Formal Groups, Witt vectors and Free Probability
Roland Friedrich and John McKay
December 6, 2019
Abstract
We establish a link between free probability theory and Witt vectors, via the theory of formal groups. We derive an exponential isomorphism which expresses Voiculescu’s free multiplicative convolution as a function of the free additive convolution . Subsequently we continue our previous discussion of the relation between complex cobordism and free probability. We show that the generic nth free cumulant corresponds to the cobordism class of the (n 1)-dimensional complex projective space. This permits us to relate several probability distributions− from random matrix theory to known genera, and to build a dictionary. Finally, we discuss aspects of free probability and the asymptotic representation theory of the symmetric group from a conformal field theoretic perspective and show that every distribution with mean zero is embeddable into the Universal Grassmannian of Sato-Segal-Wilson. MSC 2010: 46L54, 55N22, 57R77, Keywords: Free probability and free operator algebras, formal group laws, Witt vectors, complex cobordism (U- and SU-cobordism), non-crossing partitions, conformal field theory.
Contents
1 Introduction 2
2 Free probability and formal group laws4 2.1 Generalised free probability ...... 4 arXiv:1204.6522v3 [math.OA] 5 Dec 2019 2.2 The Lie group Aut( )...... 5 O 2.3 Formal group laws ...... 6 2.4 The R- and S-transform ...... 7 2.5 Free cumulants ...... 11
3 Witt vectors and Hopf algebras 12 3.1 The affine group schemes ...... 12 3.2 Witt vectors, λ-rings and necklace algebras ...... 15 3.3 The induced ring structure ...... 17
1 3.4 Faber polynomials and Adams operations ...... 17
4 Multiplicative genera and Fock spaces 19 4.1 The full Fock space and canonical random variables ...... 19 4.2 Multiplicative genera ...... 21 4.3 Genera and probability measures ...... 22
5 Physical applications 24 5.1 Fermion Fock space ...... 24 5.2 Asymptotic representation theory ...... 25 5.3 Measures, Grassmannian and τ-function ...... 26
1 Introduction
By considering the reduced free product of C∗-algebras, D.V. Voiculescu [53, 56] was led to create free probability, a non-commutative probability theory where the central notion of independence is replaced by “freeness”. Although deep connections are known to exist between C∗-algebras, and more generally operator algebras and algebraic topology, these links have not (yet) pervaded (non- commutative) probability theory, as one might expect given the close links operator theory has with probability theory. In our previous article [18], we demonstrated among other things, a direct connection of one-dimensional free probability with multiplicative genera in complex cobordism via Voiculescu’s S-transform [54]. The bridge Katsura, Shimizu and K. Ueno [33] and J. Morava [46] had previously built, permits us to establish a link between free probability, conformal field theory (CFT) and the KP hierarchy. Most notably, we observed a possible connection free probability has with the ring of Witt vectors [27] and, more generally, the theory of formal group laws. First, in Theorem 2.9 our identification of the free additive and multiplicative convolution group with Witt vectors permits us derive a natural logarithm LOG, which relates the two operations, analogously to the classical case. Second, the free additive and multiplicative convolution group define affine group schemes and formal group laws which are strongly isomorphic to those underlying Witt vectors. Then, for both the free additive and multiplicative convolution group we introduce their representing Hopf algebras. Third, the identification with Witt vectors permits us to define previously unknown operations on the space of distributions itself which are, a priori, not given by the underlying algebra, and which give rise to a commutative unital ring structure. In the case of actual probability measures positivity has to be taken into account, and then the ring structure reduces to a semiring, as shown in [23]. From a different perspective the appearance of formal group laws in probability theory can be moti- vated as follows.
2 A fundamental question in a general probability theory is, given two (non-commutative) random variables a and b, how to express the moments, if possible at all, of a + b and ab, solely from the moments of a and b. For free random variables a, b in a non-commutative probability space, Voiculescu solved the problem of expressing the nth moment mn(a + b) and mn(ab) of a + b and ab, respectively, as a function of the moments mi(a) and mj(b)[53, 54, 55] by using Lie theoretic methods. Once a notion of “independence” is present, this can always be achieved on general grounds, and it naturally leads to formal group laws, cf. e.g. [21]. Related questions appear in algebraic topology [1,7] where independence corresponds geometrically to a product structure of spaces. So, for two complex line bundles ξ1, ξ2 over a space X, with c1 the first Chern-Conner-Floyd class, the expression for c1(ξ1 ξ2) is given by a formal group law over the ∗ ⊗ ring ΩU , i.e. F (x, y) Ω [[x, y]], such that ∈ U ∞ X m n c1(ξ1 ξ2) = F (c1(ξ1), c1(ξ2)) = c1(ξ1) + c1(ξ2) + am,n c1(ξ1) c1(ξ2) ⊗ m,n=1
2 with x := c1(ξ1), y := c1(ξ2) MU (X). ∈ Since the first version of this article appeared as a preprint on the arXiv, things have considerably evolved, and the link between Witt vectors and free probability, turns out to be fruitful. First, let us mention the related but independent work of M. Mastnak and A. Nica [42], who pioneered the use of Hopf algebraic methods in order to investigate the free multiplicative convolution group of Nica and Speicher [48], thereby also establishing the relation with the Hopf algebra of symmetric functions. By now, efficient and powerful Hopf and homotopy algebraic, operadic, and Lie theoretic methods, pursued by different groups, have pervaded all the different flavours of probability theory, in order to abstractly understand, e.g. the nature of the so-called “moment-cumulant formulæ”. Let us mention the series of contributions from K. Ebrahimi-Fard and F. Patras, who started with [13], mainly from the perspective of shuffle and half-shuffle algebras, M. Sch¨urmannand his students, e.g. [41], and J.S. Park and his coauthors’ homotopy probability theory, e.g. [12]. Finally, we generalised several of the results of the present article in [19, 21, 22, 23]. Second, the connection given in Theorem 2.9 between the free additive and multiplicative convolution was subsequently further explored in the analytic category by several authors. G. C´ebron[8] studied a semigroup version of our map EXP adapted to the analytic setting for freely infinitely divisible distributions and introduced a random matrix model of it. M. Anshelevich and O. Arizmendi [4] considered exponential homomorphisms for general non-commutative probabilities, with a focus on analytic applications. Finally, in [23] the analytic counterpart is established for semirings of freely infinitely divisible measures. The content of the rest of the paper is organised as follows. In Section 2, which is composed of five short subsections, we recall first basic definitions from free probability in a generalised form and then introduce the Lie group of automorphisms of the formal line, fixing the origin. Then we introduce basic definitions and results from the theory of (commutative) formal groups. Next we interpret Voiculescu’s R- and S-transform as natural transformations and we
3 derive our first Theorem 2.9. The last part of this section is a reminder on free cumulants. In Section 3, which is subdivided into four parts, first we give the Hopf algebraic representation of the free additive and multiplicative convolution group. Then we relate these two abelian groups with Witt vectors and define a ring structure on non-commutative distributions. Finally, we discuss relations of (second order) free probability with Faber polynomials and the Grunsky matrix. In Section 4, we describe relations free probability has with multiplicative genera. First we present the Fock space versions of Voiculescu and Haagerup of the R- and S-transform in a succinct form. Then we relate multiplicative genera with the notions previously introduced and provide examples of genera as probability measures. In the final Section 5, we relate free probability to conformal field theory (CFT) and integrable systems in both the Boson and Fermion picture, which leads to a powerful unifying perspective.
2 Free probability and formal group laws
2.1 Generalised free probability
Let k be a field of characteristic zero, Algk the category of associative, not necessarily unital, k- algebras, and cAlg the associative and commutative unital k-algebras. For A cAlg , we denote k ∈ k by A× its group of units. First we recall some definitions from free probability [56] for general k-algebras, cf. [19].
