BGWM as Second Constituent of Complex Matrix Model
aa r X i v : . [ h e p - t h ] D ec BGWM as Second Constituent of Complex MatrixModel
A.Alexandrov a Blackett Laboratory, Imperial College, London SW7 2AZ, U.K. and
ITEP, Moscow, Russia
A. Mironov b Lebedev Physics Institute and
ITEP, Moscow, Russia
A.Morozov c ITEP, Moscow, Russia
FIAN/TD-8/09Imperial-TP-AA-2009-1ITEP/TH-25/09
In [1] we explained that partition functions of various matrix models can be constructed from thatof the cubic Kontsevich model, which, therefore, becomes a basic elementary building block in”M-theory” of matrix models [2]. However, the less topical complex matrix model appeared tobe an exception: its decomposition involved not only the Kontsevich τ -function but also anotherconstituent, which we now identify as the Brezin-Gross-Witten (BGW) partition function. TheBGW τ -function can be represented either as a generating function of all unitary-matrix integralsor as a Kontsevich-Penner model with potential 1 /X (instead of X in the cubic Kontsevich model). a E-mail: [email protected] b E-mail: [email protected]; [email protected] c E-mail: [email protected] ontents I The four matrix models 8 Z H ( t ) Z C ( t ) Z K ( τ ) ˜ Z K ( τ ) = Z BGW ( τ ) Z C and Z BGW . . . . . . . . . . . . . . . . . . . . . . . . . 264.5 Character phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.5.1 Virasoro constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264.5.2 Determinant representation and integrability . . . . . . . . . . . . . . . . . . . 274.6 Kontsevich phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.6.1 Virasoro constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.6.2 Determinant representation and integrability . . . . . . . . . . . . . . . . . . . 274.7 Genus expansion and the first multiresolvents . . . . . . . . . . . . . . . . . . . . . . . 28
II Decomposition formulas 30 The idea of decomposition formulas [1] 302 The basic currents, shifts and projection operators 30 Z H −→ Z K ⊗ Z K
334 Decomposition relation Z C −→ Z K ⊗ Z BGW ntroduction Matrix models [3, 4] play a very special role in modern theoretical physics. They appear regularlyand prove useful in analysis of various simplified models of concrete physical phenomena, but theirreal significance is that they somehow capture and reflect the very basic properties of string theory– and can serve to represent the universal classes of quantum field theory models. From the verybeginning matrix models were introduced to describe some very general features (eigenvalue repul-sion) of statistical distributions [5]. It was, perhaps, the first recognition of the role of group theory– the underlying theory behind matrix models – in explaining the fundamental properties of quan-tum/statistical behavior. Much later this led to discovery that integrability is the basic property of allfunctional integrals, considered as functionals on the moduli space of theories [6], and matrix modelsplayed a central role [7] in the formulation of the fundamental relationpartition function = τ -function (1)between the two central concepts of modern theory, already with a variety of applications in differentfields, from gauge theories [8] to Hurwitz theory [9] and with still many more to come. An immediateimplication of (1) is that quantitative approach – a possibility to calculate something – in stringtheory (= a theory of families of quantum mechanical models) requires extension of the standardset of special functions to a broader set of τ -functions [10, 11] – a far-going generalization of bothhypergeometric and elliptic families. A highly non-trivial step here was introduction of ”infinite-genus” τ -functions, satisfying the string equations [12] and Virasoro/ W -constraints [13]-[21], and it was onceagain inspired by the study of matrix models. Unfortunately, even the simplest of these τ -functions,associated with Hermitian [3] and Kontsevich [22, 23, 24, 25] matrix models, are not yet systematicallystudied/tabulated and still can not be included into the special-functions textbooks – see [26] for thefirst attempts in this direction. It is very important to realize that the world of such τ -functions iscognizable, and is, perhaps, actually finitely-generated: many (all?) matrix-model τ -functions areactually expressed by group-theoretical methods through a few basic ones. This decomposition resultsfrom description of genus expansion of matrix-model partition functions in terms of auxiliary ”spectral”Riemann surfaces [27, 26, 28], and explicitly relates them to representation theory of Krichever-Novikovtype deformation [29] of Kac-Moody algebras. One of spectacular byproducts of this development isthe possibility to build a ”string-field-theory-like” diagram technique [28] for the model of entirestring theory, provided by ”M-theory of matrix models” [2]. As shown in [1], the main basic block(constituent) of the matrix model partition functions in this approach is the ordinary Kontsevich τ -function Z K of [22].However, already in [1] a first counter-example was found to this (over?)-optimistic conjecture:partition function of the complex matrix model [30, 31] is made not only from Z K , but also from someother ingredient, denoted ˜ Z K in s.8 of [1]. The purpose of the present paper is to identify this ˜ Z K witha very important and well-known partition function: that of Brezin-Gross-Witten model (BGWM)[32, 33, 20]: ˜ Z K = Z BGW (2)By definition, BGWM describes correlators of unitary matrices with a non-linear Haar measure. Uni-tary correlators play a crucially important role in description of gluons in lattice gauge-theory models[34], however, unitary matrix models are more complicated than Hermitian ones, they are in inter-mediate position between eigenvalue and non-eigenvalue models and remain under-investigated, see[35]-[38] for some crucial references. A modern matrix-model-theory approach to BGWM and itsembedding into the set of generalized Kontsevich models (GKM) [23] was outlined in [20], but hasnot been developed any further since then. Hopefully reappearance of this model in the context ofmatrix-model M-theory will help to attract new attention to this unjustly-abandoned subject.We begin in s.I from reminding the definition and the main properties of the four partition functionswhich participate in our story: Z H ( t ), Z C ( t ), Z K ( τ ) and Z BGW ( τ ). They were originally introduced4s matrix integrals over Hermitian ( Z H and Z K ), complex ( Z C ) and unitary ( Z BGW ) matrices, withthe time-variables identified either with the coupling constants ( t k in Z H and Z C ) or with the Miwatransform of the background field ( τ k in Z K and Z BGW ).As functions of their parameters – the time-variables t or τ – these integrals satisfy Ward identities(or Picard-Fucks equations) [16], which have the form of the Virasoro constraints. Namely, ∂Z H ∂t k = ˆ L k − Z H , ∂Z C ∂t k = ˆ L k − Z C (3)with k ≥
1, with ∂Z H ∂t = N/gZ H , ∂Z C ∂t = N/gZ C , where N is the size of the matrix in the originalintegral representation and with ”discrete-Virasoro” operators [15]ˆ L m = ∞ X k =1 kt k ∂∂t m + k + g m X a,b =0 a + b = m ∂ ∂t a ∂t b , m ≥ − ∂Z K ∂τ k − = ˆ L k − Z K , ∂Z BGW ∂τ k − = ˆ L k − Z BGW (5)with k ≥ L m = ∞ X k =1 (cid:18) k + 12 (cid:19) τ k ∂∂τ m + k + g m − X a,b =0 a + b = m − ∂ ∂τ a ∂τ b + τ g δ m, − + 116 δ m, , m ≥ − D -module approach and define the four partition functions as solutions to thefour systems of linear differential equations (3) and (5), and original integral formulas are just integralrepresentations for the solutions.Not surprisingly, such representations are not unique, and one can instead represent the samesolutions in a very different integral form: of Kontsevich-Penner integrals over n × n Hermitian matriceswith a peculiar Penner term N tr log φ in the action. This puts all the four models in the unifyingcontext of GKM theory [23]. Direct relation between the two integral representations is provided bya version of Faddeev-Popov trick from [21].All four matrix integrals can be expressed in the form of determinants of the other matrices, whichhave ordinary single integrals as their elements. These determinant representations are very important,because they are typical for the tau-functions of integrable hierarchies [40, 24] – the generalizedcharacters of Lie algebras [39, 11]. In other words, partition functions of the matrix models are alwaysthe tau-functions [7]. Moreover, this is a general property of all partition functions – the generatingfunctions of all correlation functions in any quantum theory, – this is a consequence of the freedom tochange integration variables (fields) in the functional integral [6]. For tau-functions the whole sets ofVirasoro constraints are actually fixed by their lowest components ˆ L − or ˆ L , which therefore has itsown name: the string equation [12].Integrability means that partition function satisfies a bilinear Hirota equation [10] of the form I Z N (cid:18) t k + 1 kz k (cid:19) Z N ′ (cid:18) t ′ k − kz k (cid:19) z N ′ − N e P k ( t ′ k − t k ) z k dz (7)This equation has its origin in decomposition rule R × R ′ = P I R I for representations of Lie algebrasand this is an equation for the characters of the algebra [39, 11]. For loop algebras the characters canbe rather non-trivial, they can be actually labeled by some auxiliary (spectral) Riemann surfaces or,better, by the points of an infinite-dimensional Grassmannian (the universal moduli space of [41]) –what means that the spectral surface can actually have an infinite genus (and this is typically the casefor the matrix-model partition functions). 