A-module by L2(A, φ) will result in a change in the bimodule structure as follow:

32π : A → End(H) is unitary if π is a ∗-algebra endomorphism.

33 ∗ ∗ ∗ ([a], [b])r = a b, ([a], [b])l = ab , ([a], [b]) = τ(a b)

72 Lemma A.1.1. Let E be a left B right A Hilbert bimodule, with B-inner product B <

·, · >, and A-inner product < ·, · >A.

Using 1 ⊗ W , we can push forward from E = E ⊗ L2(A, τ) to E ⊗ L2(A, φ) a new Hilbert A A bimodule structure:

−2 ∗ B < ξ1 ⊗ [a1], ξ2 ⊗ [a2] >= B < ξ1([a1], [a2])l, ξ2 >= B < ξ1a1k a2, ξ2 >

−1 ∗ −1 < ξ1 ⊗ [a1], ξ2 ⊗ [a2] >A=([a1], < ξ1, ξ2 >A [a2])r = k a1 < ξ1, ξ2 >A a2k

Therefore, the map 1 ⊗ [1] : E → E ⊗ L2(A, φ) induces new Hilbert bimodule structure on E:

−2 B < ξ1 ⊗ [1], ξ2 ⊗ [1] >= B < ξ1k , ξ2 > (A.1.0.1)

−1 −1 < ξ1 ⊗ [1], ξ2 ⊗ [1] >A=k < ξ1, ξ2 >A k

Remark. In the above construction, we change the right multiplication on A by a modular

−1 action σ(b) = k bk. Tensoring by this new A-bimodule Hφ results in a change of the

Hilbert module structure as in (A.1.0.1). We can also change the left multiplication , and

l 34 denote resulting Hilbert space as Hφ , and tensoring from the left to any left A right B module will change the B Hilbert module structure just as Moscovici and Lesch mentioned in [36]:

−2 < [1] ⊗ ξ1, [1] ⊗ ξ2 >B=< k ξ1, ξ2 >B

34 l ∗ l l Hφ has inner product ([a], [b]) = φ(ab ) instead, and the isomorphism W : Hτ → Hφ is now multiply- ing from the left: W (a) = ka. In a similar way, the standard bimodule and Hilbert bimodule structure l l on Hτ will be pushed forward onto Hφ

73 A.2 Complex diﬀerential forms Ω(p,q)(A)

P 2 2 For the purpose of creating an operator P with principal symbol σ2(P ) = j kj ξj , it’s not enough to apply conformal twist to either side algebra of E(g, θ). One needs to consider matrix algebra over A35, which is Morita equivalent to A.

Lemma A.2.1. Let E = A⊗Cn be the A module of n×1 matrixes. The left multiplication by matrix algebra Mn(A), and right multiplication by A makes E an A − Mn(A) Morita equivalent bimodule.

There is obvious left Mn(A) Hilbert module structure

∗ (ξ1, ξ2)l = ξ1 · ξ2 , ∀ξ1, ξ2 ∈ E and right A Hilbert module structure

∗ (ξ1, ξ2)r = ξ1 · ξ2, ∀ξ1, ξ2 ∈ E

Moreover, we can apply a speciﬁc conformal twist on Mn(A) as follows: Ö è k1 0 Let k1, k2 be positive elements in A, and K = will be the Weyl factor of the 0 k2 conformal twist. Namely the map

W : E → E,W (ξ) = K · ξ

will preserve right A module structure on E, while pushing forward the standard left Mn(A) structure to the one twisted by K.

35 either Aθ or Aθ0

74 A natural place for such ﬂat bundle E will be the noncommutative diﬀerential forms

Ωp(A), thanks to the fact that the tangent bundle is parallelizable.

Deﬁnition A.2.1 (p-forms). Let A be a noncommutative m-torus, for any integer 0 ≤ Ö è m p p ≤ m, we deﬁne the space of p-forms Ωp(A) as A ⊗ C 36.