Definition 2.1. Let Alg , with unit 1A, and let φ : k be a pointed k-linear functional with A ∈ k A → φ(1A) = 1k. The pair ( , φ), is called a non-commutative k-probability space, and elements a , random variablesA . ∈ A Definition 2.2. The law or distribution of a random variable a , is the unital k-linear functional 0 n n ∈ A µa k[[X]], given by a := 1A, µa(X ) := φ(a ), n N. The coefficients (mn(a))n∈ of the power ∈ ∈ N series µa are the moments of a.
If is a C∗-algebra and the linear functional φ a state, i.e. continuous and positive (φ(a∗a) 0), A ≥ then the pair ( , φ) is called a C∗-probability space. Similarly, we have a W ∗-probability space if A is a von Neumann algebra and φ is weakly continuous. A We denote the category of algebraic k-probability spaces by AlgkP.
Definition 2.3 (Freeness). Let ( , φ) AlgkP and I be an index set. A family of sub-algebras ( i)i∈I , A ∈ ∗ A with 1A i , i I, is called freely independent if for all n N , and ak ik ker(φ), ∈ A ⊂ A ∈ ∈ ∈ A ∩ 1 k n, ≤ ≤ a1 an ker(φ), ··· ∈ holds, given that ik = ik+1, 1 k < n. 6 ≤ Remark 1. The requirement for freeness is that consecutive indices have to be distinct. Hence, ik = ik+2 is possible. Also note that the notion of freeness is purely algebraic.
Voiculescu [53, 54, 55] showed that for a pair of free random variables a, b in a non-commutative { } probability space, with distributions µa and µb, respectively, µa+b and µab depend functionally only on µa and µb.
4 The additive free convolution of µa and µb, denoted as µa µb, is equal to the distribution of µa+b. Assign to xn and yn degree n. Then the operation is given by commutative, symmetric and homogeneous universal polynomials (Pn(x1, . . . , xn, y1, . . . , yn))n∈N∗ , of degree n with Z-coefficients, such that
n n n µa µb(X ) = Pn(µa(X), . . . , µa(X ), µb(X), . . . , µb(X )) (1) n n n−1 n−1 = µa(X ) + µb(X ) + P˜n(µa(X), . . . , µa(X ), µb(X), . . . , µb(X )).
The multiplicative free convolution of µa and µb, denoted by µa µb, is equal to the distribution of µab. Assign to xn and yn degree n. Then the operation is given by commutative, symmetric and homogeneous universal polynomials (Qn(x1, . . . , xn, y1, . . . , yn))n∈N∗ , of bidegree n with Z-coefficients, such that for n 2, ≥ n n n µa µb(X ) = Qn(µa(X), . . . , µa(X ), µb(X), . . . , µb(X )) (2) n n n n n−1 n−1 = µa(X )µb(X) + µa(X) µb(X ) + Q˜n(µa(X), . . . , µa(X ), µb(X), . . . , µb(X )) and with Q1(x1, y1) = x1y1. As a consequence of the properties listed above, for n = 2, we have
2 2 2 2 Q2(x1, x2, y1, y2) = x2y + x y2 x y . 1 1 − 1 1
2.2 The Lie group Aut( ) O In the approach to conformal field theory by Kawamoto, Namikawa, Tsuchiya and Yamada [34] the spaces of formal power series, which we are going to introduce, are fundamental, cf. the discussion in the monograph [17]. They have been also linked with complex cobordism by V. Bukhshtaber and A. Shukorov [7] and J. Morava [46] in connection with the Landweber-Novikov algebra. In [18] we added and discussed its subsequent appearance in renormalisation theory, cf. [16]. For A cAlg , we define functorially the following spaces: ∈ k ( ∞ ) X n A := A[[z]] = anz : the ring of formal power series with coefficients in A. O n=0 ( ∞ ) X n mA := anz : the (unique) maximal ideal of A. O n=1
The group of automorphisms Aut( A) of the A-algebra A A[[z]], or equivalently, the auto- O O ≡ morphisms of the formal line over A which fix the origin, can be described as the set of power series 2 × a1z + a2z + ..., with a1 A . Its subgroup Aut+( A) corresponds to those automorphisms with ∈ O a1 = 1. These spaces of automorphisms carry the structure of an infinite Lie group with corresponding Lie
5 algebras:
2 d Aut+( A) Der+( A) = z A[[z]] O O dz ∩ ∩ d Aut( A) Der0( A) = zA[[z]] O O dz ∩ d Der( A) = A[[z]] O dz
n+1 d By defining `n := z , for n Z, we obtain the commutation relations of the Witt algebra − dz ∈
[`m, `n] = (m n)`m+n . −
The non-negative part of the Witt algebra corresponds to Der0( A) whereas the Lie group Aut+( A) O O is represented by the Fa`adi Bruno Hopf algebra, cf. e.g. [16], which is a graded connected, commutative but not co-commutative Hopf algebra.
2.3 Formal group laws
We recall from the theory of formal group laws several facts in order to discuss further the above results. References are, e.g. [1,7, 24, 28, 39]. Let R be a commutative ring with unit. An n-dimensional formal group law over R is an n-tuple of power series F = (F1,...,Fn) in 2n commuting variables X = (x1, . . . , xn) and Y = (y1, . . . , yn), ∗ such that for all i = 1, . . . , n, n N , ∈
Fi(X,Y ) R[[x1, . . . , xn, y1, . . . , yn]], ∈ and the following axioms are satisfied:
F (X, 0) = X,F (0,Y ) = Y, (neutral element), (3) F (F (X,Y ),Z) = F (X,F (Y,Z)), (associativity). (4)
The formal group law is commutative if
F (X,Y ) = F (Y,X) (5) holds. Relation (3) implies that for every i, one has
X k1 kn l1 ln Fi(X,Y ) = xi + yi + ck,l(i)x x y y , (6) 1 ··· n · 1 ··· n |k|,|l|≥1 where k1 + + kn = k , l1 + + ln = l , ki, lj N and ck,l(i) A. ··· | | ··· | | ∈ ∈ The axioms (3) and (4) imply the existence of a unique inverse ι(X), i.e. an n-tuple of power series ι(X) = (ι1(X), . . . , ιn(X)) with ιi(X) R[[X]], for all i, such that ∈
Fi(X, ι(X)) = 0 = Fi(ι(X),X). (7)
6 P∞ j In the one-dimensional case, the inverse ι(x) of F (x, y), is the power series ι(x) = j=1 ajx with aj A, which satisfies relation (7). ∈ If we assign in an N-dimensional formal group law F degree j to xj and yj, then we call it homoge- neous if the the power series Fj in (6) is homogeneous of degree j. Hence, every homogeneous formal group law is of the form (9) and polynomial. A homomorphism f : F G of formal groups F and G over A cAlg , is a power series → ∈ k f(x) xA[[x]] with f(0) = 0 such that ∈ f(F (x, y)) = G(f(x), f(y)) .
0 × It is an isomorphism if f(x) Aut( A), i.e. f (0) A , and it is a strict (strong) isomorphism 0 ∈ O ∈ if f(x) Aut+( A), i.e. f (0) = 1. ∈ O The formal group law F is linearisable if there exists an f(x) Aut+( A), called the logarithm of ∈ O the formal group law, such that
f(F (x, y)) = f(x) + f(y).
It can be expressed in terms of the invariant differential ω(x) as
∂F (x, y) −1 df(x) := dx, (8) ∂y y=0
∨ which defines an element of the cotangent space Der ( A) to Aut+( A). + O O The two basic formal group laws are the additive group law:
Fa(x, y) := x + y, and the multiplicative group law:
Fm(x, y) := x + y + xy.
For any Q-algebra, Fa and Fm are strongly isomorphic, with f(z) = ln(1 + z). In general, for a commutative n-dimensional formal group law we have
Theorem 2.4 (cf. e.g. [24, 39]). Let A cAlg and F = (F1,...,Fn) be an n-dimensional com- ∈ Q mutative formal group law over A. Then F is linearisable, i.e. it is (strongly) isomorphic to the n n-dimensional additive group law Fa , with n (F )i(x1, . . . , xn, y1, . . . , yn) = xi + yi, for i 1, . . . , n . a ∈ { }
2.4 The R- and S-transform
For A cAlg , we define functorially the following sets of A-valued distributions: ∈ k Σ(A) := (a1, a2, a3,... ), ai A , × { ∈ }× Σ (A) := (a1, a2, a3,... ), a1 A , { ∈ } Σ0(A) := (0, a2, a3,... ) , { } Σ1(A) := (1, a2, a3,... ) , { }
7 × which satisfy the strict inclusions: Σ1(A) Σ (A) Σ(A). ⊂ ⊂ In [[56] Theorems 3.2.3.and 3.6.3.] Voiculescu establishes the existence of two infinite complex Lie groups, induced by the free additive and multiplicative convolution. This can be generalised and will be further discussed in Section 3.1.