5or irreducible representations of finite-dimensional simple Lie algebras, the characters are givenby two determinant Weyl formulas: in the case of SL ( N ) and representation labeled by partition ~m : m ≥ m ≥ . . . ≥ m N ≥ χ ~m ( t ) = det P i + m j − ( t ) , (8)where P i ( t ) are the Shur polynomials, exp (cid:16)P k t k x k (cid:17) = P m x m P m ( t ), and, after the Miwa transform t k = k P i λ − ki , χ ~m ( t ) = det ij λ j + m j i ∆( λ ) , (9)In fact, the two Weyl formulas are mirrored in two possible representations of matrix modelspartition function: as we shall demonstrate, the Hermitian and complex matrix models, besides thestandard determinant representation of type (8), have a Kontsevich-Penner representation of type (9).Virasoro constraints can be used as recursion relations to provide the logarithm of the partitionfunctions g log Z in the form of the formal series in non-negative powers of t -variables and g . Suchformal series are unambiguously defined by the systems (3) and (5), since k ≥ ρ ( p | q ) ( z ), defined as ρ ( ·| q ) ( z , . . . , z q ) = ˆ ∇ ( z ) . . . ˆ ∇ ( z q ) log Z | t =0 = ∞ X p =0 g p − ρ ( p | q ) ( z , . . . , z q ) (10)possess an important property: they are poly-differentials on auxiliary spectral Riemann surfaces(complex curves) [26, 28], which for the four matrix models in question are all double-coverings ofthe Riemann sphere with only two ramification points (and thus Riemann spheres themselves). Thespectral curve representation arises only for the special choice of generating functions: they should be resolvents , i.e. no k -dependent coefficients are allowed in (10). There are other interesting choices ofcoefficients, when alternative generating functions possess other interesting properties, see for example[42].In s.II we proceed to decomposition formulas. Virasoro constraints can be also considered asquadratic differentials on the spectral surfaces, expanded near particular points. It turns out that”discrete” operators (4) arise in expansion near non-singular points, while ”continuous” operators (6)– in those near ramification points of degree 2. This means that a globally-defined Virasoro quadraticdifferential can be decomposed in both bases and this idea finally leads to decomposition formulas [1].The basic one is Z H = ˆ U KK ( Z K ⊗ Z K ) , (11)it expresses Z H ( t ) for Hermitian matrix model through Z C ( τ ) for Kontsevich model. It is explained infull detail in s.3. Ingredients of the construction are: explicit parametrization of the spectral curve andof the singular differentials in the vicinities of particular points on it, explicit formula for the globalˆ U (1) current and the Virasoro differential, its projection onto ”canonical” quadratic differentials in thevicinities of the particular points and Bogoliubov transform of the time-variables with the help of theconjugation ˆ U -operator, and, finally, projection from generic Laurent series for the Virasoro operatorto a Taylor series, provided by peculiar projection operator P , which picks up a triangular subalgebrafrom entire Virasoro (Krichever-Novikov) algebra. Generators of this triangular subalgebra can beimposed as constraints on partition function and form a consistent and resolvable set of constraints.From the point of view of D -module approach the difference between Z H and Z C in (3) looksminor: both are annihilated by the same discrete-Virasoro operators ˆ L n , only n ≥ − Z H , but n ≥ Z C . The second difference is that the shift of time-variables, which generates the l.h.s.in (3), is also different: t is shifted in the case of Z H , but t is shifted in the case of Z C – this isimportant to explain why both sets of equations (3) have unambiguous formal-series solution, despitethe set of constraints contains one less equation in the case of Z C . However, this second difference6s not essential for comparison of (3) and (5). Indeed, the relation between Z K and ˜ Z K = Z BGW in (5) is exactly the same: both partition functions are annihilated by the same continuous-Virasorooperators L n , but n ≥ − Z K , while n ≥ Z BGW . This time shifted are τ in Z K and τ in Z BGW , what guarantees that the formal-series solutions are unambiguously defined in both cases. Allthis implies that in the spectral-surface formalism the difference between Z H and Z C is concentratedin the choice of projection operator P at the last stage. Since projection operator can be realized as acontour integral, its modification can actually be shifted from Z H (where it transformed Z H into Z C )to one of the two Z K at the r.h.s. of (11) and convert it into ˜ Z K = Z BGW . In other words, instead of(11) we obtain Z C = ˆ U K ˜ K ( Z K ⊗ ˜ Z K ) i . e . Z C = ˆ U K ˜ K ( Z K ⊗ Z BGW ) (12)This is the main result of the present paper and it is discussed in full detail in s.4. This formula wassupported in [1] by explicit comparison of the first terms of expansions of partition functions at bothsides of (12).Some concluding remarks are given in s.4. 7 art I
The four matrix models Z H ( t ) Partition function of Hermitian matrix model [3] is defined by the integral over hermitian N × N matrix Z H ( t ) = Z N × N exp − g Tr H + 1 g ∞ X k =0 t k Tr H k ! dH (I.1)where the measure dH = Q Ni,j =1 dH ij . This is nothing but the generating function of all GL ( N )-invariant Gaussian correlators of Hermitian matrix H . The Hermitian matrix Φ can be diagonalized by a unitary transformation, H = U DU † , where D =diag( H , . . . , H N ) is the diagonal matrix made from the eigenvalues of H . The norm of H decomposesas Tr ( δH ) = Tr ( δD ) + Tr (cid:16)h U † δU, D i(cid:17) = N X i =1 ( δH i ) + N X i
1. This invariance implies that [16] ∂Z H ( t ) ∂t n +2 = ˆ L n Z H ( t ) , n ≥ − L n defined in (4) and ∂Z H ( t ) ∂t = Ng Z H ( t ) (I.6)The l.h.s. in (I.5) is produced by the shift of the t variable t → t − / Z H ( t ) term-by-term as a formal series in non-negative powers of t -variables.8 .4 Determinant representations and integrability As we shall explain now, the properly normalized matrix integral Z N ≡ V N N ! Z H ( t ) is a τ -function ofthe Toda-chain integrable hierarchy [15, 7]. In this paragraph we rescale the time variables to cancelthe coefficient g in front of them in order to have the standard definition of integrable hierarchies.One of the technical ways to deal with integrals of form (I.4) was proposed in [3]. The authorsintroduced a system of orthogonal polynomials with the orthogonality condition Z P i ( H ) P j ( H ) e − V ( H ) dH = δ ij e ϕ i ( t ) , (I.7)where e ϕ i ( t ) are norms defined by integral (I.7) and the normalizing condition for the polynomials is P i ( H ) = X j ≤ i γ ij H j , γ ii = 1 , (I.8)i.e., the coefficient of the leading term is put equal to unity.Using polynomials (I.7), (I.8), we can rewrite (I.4) as Z N = ( N !) − Z Y i dH i det P k − ( H j ) det P l − ( H m ) exp {− X i H i + X i,k t k H k i } = N − Y i =0 e ϕ i ( t ) . (I.9)In order to get the determinant representation, we rewrite orthogonality condition (I.7) in a “ma-trix” form. That is, we introduce the matrix Γ with matrix elements γ mn defined by (I.8), the so-called moment matrix C with the matrix elements C ij = Z dHH i + j − e − V ( H ) , (I.10)and the diagonal matrix J with diagonal elements e ϕ n . Then, (I.7) can be written as a matrix relation Γ C Γ T = J (I.11)where Γ T is the transposed matrix. Evaluating the determinant of the both sides of this relation andusing (I.9), we obtain Z N = det N × N C ij . (I.12)The moment matrix satisfies a number of relations, which follow directly from its explicit form (I.10), ∂C ∗ ( t ) ∂t k ≡ ∂ k C ∗ ( t ) = ∂ k C ∗ ( t ) ∂t k ≡ ∂ k C ∗ ( t ) , (I.13) C ij = C i + j (I.14)and C N = ∂ N − C ≡ ∂ N − C (I.15)Finally, the partition function of the one-matrix model is Z N = det ∂ i + j − C, (I.16)which results in the Toda chain [24, 40, 7, 15]. Note that conditions (I.13) and (I.15) are satisfiedfor the whole hierarchy of the two-dimensional Toda lattice and for the KP hierarchy, while (I.14) isspecific for the Toda chain. This relation is nothing but the Riemann–Hilbert problem also known as the factorization problem, see [43, 15, 44]for the details. .5 Kontsevich-Penner representation The system (I.5) is solved by another integral, very different from (I.1) [45]: Z H ( t ) = π N − n g ( N − n )22 i − Nn e g tr L Z n × n exp (cid:18) − g h + N tr log h − i tr hL (cid:19) dh (I.17)Integral is now over n × n Hermitian matrix h , dh = Q na,b =1 dh ab and depends on additional n × n matrix (background field) L . To emphasize the difference between N and n we use small letters for h and tr instead of H and Tr in (I.1). If expanded around a saddle-point h = L this integral is a formalseries in positive powers of variables t k = − gk tr L − k , k ≥ t = g tr log L (I.20)This is a model from the GKM family and peculiar logarithmic term in the action is often namedPenner term [46], so that (I.17) is known as Gaussian Kontsevich-Penner model . In order to check that (I.17) satisfies (3) one begins with the Ward identity for this integral [23],associated to the shift h → h + ǫ of the integration variable by a small arbitrary matrix ǫ : g ∂∂L tr + N (cid:18) ∂∂L tr (cid:19) − + L ! Z n × n exp (cid:18) − g h + N tr log h − i tr hL (cid:19) = 0 (I.21)which gives g (cid:18) ∂∂L tr − Lg (cid:19) + N + n + L (cid:18) ∂∂L tr − Lg (cid:19)! Z = g ∂ ∂L tr + N − L ∂∂L tr ! Z = 0 (I.22)It remains to substitute a function Z H ( t ) with t ’s expressed through L by the Miwa transform (I.18).Then (I.21) becomes (I.5), see [45, 24, 7] for technical details.
The two integral representations (I.1) and (I.17) can be related directly, without a reference to Virasoroconstraints (I.5). One procedure, making use of orthogonal polynomials (Hermite polynomials in thisparticular case) is described in details in [24, 7]. One can rewrite (I.9) with time variables substituted It is possible to introduce dependence on t implicitly, namely to define the partition function as follows: Z H ( t ) = π N − n g ( N − n )22 i − Nn e t Ng + g tr L − Ntr log L Z n × n exp (cid:16) − g h + N tr log h − i tr hL (cid:17) dh (I.19)this slightly change all calculations but leave the partition function unchanged.