Remark. Since Ωp(A) is an A − MÖ è(A) Morita equivalent bimodule, it is natually m p associated with a inner product. By a sligth abuse of notation, we will also denote the

Hilbert space completion as Ωp(A).

Definition A.2.2 (differential d). The differential d :Ωp(A) → Ωp+1(A) can defined using Ö è δ1(a) equations in the classical case. For example on a two torus, when p = 0, da = δ2(a) Ö è a 0 0 under intuition that df = f1dx1 + f2dx2; when p = 1, d = −δ2(a) + δ1(b) under b 0 0 intuition d(fdx1 + gdx2) = (−f2 + g1)dx1 ∧ dx2. It’s easy to see that d2 = 0.

Remark. The adjoint of d, d∗ :Ωp → Ωp−1 can be computed using the Hilbert space â ì a1 ∗ . structure. For example, when p = 1, d . = δ1(a1) + ··· + δm(am).

Finally, there is a standard complex structure dz = dx1 + idx2, dz¯ = dx1 − idx2 which splits Ω1(A) into holomorphic and antiholomorphic parts. Just as in the commutative case, basis dzI dz¯J , |I| = p, |J| = q deﬁnes (p, q)-forms Ω(p,q)(A).

36 p P I An element (aI )|I|=p in Ω (A) is thought of as |I|=p aI dx

75 Deﬁnition A.2.3 (∂, ∂¯ operators). The partial ∂ :Ω(p,q)(A) → Ω(p+1,q)(A) and partial

¯ (p,q) (p,q+1) bar ∂ :Ω (A) → Ω (A) operators can also be deﬁned, using z = x1 + ix2 as the complex variable.

For example on a two torus, and when p = q = 0,

Ö è Ö è 1 1 (δ1 + iδ2)(a) (δ1 − iδ2)(a) ∂a¯ = 2 , ∂a = 2 (A.2.0.1) −i i 2 (δ1 + iδ2)(a) 2 (δ1 − iδ2)(a)

Ö è Ö è 1 1 Therefore Ω(0,1)(A) = ·A,Ω(1,0)(A) = ·A. −i i

A.3 Construction of asymmetric Dirac

Now we have enough preparation for the construction of asymmetric Dirac on E(g, θ).

The plan is to take the ∂¯ on a ”twisted” Dolbeault complex, under the intuition of changing metric on Aθ0 . To motivate the deﬁnition, let’s recall some facts Riemannian geometry on Surfaces.

A.3.1 motivation from Riemannian geometry

2 Ifg ˜ be a new metric on T , such that under an orthornormal basis {dx1, dx2} of the

−2 cotangent bundle,g ˜ij = 1ijkj . The new orthornormal basis k1dx1, k2dx2 determine the inner product structure on diﬀerential forms.

0 2 The new volume forms dV (˜g) = k1k2dx1dx2 determines inner product on Ω (T ) as

Z (f, g) = fgk¯ 1k2dx1dx2.

76 Inner product on forms are given by integrating the point wise inner product with respect to the new volume form dV (˜g). For example on Ω1(T2),

Z Z −1 −1 (dxi, dxj) = g˜(dxi, dxj)k1k2dx1dx2 = ki kj k1k2dx1dx2.

Moreover, as there is a bijection between conformal class of metric on a surface and the complex structure on the surface. If the change of metric is not conformal, there will be a change of complex structure, and therefore a change of splitting of Ω1 into holomorphic part Ω(1,0) and antiholomorphic part Ω(0,1). The new almost complex structure that is compatible withg ˜ should maps k1dx1 to k2dx2, and therefore a (0, 1)-form should be

37 f(x1, x2)(k1dx1 + ik2dx2)

A.3.2 twisted Ω0

0 2 0 0 Let ΩK (Aθ ) = L (Aθ , φk1k2 ) be the space of functions equipped with inner product

∗ (a, b) = τ(a k1k2b). As explained in §A.1, there is a Hilbert space isomorphism

−1 2 0 » W0 : L (Aθ0 , τ) → ΩK (Aθ0 ),W ([x]) = [ k1k2 x]