× Theorem 2.5. The pairs (Σ, ) and (Σ , ) define pro-affine commutative group schemes over k, with the group laws and given by the universal polynomials P := (Pn)n∈N∗ in (1) and Q := (Qn)n∈N∗ in (2), respectively.
For every A cAlg the group Σ×(A) is the semi-direct product of the maximal torus A× and the ∈ k connected subgroup Σ1(A), viz. × × Σ (A) = A n Σ1(A). ∗ Proposition 2.6. For every A cAlgk and (fixed) N N , the following holds: ∈ ∈ N 1. the polynomials (P ) ∗ in (1) define a commutative N-dimensional formal group law F := n n∈N (P1,...,PN ) over Z, such that Pj(X,Y ) Z[x1, . . . , xj, y1, . . . , yj] and ∈
X k1 kj−1 l1 lj−1 Pj(X,Y ) = xj + yj + ck,l(j)x x y y , (9) 1 ··· j−1 1 ··· j−1 |k|,|l]≥1
which satisfy the homogeneity condition (k1 + l1) + 2(k2 + l2) + + (j 1)(kj−1 + lj−1) = j. ··· − 2. For every A cAlg , the formal group law F N over A is strongly isomorphic to the N- Q ∈ N dimensional additive group law Fa .
The actual construction of the logarithm for the free additive convolution was done by Voiculescu [[53], [54] and [56] Section 3.2], and it is called the R-transform. We (re)state its content in intrinsic terms N first. Let (A , +) denote the N-dimensional additive group scheme. Definition / Theorem 2.7 (Voiculescu’s R-transform). The R-transform is a natural isomorphism, i.e. for every A cAlg there exists an isomorphism of abelian groups ∈ k ∗ N RA : (Σ(A), ) (A , +), → such that for all f, g Σ(A) ∈ RA(f g) = RA(f) + RA(g), holds.
Its definition is as follows, cf. [[56] p. 23-24]. For an a Σ(A), with a = (a1, a2, a3,... ) and a0 := 1, ∈ its Cauchy transform is the formal power series
∞ X −(n+1) Ga(z) := an z . n=0
The compositional inverse of Ga(z), is the Laurent series
∞ 1 X G−1(z) = + α zn. a z n n=0
8 Then the -transform of a is the formal power series R ∞ −1 1 X n a(z) := G (z) = αnz . R a − z n=0 Often, it is more convenient to consider instead the series
Ra(z) := z a(z), R where we use a simplified notation, i.e. we do not indicate the dependence on the k-algebra A. ∗ Further, depending on the context, we shall perceive the R-transform either as an AN -valued natural transformation or as a formal power-series valued functor.
Remark 2. If one expresses initial coefficients bk in the Lagrange inversion of series of the form P∞ k { } P∞ k+1 f(q) = 1/q + k=0 akq into the reverted series q = k=0 bk/f , then each bk is a polynomial over N in the original coefficients. This follows from an interpretation of the products in the isobaric bk in terms of Dyck paths. Further, if the original series has coefficients that are values of a character on some group element, g, then so are the bk-polynomials, [15].
For A cAlg , the units of the ring A[[z]] are given by ∈ k A[[z]]× = A×(1 + zA[[z]]).
Let [[z]]× denote the functor A A[[z]]× and Λ(A) A[[z]]× the subset Λ(A) := 1 + zA[[z]]. • 7→ ⊂ × To every f Σ (A), with f = (a1, a2, a3,... ) corresponds a power series f(z) Aut( A) of the form ∈ ∈ O 2 3 × f(z) = a1z + a2z + a3z + . . . , a1 A , ∈ −1 with compositional inverse f (z) Aut( A). ∈ O Definition / Theorem 2.8 (Voiculescu’s S-transform). The S-transform is, for every A cAlg ∈ k and f Σ×(A), a natural isomorphism of abelian groups, ∈ × × SA : (Σ (A), ) (A[[z]] , ) → · which is given by 1 + z S (f) := f −1(z). A z For f, g Σ×(A) it satisfies ∈ SA(f g) = SA(f) SA(g). · In concrete terms the S-transform can be calculated as follows, cf. [[56] Theorem and Definition 3.6.3]: × × for a Σ (A), with a = (a1, a2, a3,... ), a1 A the associated moment series or -transform ∈ ∈ M is ∞ X n a(z) := an+1 z , (10) M n=0 −1 and Ma(z) := z a(z) Aut( A), with compositional inverse M (z). The S-transform of a, is the M ∈ O a formal power series 1 + z S (z) := M −1(z), (11) a z a
9 where again we do not indicate the dependence on the k-algebra A. Let us now compare the R- and S-transform with the integral transforms in classical probability theory. For of a positive real random variable X, i.e. X 0, the Mellin transform M, cf. e.g. [25], ≥ is defined as Z ∞ it it MX (t) := E[X ] = x dµX (x), 0 it with t R and the convention 0 := 0, for all t. ∈ Then, the following relations hold for X,Y classically independent and a, b free non-commutative random variables, respectively:
X,Y independent convolution transform (Fourier / Mellin) additive µX+Y = µX µY X+Y (t) = X (t) Y (t) ∗ F F ·F multiplicative µX·Y = µX m µY MX·Y (t) = MX (t) MY (t) ∗ · a, b free convolution transform (R / S) additive µa+b = µa µb Ra+b(z) = Ra(z) + Rb(z) multiplicative µa·b = µa µb Sa·b(z) = Sa(z) Sb(z) · where we used the notation and m for the respective convolution operations and Ra(z) := Rµ (z) ∗ ∗ a and Sa(z) := Sµa (z). Furthermore, by taking the natural logarithm of the Fourier, Mellin and S-transform, which are maps from the convolution group to the (co)-tangent space, the corresponding operations are linearised.
In fact, in the free case there exists also a functional relation between and , analogous as for classically independent random variables between and +. · Theorem 2.9. Let k be a field of characteristic zero and A cAlg . There exists a natural group ∈ k isomorphism, LOG : (Σ1, ) (Σ0, ), such that for f, g Σ1(A) → ∈ LOGA(f g) = LOGA(f) LOGA(g), holds.
Proof. One possibility to proof the statement is by composing the group isomorphisms in the diagram:
SA (Λ(A), ) o (Σ1(A), ) · d z dz ln =:LOGA −1 RA (zA[[z]], +) / (Σ0(A), )
Remark 3. If we are dealing with actual measures positivity matters, and then one obtains semi-groups instead of groups, cf. [23].
10 2.5 Free cumulants
Let us recall the groundbreaking combinatorial interpretation of the R- and S-transform given by R. Speicher [50] and then jointly further developed with A. Nica [48].
Let S be a finite set, e.g. 1, . . . , n .A partition π of S is a collection π = V1,...,Vr of non-empty, { } { } `r pairwise disjoint subsets Vi S, called the blocks of π, whose union is again S, i.e. S = Vi. ⊂ i=1 Let (S) denote the set of all partitions of S, and write (n) if S = 1, . . . , n . On 1, . . . , n there P P { } { } exists an equivalence relation π: namely, let 1 p, q n, and set ∼ ≤ ≤
p π q : i : p, q Vi, i.e. p, q are in the same block Vi of π. ∼ ⇔ ∃ ∈
A partition π of S = 1, . . . , n is called crossing if there exist p1 < q1 < p2 < q2 in S such that { } p1 π p2, q1 π q2 but p2 π q1.A non-crossing partition, as introduced by Kreweras [50], is a ∼ ∼ partition π which is not crossing. The set of all non-crossing partitions of 1, . . . , n is denoted by { } NC(n). These notions can be generalised to any finite, totally ordered set.