10y (I.18), (I.20) in terms of orthogonal polynomials in the following way: Z N = ( N !) − Z Y i dH i ∆ ( H ) exp − X i ˜ V ( H i ) + 1 g X i,k =0 t k H k i == ( N !) − Z Y i dH i ∆ ( H ) exp − X i ˜ V ( H i ) ! Y i,a ( L a − H i ) == ( N !) − Z Y i dH i exp − X i ˜ V ( H i ) ! ∆( H ) ∆( H, L )∆( L ) == ( N !) − ∆ − ( L ) Z Y i dH i exp − X i ˜ V ( H i ) ! ×× det N × N ˜ P i − ( H j ) det ( N + n ) × ( N + n ) ˜ P i − ( H j ) ... ˜ P N + b − l ( H j ) . . . ... . . . ˜ P i − ( L a ) ... ˜ P N + b − l ( L a ) , (I.23)where i, j = 1 , ..., N , a, b = 1 , ..., n and the orthogonal polynomials ˜ P k ( H ) are orthogonal with themeasure exp( − ˜ V ( H )). In the case under consideration, it is equal to exp (cid:16) − g H (cid:17) . Now calculatingthe determinants and using the orthogonality condition (I.7), one arrives at Z N = ∆ − ( L ) det n × n ˜ P N + a − ( L b ) Y i e ˜ ϕ i ( s ) == "Y i e ˜ ϕ i ( s ) det ( ab ) φ ( N ) a ( L b )∆( L ) = Z N | L a = ∞ × det ( ab ) φ ( N ) a ( L b )∆( L ) , (I.24)with φ ( N ) a ( L ) = ˜ P N + a − ( L ) (I.25)Now let us see that integral (I.17) can be also transformed to this form. To this end, we need theItzykson-Zuber formula, [47] Z U ( n ) dU e tr AUBU † = V n det e a i b j ∆( x )∆( y ) (I.26)where integral runs over unitary n × n matrices with the Haar measure dU , and a i , b j are eigenvalues ofHermitian matrices A and B . Now, using (I.3), we can perform integration over the angular variablesand rewrite (I.17) as e g tr L Z n × n dh exp (cid:18) − g h + N tr log h − i tr hL (cid:19) ∼∼ e g P i L i Z Y i dh i ∆( h )∆( L ) exp − g X i h i + N X i log h i − i X i h i L i ! == e g P i L i Z Y i dh i det h j − i ∆( L ) exp − g X i h i + N X i log h i − i X i h i L i ! = det Φ i ( L j )∆( L ) (I.27)where Φ i ( L ) ≡ e g L Z dxx i − exp (cid:18) − g x + N log x − ixL (cid:19) (I.28)11t remains to note that the orthogonal polynomials with the measure exp (cid:16) − g H (cid:17) are the Hermitpolynomials which have the integral representation˜ P k ( x ) = g k/ √ π Z + ∞−∞ dy x √ g + iy ! k e − y / = i n g k + √ π e x g Z + ∞−∞ dyy k e − gy − ixy (I.29)This finally reduces (I.24) to (I.17). Another way to connect the two matrix integrals, suggested recently in [21] is by using the Faddeev-Popov trick. In order not to make calculations with Grassmann variables, we choose the opposite signin (I.18), (I.20) t k = gk tr L − k , k ≥ t = − g tr log L (I.31)Then, (I.17) should be substituted with (the results for the two choices of sign can be also related bycontinuation) ∼ e − g tr L Z n × n exp (cid:18) − g h − N tr log h + tr hL (cid:19) dh (I.32)If Miwa transform (I.30) is made in the original integral (I.1), it becomes [7](I.1) = (det L ) − N Z N × N e − g Tr H dH det( I ⊗ I − H ⊗ L − ) = Z Z Z e − g Tr H + B ( I ⊗ L − H ⊗ I ) C dHd B, (I.33)where Faddeev-Popov trick is applied to substitute the determinant in the denominator by an integralover auxiliary rectangular N × n complex matrix fields B and C = B † . Here d B = dBdC = Q Ni =1 Q na =1 d B ia and B ( H ⊗ I − I ⊗ L ) C = B ai H ij C ja − B ia L ab C bi = tr B † HB − Tr BLB † (I.34)Taking the Gaussian integral over H we finally obtain:(I.1) = g N Z Z e g Tr BB † BB † + Tr BLB † d B (I.35)At the same time the Kontsevich-Penner integral (I.32) is equal to(I.32) = g N + n Z e − g tr h dh det ( L + gh ) N = g N + n Z e − g tr h dh det( I ⊗ gh + I ⊗ L ) == g N + n Z Z Z e − g tr h + B ( I ⊗ gh + I ⊗ L ) C dhdBdC (I.36)where the fields B, C are exactly the same as in (I.33), while B ( I ⊗ gh + I ⊗ L ) C = gB ia h ab C bi + B ia L ab C bi = g Tr BhB † + Tr BLB † (I.37)Again we can take the Gaussian integral over h and obtain:(I.32) = g N Z Z e g Tr BB † BB † + Tr BLB † d B (I.38)i.e. exactly the same expression as at the r.h.s. of (I.35). Thus we conclude that(I.1) = (I.32) (I.39)12he two integral representation for Z H ( t ) coincide.Inverting the argument, the two matrix-integral representations (I.1) and (I.32) for Z H ( t ) areassociated with two ways to decompose the quartic vertex Tr BB † BB † = tr B † BB † B with the helpof auxiliary fields H and h , coupled respectively to BB † and B † B and thus having the different sizes: N × N and n × n .As a word of precaution we remind only that for any finite n the Miwa transform (I.18) definesonly an n -dimensional subset in the infinite-dimensional space of t -variables: when expressed through L the higher t k with k > n are actually algebraic functions of the lowest t , . . . , t n . Thus Z H ( t ) in thiscontext should be interpreted as a projective limit at n → ∞ . Multiresolvents for Hermitian model are described in detail in the reference-paper [26]. Here we remindonly the simplest of the relevant formulas.Multiresolvents are defined by eq.(10) and the first step is to rewrite Virasoro constraints asrecurrent relations for ρ ( p | q ) . Such recursive reformulation is possible only if the genus expansion ofthe free energy F = log Z is performed, F = P ∞ p =0 g p − F ( p ) . As explained in some detail in [26], thisrequirement picks up some rather special solutions to Virasoro constraints, and only such solutionspossess well-defined multiresolvents and are associated with the bare spectral curves of finite genera. The bare spectral curve Σ is defined from non-linear equation for ρ (0 | – the starting point of therecursion. The next step provides ρ (0 | , which appears to be easily connected with Bergman kernelbi-differential on Σ [48]. At each step of recursion there exists a certain arbitrariness in the choice ofsolutions, which is, however, absent for the Virasoro constraint (I.5) – crucial for this unambiguity isthe form of the l.h.s. of (I.5): the fact that it is obtained by the shift of the time variable t → t − / Shift is parameter which is allowed to stand in denominators when we build up a formal-series solutionto Virasoro constraints. The shift of t is obviously associated with the integral (I.1) and defines whatis naturally called the Gaussian phase of Hermitian model. If other time or many times are shifted,then arbitrariness is unavoidable, see [49] for its full description. If partly fixed and parameterized byseveral arbitrary variables, it provides the family of Dijkgraaf-Vafa non-Gaussian partition functions[50, 51, 48]. Gaussian phase.
The bare spectral curve for Gaussian phase of Hermitian model isΣ HG : y = z − S (I.40)where S = gN and a few first Gaussian multiresolvents are: ρ (0 | H = z − √ z − S ρ (0 | H ( z , z ) = 12( z − z ) (cid:18) z z − Sy ( z ) y ( z ) − (cid:19) (I.42) ρ (1 | H ( z ) = Sy ( z ) (I.43) ρ (0 | H ( z , z , z ) = 2 S ( z z + z z + z z + 4 S ) y ( z ) y ( z ) y ( z ) (I.44) It deserves emphasizing that genus p in ”genus expansion” refers to the genus of the fat-graph Feynman diagramscontributing to F ( p ) . It has nothing to do with the genus of the bare spectral curve, throughout this text this genuswill be only zero, while the genus of the full spectral curve (which defines the point of the Universal Grassmannian [41]underlying the matrix-model τ -function) is infinite. Relation between bare and full spectral curves is rather tricky andis not yet fully clarified in the literature. Relation between generic genus-expansion-possessing solutions of [49] and the Dijkgraaf-Vafa family is very similarto that between the ”general” and ”total” solutions to the Hamilton-Jacobi equation, see [52, sect.7]. (1 | H ( z , z ) = Sy ( z ) y ( z ) (cid:16) z z (5 z + 4 z z + 3 z z + 4 z z + 5 z )++4 S (cid:16) z − z z ( z + z z + z ) + z (cid:17) + 16 S ( − z + 13 z z − z ) + 320 S (cid:17) (I.45) ρ (2 | H ( z ) = 21 S (cid:0) z + S (cid:1) y ( z ) (I.46)They are deduced from the recurrent relations y ( z ) ρ (0 | H ( z , z ) = ∂ z ρ (0 | H ( z ) − ρ (0 | H ( z ) z − z (I.47) y ( z ) ρ (1 | H ( z ) = ρ (0 | H ( z , z ) (I.48) y ( z ) ρ (0 | H ( z , z , z ) = 2 ρ (0 | H ( z , z ) ρ (0 | H ( z , z )++ ∂ z ρ (0 | H ( z , z ) − ρ (0 | H ( z , z ) z − z + ∂ z ρ (0 | H ( z , z ) − ρ (0 | H ( z , z ) z − z (I.49) y ( z ) ρ (1 | H ( z , z ) = 2 ρ (0 | H ( z , z ) ρ (1 | H ( z ) + ρ (0 | H ( z , z , z ) + ∂ z ρ (1 | H ( z ) − ρ (1 | H ( z ) z − z (I.50) y ( z ) ρ (2 | H ( z ) = (cid:16) ρ (1 | H ( z ) (cid:17) + ρ (1 | H ( z , z ) (I.51) Non-Gaussian phases.