0 that pushes forward the standard Aθ0 − Aθ0 bimodule structure onto ΩK as

» −1 » a[x]b = [ k1k2 a k1k2xb]

0 and the standard Hilbert bimodule structure onto ΩK as

∗ ([a], [b])Aθ0 = a k1k2b

» ∗» Aθ0 ([a], [b]) = k1k2ab k1k2

37 k1dx1 + ik2dx2 itself may not be integrable, so may not equal to dz¯, but will still be proportional to dz¯ up to a complex valued function

77 A.3.3 twisted Ω1

1 1 By ΩK (Aθ0 ) we denote the set Ω (Aθ0 ) equipped with inner product

Ö è Ö è a a0 k k ( , ) = τ(a∗ 2 a0 + b∗ 1 b0). b b0 k1 k2

Ö » è k1/k2 0 Let K = be a positive diagonal matrix in M (A 0 ). There is 1 » 2 θ 0 k2/k1 a Hilbert space isomorphism

Ö è Ö è Ö » è 1 1 a a k1/k2a W :Ω (A 0 ) → Ω (A 0 ),W = K = 1 θ K θ 1 1 » b b k2/k1b

1 that will push forward the standard bimodule structure onto ΩK as

Ö è Ö è a −1 ac M · · c = K1MK1 , ∀M ∈ M2(Aθ0 ) b bc

A.3.4 twisted Ω(0,1) and twisted Dolbeault operator

(0,1) By ΩK (Aθ0 ) we denote the subspace

Ö » è Ö è (0,1) k1/k2 k1 1 W (Ω (A 0 )) = ·A 0 = ·A 0 ⊂ Ω (A 0 ) 1 θ » θ θ K θ −i k2/k1 −ik2

(1,0) (1,0) And similarly ΩK (Aθ0 ) is W1(Ω ).

1 Splitting of ΩK together with diﬀerential d deﬁne our twisted Dolbeault operator

Ö è k1 1 ∂¯ :Ω0 → Ω(0,1), ∂¯ (a) = (k−1δ + ik−1δ )(a) K K K K 2 1 1 2 2 −ik2

78 0 (0,1) The inner product on ΩK and ΩK is not the standard one, it will take some work

¯ ¯ 0 to compute the operator adjoint of ∂K . A standard trick is to pull back ∂K onto Ω and

(0,1) Ω by W0 and W1 respectively.

¯ 0 ∂K (0,1) ΩK ΩK

W0 W1 ∂¯0 Ω0 K Ω(0,1)

¯0 ∗ ¯ ¯ Proposition A.3.1. Let ∂K = W1 ◦ ∂K ◦ W0 be the pulled back operator of ∂K . Then

Ö è 1 −1 ¯0 1» −1 −1 » ∂K (a) = k1k2(k1 δ1 + ik2 δ2) k1k2 a −i 2

Ö è a √ (0,1) Further more, we can identify Ω as Aθ0 by 7→ 2a, which is an Aθ0 −Aθ0 −ia bimodule isomorphism and Hilbert space isomorphism. Then we have

−1 ¯0 ¯0 1 » −1 −1 » ∂ : Aθ0 → Aθ0 , ∂ = √ k1k2(k δ1 + ik δ2) k1k2 K K 2 1 2

Therefore the adjoint is

−1 ¯0∗ 1 » −1 −1 » ∂ = √ k1k2 (δ1k − iδ2k ) k1k2 : Aθ0 → Aθ0 K 2 1 2

Remark. we have therefore obtained a spectral triple

Ö è ¯∗ op 0 (0,1) ∂K (A 0 , Ω ⊕ Ω ,D = ) θ K K ¯ ∂K

and it’s equivalent spectral triple

Ö è ¯0∗ op 0 (0,1) ∂K (Aθ0 , Ω ⊕ Ω ,D = ) ¯0 ∂K

79 which can be explicitly written as

Ö √ −1 √ è √1 k k (δ k−1 − iδ k−1) k k op 2 2 1 2 1 1 2 2 1 2 (A 0 , A 0 ⊗ ,D = ) θ θ C √ √ −1 √1 k k (k−1δ + ik−1δ ) k k 2 1 2 1 1 2 2 1 2

A.4 Twisted Dolbeault operator on E(g, θ)

The above construction can be copied and pasted onto Heisenberg module, if we tensor by E(g, θ) over Aθ0 from the right, to the above spectral triples. Note that by construction,

0 1 the right Aθ0 multiplication on ΩK and ΩK is always standard, since we only twist the left multiplications.