For Algk, let ( , φ) be a non-commutative probability space. The free cumulants κn, are k- A ∈ A n n multilinear functionals κ : k (n N) which are inductively defined by the moment-cumulant A → ∈ formula: X κ1(a) := φ(a), φ(a1 an) = κπ[a1, . . . , an] , ··· π∈NC(n) where for π = V1,...,Vr NC(n) { } ∈ r Y κπ[a1, . . . , an] := κVj [a1, . . . , an] , j=1 and for V = (v1, . . . , vs)
κV [a1, . . . , an] := κs[av1 , . . . , avs ] .
∗ For a and n N , one sets κn(a) := κn(a, . . . , a) and calls the resulting series (κn(a))n∈ ∗ the ∈ A ∈ N series of free cumulants of a. Remark 4. The above construction continues to hold for algebraic probability spaces with values in cAlgk, however, the original notion of freeness has to be appropriately redefined, as a generic random variable can not be centred, cf. [19, 21]. Free cumulants are intrinsically Lie theoretic quantities [53], which is also true for cumulants in other non-commutative probability theories, cf. [13, 21].
For m Σ(A), with m = (m1, m2, m3,... ) the free cumulants, κ(m) = (κ1(m), κ2(m), κ3(m),... ) are ∈ related according to [[48] p. 209, equation (12.2)] by the algebro-combinatorial formula X mn = κπ(m). π∈NC(n)
For a Σ(A), the coefficients of its R-transform are given by the free cumulants as ∈ ∞ X n Ra(z) = κn(a)z . (12) n=1
11 −1 If k1(a) = 0, the compositional inverse R (z) of (12) exists as an element of Aut( A). 6 a O Following [[48] Definition 18.15.], the S-transform of an a with κ1(a) = 0, can be expressed in terms 6 of the R-transform as R−1(z) S (z) = a , (13) a z which according to [[48] Formula(18.11)] is called the -transform. F
3 Witt vectors and Hopf algebras
3.1 The affine group schemes
We use two basic observations, namely commutative Hopf algebras are dual to affine group schemes (Cartier duality) and an n-dimensional formal group law over k defines a smooth formal k-group, and vice versa. References are [10, 28, 58] In detail, for k a commutative unital ring, an affine group scheme over k or a k-group (functor), is a covariant functor G : cAlg Grps which is representable, i.e. there exists a commutative but k → not necessarily co-commutative Hopf algebra H over k, such that for all A cAlg , ∈ k
G(A) HomcAlg (H,A),G = Spf(H) ' k holds, as convolution groups. Let us now assume that k is a field of characteristic zero. The coordinate algebra H = k[G] (or (G)), is co-connected if there exists a chain (Cn)n∈ of subspaces Cn H (filtration), such that O S N ⊂ C0 = k, C0 C1 C2 ... , Cn = H and ⊆ ⊆ ⊆ n∈N n X ∆(Cn) Ci Cn−i. ⊂ ⊗ i=0 An affine group scheme G over k is (pro)-unipotent if its representing Hopf algebra k[G] is co- connected, cf. [[58] Theorem p.64]. The formal scheme G = Spf(H) is connected if H is a local ring and a connected formal k-group G = Spf(H) is smooth if H is isomorphic to a power-series algebra k[[x1, . . . , xn]] in n commuting variables. Every smooth formal k-group is connected, because k[[x1, . . . , xn]] is a local ring. An n-dimensional formal group law F (X,Y ) over a k-algebra A defines a co-product by:
∆F : A[[x1, . . . , xn]] A[[x1, . . . , xn]] ˆ A[[x1, . . . , xn]] → ⊗ xi Fi(x1 1, . . . , xn 1, 1 x1,..., 1 xn) = 7→ ⊗ ⊗ ⊗ ⊗ xi 1 + 1 xi + (terms of degree 3). ⊗ ⊗ ≥ Together with the co-identity (augmentation)
: A[[x1, . . . , xn]] A, xi 0, → 7→ and the co-inverse (antipode) α, given by X ι(X), where ι(X) is the inverse (7), the A-algebra 7→ A[[x1, . . . , xn]] becomes a commutative but not necessarily co-commutative Hopf algebra. For a smooth
12 formal group G = Spf(H), H = k[[x1, . . . , xn]], the co-multiplication ∆ : H H ˆ H is determined by → ⊗ the values ∆(xi) = Fi(x1 1, . . . , xn 1, 1 x1,..., 1 xn), ⊗ ⊗ ⊗ ⊗ and hence by an n-tuple of power series F = (F1,...,Fn) in 2n variables.
For Xi := x1 1, Yi := 1 xi, X = (X1,...,Xn) and Y = (Y1,...,Yn), we have ⊗ ⊗
k[[x1, . . . , xn]] ˆ k[[x1, . . . , xn]] = k[[X,Y ]]. ⊗ ∼ Then F (X,Y ) k[[X,Y ]] and it has to satisfy the axioms of a formal group law (3) (neutral element), ∈ (4) (associativity) but not necessarily (5) (commutativity).
The free polynomial algebra Z[x1, x2, x3,... ], in countably many commuting variables xi, becomes a ∗ graded connected algebra if we define deg(xi) := i, i N , with the direct sum decomposition ∈ ∞ M Z[x1, x2, x3,... ] = Hn, n=0 into homogeneous subspaces Hj, where H0 = Z 1 and for n 1, Hn is the Z-linear span of all · Z ≥ monomials xi xi of degree i1 + + im = n, satisfying Hi Hj Hi+j. 1 ··· m ··· · ⊆ ∗ We obtain a Hopf algebra H , by defining the co-unit ε as ε (xi) := 0 for i N , the co-multiplication ∈ ∆ as ∆ (1) = 1 1, and on generators xn, as ⊗ n M ∆ (xn) := Pn(x1 1, . . . , xn 1, 1 x1,..., 1 xn) Hj Hn−j (14) ⊗ ⊗ ⊗ ⊗ ∈ ⊗ j=0 with the universal polynomials (Pn)n∈N∗ from (1), and where the last inclusion follows from expres- sion (9). The antipode α : H H is given by the inverse ι of the formal group law F , and is → recursively determined, starting with α (x1) = ι (X) = x1. − The following holds in general:
∗ Proposition 3.1. Let k be a field of characteristic 0, and F an N-dimensional, N N + , homogeneous formal group law over k. Then the associated Hopf algebra is (pro)-unipotent,∈ ∪ and { ∞} the co-product satisfies X ∆F (xi) = xi 1 + 1 xi + pij qij, ⊗ ⊗ ⊗ j for i = 1,...,N, and polynomials pij, qij k[x1, . . . , xi−1]. ∈
We note that for any field k of characteristic zero, one has Z k. ⊂
Proposition 3.2 (Additive case). The Hopf algebra (k[x1, x2, x3,... ], ε, ∆, α), induced by the free additive convolution , is graded connected, co-connected and co-commutative. It represents the pro- unipotent formal group scheme (Σ, ), and its C-points correspond to Voiculescu’s original infinite Lie group (Σ(C), ).
Let k[x−1, x]] denote the ring of formal Laurent series.