For the sake of completeness we also give some formulas for non-Gaussianphases. If instead of the Gaussian shift t → t − we apply t k → t k − T k with W ( z ) = P n +1 k =0 T k z k and rewrite the shifted Virasoro constraints in terms of the multiresolvents we get: W ′ ( z ) ρ ( z ) = ρ ( z ) + f ( z ) + g ˆ ∇ ( z ) ρ ( z ) + ˆ P − z (cid:2) v ′ ( z ) ρ ( z ) (cid:3) (I.52)where ˆ ∇ ( z ) = ∞ X k =0 z k +1 ∂∂t k (I.53)and ρ ( z ) = ˆ ∇ ( z ) g log Z (I.54) f ( z ) = ˆ P + z (cid:2) W ′ ( z ) ρ ( z ) (cid:3) = ˆ R H ( z ) g log Z (I.55) W ′ ( z ) ρ ( p | m +1) W ( z, z , . . . , z m ) − f ( p | m +1) W ( z | z , . . . , z m ) == X q X m + m = m ρ ( q | m +1) W ( z, z i , . . . , z i m ) ρ ( p − q | m +1) W ( z, z j , . . . , z j m )++ m X i =1 ∂∂z i ρ ( p | m ) W ( z, z , . . . , ˇ z i , . . . , z m ) − ρ ( p | m ) W ( z , . . . , z m ) z − z i + ˆ ∇ ( z ) ρ ( p − | m +1) W ( z, z , . . . , z m ) (I.56) f ( p | m +1) H ( z | z , . . . , z m ) = ˇ R H ( z ) ρ ( p | m ) ( z , . . . , z m ) (I.57)ˇ R H ( z ) = P + z h W ′ ( z ) ˇ ∇ ( z ) i = − n − X a =0 n − a − X b =0 ( a + b + 2) T a + b +2 z a ∂∂T b (I.58)For further details we refer to [26, 49] and references therein.14 Complex-matrix model Z C ( t ) Complex matrix model was originally defined as an integral over N × N complex matrices Φ Z C ( t ) = Z N × N exp − Tr ΦΦ † g + ∞ X k =0 t k g Tr (ΦΦ † ) k ! d Φ (I.59)where d Φ = (cid:16) i (cid:17) N Q Ni,j =1 d Φ ij = Q Ni,j =1 d Re(Φ ij ) d Im(Φ ij ) = Q Ni,j =1 i d Φ ij d Φ † ij . One can express a complex matrix Φ through Hermitian H and unitary U matrices,Φ = U H, Φ † = HU † (I.60)and, further, through diagonal matrix D and two unitary matrices U and V :Φ = U DV † , Φ † = V DU † (I.61)The norm of Φ decomposes asTr δ Φ δ Φ † = Tr ( δH ) − Tr H ( U † δU ) + Tr [ H, δH ]( U † δU ) == Tr ( δD ) − Tr ( U † δU ) D − Tr D ( V † δV ) + 2Tr ( U † δU ) D ( V † δV ) D, (I.62)so that the measure d Φ = [ dU ][ dV ] N Y i HdHdθ .Comparing with (I.2), (I.3), one can see that for complex matrices the measure is actually thesame as for Hermitian matrix H , such that ΦΦ † = U H U † . Since action in the model (I.59) alsorespects this substitution, we obtain: Z C ( t ) = V N Z ∞ Y i dH i ∆ ( H ) exp − g X i H i + 1 g X i,k t k H ki ∼∼ Z [ dU ] Z exp − g Tr H + 1 g ∞ X k =0 t k Tr H k ! d ( H ) (I.64)This integral looks just the same as (I.1) for Hermitian matrix model, however, there is a differencein the integration contour: H is not an arbitrary Hermitian matrix. Relation between (I.1) and (I.64)is like between an integral over entire real axis and over its positive ray: the answers are different andeven invariance properties – and thus the Picard-Fucks equations (Ward identities) are not exactlythe same. Since the Eq.(I.64) means that the Virasoro constraints are the same as the ”discrete Virasoro con-straints” for Hermitian matrix model. However, there are two differences.15irst, L − -constraint, associated with the shift δ ( H ) = ǫ , is excluded, because it would correspondto a singular transform δH ∼ H − . This exclusion can also be considered as a result of the above-mentioned change of integration contour: from entire real line in the case of Hermitian model to apositive ray 0 ≤ H < ∞ in the case of (I.64).Second, the shift of time variables is t k = ˜ t k − δ k, in the Gaussian phase of Hermitian model,but it is rather t k = ˜ t k − δ k, in the Gaussian phase of (I.64).The two changes together make the seemingly diminished set of Virasoro constraintsˆ L m Z C = 0 , m ≥ , (I.65)with L m ( t ) = − ∂∂t m + ∞ X k ≥ kt k ∂∂t k + m + g X a + b = m ∂ ∂t a ∂t b (I.66)and ∂Z C ∂t = Sg Z C (I.67)fully exhaustive: despite the lack of ˆ L − , these equations are enough for unambiguous recursive re-construction of all terms in the formal series Z C ( t ) for the Gaussian branch of the complex matrixmodel. The integrable properties of the complex matrix model are practically identical to those of the Her-mitian model. In particular, the partition function Z C ∼ Z CN = det ∂ i + j − C (I.68)where the moment matrix is given a bit different integral as compared with the Hermitian case, C ij ≡ Z ∞ dx exp − g x + 1 g X k t k x k ! (I.69)This is still a Toda chain τ -function, however, it corresponds to another solution to the hierarchy givenby the Virasoro constraints (I.65). Like in the case of Hermitian model, the set of constraints (I.65) has another matrix-integral solution[31], different from (I.59): Z C ( t ) = π N − n e − tr ηη † Z exp (cid:16) − tr φφ † + tr η † φ + tr φ † η + N tr log( φφ † ) (cid:17) d φ (I.70)This time integral is over complex matrices φ , but their size n is, like in (I.17), independent of N ,which appears only as a parameter in the Penner term. For the sake of simplicity, we put here g = 1,the g -dependence being easily restorable. The time variables are related to the external matrices η and η † as t k = − k tr ( ηη † ) − k , k ≥ ,t = log( ηη † ) (I.71)16 .5.1 Proof I: Ward identities The Ward identity associated with the shifts φ → φ + ǫ of the integration variable, is now − ∂∂η tr + N ∂∂η † tr ! − + η † Z exp (cid:16) − tr φφ † + tr η † φ + tr φ † η + N tr log( φφ † ) (cid:17) d φ = 0 (I.72)Therefore, one gets " − ∂∂η † tr + η ! (cid:18) ∂∂η tr + η † (cid:19) + N + ∂∂η † tr + η ! η † Z C = " − ∂ ∂η∂η † + N − η ∂∂η − η † ∂∂η † Z C = 0(I.73)and, substituting Z C as a function of the Miwa variables (I.71) one reproduces (I.65), see [31]. Let us put L ≡ ηη † . Then, one can immediately repeat the calculation of s.1.5.2 in order to obtain that(I.59) is equal to (I.24) and (I.25), where the polynomials ˜ P k ( H ) are now orthogonal with the weightexp( − x ) on the positive real semi-axis. Such orthogonal polynomials are nothing but the Laguerrepolynomials [53], which have the following integral representation˜ P k ( x ) = ( − k e x Z ∞ dyy k e − y J (2 √ xy ) (I.74)where J ( x ) is the zero order Bessel function.Now let us rewrite (I.70) in the determinant form. This time in order to integrate over angularvariables, we need to use instead of the Itzykson-Zuber formula the following very nice formula ofintegration over two unitary n × n matrices, [54] Z U ( n ) dU Z U ( n ) dV exp (cid:18) 12 tr h U AV B + B † V † A † U † i(cid:19) = 2 n ( n − V n det J ( x i y j )∆( x )∆( y ) (I.75)where x i and y j are the eigenvalues of A † A and BB † respectively, A and B being arbitrary n × n complex matrices. Using this formula and (I.63) and denoting eigenvalues of φφ † and ηη † through y i and x i respectively, one obtains e − tr ηη † Z exp (cid:16) − tr φφ † + tr η † φ + tr φ † η + N tr log( φφ † ) (cid:17) d φ ∼∼ e − P i x i Z ∞ Y i dy i ∆( y )∆( x ) exp − X i y i + N X i log y i ! J (2 √ x i y i ) = det Φ ( C ) i ( x j )∆( x ) (I.76)where Φ ( C ) i ( x ) = e − x Z ∞ dyy i − exp ( − y + N log yJ (2 √ xy )) (I.77)Comparing this with (I.74) we ultimately identify (I.59) and (I.70). Direct equivalence of the two integrals (I.59) and (I.70) can be proved by a somewhat tricky general-ization of Faddeev-Popov argument from [21], which we applied in s.1.5 above.As before, we make the other choice of the sign in the Miwa transform, t k = 1 k ( ηη † ) − k , k ≥ Z exp − Tr ΦΦ † + ∞ X k =0 k tr ( ηη † ) − k Tr (ΦΦ † ) k ! d Φ = det( − ηη † ) N Z e − Tr ΦΦ † d Φdet(ΦΦ † ⊗ I − I ⊗ ηη † ) == det( − ηη † ) N Z Z e − Tr ΦΦ † + tr B † ΦΦ † B − Tr Bηη † B † d Φ d B = det( − ηη † ) N Z e − tr ηη † B † B d B det N × N ( I − BB † ) N (I.79)Note that the last determinant is raised to the power N , this is because we integrate over N complex-valued variables Φ ij : the relevant piece of the action is P Ni,j,k =1 Φ ij ¯Φ ik (cid:0) δ jk − P na =1 B ja ¯ B ka (cid:1) .At the same time integral (I.70) is:(I.70) = Z e − tr φφ † d φ det Nn × n (cid:16) ( φ + η )( φ † + η † ) (cid:17) = Z Z e − tr φφ † − Tr b † ( φ − η )( φ † − η † ) b d φd b = Z e − tr ηη † b † b b † b d b det n × n (1 + b † b ) n (I.80)Like B , b is rectangular N × n matrix, but the integrals are not literally equal as it was in the case ofHermitian model: one still needs to relate B and b .Let us begin with a few examples. Examples. N = n = 1: In this case we can introduce new variables: ρ = B † B = | B | and σ = b † b = | b | . Denoting also K = ηη † = | η | , we obtain our two integrals in the form: N = n = 1 : (I.59) = Z e − ρK dρ − ρ , while (I.70) = Z e − σ σ K dσ σ (I.81)and integrands coincide because for ρ = σ σ we have dρ = − dσ (1+ σ ) , while 1 − ρ = σ . n = 1, N arbitrary: In this case B is a complex N -vector ( B , . . . , B N ), the N × N matrix BB † has rank 1 and det( I − BB † ) = 1 − | B | − . . . − | B n | = 1 − ρ − . . . − ρ n = 1 − ρ + , so that(I.59) = Z e − ρ + K dρ . . . dρ N (1 − ρ + ) N ∼ Z ρ N − e − ρ + K dρ + (1 − ρ + ) N (I.82)and similarly (I.70) = Z e − σ +1+ σ K dσ . . . dσ N σ + ∼ Z σ N − e − σ +1+ σ K dσ + σ + , (I.83)where we used the fact that the volume of a simplex ρ > , . . . , ρ N > , ρ + . . . + ρ N = ρ + isproportional to ρ N − and similarly for σ ’s. Making the same transformation as above, ρ + = σ + σ + , wesee again that the integrals coincide. N = 1, n arbitrary: This time B and b