If we denote by ∇1, ∇2 the Chern connection on E(g, θ) that is compatible with the standard Hilbert bimodule structure, namely if by abuse of notation δj is the standard derivation on either Aθ or Aθ0 , and then

δj((ξ1, ξ2)r) = (∇jξ1, ξ2)r + (ξ1, ∇jξ2)r, 1 δ0 ((ξ , ξ ) ) = (∇ ξ , ξ ) + (ξ , ∇ ξ ) , δ0 = δ j 1 2 l j 1 2 l 1 j 2 l j cθ + d j

After tensoring by the Heisenberg module, the twisted Dolbeault operator is (upto constant multiple cθ + d)

0 (0,1) ∇K :ΩK ⊗ E(g, θ) → ΩK ⊗ E(g, θ) (A.4.0.1) Ö è k1 1 » −1 ∇ (a ⊗ ξ) = (k−1∇ + ik−1∇ )( k k aξ) (A.4.0.2) K 2 1 1 2 2 1 2 −ik2

80 Ö è (∇ )∗ op 0 (0,1) K We have obtained our target spectral triple, (Aθ , (ΩK ⊕ΩK )⊗E(g, θ),D = ), ∇K ˜ ˜ 38 ∗ which will be much easier to study, if pulled by W0, W1 . We will abbreviate ΩE = Ω∗ ⊗ E(g, θ) from now on.

0 ˜ ∗ Corollary A.4.1. Let ∇K = W1 ∇K W0 be the pulled back operator of ∇K onto untwisted

0 (0,1) (0,1) forms (ΩE ⊕ΩE ),and identify Ω with Aθ0 as in prop A.3.1, then we obtain the spectral

op 0 (0,1) triple (Aθ ,H = ΩE ⊕ ΩE ,D), where

Ö è 1 √ −1 −1 −1 √ √ k1k2 (∇1k − i∇2k ) k1k2 D = 2 1 2 √ √ −1 √1 k k (k−1∇ + ik−1∇ ) k k 2 1 2 1 1 2 2 1 2 (A.4.0.3)

0 Remark. In the case when k1 = k2, on functions ΩK the Weyl factor is just normal

1 conformal change by k, as for space of one forms, the inner product on ΩK becomes Ö » è k1/k2 0 untwisted as the Weyl factor on one forms K = = I. Moreover, 1 » 0 k2/k1 (0,1) (0,1) the complex structure is the same as the original complex structure, as ΩK = Ω . Therefore our model coincides with the model of Moscovici and Lesch in [36].

It is not surprising that when we set k1 = k2, our spectral triple also recover the spectral triple of Moscovici and Lesch in [36]. More speciﬁcally the Dirac operator becomes

Ö è 1 −1 −1 −1 √ k (∇1k − i∇2k )k D = 2 √1 k(k−1∇ + ik−1∇ )k−1 2 1 2 Ö è 1 −1 √ k (∇1 − i∇2) = 2 √1 (∇ + i∇ )k−1 2 1 2

38 ˜ Wj = Wj ⊗ 1E(g,θ)

81 Appendix B: Gaussian integral

This section aims at proving proposition B.0.4, and its corollaries such as alternative

˜ ˜ l deﬁnition for Fα;I through equation B.0.0.3, and the relation between Fα;I and Fα as in prop. B.0.5.

More detail can be found in [47], from which we get the method of using Gaussian average to help evaluate T (α; I).