13 × Proposition 3.3 (Multiplicative case). The pro-affine k-group scheme (Σ , ) is represented by the −1 Hopf algebra k[x1 , x1, x2, x3,... ], with the co-commutative co-product given by n n ∆ (xn) := xn x + x xn + Q˜n(x1 1, . . . , xn−1 1, 1 x1,..., 1 xn−1), ⊗ 1 1 ⊗ ⊗ ⊗ ⊗ ⊗ ∗ ˜ ±1 for n N with n 2, the polynomial Qn from (2), and x1 being group-like. The co-unit satisfies ±∈1 ≥ ε(x1 ) = 1k and ε(xn) = 0, for n 2. The antipode α is calculated recursively, starting with ±1 ±∓1 ≥ α (x ) = x , and from the (general) identity (α (xn))( ) = xn(ι ( )), where ι ( ) is the group 1 1 • • • inverse of a group element , and xn the nth coordinate function. • N ∗ N Let A , N N + , denote the (pro) affine N-space over k, i.e. the functor A : cAlgk ∈ ∪ { ∞} → Sets, with A AN . 7→ The natural isomorphism N N+1 ϕA : A (A) Σ (A), (15) → 1 a (1, a1 + 1, a2 + 1, a3 + 1, . . . , aN + 1), 7→ for A cAlgk, induces, via the pullback, an universal commutative polynomial formal group law ∈ ∗ N ∗ F˜ := ϕ ( ) over k, such that (A ,F˜ ) defines a k-group functor for all N N + and ∈ ∪ { ∞} 1 n N, with ≤ ≤ F˜ (x1, . . . , xn, y1, . . . , yn) = Qn+1(1, x1 + 1 . . . , xn + 1, 1, y1 + 1, . . . , yn + 1) 1. ,n − If we set deg(xi) = deg(yi) := i, then F˜ is homogeneous of degree n, which follows from the ,n relation non-crossing partitions have with their Kreweras complement, cf. [21, 42]. Further, according to Theorem 2.4, the logarithm N N logF ˜ :(A ,F˜ ) (A , +) (16) → exists.
Proposition 3.4 (Linearisation of ). There exists a natural isomorphism of k-group functors ∗ N log : (Σ1, ) (A , +), → which for A cAlg , is given by the diagram: ∈ k −1 ϕA ∗ (Σ1(A), ) / ( N (A),F˜ ) A log F ˜ log :=log ◦ϕ−1 F ˜ A ' ∗ (AN (A), +)
−1 with ϕA the inverse morphism of (15) and logF ˜ in (16).
In summary, for ( , φ), a non-commutative probability space with values in A, A cAlg , define the A ∈ k affine subspace 1 := a φ(a) = 1 . Then we have the following sequence of pullback cones: A { ∈ A | } M LOG 1 / Σ1(A) / Σ0(A) A R S R d F z dz ln ∗ Σ1(A) / Λ(A) / AN (A) with as in (13). F
14 3.2 Witt vectors, λ-rings and necklace algebras
The algebraic structures we shall encounter are of the same nature as in (1) and (2), despite the fact that they arise in quite different contexts. References for this section are [11, 27, 32, 40, 44].
Let cRingk be the category of commutative unital rings over k. Let W be the functor W : cAlgk ∗ N → cRingk, with underlying set W(A) := A . Proposition 3.5 (e.g. [27] p. 12). (W(A), + , , (0, 0, 0,... ), (1, 0, 0, 0,... )) is a commutative uni- W ·W tal ring, called the ring of Witt vectors, with multiplicative unit (1, 0,..., 0,... ). Addition and multiplication are given in terms of universal polynomials SW = (SWn)n∈N∗ , PW = (PWn)n∈N∗ , i.e.
(x +W y)n := SWn(x, y), (x y)n := P (x, y), ·W Wn where S n,P n Z[x1, . . . , xn, y1, . . . , yn], and which satisfy: W W ∈
S (x1, y2) = x1 + y1 • W1 S (x1, . . . , xn, y1, . . . , yn) = S (y1, . . . , yn, x1, . . . , xn) • Wn Wn S (x, y) = xn + yn + S˜ (x1, . . . , xn−1, y1, . . . , yn−1) • Wn Wn P (x1, y1) = x1 y1 • W1 · P (x1, . . . , xn, y1, . . . , yn) = P (y1, . . . , yn, x1, . . . , xn) • Wn Wn
The polynomials SW and PW can be determined by the requirement that the ghost map ∗ N wA : W(A) A → ∗ is a natural isomorphism of unital commutative rings, where AN is given the ring structure of component-wise addition and multiplication.
The map w = (wn)n∈ ∗ , wn(x1, . . . , xn) Z[x1, . . . , xn], composed of the Witt polynomials, has N ∈ components
wn : W(A) A, → X n/d x wn(x1, . . . , xn) := dx , 7→ d d|n such that all of them are ring homomorphisms.
The set Λ(A) is a commutative ring, the Grothendieck λ-ring, with the addition +Λ defined by the usual multiplication of power series, i.e. +Λ := and with “zero” 0Λ = 1. · The “multiplication” Λ, is defined as follows: for f, g Λ(A) consider the formal factorisations Q∞ −·1 Q∞ −1 ∈ f(z) = (1 xiz) and g(z) = (1 yjz) , and define i=1 − j=1 − −1 −1 −1 (1 az) Λ (1 bz) := (1 abz) , − · − − which is then bilinearily extended , i.e. one uses
f Λ (g +Λ h) = f Λ g +Λ f Λ h. · · ·
15 Then for every A cAlg , the map ∈ k d ∗ N z ln : (Λ(A), +Λ, Λ) (A , +, , pointwise) dz · → · is an isomorphism of commutative unital rings.
The natural isomorphism E : W Λ, the Artin-Hasse exponential map, is, for A cAlgk, given → ∈ by ∞ Y 1 ∗ EA : W(A) Λ(A), (xn)n∈N n . → 7→ 1 xnz n=1 −
N. Metropolis and G.-C. Rota [44], introduced the functor Nr : cRingk cRingk, called the neck- ∗ → lace algebra, which has as underlying set Nr(A) := AN .
For a, b Nr(A), addition is given by component-wise addition, and the multiplication MR is defined ∈ ∗ as X (a MR b)n := gcd(i, j) aibj, ∗ lcm(i,j)=n where lcm(i, j) and gcd(i, j) denote the ‘least common multiple’ and the ‘greatest common divisor’ of ∗ i and j, respectively. The ghost map gA : Nr(A) AN is given by → X (an)n∈ ∗ d ad. N 7→ · d|n
A. Dress and B. Siebeneicher [11] gave a combinatorial interpretation of the integer Witt ring W(Z) by viewing the necklace algebra Nr(Z) as the completed Burnside-Grothendieck ring Ω(ˆ C) of almost finite cyclic sets, where C is the infinite cyclic group.
Theorem 3.6. For every A cAlgk, the abelian groups (W(A), +W), (Nr(A), +Nr), (Λ(A), +Λ), ∈ ∗ (Σ0(A), ) and (Σ1(A), ) are naturally isomorphic to (AN , +), as shown in the diagram below:
E S (W, +W ) / (Λ, ) o (Σ1, ) ·
d z dz ln
g ∗ R (Nr, +) / (AN , +)o (Σ0, )
−1 The natural isomorphism ϕ :(W, + ) (Σ1, ), ϕ := E S , is, for A cAlgk, given by W → ◦ ∈ ∞ ! z Y 1 ϕA = inv n , 1 + z 1 anz n=1 − where inv denotes the compositional inverse of power series. Therefore, the equality
ϕA(a +W b) = ϕA(a) ϕA(b). holds. Remark 5. In [40] C. Lenart summarises various previously known classical isomorphisms between the different rings.
16 3.3 The induced ring structure
∗ The natural ring structure on (AN , +, ) induces via the R- and S-transform additional operations. ∗ · For a, b AN , define ∈ −1 a b := R (R(a) H R(b)), · where H is the point-wise Hadamard multiplication, and for a, b Σ1(A), define · ∈ −1 a b := S (S(a) Λ S(b)). ∗ ·
Proposition 3.7. For every A cAlg , (Σ(A), , ) and (Σ1(A), , ) are commutative unital rings ∈ k ∗ with neutral element 1 and S−1(1 z), respectively. − The natural ring isomorphism LOG : (Σ1, , ) (Σ0, , ), is given by ∗ → d LOG := −1 (z ln) S. R ◦ dz ◦
For compactly supported freely infinitely-divisible probability measures on the real line, the struc- ture resulting from is a semiring, with a similar statement for , as shown in [20, 23]. ∗
3.4 Faber polynomials and Adams operations
In this section we show the relations Faber polynomials and the Grunsky matrix have with (second order) free probability, Voiculescu’s free entropy [57] and integrable systems [51, 52]. The Faber polynomials appear 1903 in complex analysis in approximation theory, see [43, 49]. In slightly modified versions they appear later in other contexts. The reason for putting the emphasis on the complex variables approach is that the original Faber polynomials play a profound role in diverse mathematical fields, and in particular in the theory of univalent functions, see [43, 49, 52] and references therein. The explicit connection with Witt vectors reveals new perspectives. 2 Let h(z) = 1 + b1z + b2z + Λ(A) with coefficients bi C and let g(z) := zh(1/z). Then · · · ∈ ∈ ∞ X −n g(z) = z + bn+1z , n=0 which in complex analytic terms represents the germ of a univalent function around the point at 0 1 infinity, with g( ) = and g ( ) = 1. The transformation g(z) g(1/z) gives a bijection with the ∞ ∞ ∞ 2 7→ 3 locally univalent functions around the origin of the form z + a2z + a3z + ... .