are complex n -vectors. If we perform an SU ( n ) rotation todiagonalize ηη † → diag( K , . . . , K n ), then both integrals (I.59) and (I.70) still contain B and b onlyin the form of squared modules ρ a = | B a | and σ a = | b a | :(I.59) = Z e − P na =1 ρ a K a dρ . . . dρ n − ρ + and (I.70) = Z e − P na =1 σ a K a / (1+ σ + ) dσ . . . dσ n (1 + σ + ) n (I.84)The integrals are related by our usual change of variables ρ a = σ a σ + , only the measure transform getsa little trickier: ∧ na =1 dρ a = ∧ na =1 (cid:18) dσ a σ + − σ a dσ + (1 + σ + ) (cid:19) = ∧ na =1 dσ a (1 + σ + ) n − P na =1 σ a ∧ na =1 dσ a (1 + σ + ) n +1 = 1(1 + σ + ) n +1 ∧ na =1 dσ a (I.85)Substituting also 1 − ρ + = (1 − σ + ) − , we see that the two integrals are in fact the same.18 .6 Genus expansions and the first multiresolvents Gaussian phase. The first resolvent [26] in this case is ρ (0 | C ( z ) = 12 − s − Sz (I.86)and the bare spectral curve Σ C : y = z ( z − S ) (I.87)A few next multiresolvents are: ρ (0 | C ( z , z ) = 12( z − z ) (cid:18) z z − S ( z + z ) y y − (cid:19) (I.88) ρ (1 | C ( z ) = zS y C ( z ) (I.89) ρ (0 | C ( z , z , z ) = 2 z z z S y C ( z ) y C ( z ) y C ( z ) (I.90) ρ (1 | C ( z , z ) = z z S y y (cid:16) z z + 3 z z + 2 z z − z + 17 z z + 17 z z + z ) S + 8(3 z + 19 z z + 3 z ) S − z + z ) S + 128 S (cid:17) (I.91) ρ (2 | C ( z ) = (9 S − zS + 8 z ) S z y ( z ) (I.92)They are deduced from the recurrent relations y ( z ) z ρ (0 | C ( z , z ) = 1 z ∂ z z ρ (0 | C ( z ) − z ρ (0 | C ( z ) z − z (I.93) y ( z ) z ρ (1 | C ( z ) = ρ (0 | C ( z, z ) (I.94) y ( z ) z ρ (0 | C ( z , z , z ) = 2 ρ (0 | C ( z , z ) ρ (0 | C ( z , z )++ 1 z ∂ z z ρ (0 | C ( z , z ) − z ρ (0 | C ( z , z ) z − z + 1 z ∂ z z ρ (0 | C ( z , z ) − z ρ (0 | C ( z , z ) z − z (I.95) y ( z ) z ρ (1 | C ( z , z ) = 2 ρ (0 | C ( z , z ) ρ (1 | C ( z ) + ρ (0 | C ( z , z , z ) + 1 z ∂ z z ρ (1 | C ( z ) − z ρ (1 | C ( z ) z − z (I.96) y ( z ) z ρ (2 | C ( z ) = (cid:16) ρ (1 | C ( z ) (cid:17) + ρ (1 | C ( z , z ) (I.97) Non-Gaussian phases. A generic phase of the complex matrix model is given by first several timevariables shifted, t k → T k + t k . Then, for a generic polynomial potential W ( z ) = P n +1 k =0 T k z k theVirasoro constraints for the complex-matrix model look like W ′ ( z ) ρ ( z ) = ρ ( z ) + f ( z ) + g ˆ ∇ ( z ) ρ ( z ) + 1 z P − z (cid:2) zv ′ ( z ) ρ ( z ) (cid:3) (I.98)where f ( z ) = 1 z P + z (cid:2) zW ′ ( z ) ρ ( z ) (cid:3) (I.99)19igher resolvents can be extracted from the equations W ′ ( z ) ρ ( p | m +1) W ( z, z , . . . , z m ) − f ( p | m +1) C ( z | z , . . . , z m ) == X q X m + m = m ρ ( q | m +1) W ( z, z i , . . . , z i m ) ρ ( p − q | m +1) W ( z, z j , . . . , z j m )++ m X i =1 z ∂∂z i zρ ( p | m ) W ( z, z , . . . , ˇ z i , . . . , z m ) − z ρ ( p | m ) W ( z , . . . , z m ) z − z i + ˆ ∇ ( z ) ρ ( p − | m +1) W ( z, z , . . . , z m ) . (I.100)where f ( p | m +1) C ( z | z , . . . , z m ) = ˇ R C ( z ) ρ ( p | m ) ( z , . . . , z m ) (I.101)with ˇ R C ( z ) = 1 z P + z h zW ′ ( z ) ˇ ∇ ( z ) i = − n − X a = − n − a − X b =0 ( a + b + 2) T a + b +2 z a ∂∂T b (I.102)Quadratic equation for simplest resolvent leads to the answer ρ (0 | C ( z ) = W ′ ( z ) − y C ( z ) z y C ( z ) = z (cid:16) W ′ − R C ( z ) F (0) C (cid:17) (I.104)For the Gaussian complex model T k = δ k, , so W ( z ) = z , ˇ R C ( z ) = − z ∂∂T Kontsevich model Z K ( τ ) Kontsevich model was originally defined in [22] as a generating function of topological indices of themoduli space of Riemann surfaces. M.Kontsevich represented this generating function in form ofthe now-famous matrix integral over auxiliary n × n dimensional Hermitian matrices (one can easilyintroduce into this integral the parameter g similarly to (I.17), [1] however, for the sake of simplicity,we put here g = 1): Z K = exp (cid:18) − 23 tr L (cid:19) R n × n dh exp (cid:16) − tr h + tr L h (cid:17)R n × n dh exp ( − tr Lh ) (I.105)where time-variables are Miwa-transformed: τ k = 1 k tr L − k (I.106)If expressed through the τ -variables, Z K ( τ ) is actually independent of auxiliary parameter n . Thismodel can be further generalized to Generalized Kontsevich Model (GKM) [23], Z GKM = R n × n e − U ( L,h ) dh R n × n e − U ( L,h ) dh (I.107)where U ( L, h ) ≡ tr [ V ( L + h ) − V ( L ) − V ′ ( L ) h ] (I.108)and U ( L, h ) = lim ǫ → ǫ U ( L, ǫh ) (I.109)is an h -term in U . W is here an arbitrary (power series) potential and Z GKM is a function of thesame ( W -independent) Miwa transform (I.106). Many properties of GKM are in fact independent ofthe choice of V ( h ).In fact, one can also consider the matrix model (I.107) with a different normalization as a functionof time variables T k = 1 k tr Λ k (I.110)where the matrix Λ = V ′ ( L ) enters in positive powers. This function is called the character phaseand is considered in detail in [20]. In this paper we restrict ourselves with the Kontsevich phase only,where the time variables are given by (I.106).Note also that it is often convenient to fix V ( h ) to be a polynomial of h (polynomial Kontsevichmodel, [23]) or that of h − (antipolynomial Kontsevich model [20]). In this section we consider onlythe polynomial case, leaving the antipolynomial one until the next section (where it emerges withinthe context of the unitary matrix model). Shifting the integration variable h → h − L one obtains that Z GKM ∼ Z n × n dh exp (cid:0) − tr V ( h ) − tr V ′ ( L ) h (cid:1) (I.111)Now using the Itzykson-Zuber formula and (I.3), one can perform integration over the angular variablesin this integral: Z n × n dh exp (cid:0) − tr V ( h ) − tr V ′ ( L ) h (cid:1) ∼ det ij F i ( λ j )∆( λ ) (I.112)where λ i are the eigenvalues of the matrix V ′ ( L ) and F i ( λ ) = Z dxx i − exp ( −V ( x ) + λx ) (I.113)21 .3 Virasoro constraints Straightforward Ward identities for Z K are as previously associated with the shift h → h + ǫ ofintegration variable h : "(cid:18) ∂∂L tr (cid:19) − L n × n dh exp (cid:18) − 13 tr h + tr L h (cid:19) = 0 (I.114)Now one should take into account the normalization factor and come to the τ -variables. Conversionto the τ -variables is highly non-trivial, it was first performed in [18] and leads to the celebrated result[13, 14]: ˆ L n Z K = 0 , ˆ L n = 12 X k ≥ δn +1 , k odd kτ k ∂∂T k +2 n + 14 X a + b =2 na,b ≥ a,b odd ∂ ∂τ a ∂τ b ++ δ n +1 , · τ δ n, · − ∂∂τ n +3 (I.115)This proved equivalence of Witten’s topological 2 d gravity [12] to Z K and – since Z K is trivially a KPtau-function – proved that partition function of 2 d gravity is indeed a tau-function (as anticipated in[55]). Analogous conversion to T -variables of the Ward identities for Z GKM is even more sophisticatedand give rise to W -constraints (or ˜ W -constraints in the character phase) [19, 20]. Now one can take into account all the normalization factors and further transform this determinant(after quite tedious calculation, [23]) to Z GKM = det ij φ i ( L j )∆( L ) (I.116)where φ i ( L ) = q V ′′ ( L ) e V ( L ) − L V ′ ( L ) F i ( V ′ ( L )) (I.117)Formula (I.116) if true for any number of Miwa variables (size of the determinant) fixes a KP hierarchy τ -function [10, 24] that depends on times τ k = 1 k X i L − ki (I.118)provided the asymptotics of φ i ( L ) at large L is φ i ( L ) L →∞ ∼ L i − (1 + O (1 /L )) (I.119)In particular, (I.119) guarantees that Z GKM is a function of variables τ k (I.118) and does not dependon their number.Now if one takes the monomial potential V ( h ) = h p +1 (the case of a polynomial potential of degree p + 1 describes a hierarchy equivalent to the monomial case, see details in [56]), the partition function Z GKM is a τ -function of the p -reduced KdV hierarchy, which does not depend on times τ pk for all k .In particular, the Kontsevich partition function (I.105) is a KdV τ -function depending only on oddtimes τ k +1 . A concrete solution of the KdV hierarchy is fixed by the Virasoro constraints (I.115) (infact, it is enough to use only the lowest constraint in addition to the KdV hierarchy equations in orderto fix the partition function unambiguously).Note that one could starts from the Virasoro constraints (I.115) instead of the matrix integral.Then, there are much more solutions, the KdV one corresponding only to distinguished solutions ofthe Dijkgraaf-Vafa type [26, 2nd paper]. 