Let V be a m-dimensional real vector space. Given H ∈ GL(V ) , we will denote » h·, ·iH := hH·,H·i, and |v|H = hv, viH .

Note that given a Riemanian metric g on V, one can always choose H to be self-adjoint, positively deﬁnite, while full-ﬁlling equation g(·, ·) = h·, ·iH .

Deﬁnition B.0.1. Let f ∈ C(V ), the Gaussian average of f with respect to a back- ground metric h·, ·iH , is given by

Z Z −|ξ|2 −|η|2 −1 −1 GH (f) := e H f(ξ)dξ = e f(H η)det(H )dη V V

We will only be interested in the special case G(H,I) when H is diagonal, say,

H = diag(H11, ··· ,Hmm).

82 2 ix·ξ R −|ξ|2 ix·ξ m − |x| Lemma B.0.1. G(e ) = e e dξ = π 2 e 4

˜ i Proof. After substitution ξj = ξj − 2 xj, it reduces to equation

Z −|ξ|2 m e dξ = π 2

−1 2 ix·ξ R −|ξ|2 ix·ξ m −1 − |H x| Corollary B.0.2. GH (e ) = e H e dξ = π 2 det(H )e 4

Proof.

Z ix·ξ −|Hξ|2 iH−1x·Hξ −1 GH (e ) = e e det(H )d(Hξ) V Z = det(H−1) e−|η|2 eiH−1x·ηdη V

Now if one look into the Taylor series in x on left-handside of Lemma B.0.2, one gets the following:

Lemma B.0.3.

I ix·ξ |I| ∂x|x=0(GH (e )) = i G(H,I)

I ix·ξ I (ix·ξ)|I| |I| I Proof. ∂x|x=0(e ) = ∂x|x=0( |I|! ) = i ξ

ˆ Remark: It’s obvious from symmetry in ξ that when not all Ij are even, G(H,I) = 0.

From now on, assume |I| = 2k.

83 If one look into the Taylor series in x on right-handside of Lemma B.0.2, and apply

I ∂x |x=0 one gets:

m −1 2 |I| 2 −1 I −|H x| /4 i G(H,I) = π det(H )∂x |x=0 (e )

m 1 1 X 2 −1 I |I|/2 −2 |I|/2 = π det(H )∂x |x=0 [(− ) ( (H )i,jxixj) ] 4 (|I|/2)! i,j

1 m X 2 −1 I −2 k G(H,I) = 2k π det(H )∂x |x=0 ( (H )i,jxixj) (B.0.0.1) 2 (k)! i,j

By Leibniz rule, for any 2-partition σ ∈ P2(I), there correspond k! many ways to

σ1,σ2 σ2k−1,σ2k P −2 distribute each ∂x , ··· , ∂x into one of the i,j(H )i,jxixj factor.

Qk And also recall the deﬁnition Aσ = j=1 Aσ2j−1,σ2j , given symmetric bilinear form A

m on R , and σ ∈ P2(I) for some multiindex I with |I| = 2k. Here is the main conclusion for Gaussian average of ξI with respect to H.

Proposition B.0.4.

1 m −1 X −2 G(H,I) = π 2 det(H ) H (B.0.0.2) 2k σ σ∈P2(I)

P −2 −2 −2 −2 Proof. ∂xi ∂xj ( k,l Hk,l xkxl) = Hi,j + Hj,i = 2Hi,j .

I −2 k k P −2 ∂x((Hi,j xixj) ) = 2 k! σ∈P2(I) Hσ

˜ With proposition B.0.4, we can rewrite Fα;I (Z) in a much more natural way as a

Hypergeometric integral.

Z ˜ −1 X −2 Fα0,··· ,αn;I (Z) = det(H ) Hσ dµ(α)(s) (B.0.0.3) Qn−1 σ∈P2(I)

84 ˜ l the relation between Fα;I as in deﬁnition 2.6.2 and Fα as in deﬁnition 2.6.3 follows

Proposition B.0.5. 1. There is a constant CI that only depends on multiindex I,

such that

ˆ ˆ ( 1+I1 ,..., 1+Im ) ˜ 2 2 Fα;I = CI Fα

 0 if ∃j, Iˆ is odd  j C = (B.0.0.4) I Q (Iˆ )!  j j ˆ  Iˆ if Ij is even ∀j  |I|/2 Q j 2 j ( 2 )!