The nth Faber polynomial Fn(w) of g(z) is defined by the expansion at infinity
∞ g(z) w X Fn(w) ln − = z−n . z − n n=1
Fn(w) depends on the coefficients b1, . . . , bn and the Fn(w) satisfy the recursion relations
n−1 X Fn+1(w) = (w b1)Fn(w) bn−k+1Fk(w) (n + 1)bn+1 . − − − k=1
17 The Grunsky matrix is defined as
∞ ∞ g(z) g(w) X X −m −n ln − = βmnz w , (17) z w − − m=1 n=1 with the Grunsky coefficients βmn being symmetric, i.e. βmn = βnm. The relation between the Faber polynomials and the Grunsky coefficients is given by
∞ n X −m Fn(g(z)) = z + n βmnz . m=1
Considering Fn(0) =: Fn(b1, . . . , bn) leads to an alternative definition. The nth Faber polynomial Fn(b1, . . . , bn) for n N is defined by ∈ ∞ d X z ln g(z) = F (b , . . . , b )z−n , dz n 1 n n=0
2 3 with F0 = 1, F1 = b1, F2 = b 2b2, F3 = b + 3b1b2 3b3,... . − 1 − − 1 − Equivalently we have, cf. [6],
∞ ! 2 3 X Fn(b1, b2, . . . , bn) n 1 + b1z + b2z + b3z + = exp z . ··· − n n=1
Proposition 3.8. For a distribution µ Σ1, the linearisation of its S-transform Sµ(z), is given in ∈ terms of the Faber polynomials Fn n∈ as: { } N ∞ X 1 n z ln(Sµ(z)) = Fn(µ)z . (18) − n n=1
Alternatively, cf. [43], the Faber polynomials Fn(w) are given by:
Fn(w) = det(w1 An) , − where b1 1 0 0 ··· .. . 2b2 b1 1 . . A .. .. n := 3b3 b2 . . 0 . . .. . . . b1 1 nbn bn−1 b2 b1 ··· The nth Adams operator Ψn is given in terms of the operations λj by λ1 1 0 0 ··· . . 2λ2 λ1 1 .. . n 3 2 .. .. Ψ = det 3λ λ . . 0 . . .. 1 . . . λ 1 nλn λn−1 λ2 λ1 ··· 18 which corresponds to the determinant formula linking the power sums and the elementary sym- metric functions [27]. We have the following relation of Faber polynomials with Adams operations.
Proposition 3.9. The ghost components of Λ(k) are given by the Faber polynomials Fn(0), n N. ∈ The nth Adams operator is, up to a sign, equal to the nth Faber polynomial for w = 0,
n n 1 n Ψ = ( 1) Fn(λ , . . . , λ ) , − j with the coefficient bj replaced by the operation λ .
In Voiculescu’s approach to the microstate-free entropy [57], in the one-variable case, he introduced the difference quotient derivation ∂, as
g(x) g(y) g ∂g := − . 7→ x y − This motivates to define the generalised Grunsky matrix as
g(x) g(y) g log(∂g) = log − . 7→ x y − We obtain an identity which equates Voiculescu’s free entropy [57] with the free Boson [[51] Theorem 3.9. and Corollary 3.12.], via second-order freeness [[9] Equation (7)], as
∞ ∂2 X ∂2 log(τ) log(∂g) = x−m−1y−n−1 = G(x, y), (19) ∂x∂y ∂t ∂t m,n=1 m n where τ is the τ-function of a smooth Jordan contour C as in [51], cf. also [52], and G(x, y) the second order Cauchy transform in the case of vanishing second order free cumulants, as for Gaussian or Wishart random matrices [9]. The generalised energy-moment tensor or generalised Schwarzian derivative, cf. for confor- mal field theory [34] and for the connection with replicable functions [43], is then
g(x) g(y) g lim dzdw log(∂g) = lim dzdw log − . 7→ z→w z→w x y −
4 Multiplicative genera and Fock spaces
4.1 The full Fock space and canonical random variables
Let us recall from [26, 48, 54, 56] some facts which are relevant in connection with multiplicative genera but also with Faber polynomials and the Grunsky matrix.
For a C-Hilbert space (finite or infinite), the full Fock space over is H H ∞ ∞ M ⊗n M ⊗n F( ) := T ( ) = = CΩ , H H H ∼ ⊕ H n=0 n=1
19 with Ω the vacuum vector, a distinguished unit vector, and B(F( )) the algebra of bounded linear H operators. F( ) is a Hilbert space with the different summands ⊗n being pairwise orthogonal, and H H with x1 xn, y1 yn = x1, y1 xn, yn . h ⊗ · · · ⊗ ⊗ · · · ⊗ i h i · · · h i For T B (F( )), the vacuum expectation value τΩ is given by ∈ H τΩ(T ) := T (Ω), Ω , h i ∗ which induces the vacuum state τΩ : B (F( )) C, such that (B(F( )), τΩ) becomes a C - H → H probability space. For h , the associated left-creation operator l(h): F( ) F( ) is defined as ∈ H H → H l(h)Ω := h,
l(h)x1 xn := h x1 xn, ⊗ · · · ⊗ ⊗ ⊗ · · · ⊗ with the adjoint operator l∗(h) satisfying
l∗(h)Ω = 0 ( ∗ h, x1 Ω, for n = 1, l (h)x1 xn = h i ⊗ · · · ⊗ h, x1 x2 xn for n 2. h i ⊗ · · · ⊗ ≥
For e1, . . . , en, pairwise orthonormal vectors in , let li := l(e1), for i = 1, . . . , n. H Voiculescu [[56] Lemma and Definition 3.2.2.] gave a functional analytic representations of the R- transform in terms of “canonical random variables”, as did U. Haagerup [[26] Theorem 2.2. and Theorem 2.3], who additionally found such a representation for the S-transform, which we both recall now. One should also note the parallels with the derivation of the Faber polynomials in Section 3.4. Theorem 4.1 (Fock space models for the R- and S-transform). For every µ Σ, with moment series ∈ µ(z), moments (mn(µ))n∈N∗ and free cumulants (κn(µ))n∈N∗ , define for the canonical random Mvariables ∞ ∗ X n ∗ Tµ := l + κn+1(µ)l1 = l + µ(l1), 1 1 R n=0 ∞ ∗ X ∗ n ∗ T := l1 + mn+1(µ)(l ) = l1 + µ(l ). µ 1 M 1 n=0
Then the identities l∗+R (l )(z) = µ(z) = l +M (l∗)(z) (20) M 1 µ 1 M R 1 µ 1 hold.
× For every ν Σ , with moment series ν(z), define the operator ∈ M ∗ aν := (id +l1) ν(l ). M 1 Then the identities 1 Saν (z) = (21) ν(z) M holds.