22ote that one can easily continue the (Generalized) Kontsevich matrix integral to the whole Todalattice hierarchy adding to U ( L, h ) (but not to U ( L, h )) the term∆ U ( L, h ) = ∆ V ( L + h ) − ∆ V ( L ) , ∆ V ( h ) = ℵ log h − X k ¯ τ k h − k (I.120)Here ℵ is the zeroth (discrete) Toda time and ¯ τ k are the negative Toda times. In the special case ofquadratic potential V ( h ) = h this matrix integral reduces to the Toda chain, as we observed in s.2. Of course, Kontsevich model is already in the Kontsevich form. No parameter N is obligatory presentand no Penner term is needed (until one wants to deal with the (Generalized) Kontsevich integral aswith the Toda lattice hierarchy). Generic phase. Similarly to the Hermitian and complex matrix models, a generic phase of theKontsevich model is given by first several time variables shifted τ k +1 → τ k +1 + T k . In this case,ˆ ∇ ( z ) = ∞ X k =0 z k +1 ∂∂τ k +1 (I.121) W ′ ( z ) = n +1 X k =0 (cid:18) k + 12 (cid:19) T k z k (I.122) v ′ ( z ) = ∞ X k =0 (cid:18) k + 12 (cid:19) τ k +1 z k (I.123) W ′ ( z ) ρ ( z ) = ρ ( z ) + f K ( z ) + g ˆ ∇ ( z ) ρ ( z ) + zP − z (cid:20) v ′ ( z ) ρ ( z ) z (cid:21) + g z + ( τ − T ) 16 (I.124) f K ( z ) = zP + z (cid:20) W ′ ( z ) ρ ( z ) z (cid:21) = g ˆ R K ( z ) log Z (I.125)ˆ R K ( z ) = − n +1 X k =2 k − X m =0 (cid:18) k + 12 (cid:19) T k z k − m − ∂∂T m (I.126) ρ ( z ) = g ˆ ∇ ( z ) log Z (I.127) W ′ ( z ) ρ ( p | m +1) W ( z, z , . . . , z m ) − f ( p | m +1) W ( z | z , . . . , z m ) == X q X m + m = m ρ ( q | m +1) W ( z, z i , . . . , z i m ) ρ ( p − q | m +1) W ( z, z j , . . . , z j m )++ m X i =1 (cid:18) ∂∂z i − z i (cid:19) z i ρ ( p | m ) W ( z, z , . . . , ˇ z i , . . . , z m ) − zρ ( p | m ) W ( z , . . . , z m ) z − z i + ˆ ∇ ( z ) ρ ( p − | m +1) W ( z, z , . . . , z m )+ δ p, δ m, z + δ p, δ m, T − δ m, T z + δ m, z z ! (I.128)23 aussian case. In this case, W ′ ( z ) = z + T , f ( k | m ) = 0 ρ (0 | ( z ) = z + T − √ z + T z ρ (0 | ( z , z ) = 14( z − z ) (cid:18) ( z + z + 2 T ) z z y ( z ) y ( z ) − ( z + z ) (cid:19) (I.130) y ( z ) ρ (0 | ( z , z ) = (cid:18) ∂∂z − z (cid:19) z ρ (0 | ( z ) − z ρ (0 | ( z ) z − z − T z (I.131) ρ (0 | ( z , z , z ) = z z z y ( z ) y ( z ) y ( z ) (I.132) y ( z ) ρ (0 | ( z , z , z ) = (cid:18) ∂∂z − z (cid:19) z ρ (0 | ( z , z ) − z ρ (0 | ( z , z ) z − z + (cid:18) ∂∂z − z (cid:19) z ρ (0 | ( z , z ) − z ρ (0 | ( z , z ) z − z + 18 z z (I.133) y ( z ) ρ (1 | ( z ) = ρ (0 | ( z, z ) + 116 z (I.134) ρ (1 | ( z ) = z y ( z ) (I.135) y ( z ) ρ (1 | ( z , z ) = 2 ρ (0 | ( z , z ) ρ (1 | ( z ) + ρ (0 | ( z , z , z ) + (cid:18) ∂∂z − z (cid:19) z ρ (1 | ( z ) − z ρ (1 | ( z ) z − z (I.136) y ( z ) ρ (2 | ( z ) = (cid:16) ρ (1 | ( z ) (cid:17) + ρ (1 | ( z, z ) (I.137) ρ (2 | ( z , z ) = z z 32 5( z + T ) + 3( z + T )( z + T ) + 5( z + T ) y ( z ) y ( z ) (I.138) ρ (2 | ( z ) = 105254 z y ( z ) (I.139)24 BGW model ˜ Z K ( τ ) = Z BGW ( τ ) Brezin-Gross-Witten (BGW) model is defined as a generating function for all correlators of unitarymatrices with Haar measure [ dU ]: Z BGW = Z N × N [ dU ] exp (cid:16) Tr J † U + Tr J U † (cid:17) (I.140)The integral actually depends only on eigenvalues of Hermitian matrix M = J J † , i.e. on the time-variables of the form τ k = Tr ( J J † ) k .Haar measure [ dU ] for unitary matrices is non-linear, it can be reduced to a flat measures in differentways. One possibility is to express U through Hermitian matrices, U = iH − iH [57], which defines [ dU ]as the flat Hermitian measure dH = Q Ni,j =1 dH ij with additional Jacobian factor, [ dU ] = J ( H ) dH , J = det(1 + H ) − N . Another possibility [20, 21] is to impose the constraints on the complex matrices:[ dU ] = Z d Φ δ (ΦΦ † − I ) = Z N × N dhe − Tr h Z N × N d Φ e Tr h ΦΦ † (I.141)For certain actions the integral over d Φ can be explicitly taken and this gives rise to reformulation oforiginal unitary-matrix model. Since technically the most simple way to obtain eigenvalue representations is to start with theKontsevich-Penner representation of the BGW model, we first consider this representation. In variance with all other Kontsevich-Penner representations, that of the BGW model connects thetwo integrals over the two matrices (unitary and Hermitian ones) of the same size, [20]: Z BGW = Z N × N dh exp (cid:16) − tr h − + tr M h − N tr log h (cid:17) (I.142)This makes theory of the BGW model somewhat harder and one sometimes embeds it into the universalBGW model [20] with an arbitrary coefficient in front of the logarithmic term. However, in order tomake contact with the BGW model (I.140) one ultimately has to put this coefficient equal to − N . The simplest Ward identity for Z BGW has the form ∂∂J † tr · ∂∂J tr Z BGW ( J, J † ) = I · Z BGW ( J, J † ) . (I.143)or, equivalently, [20] ∂∂M tr M ∂∂M tr Z BGW ( M ) = I · Z BGW ( M ) . (I.144)At the same time, integral (I.142) satisfies the equation " ∂∂M tr M ∂∂M tr + ( N − N ) ∂∂M tr + (cid:18) ∂∂M tr (cid:19) V ′ (cid:18) ∂∂M tr (cid:19) N × N dhe tr ( Mh −N log h + V ( h )) = 0 . (I.145)At N = N and V ′ ( h ) = 1 /h (I.144) and (I.145) coincide, which establishes (I.142).25 .3.2 Proof II: Faddeev-Popov trick Another simple way to derive the Kontsevich-Penner representation for the BGW model is to use thetrick (I.141), [21]. Indeed, Z BGW = Z dhe − Tr h Z d Φ exp (cid:16) tr h ΦΦ † + Tr J † Φ + tr J Φ † (cid:17) == Z N × N dh exp (cid:16) − tr h + tr M/h − N tr log h (cid:17) h → /h = h → /h = Z N × N dh exp (cid:16) − tr h − + tr M h − N tr log h (cid:17) (I.146)The BGW model has two phases [20]: the Kontsevich phase , where partition function is expandedin negative powers of M and the character phase where expansion goes in positive powers of M . Belowwe describe them separately. Z C and Z BGW In the Kontsevich-Penner form (I.70) the complex matrix model looks somewhat similar to originalform (I.140) of the BGW model. From (I.70) one obtains (representing φ = HU † and φ † = U H ): Z C = π N − n e − tr ηη † Z dHe − tr H +2 N tr log H Z BGW ( η † H η ) (I.147)This tricky formula is the best direct relation known at present. A more transparent relation is stilllacking. In this phase the BGW partition function is considered as a function of the variables T k = 1 k tr M k (I.148)and one has to consider the Universal BGW model, i.e. the Kontsevich integral (I.142) with anarbitrary coefficient of the logarithm, which is a free parameter and not the size of the unitary matrix. Performing the change of variables in (I.145) from M to T k , one can directly obtain the Virasoroconstraints satisfied by the BGW partition function Z + BGW in the character phase:ˆ L m ( N, T ) Z + BGW = δ m, Z + BGW , m ≥ L m ( α, T ) = α ∂∂ T m + ∞ X k ≥ k T k ∂∂ T k + m + X a + b = m ∂ ∂ T a ∂ T b (I.149)Therefore, the Ward identity (and its solutions) depends on the size of matrix N . This means thatthe integral (I.142) is not just a function of variables T , but also depends on N . The way out is toconsider the Universal BGW model given by the integral Z UBGW = Z N × N dh exp (cid:16) − tr h − + tr M h − N tr log h (cid:17) (I.150)Then, the Virasoro constraints becomeˆ L m (2 N − N , T ) Z + UBGW = δ m, Z + UBGW , m ≥ N = 2 N − ℵ , one arrives at the partition function Z + BGW that does not depend on N (but only on the parameter ℵ ) though the integrand in (I.150) does!26 .5.2 Determinant representation and integrability One can easily integrate over the angular variables in (I.150) as in the previous section to obtain Z UBGW = det F i ( M j )∆( M ) (I.152)where F i ( M ) = Z dhh i − exp (cid:16) − h + M H − N log h (cid:17) = 2 πi (cid:16) √ M (cid:17) N − i I N − i (cid:16) √ M (cid:17) (I.153)where I k ( z ) are the modified Bessel functions. After some work [20], this formula can be recast to theform (I.116) with the asymptotics (I.119), where L = 1 /M and φ i ( M ) = Γ (2 N − N − i + 2)2 − i πi (cid:18) √ µ (cid:19) N −N − I N −N / − i √ µ ! (I.154)Again in order to make these functions independent of N , one has to choose N = 2 N − ℵ . At thesame time, this proves that, under such a choice, Z + UBGW is a τ -function of the KP hierarchy. In the Kontsevich phase the unitary matrix integral is considered as a function of variables τ k = − k tr M − k (I.155)This time the integral (I.142) does not depend on N provided it is properly normalized: Z + BGW = e − tr M / vuut det (cid:0) M / ⊗ M + M ⊗ M / (cid:1) (det M ) N Z BGW (I.156)One can check by a direct (quite involved) calculation [20] that Z + BGW depends only on odd times τ k +1 . Using the Ward identity (I.144) one can now make the change to variables (I.155) to obtain theVirasoro constraints satisfied by Z + BGW [33, 20]:ˆ L m Z + BGW = 0 , m ≥ L m = − X odd k kτ k ∂∂τ k − m + 12 X odd a,ba + b =2 m ∂ ∂τ a ∂τ b + ∂∂τ m +1 + δ m, 16 (I.157) In the Kontsevich phase, performing integration over the angular variables in (I.142) and taking intoaccount the normalization factor, one obtains [20] the determinant representation (I.116) with theasymptotics (I.119), where φ i − N ( M ) = 2 √ πe − M M i − N − / I i − N − (2 M ) (I.158)This means that the partition function Z + BGW of the unitary matrix model is a τ -function of the KPhierarchy. Moreover, as it was already noted, it does not depend on odd times and is, in fact, a τ -function of the KdV hierarchy. 