Proof. By deﬁnition, for any σ ∈ P (I), H−2 = H−2 ··· H−2 . 2 σ σ1σ2 σ2k−1σ2k

−2 −2 I Since H is diagonal, Hσ will vanish unless σ1 = σ2, ··· , σ2k−1 = σ2k, namely when ξ is even.

I I Q 2aj Now assume ξ is even, let’s say ξ = ξj .

  0 if ∃j, σ2j−1 6= σ2j −2  Hσ = (l) Q −2 aj Q Pn −aj  j(Hjj ) = j(1 − l=1 zj sl) otherwise

Also, n −1 Y X (l) −1/2 det(H ) = (1 − zj sl) l=1   0 if ∃j, σ2j−1 6= σ2j Therefore R det(H−1)H−2dµ (s) = . Qn−1 σ (α) 1 1  (a1+ 2 ,··· ,am+ 2 ) Fα otherwise

1 1 (a1+ 2 ,··· ,am+ 2 ) To understand the multiplicity of Fα when we take summation over P2(I), we need to count the number of 2-partition σ ∈ P2(I), such that σ2j−1 = σ2j, ∀j.

85 Qm Let’s view P2(I) as P2({ξ1, ··· , ξ1, ..., ξm, ··· , ξm}). The subset j=1 P2({ξj, ··· , ξj}) | {z } | {z } | {z } 2a1 copies 2am copies 2aj copies that consists of partition that splits into product of partitions of {ξj, ··· , ξj}. It’s not | {z } 2aj copies Qm diﬃcult to see that σ2j−1 = σ2j, ∀j if and only if σ ∈ j=1 P2({ξj, ··· , ξj}). | {z } 2aj copies

By abuse of notation, let’s abbreviate P2({ξj, ··· , ξj}) as P2(2aj), and | {z } 2aj copies

Ä2aj äÄ2aj −2äÄ2aj −4ä Ä2ä 2 2 2 ··· 2 (2aj)! |P2(2aj)| = = a aj! 2 j aj!

1 1 (a1+ 2 ,··· ,am+ 2 ) Therefore the multiplicity of Fα is

(2a )! Q(2a )! | Y P (2a )| = Y j = j 2 j a a +···+a Q 2 j aj! 2 1 m aj!

Z (2a )! ··· (2a )! (a + 1 ,··· ,a + 1 ) ˜ −1 X −2 1 m 1 2 m 2 Fα0,...,αn;I = det(H ) Hσ dµ(α)(s) = k Fα0,··· ,αn Qn−1 2 a1! ··· am! σ∈P2(I)

86 Appendix C: Generalized hypergeometric function

(l) In this section, we will justify our deﬁnition of Fα as ”generalized hypergeometric function”.

Before stepping forward to the generalized version, let’s ﬁrst recall the deﬁnition of some hypergeometric functions.

Deﬁnition C.0.1 (Gauss hypergeometric function 2F1). The Gauss hypergeometric function 2F1(a, b; c; z) is an analytic function in z, that has a power series expansion as the Gauss hypergeometric series

X (a; n)(b; n) n 2F1(a, b; c; z) = z , when |z| < 1 n≥0 (c; n)n!

Γ(a+n) (a; n) := Γ(a) is the Pochhammer Symbol.

P.Appell introduced four power series, F1,F2,F3,F4, as two variables generalization of 2F1 Other than Appell hypergeometric functions, there are many other two variables

39 hypergeometric series . We will only focus on Appell hypergeometric function F1.