20 4.2 Multiplicative genera
Previously, in [18] we described relations between free probability and complex cobordism, which we shall now extend further. As references we use [1,7, 29, 33]. Let MU ∗ denote the complex cobordism ring and A cAlg . ∈ Q A genus with values in A, is a ring homomorphism
∗ ϕ : MU Q A, with ϕ(1MU ∗⊗ ) = 1A. ⊗Z → ZQ
n 0 For CP , the n-dimensional complex projective space, with CP := 1, according to A. Mishchenko [7], the logarithm of the genus ϕ, is defined as the formal power series ∞ X 1 n−1 n logϕ(z) := ϕ(CP )z Aut+( A), (22) n ∈ O n=1 −1 with inverse logϕ (z). According to Novikov [7], every genus ϕ determines uniquely a Hirzebruch multiplicative sequence Kn , cf. [29], via the associated characteristic power series Qϕ(z) Λ(A), through the identities { } ∈ z K(1 + z) = Qϕ(z) := −1 . (23) logϕ (z) Further, over Q, all Hirzebruch genera yield formal group laws which are strongly isomorphic to the additive group law, cf. [7]. Now the following bijections, as shown in the commutative diagrams below, hold. For complex cobor- dism we have ∗ Q-series HomcRing (MU Q,A) / Λ(A) Q ⊗Z 4 log z/ log−1 Aut+( A) O and for free probability
A A x ,A S A Σ1( ) = HomQ,•,1( [ ] ) /4 Λ( ) R R−1/z Aut+( A) O holds. Haagerup’s representation of the S-transform (21) induces a natural operator-valued representation of complex genera. ∗ Proposition 4.2. There exists a lift σ : HomcRing(MU Z Q, C) (B(F(H)), τΩ), such that the diagram ⊗ → (B(F( ), τΩ) 3 H σ S
∗ Q HomcRing(MU Q, C) / Λ(k) ⊗Z
21 commutes. For a genus ϕ, define
1 ∗ aϕ := (1 + `) (` ) B(F( )), Qϕ ∈ H with Qϕ the characteristic power series associated to ϕ, from which we obtain the identity:
Saϕ (z) = Qϕ(z).
From the bijection ∗ HomcAlg (MU Q,A) Σ1(A), Q ⊗Z → we obtain a relation, already deduced in [18], between a genus ϕ and a distributionµ ˜, namely
Rµ˜(z) = logϕ(z). (24) Then the identities (12), (13) and (22) yield 1 κ (˜µ) = ϕ[ n−1]. n n CP
4.3 Genera and probability measures
In order to relate genera with probability measures, a gauge choice is needed. This is permitted, as cumulants are defined only up to a linear transformation, given by an infinite lower-triangular matrix.
Namely, the linearising map from “moments” (xn)n∈N∗ to “cumulants” (yn)n∈N∗ is given by universal polynomials (pn)n∈ ∗ such that yn = pn(x1, . . . , xn). Therefore every linear automorphism of the N∗ additive group (AN , +) of cumulants which preservs the causal structure, is given by a projective limit of invertible lower-triangular matrices. So, let us reformulate equation (24) in terms of the invariant differential (8) as follows. Consider the map ϕ µ, given by 7→ µ(z)dz = d log (z), R ϕ from which the identity n−1 κn(µ) = ϕ[CP ] follows. Now, T. Honda’s [30] definition of a zeta-function for group laws, motivates the following formal L-series for measures.
Definition 4.3. For µ Σ1, with free cumulants (κn(µ))n∈N∗ , the ζ-function of the distribution µ is given by: ∈ ∞ X −s ζµ(s) := κn+1(µ)n . (25) n=1
Let us discuss now three particular examples. First, we recall from [48, 56] the following facts.
Definition 4.4. Let λ R+ and α R. The free Poisson distribution with rate, λ, and jump size, α, is given by the limit∈ in distribution∈ λ λ N ν∞,λ,α := lim 1 δ0 + δα . N→∞ − N N
22 ∗ Definition 4.5. The Marchenko-Pastur measure, with parameters λ R+ and α R, is the ∈ ∈ probability measure µMP,λ,α on R, with support the interval
[α(1 √λ)2, α(1 + √λ)2], − and which is given by ( (1 λ)δ0 + λν for 0 λ 1 , µMP,λ,α := − ≤ ≤ ν for λ > 1, where ν is has the density
1 q dν(x) = 4λα2 (x α(1 + λ))2dx. 2παx − −
The following statement summarises the content of [[56] (c) pp. 34-35] and [[48] pp. 218, 380].
∗ Proposition 4.6. For λ R+ and α R, the equality ∈ ∈
ν∞,λ,α = µMP,λ,α
n ∗ holds in distribution, and the corresponding free cumulants are κn(ν∞,λ,α) = λα , n N . Its R- transform is ∈ z Rν (z) = λα . ∞,λ,α 1 αz − Definition 4.7. The semi-circular distribution with centre a and radius r, is the probability measure with support the compact interval [ r + a, a + r] and with density − ( 2 √r2 x2 for r + a < x < a + r, f(x) = πr2 − − 0 otherwise.
Proposition 4.8. Let γ1,2 be the semi-circular distribution centred at 1 and of radius 2, and Rγ1,2 (z) = 2 z + z its R-transform. Then the S-transform of γ1,2 is equal to
∞ 1 X n−1 n−1 Sγ (z) = √4z + 1 1 = ( 1 )Cnz , 1,2 2z − − n=1 where Cn is the nth Catalan number.
Let us discuss the following examples and summarise the corresponding results in a table.
1. The trivial genus, with characteristic power series Q(z) = 1, corresponds to the Dirac distri- bution at 1.
2. The Todd genus, with logarithm log(1 x), corresponds to the free Poisson distribution with − − parameters λ = α = 1, i.e. d z log (z) = R (z). dz T d ν∞,1,1
3. For the semi-circular distribution, γ1,2, the corresponding characteristic power series Q(z) is equal to z/( 1 + √2z + 1). −
23 Distribution: Name: multiplicative genus name δ1 Dirac 1 trivial genus z ν∞,1,1 free Poisson 1−e−z Todd genus γ √z 1,2 semi-circular −1+ 2z+1 ??? In the Planck’s radiation law, cf. [59], according to the Bose-Einstein statistics, the average energy of quantised modes is given by the formula
~ν E(ν, T ) = , (26) ~ν e kB ·T 1 − where ~ is Planck’s Wirkungsquantum, ν frequency, T temperature and kB the Boltzmann constant. By performing in (26) the formal substitution ν ν and by setting T = kB = ~ = 1, we obtain 7→ −
E( ν, 1) = ϕT d(ν). −
5 Physical applications
We continue the discussion of free probability from a conformal field theoretic perspective, as started in [18]. Novel aspects will include S. Kerov’s [35] asymptotic representation theory of the symmetric group. Relations of the Hopf algebra Symm with the Fock space have previously appeared in [38, 46], and as we show, these results are natural from the point of view of Witt vectors and the Sato-Segal- Wilson Grassmannian. Background material covering the topics involved can be found in [2, 33, 34, 45, 46].
5.1 Fermion Fock space
Generally speaking, the Boson-Fermion correspondence [2, 34, 45] is a rich source of algebraic identities, with analogs in non-commutative probability theory.
Definition 5.1 ([34, 45]). A Maya diagram consists of a subset M Z+1/2 such that M (Z+1/2)>0 ⊂ ∩ and M (Z + 1/2)<0 are finite sets, where (Z + 1/2)>0 := u Z + 1/2 u > 0 etc., and M is the { ∩ { ∈ | } { complement of M in Z + 1/2. The charge or Euler characteristic of a Maya diagram M, is the integer number
χ(M) := #(M (Z + 1/2)>0) #( M (Z + 1/2)<0). ∩ − { ∩
We denote the set of all Maya diagrams by and the subset of diagrams of charge p, p Z, by p. M ∈ M The Fermion Fock space is the C-vector space spanned by all Maya diagrams. A pictorial definition of a Maya diagram, as given in [34, 45], consists of an infinite sequence of black and white dots (Go stones) positioned at the half integers Z + 1/2, such that on the positive • ◦ axis there are only finitely many white and on the negative axis only finitely many black dots, as ◦ • shown in Figure1. The Dirac vacuum 0 corresponds to the sequence with all white stones lined up on the negative | i axis and the black stones on the positive axis.