27 .7 Genus expansion and the first multiresolvents Generic phase. As before, we shift first time variables τ k +1 → τ k +1 + T k ,ˆ ∇ ( z ) = ∞ X k =0 z k +1 ∂∂τ k (I.159) W ′ ( z ) = n +1 X k =0 (cid:18) k + 12 (cid:19) T k z k (I.160) v ′ ( z ) = ∞ X k =0 (cid:18) k + 12 (cid:19) τ k z k (I.161)and obtain the loop equations W ′ ( z ) ρ ( z ) = ρ ( z ) + f BGW ( z ) + g ˆ ∇ ( z ) ρ ( z ) + P − z (cid:2) v ′ ( z ) ρ ( z ) (cid:3) + g z (I.162) f K ( z ) = P + z (cid:2) W ′ ( z ) ρ ( z ) (cid:3) = g ˆ R K ( z ) log Z (I.163)ˆ R K ( z ) = − n +1 X k =1 k − X m =0 (cid:18) k + 12 (cid:19) T k z k − m − ∂∂T m (I.164) ρ ( z ) = g ˆ ∇ ( z ) log Z (I.165) W ′ ( z ) ρ ( p | m +1) W ( z, z , . . . , z m ) − f ( p | m +1) W ( z | z , . . . , z m ) == X q X m + m = m ρ ( q | m +1) W ( z, z i , . . . , z i m ) ρ ( p − q | m +1) W ( z, z j , . . . , z j m )++ m X i =1 (cid:18) z i ∂∂z i + 12 (cid:19) ρ ( p | m ) W ( z, z , . . . , ˇ z i , . . . , z m ) − ρ ( p | m ) W ( z , . . . , z m ) z − z i + ˆ ∇ ( z ) ρ ( p − | m +1) W ( z, z , . . . , z m )+ δ p, δ m, z (I.166) Gaussian phase. In this case, W ′ ( z ) = 1, f ( k | m ) = 0 ρ (0 | ( z ) = 0 (I.167) ρ (0 | ( z , z ) = 0 (I.168) ρ (0 | ( z , z , z ) = 0 (I.169) ρ (1 | ( z ) = 116 z (I.170) ρ (1 | ( z , z ) = (cid:18) z ∂∂z + 12 (cid:19) ρ (1 | ( z ) − ρ (1 | ( z ) z − z (I.171) ρ (2 | ( z ) = (cid:16) ρ (1 | ( z ) (cid:17) + ρ (1 | ( z, z ) (I.172) ρ (1 | ( z , z ) = 132 z z (I.173) ρ (2 | ( z ) = 9256 z (I.174)28 ree energy in the Gaussian case. As a direct corollary of Virasoro constraints (I.157), one cancalculate the free energy expansion in the parameter g , log Z BGW = P k =0 g k − F ( k ) BGW : F (0) BGW = 0 F (1) BGW = − / τ − F (2) BGW = − τ ( τ − F (3) BGW = − τ ( τ − + 56764 τ ( τ − F (4) BGW = − τ ( τ − + 388125512 τ τ ( τ − − τ ( τ − (I.175)In general F ( p ) BGW is a polynomial F ( p ) BGW = X k + ... + k m = p − c k ,...,k m Q k . . . Q k m (I.176)of the variables Q k = τ k ( τ − k +1 . This is the best illustration of drastic simplicity of the BGWpartition function as compared to the Kontsevich and Hermitian cases, where all F ( p ) are sophisticatedtranscendental functions, and are simplified only in terms of moment variables. One may say in theBGW case the moment variables are extremely simple.29 art II Decomposition formulas The key observation is that multiresolvents – if defined according to the rule (10) – are polydifferentialson the bare spectral curve Σ, intimately related to the d U (1) current ˆ J ( z ) on Σ, with prescribedsingularities: usually they are allowed at some fixed points ( punctures ) on Σ. In this approach theVirasoro constraints on partition function are written asˆ P − (cid:16) ˆ J ( z ) (cid:17) Z = I C K ( z, z ′ ) (cid:16) ˆ J ( z ) (cid:17) Z = 0 (II.1)with a certain kernel K ( z, z ′ ), made out of the free-field Green function on Σ. The current is also”shifted”: ˆ J ( z ) −→ ˆ J ( z ) + ∆ ˆ J ( z ) and partition function Z depends on the choice of: • the complex curve (Riemann surface) Σ, • the Green function K ( z, z ′ ), i.e. projection operator ˆ P − , • the punctures on Σ and associated loop operator ˆ J ( z ), • the local coordinates in the vicinity of the punctures, • the involution of the curve with punctures and loop operator, • the shift ∆ J ( z ) on Σ, • the contour C which separates two sets of punctures.If contour C goes around an isolated puncture, Z is actually defined by its infinitesimal vicinityand depends on behavior (the type of singularity) of ˆ J ( z ) at this particular puncture. Coordinatedependence is reduced to the action of a unitary operator (Bogoliubov transform, and exponentialof bilinear function of ˆ J ) on Z . Types of singularities and associated Z ’s can be classified, and ourquartet Z H , Z C , Z K and Z BGW are the lowest members of this classification. The former two areassociated with a puncture at regular point of Σ, while the latter two – with that at a second-orderramification point. Z C and Z BGW differ from Z H and Z K by the choice of projection operator ˆ P − ,i.e. the kernel K ( z, z ′ ).If contour C is moved away from the vicinity of the puncture, it can be decomposed into contoursencircling all other punctures: this provides relations between Z ’s of different types, associated withdifferent punctures. If Σ has handles or boundaries, there will be additional contributions, associatedwith non-contractible contours – the corresponding elementary partition functions are not yet identifiedand investigated – this seems to be a very interesting problem of its own.In what follows we present the two simplest examples of this procedure, both associated withΣ, represented as a double-covering of the Riemann sphere with two ramification points. Such Σis of course also a Riemann sphere, however, representing it as a double-covering provides a simpledescription of behavior, which we allow ˆ J ( z ) to have at the two ramification points. The other pairof punctures are chosen at preimages of a regular points ( z = ∞ ± in what follows). After that,depending on the choice of projection operator P − we obtain either a relation between Z H and thetwo Kontsevich models, Z H = ˆ U KK (cid:16) Z K ⊗ Z K (cid:17) , or between Z C and the pair: Kontsevich model andBGW model, Z C = ˆ U KBGW (cid:16) Z K ⊗ Z BGW (cid:17) . These both examples were already described in [1], buthere we provide a more targeted and, hopefully, more clear presentation of the subject. Some mistakesof original version are also corrected, in the case of discrepancies from [1] the present version shouldbe trusted more. These are the data, defining the standard Virasoro constraints (3) and (5) and thus the four models,discussed in the section I above. All the four are defined in vicinity of a particular puncture and donot depend on the global properties of the bare spectral curve Σ.30 .1 Hermitian current This one is used in the definition of (4) and thus of partition functions Z H and Z C .ˆ J H ( z | g ) = d ˆΩ H ( z ) = ∞ X k =0 (cid:18) k t k z k − dz + g dzz k +1 ∂∂t k (cid:19) (II.2)With this current one can immediately associate a bi-differential f J ( z, z ′ ) = ˆ J ( z ) ˆ J ( z ′ ) − : ˆ J ( z ) ˆ J ( z ′ ) :where the normal ordering means all t k placed to the left of all t -derivatives. It is related to the centralextension d U (1) and is equal to f H ( z, z ′ | g ) = g dzdz ′ z − z ′ ) (II.3)This bi-differential will play an important role in comparison of global and local currents and, therefore,in construction of conjugation operators in the next subsections 3 and 4.The further difference between various partition functions comes from different choices of the shiftfunctions W ( z ) [26] and projector operators [1], P m ∞ X k = −∞ a k dz z k +2 = ∞ X k = m a k dz z k +2 (II.4)The two correlated choices lead to the two simplest models, associated with (II.2): to Z H an Z C .1. Gaussian Hermitian model [26] This model corresponds to the shift ∆ ˆ J H ( z ) = − zdz T H ( z ) Z H = 0 (II.6)where ˆ T H ( z ) = P − h : ( ˆ J H ( z ) + ∆ ˆ J H ( z )) : i = g X n = − ( dz ) z n +2 ˆ L n , ˆ L n = ∞ X k =1 k (cid:18) t k − δ k, (cid:19) ∂∂t k + n + g n X k =0 ∂ ∂t k ∂t n − k (II.7)with ∂∂t Z H ( t ) = Sg Z H ( t ) (II.8)Given (II.5), the choice of P − from all the P m is distinguished: with this choice only partitionfunction is unambiguously defined by (II.6). There are interesting situations, when the choice of P m is not adjusted to the shift in this way: the best known example is provided by Dijkgraaf-Vafa partition functions [50, 51], where projector is the same P − as in Gaussian model, but theshift ∆ ˆ J H ( z ) = dW ( z ) is generated by polynomial W ( z ) of degree higher than two.2. Gaussian complex model [30] This model corresponds to the shift ∆ ˆ J C ( z ) = − dz T C ( z ) Z C = 0 (II.10)31here projector is taken to be P – again, to guarantee the uniqueness of the solution to (II.10),– and ˆ T C ( z ) = P h : ( ˆ J H ( z ) + ∆ ˆ J C ( z )) : i = g X n =0 ( dz ) z n +2 ˆ L n , ˆ L n = ∞ X k =1 k ( t k − δ k, ) ∂∂t k + n + g n X k =0 ∂ ∂t k ∂t n − k (II.11)with ∂∂t Z C ( t ) = Sg Z C ( t ) (II.12) This one is used in the definition of (6) and thus of partition functions Z K and Z BGW ,ˆ J K ( ξ | g ) = d ˆΩ K ( ξ ) = ∞ X k =0 (cid:18) k + 12 (cid:19) τ k ξ k dξ + g dξξ k +2 ∂∂τ k (II.13)is – up to traditional but unimportant change of time-variables – the even part of the current (II.2).However, associated central term bi-differential looks more sophisticated (being the symmetric partof the bi-differential): f K ( ξ, ξ ′ | g ) = g ( ξ + ξ ′ ) dξdξ ′ ξ − ξ ′ ) (II.14)The simplest partition functions, associated with this current, are Kontsevich τ -function and BGWmodel.1. Kontsevich τ -function [22, 23] This time the shift is ∆ ˆ J K = − ξ dξ P − :ˆ T K ( ξ ) = P − (cid:20) : (cid:16) ˆ J K + ∆ ˆ J K (cid:17) : (cid:21) = g ∞ X n = − ( dξ ) ξ n +2 ˆ L n , ˆ L n = ∞ X k =0 (cid:18) k + 12 (cid:19) (cid:18) τ k − δ k, (cid:19) ∂∂τ k + n + g n − X k =0 ∂ ∂τ k ∂τ n − − k + δ n, 16 + δ n, − τ g (II.16)Then Z K is uniquely defined by ˆ T K ( ξ ) Z K = 0 (II.17)2. Brezin–Gross–Witten model [20] Now the shift is ∆ J BGW = − cdξ T BGW ( ξ ) = P h : ( J K + ∆ J BGW ) : i = g ∞ X n =0 ( dξ ) ξ n +2 L n , L n = ∞ X k =0 (cid:18) k + 12 (cid:19) ( τ k − δ k, ) ∂∂τ k + n + g n − X k =0 ∂ ∂τ k ∂τ n − − k + δ n, 16 (II.19)with projector P unambiguously specify Z BGW byˆ T BGW ( ξ ) Z BGW = 0 (II.20)32 Decomposition relation Z H −→ Z K ⊗ Z K Now we can select a bare spectral curve Σ: y = z − a (II.21)select the punctures: at z = ± a and z = ∞ ± , and select the global current by allowing specificsingularities at punctures:ˆ J ( z | g ) = X k =0 (cid:18) k + 12 (cid:19) ( A