39such as the other 30 functions that Horn enumarated in 1931

87 Deﬁnition C.0.2 (Appell hypergeometric functions).

X (a; m + n)(b1; m)(b2; n) m n F1(a, b1, b2; c; z, w) = z w , |z|, |w| < 1 (C.0.0.1) m,n (c; m + n)m!n!

X (a; m + n)(b1; m)(b2; n) m n F2(a, b1, b2; c1, c2; z, w) = z w , |z| + |w| < 1 m,n (c1; m)(c2; n)m!n!

X (a1; m)(a2; n)(b1; m)(b2; n) m n F3(a1, a2, b1, b2; c; z, w) = z w , |z|, |w| < 1 m,n (c; m + n)m!n!

X (a; m + n)(b; m + n) m n 1/2 1/2 F4(a, b; c1, c2; z, w) = z w , |z| + |w| < 1 m,n (c1; m)(c2; n)m!n!

The Appell hypergeometric functions was generalized to higher dimension by Lauri- cella, and later completed by Appell and Kamp´ede F´eriet.

Deﬁnition C.0.3 (Lauricella hypergeometric functions). Lauricella hypergeometric function of type A,B,C,D respectively, are deﬁned as follows

(a; P m ) Q (b ; m ) (n) X j j j j j m1 mn FA (a, b1, ..., bn; c1, ..., cn; z1, ..., zn) = Q Q z1 ...zn m1,...,mn j(cj; mj) j mj! Q (a ; m ) Q (b ; m ) (n) X j j j j j j m1 mn FB (a1, ...an, b1, ..., bn; c; z1, ..., zn) = P Q z1 ...zn m1,...,mn (c; j mj) j mj! (a; P m )(b; P m ) (n) X j j j j m1 mn FC (a, b; c1, ..., cn; z1, ..., zn) = Q Q z1 ...zn m1,...,mn j(cj; mj) j mj! (a; P m ) Q (b ; m ) (n) X j j j j j m1 mn FD (a, b1, ..., bn; c; z1, ..., zn) = P Q z1 ...zn (C.0.0.2) m1,...,mn (c; j mj) j mj!

Remark. When n = 2, the above Lauricella hypergeometric functions reduces to Appell

(2) (2) (2) (2) functions. FA = F2,FB = F3,FC = F4,FD = F1.

(n) Finally we will generalized the Lauricella type D hypergeometric function FD to an hypergeometric function Fm×n(αα,ββ; c; Z) on Z = (zjl) ∈ Mm×n(C), with indexes

α = (α1, ..., αm), β = (β1, ..., βn).

In the following deﬁnition, we will use index j for {1, 2, ..., m} and index l for {1, 2, ..., n}.

P Pm Q Qn For example j means j=1, and l means l=1.

88 Deﬁnition C.0.4 (Generalized hypergeometric functions of type (m + 1, m + n + 1),[1]).

Q P Q P X j(αj : l Mjl) l(βl; j Mjl) F (α , ..., α , β , ..., β ; c;(z )) = zM11 ...zMmn m×n 1 m 1 n jl (c; P M ) Q M ! 11 mn M11,...,Mmn≥0 j,l jl j,l jl

Remark. In some reference Fm×n is refered to as hypergeometric function on Grassman- nian, of type (m + 1, m + n + 1).

Remark. There is an obvious identiﬁcation between a type (m + 1, m + n + 1) general hypergeometric function Fm×n(α1, ..., αm, β1, ..., βn; c;(zjl)) and a type (n + 1, m + n + 1) general hypergeometric function Fn×m(β1, ..., βn, α1, ..., αm; c;(zlj)).

If m = 1 or n = 1, a type (m + 1, m + n + 1) general hypergeometric function reduces to Lauricella type D function.

Proposition C.0.1.