24 5 3 1 1 3 5 7 9 11
Figure 1: Maya diagram of charge 0 corresponding to the state 9 , 7 , 5 ; 3 , 9 , 11 . | − 2 − 2 − 2 2 2 2 i
Any Maya diagram, up to a sign, can be obtained by successively applying fermionic creation ∗ operators ψm, ψn, m, n < 0, to the Dirac vacuum, i.e.
∗ ∗ n1 . . . ns; m1 . . . mr := ψm ψm ψ ψ 0 , | i 1 ··· r n1 ··· ns | i
for m1 < < mr < 0 and n1 < < ns < 0. The first r entries correspond to particle states and ··· ··· the last s entries to anti-particle states. The bijection between Maya diagrams and charged Young diagrams, cf. e.g. [[34] (1.8) Lemma] permits us to give fermionic interpretations of several (free) probabilistic results.
5.2 Asymptotic representation theory
S. Kerov [35] defined maps from the set Y0 of Young diagrams of charge zero to the space of probability measures on the real line, i.e.
Kerov : Y0 P-measures on R, (27) →
by associating to λ Y0, the transition ωˆλ and the co-transition ωˇλ measure, respectively. ∈ These are defined as follows, cf. [31, 35, 37]:
r X ωˆλ := qˆλ(i) δxi , i=1 r−1 X ωˇλ := qˇλ(i) δyj , i=1
where x1 < y1 < xr−1 < yr−1 < xr is an interlacing sequence of integers xi, yj, and δx , δy Dirac ··· i j measures supported at xi and yj, respectively. The weights
i−1 r Y xi yj Y xi yl−1 qˆλ(i) := − − , xi xj xi xl j=1 − l=i+1 − i−1 r (xr yi)(yi x1) Y yi yj+1 Y yi xl qˇλ(i) := P − − − − , (yj xj)(xl yl−1) yi yj yi yl j 25 Kerov [35] and Ph. Biane [5] proved that the limit shape Ω of rescaled Young diagrams with respect to the Plancherel measure is equal to, cf. [[31] pp. 267], ( 2 (x arcsin( x ) + √4 x2), x 2, Ω(x) = π 2 − | | ≤ x , x > 2. | | | | The associated transition probability measureω ˆΩ is the Wigner semi-circular law dωˆ 1 p Ω = 4 x2, dx 2π − which corresponds exactly to the distribution from Proposition 4.8, when translated by 1. 5.3 Measures, Grassmannian and τ-function Katsura, Shimizu and Ueno [33] established a link between CFT and the complex cobordism ring which we extend now further to free probability theory. H P a zn a Consider the complex Hilbert space of Laurent series := n∈Z n n C with the polarisation n m{ | ∈ } H = H+ H−, and with bases z n 0 for H+ and z m < 0 for H−, respectively. ⊕ { | ≥ } { | } The associated (Sato)-Segal-Wilson Grassmannian [47], Gr(H), is the set of subspaces W H, ⊂ such that the orthogonal projection pr : W H+ is a Fredholm operator and pr : W H− a + → − → Hilbert-Schmidt operator. The big cell Gr0(H) is W Gr(H) pr : W H+ isomorphism . { ∈ | + → } A. Kirillov and D. Yuriev [36] showed that every Grunsky matrix (17) defines a subspace W Gr(H), ∈ spanned by elements of the form ∞ −n X m−1 wn := z + Bnmz . m=1 Physically, as we observe, this corresponds to a Bogoliubov transformation of the Dirac vacuum [2], which is represented in terms of fermionic operators as: ∞ X ∗ B(W ) = exp Bnmψ−m+1/2ψ 0 . | i − −n+1/2 | i n,m=1 Every complex multiplicative sequence K = Kn(p1, . . . , pn) n∈ , as introduced by Hirzebruch [29], { } N is a sequence, for deg(pi) = j, of homogeneous polynomials with complex coefficients which gives rise to a monoid isomorphism ∞ ∞ X j X j K : Λ(C) 1 + pjz Kj(p1, . . . , pj)z C[p1, p2, p3,... ][[z]], 3 7→ ∈ j=1 j=0 with K0 := 1. Let bn C be the coefficient of pn in the polynomial Kn(p1, . . . , pn). Then according to [[33] Theo- ∈ rem 5.3, 1], the expression ∞ X n+1 τK (t) := exp ( 1) bntn ˆ0(C), − ∈ H n=1 26 with t = (t1, t2, t3,... ) and ˆ0(C), the completion of the Boson Fock space 0 of central charge H H zero, is a tau-function of the KP hierarchy, cf. [34, 45]. In particular, the coefficients bj can be expressed as polynomials in the free cumulants κi, i.e. bj = bj(κ2, . . . , κj). By [[33] Theorem 5.3] there exists an injective map from the ring of Witt vectors (Lambda ring) into the big cell of the Grassmannian: KSU : Λ Gr0, (28) → a Wa, 7→ where aK = (a1(K), a2(K), a3(K),... ) is the characteristic power series corresponding to a multi- plicative sequence K, as in equation (23). In summary, we obtain Theorem 5.2. Every distribution with mean zero (probability measure with mean zero on the real line) can be embedded into the big cell of the Sato-Segal-Wilson Grassmannian, as shown in the diagram: Kerov EXP S KSU Y0 / Σ0 / Σ1 / Λ / Gr0 with the injection Kerov (27), bijection EXP, which is the inverse to (2.9), bijection S (2.8) and injection KSU (28). The Virasoro algebra of central charge one, as part of a bigger algebra, acts on the determinant line bundle Det over Gr0(H), cf. [34, 36, 45], and hence by our embedding on the set of probability measures. First, this relates to the work of, H. Kvinge, A. Licata and S. Mitchell [37] and A. Lascoux and JY. Thi- bon [38] and others naturally, who studied the possible links Khovanov’s Heisenberg category has with Jones’ planar algebras and free probability or the action of vertex operators on (shifted) sym- metric functions, and in particular polynomial representations of GLn. From the infinite dimensional perspective we have considered, either Segal and Wilson’s [47] functional analytic restricted general 0 ∗ linear group GL1(H) and its C -central extension GLˆ 1 or the Kyoto School’s [[45] pp. 53] vertex operator representation of the infinite Lie algebra gl( ) and its corresponding Lie group G have to ∞ appear. Second, the Krichever embedding [2, 34, 36, 47], (C, L, p, z, ϕ) W Gr(H), of the datum 7→ ∈ consisting of a compact Riemann surface C (a curve) of genus g, a complex line bundle L over C, p C a marked point, z a local parameter around p, and ϕ : L D C a local trivialisation ∈ |U p → × over the closure of a neighbourhood Up of p, where D is the unit disc, defines a point W in the Sato- Segal-Wilson Grassmannian. If the Euler characteristic, χ(L) := dim H0(C; L) dim H1(C; L), C − C is equal to 1, then the corresponding subspace W belongs to the big cell Gr0(H) [[47] Proposition 6.1.]. Therefore we obtain for the moduli space ˆ g,1 of triples (C, p, z), with (C, p) g,1, for g,1 M ∈ M M the moduli space of Riemann surfaces C of genus g with p C, a marked point, and z a formal ∈ coordinate at p, the fibred product, ˆ g Gr Λ, cf. [[34] (2.28) Theorem.]. So, it would be interesting M × 0 to characterise those curves which arise from complex genera / probability measures and to describe the relation with the genus expansion of random matrices. Third, as outlined in the work with T. Amaba [3], besides the known relations between random matrices, integrable hierarchies and growth models, cf. e.g. [51, 52], there exists, as we show, a link with higher order free probability [9]. The dictionary we started to compile, gives yet another direction, in particular in connection with B. 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Waterhouse, Introduction to Affine Group Schemes, Springer (1979) [59] Wikipedia, https://de.wikipedia.org/wiki/Plancksches_Strahlungsgesetz Authors addresses Roland Friedrich, Fachrichtung Mathematik, Universit¨atdes Saarlandes, 66123 Saarbr¨ucken, Germany [email protected] John McKay, Dept. of Mathematics, Concordia University, Montreal, Canada [email protected] 31