k + zB k ) y k − dz + g ( C k + zD k ) dzy k +3 (II.22)Bi-differential for this current f J = g ( zz ′ − a ) dzdz ′ z − z ′ ) y ( z ) y ( z ′ ) (II.23)is defined by commutation relations C k = a ∂∂A k + k + 1 k + ∂∂A k +1 , D k = ∂∂B k , (II.24)At punctures it is equivalent to the bi-differentials of the basic currents: f J ( z, z ′ ) z →∞ ± ∼ f H ( z, z ′ ) f J ( z, z ′ ) z →± a ∼ f K ( ξ ± , ξ ′± ) (II.25)where ξ ± are some local coordinates in the vicinity of ramification points a and − a , defined respectivelyby z = a + ∞ X k =1 α + k ξ k + (II.26)and z = − a + ∞ X k =1 α − k ξ k − (II.27)The current (II.22) itself is equivalent to the currents from s.2:ˆ J ( z ) z →∞ ± ∼ ˆ J H ( z )ˆ J ( z ) z →± a ∼ J K ( ξ ) (II.28)Time-variables in parametrization of the global current are related to local time as follows: t k ∼ k I ˆ J z k ∂∂t k ∼ g I z k ˆ J (II.29) τ k ∼ k + 1 I ˆ J ξ k +1 ∂∂τ k ∼ g I ξ k +1 ˆ J (II.30)Global current is related to local currents by conjugation operators. Conjugation operator at infinityis U H = 2 g I ∞ I ∞ (cid:0) f J ( z, z ′ ) − f H ( z, z ′ ) (cid:1) ˆΩ H ( z ) ˆΩ H ( z ′ ) == 12 g I ∞ I ∞ ρ (0 | H ( z, z ′ ) v ( z ) v ( z ′ ) (II.31)33here ρ H is a bi-differential counterpart of the two-point function of Gaussian Hermitian model ρ (0 | H ( z, z ′ ) = 1 g (cid:0) f J ( z, z ′ ) − f H ( z, z ′ ) (cid:1) = 12( z − z ) z z − a y ( z ) y ( z ) − ! (II.32)At ramification points the conjugation operator is as followsˆ V H = 2 g X i,j = ± I a i I a j (cid:0) f J ( z, z ′ ) − δ ij f K ( ξ i , ξ j ) (cid:1) ˆΩ K ( ξ i ) ˆΩ K ( ξ j ) (II.33)To establish required Virasoro constraints one should shift the global currentˆ J → ˆ J − y ( z ) dz U H → U H + 2 g I z − y ( z )2 ˆΩ H ( z ) dz = U + 1 g I ρ (0 | GH ( z ) v ( z ) dz ˆ V H → ˆ V H + 2 g I ξ + =0 ξ dξ + − y ( z ) dz ! ˆΩ K ( ξ + ) + I ξ − =0 ξ − dξ − − y ( z ) dz ! ˆΩ K ( ξ − ) ! (II.35)Then the projector I C z − z ′ ) dz ′ : ( ˆ J ( z ′ ) + ∆ ˆ J ( z ′ )) : (II.36)with contour C encircles the segment ramification points ± a on the spectral curve (but not the point z ! so that always | z | > | z ′ | ) do the job: since1( z − z ′ ) dz ′ = X k ≥ ( z ′ ) k z k +1 dz ′ (II.37)it picks up the terms with n ≥ − z − z ′ ) dz ′ = X k ≥ ( ξ ′ ) k − ξ k +2 dξ ′ (II.38)it picks up the terms with n ≥ − Z H ( t ) = e U H e ˆ V H Z K ( τ + ) Z K ( τ − ) (II.39) Z C −→ Z K ⊗ Z BGW This decomposition formula was the topic of s.8 of ref.[1], however, it is described there in a toosketchy and partly misleading form. Thus we provide here a more detailed and careful presentation. Actually, as it was already indicated in [1], we get a whole family of such formulas, with infinite set of free parametersgiven by coefficients α ± k in (II.26), (II.27). In [1] we considered decomposition formula for the complex model, starting from the same spectral curve (II.21) asfor the Gaussian Hermitian matrix model y H = z − S (II.40)with additional puncture in z = 0 ± . However, the global current which we introduced was singular at 0 ± . Actually, innotations of [1], the proper global current should be defined on the curve y c = z ( z − S ) (II.41)It is more natural to consider instead the current on y C = ξ ( ξ − S ) (II.42)of which the previous one is a double covering z = √ ξ as we do in the present text. y C = z ( z − S ) (II.43)and the four punctures are chosen at z = 0 , S, ∞ ± . Accordingly on this curve we define the globalcurrent J ( z ) = ∞ X k =0 (cid:18) k + 12 (cid:19) ( A k + zB k ) y k − C dz + g dzy k +3 C ( C k + zD k ) (II.44)with commutation relations C k = 8 S ∂∂A k − S ∂∂B k + k + 1 k + ∂∂A k +1 D k = ∂∂B k − S ∂∂A k (II.45)and the global bi-differential f J ( z, z ′ ) = g ( zz ′ − S ( z + z ′ )) dzdz ′ z − z ′ ) y C y ′ C == g y C y ′ C + ( zz ′ − S ( z + z ′ ) + 8 S )( y C + y ′ C )2( y C − y ′ C ) y C y ′ C dzdz ′ (II.46)At punctures this bi-differential is equivalent to the following canonical bi-differentials from s.2: f J ( z, z ′ ) z →∞ ± ∼ f H ( z, z ′ ) f J ( z, z ′ ) z → S ∼ f K ( z, z ′ ) f J ( z, z ′ ) z → ∼ f K ( z, z ′ ) (II.47)The current has the following behavior: J ( z ) z →∞ ± ∼ J H ( z ) J ( z ) z → S ∼ J K ( ξ + ) J ( z ) z → ∼ J K ( ξ − ) (II.48)with ξ + , ξ − – some local coordinates in the vicinities of 4 S , 0 respectively: z = 4 S + ∞ X k =1 α + k ξ k + z = ∞ X k =1 α − k ξ k − (II.49)Time-variables of the local currents are expressed through those of the global one in the same way asin s.3: t k ∼ k I J z k ∂∂t k ∼ g I z k J (II.50) τ k ∼ k + 1 I J ξ k +1 ∂∂τ k ∼ g I ξ k +1 J (II.51)35lobal current is related to local currents through conjugation operators. Conjugation operator atinfinity U C = 2 g I ∞ I ∞ (cid:0) f J ( z, z ′ ) − f H ( z, z ′ ) (cid:1) Ω H ( z )Ω H ( z ′ ) == 12 g I ∞ I ∞ ρ (0 | C ( z, z ′ ) v ( z ) v ( z ′ ) (II.52)where ρ C is a bi-differential counterpart of the two-point function of Gaussian Hermitian model ρ (0 | C ( z, z ′ ) = 1 g (cid:0) f J ( z, z ′ ) − f H ( z, z ′ ) (cid:1) (II.53)At ramification points the conjugation operator looks as follows V C = 2 g X i,j = ± I a i I a j (cid:0) f J ( z, z ′ ) − δ ij f K ( ξ i , ξ j ) (cid:1) Ω K ( ξ i )Ω K ( ξ j ) (II.54)The shift of the current J → J − y C ( z ) dz z (II.55)corresponds to the shift of conjugation operators U H → U H + 2 g I z − y ( z )2 z Ω H ( z ) dz = U + 1 g I ρ (0 | C ( z ) v ( z ) dzV H → V H + 2 g I ξ + =0 (cid:18) ξ dξ + − y ( z ) dz z (cid:19) Ω K ( ξ + ) + I ξ − =0 (cid:18) cdξ − − y ( z ) dz z (cid:19) Ω K ( ξ − ) ! (II.56)The difference from the case of Hermitian model is that now we should get L n , n ≥ ∞ , L n , n ≥ a, L n , n ≥ − − a (II.57)Thus this time the proper projector is I C z ′ ( z − z ′ ) dz ′ : ( J ( z ′ ) + ∆ J ( z ′ )) : (II.58)an we finally obtain the decomposition formula for complex model: Z C ( t ) = e V C e U C Z K ( τ + ) ˜ Z K ( τ − ) = e V C e U C Z K ( τ + ) Z BGW ( τ − ) (II.59) Conclusion In this paper we demonstrated that decomposition formula Z H → Z K ⊗ Z K of partition functionfor Gaussian Hermitian model into two cubic Kontsevich models has as its closest analogue anotherdecomposition: Z C → Z K ⊗ Z BGW of the Gaussian complex model into the cubic Kontsevich andBrezin-Gross-Witten models. Thus all the four models are indeed the very close relatives, though thisis not quite so obvious from their original matrix-integral representations. This paper is thereforean important outcome and summary of many different approaches, worked out during the years ofdevelopment of matrix-model theory. It brings us one-step closer to providing a unified look at thewhole variety of eigenvalue models and building up the M-theory of matrix models , suggested in [2].Technically it adds to content of [1] an identification of partition function, denoted there by ˜ Z K ,with that of the very important BGW model – the generating function of all unitary-matrix correlators.36rom technical point of view the road is now open for search of two different generalizations: toDijkgraaf-Vafa models [50, 51], which are not fully specified by the Virasoro constraints alone and relyupon intriguing and under-developed theory of check-operators [49], and to more interesting unitary-matrix models with Itzykson-Zuber measures and further to Kazakov-Migdal multi-matrix models[34]-[37], important both for Yang-Mills theory and for the theory of integer partitions. Putting allthese very different problems into the same context, moreover, underlined by the well establishedtheory of free fields on Riemann surfaces [58], is a challenging and a promising perspective.Another, but, perhaps, related, open problem is direct derivation of decomposition formula (II.59)from integral representations of all the models, bypassing the Virasoro constraints and D -modulerepresentations. Note that this kind of problem remains unsolved even for the crucially importantdecomposition Z H = ˆ U ( Z K ⊗ Z K ), describing the double-scaling continuum limit of Hermitian matrixmodel. Acknowledgements A.A. is grateful to Denjoe O’Connor for his kind hospitality while this work was in progress. Our workis partly supported by Russian Federal Nuclear Energy Agency, by the Dynasty Foundation (A.A.),by the joint grants 09-02-91005-ANF, 09-02-90493-Ukr, 09-02-93105-CNRSL and 09-01-92440-CE, bythe Russian President’s Grant of Support for the Scientific Schools NSh-3035.2008.2, by RFBR grants08-01-00667 (A.A.), 07-02-00878 (A.Mir.) and 07-02-00645 (A.Mor.). References [1] A.Alexandrov, A.Mironov and A.Morozov, Physica D235 (2007) 126-167, hep-th/0608228[2] A.Alexandrov, A.Mironov and A.Morozov, hep-th/0605171[3] E.Brezin, C.Itzykson, G.Parisi and J.-B.Zuber, Comm.Math.Phys. (1978) 35;D.Bessis, Comm.Math.Phys. (1979) 147;D.Bessis, C.Itzykson and J.-B.Zuber, Adv. Appl. Math. (1980) 109;M.-L. Mehta, Comm. Math. Phys. 79 (1981) 327; Random Matrices , 2nd edition, Acad. Press.,N.Y., 1991;D.Bessis, C.Itzykson and J.-B.Zuber, Adv.Appl.Math. (1980) 109[4] A.Migdal, Phys.Rep. (1983) 199;F.David, Nucl. Phys. B257 [FS14] (1985) 45, 543;J. Ambjorn, B. Durhuus and J. Frohlich, Nucl. Phys. 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