(m+1) Fm×1(α1, ..., αm, β; c;(z1, ...zm)) = FD (β, α1, ..., αm; c; z1, ...zm)

= F1×m(β, α1, ..., αm; c;(z1, ..., zm))

F2×1(α1, α2, β; c;(z1, z2)) = F1(β, α1, α2; c; z1, z2)

= F1×2(β, α1, α2; c;(z1, z2))

F1×1(α, β; c;(z)) = 2F1(α, β; c; z)

All hypergeometric functions mentioned above can be deﬁned through some kind of hypergeometric integral,cf [1]. In this paper we will only focus on the integral representation of Fm×n. For consistency of notation with the main theorem (Prop. 3.1.1), we will work with Fn×m(α1, ..., αn, l1, ..., lm; c;(zjl)) instead.

89 P Proposition C.0.2 (Integral representation of Fn×m). For α1, ..., αn > 1, c > l αl + 1, and l1, ..., lm ∈ C

Fn×m(α1, ..., αn, l1, ..., lm; c;(zlj)) n m Γ(c) Z n c−P αl Y αl−1 Y −lj = P Qn (1 − s1 − ... − sn) l=1 sl (1 − z1js1 − ... − znjsn) ds1...dsn Γ(c − αl) Γ(αl) l l=1 Q l=1 j=1 (C.0.0.3) sketch proof. The proof should be quite straight forward with the following two identities.

The ﬁrst one is multinomial series

−lj (1 − z1js1 − ... − znjsn) Ö è Ö è Ö è n −lj −lj − M1j −lj − M1j − ... − M(n−1)j X Y Mlj = ... (−zljsl) M1j ,...,Mlj ≥0 M1j M2j Mnj l=1 P P X Γ(lj + l Mlj) Y Mlj Mlj X (lj; l Mlj) Y Mlj Mlj = Q zlj sl = Q zlj sl Γ(lj) Mlj! Mlj! M1j ,...,Mlj ≥0 l l M1j ,...,Mlj ≥0 l l

Ö è −A (−A)(−A−1)...(−A−B+1) B Γ(A+B) note that we use = B! = (−1) Γ(A)B! in the last equation B

the second identity is the generalized Beta function

Z Γ(a )Γ(a )...Γ(a ) a0−1 a1−1 an−1 0 1 n (1 − s1 − ... − sn) s1 ...sn ds1...dsn = Qn−1 Γ(a0 + a1 + ... + an) from which we have

P P Z P α1+ M1j −1 αn+ Mnj −1 c− αl−1 j j (1 − s1 − ... − sn) l s1 ...sn ds1...dsn Qn−1 P P P Γ(c − l αl)Γ(α1 + j M1j)...Γ(αn + j Mnj) = P Γ(c + j,l Mlj)

90 Q Mlj It’s not diﬃcult to see that the coeﬃcient of lj zlj in (C.0.0.3) is exactly

Q P Q P l(αl; j Mlj) j(lj; l Mlj) P Q (c; j,l Mlj) j,l Mlj!

For more detail about thie general hypergeometric function of type (n + 1, m + n + 1), we refer our reader to the great book by K.Aomoto and M.Kita [1] ,section 3.4.

˜ (1) (n) Now we can apply the integral representation to our function Fα0,...,αn;I (z1 , ..., zm ). The following corollary shows that all of the involved function in the rearrangement lemma are in fact generalized hypergeometric function on Grassmannian.

ˆ Corollary C.0.3. For any multiindex I, recall that Ij is the multiplicity of j in I, and

F˜α;I C = ˆ ˆ is the combinatorial constant in ( (B.0.0.4)). I (1/2+I1,...,1/2+Im) Fα

˜ (1) (l) (n) Fα0,...,αn;I (z1 , ..., zj ..., zm ) 1 1 ˆ 1 ˆ =CI Fn×m(α1, ..., αn, + I1, ..., + Im; α0 + ... + αn;(..., zlj, ...))|z =z(l) Γ(α0 + ... + αn) 2 2 lj j

ˆ ˆ ˜ (1/2+I1,...,1/2+Im) Proof. By deﬁnition of CI as in B.0.0.4, Fα;I = CI Fα . By deﬁnition from formula (2.6.3),

l 1 Fα = Fn×m(α1, ...αn, l1, ..., lm; α0 + ... + αn) Γ(α0 + ... + αn)

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