Exact correlators in the Gaussian Hermitian matrix model
aa r X i v : . [ h e p - t h ] S e p Exact correlators in the Gaussian Hermitian matrix model
Bei Kang a , Ke Wu a , Zhao-Wen Yan b , Jie Yang a , Wei-Zhong Zhao a a School of Mathematical Sciences, Capital Normal University, Beijing 100048, China b School of Mathematical Sciences, Inner Mongolia University, Hohhot 010021, China
Abstract
We present the W ∞ constraints for the Gaussian Hermitian matrix model, where theconstructed constraint operators yield the W ∞ n -algebra. For the Virasoro constraints, wenote that the constraint operators give the null 3-algebra. With the help of our Virasoroconstraints, we derive a new effective formula for correlators in the Gaussian Hermitian matrixmodel.Keywords: Conformal and W Symmetry, Matrix Models, n -algebra The various constraints for matrix models have attracted remarkable attention, such as Virasoro/ W -constraints [1]-[5] and Ding-Iohara-Miki constraints [6, 7]. Due to the Bagger-Lambert-Gustavsson(BLG) theory of M2-branes [8, 9], n -algebra and its applications have aroused much interest [10]-[17]. In the context of matrix models, usually the Virasoro/W-constraint operators do not yieldthe closed n -algebra. Whether there exist such kind constraint operators leading to the closed n -algebra has recently been investigated for the (elliptic) Hermitian one-matrix models. By insertingthe special multi-variable realizations of the W ∞ algebra under the integral, it was found thatthe derived constraint operators for the Hermitian one-matrix model may yield the closed W ∞ ( n -)algebras [18]. For the case of the elliptic matrix model, one can obtain the constraint operatorsassociating with the q -operators [19, 20]. The situation is different from that of the Hermitianone-matrix model, since the derived constraint operators do not yield the closed algebra. However,it was shown that the ( n -)commutators of the constraint operators are compatible with the desiredgeneralized q - W ∞ ( n -)algebras once we act on the partition function [20].The partition functions of various matrix models can be obtained by acting on elementaryfunctions with exponents of the given operators. For the Gaussian Hermitian matrix model, its Corresponding author: [email protected] W − [21]. This operator is also the constraintoperator for the Hermitian one-matrix model which associates with the Lassalle operator and thepotential of the A N − -Calogero model [18]. The correlators in the Gaussian Hermitian matrixmodel have been well investigated [22]-[28]. A compact formula for correlators has been given byfinite sums over Young diagrams of a given size, which involve also the well known characters ofsymmetric group [27]. Moreover, the 2 m -fold Gaussian correlators of rank r tensors have been givenby r -linear combinations of dimensions with the Young diagrams of size m [28]. In this letter, wereinvestigate the Gaussian Hermitian matrix model and present its Virasoro/ W -constraints. Weintend to further explore the properties of the constraints and derive a new formula for correlatorsin this matrix model. W ∞ constraints for the Gaussian Hermitian matrix model Let us consider the Gaussian Hermitian matrix model Z G = Z N × N dφexp ( − trφ / ∞ X k =0 t k trφ k )= e Nt (1 + C i ( N ) t i + 12! C i i ( N ) t i t i + 13! C i i i ( N ) t i t i t i + · · · ) , (1)where the coefficients C i ··· i l ( N ) are the so-called l -point correlators, which are given by the Gaus-sian integrals C i ··· i l ( N ) = h trφ i · · · trφ i l i = Z N × N dφtrφ i · · · trφ i l exp ( − trφ ) . (2)Due to the reflection symmetry of the action trφ , when i + · · · + i l is odd, we have C i ··· i l ( N ) = 0.The partition function of the Gaussian model (1) can also be expressed as [21] Z G = ∞ X s =0 Z ( s ) G = e ˆ W − / e Nt , (3)where Z ( s ) G = e Nt ∞ X l =0 X i + ··· + i l = si , ··· ,i l > h trφ i · · · trφ i l i t i · · · t i l l ! , (4)and the operator ˆ W − is given byˆ W − = ∞ X j ,j =0 ( j j t j t j ∂∂t j + j − + ( j + j + 2) t j + j +2 ∂∂t j ∂∂t j ) . (5)2t indicates that the partition function (1) can indeed be generated by the operator ˆ W − .The action of the operator ˆ W − on Z ( s ) G leads to increase the grading in the following sense:ˆ W − Z ( s ) G = ( s + 2) Z ( s +2) G . (6)The operator preserving the grading is given by [21]ˆ D = ∞ X j =0 jt j ∂∂t j , (7)which acting on Z ( s ) G gives ˆ DZ ( s ) G = sZ ( s ) G . (8)The commutation relation between ˆ D and ˆ W − is[ ˆ D, ˆ W − ] = 2 ˆ W − . (9)Note that the actions of ˆ D and ˆ W − on Z G giveˆ DZ G = ˆ W − Z G . (10)For the operators ∂∂t and ˆ D , there is the similar commutation relation as (9)[ ˆ D, ∂∂t ] = − ∂∂t . (11)The actions of ∂∂t and ˆ D on Z G give ∂∂t Z G = ( ˆ D + N ) Z G . (12)By means of (11) and (12), it is easy to show that ∂∂t Z ( s ) G = ( s − N ) Z ( s − G . (13)In contrast with the operator ˆ W − , we see that the operator ∂∂t decreases the grading in the sense(13).Let us introduce the operators W rm = ( −
12 ) r − ( ˆ W − ) m ( ˆ W − − ˆ D ) r − , m, r ∈ N , r > , (14)which obviously satisfy W rm Z G = 0 . (15)3he remarkable property is that these constraint operators yield[ W r m , W r m ] = ( r − X k =0 C kr − m k − r − X k =0 C kr − m k ) W r + r − − km + m , (16)and n -algebra[ W r m , W r m , . . . , W r n m n ] := ǫ i i ··· i n ··· n W r i m i W r i m i · · · W r in m in = ǫ i i ··· i n ··· n β X α =0 β X α =0 · · · β n − X α n − =0 C α β C α β · · · C α n − β n − · m α i m α i · · · m α n − i n W r + ··· + r n − ( n − − α −···− α n − m + ··· + m n , (17)where C kr = r ( r − ··· ( r − k +1) k ! , β k = r i − , k = 1 , k X j =1 r i j − k − k − X i =1 α i , k n − , and ǫ i i ··· i n ··· n is given by ǫ i ··· i p j ··· j p = det δ i j · · · δ i j p ... ... δ i p j · · · δ i p j p . It is noted that (16) and (17) completely match with the W ∞ ( n -)algebras presented inRef.[18]. The W ∞ n -algebra (17) with n even is a generalized Lie algebra (or higher order Liealgebra), which satisfies the generalized Jacobi identity ǫ i i ··· i n − ··· (2 n − [[ A i , A i , · · · , A i n ] , A i n +1 , · · · , A i n − ] = 0 . (18)For the constraint operators W rm (14) with fixed r = n + 1, by taking the appropriate scalingtransformations, it is not difficult to show that these operators constitute the subalgebras[ W n +1 m , W n +1 m , . . . , W n +1 m n ] = Y j 12 ) r − ( ∂∂t ) m ( ˆ D − ∂∂t + N ) r − , m, r ∈ N , r > . (21)4traightforward calculation shows that they also yield the W ∞ algebra (16) and n -algebra (17).By carrying out the action of the operators (21) on the partition function of the Gaussian model,it gives another W ∞ constraints ˜ W rm Z G = 0 . (22) Let us first recall the correlators in the Gaussian Hermitian matrix model. Harer and Zagierpresented a generating function for exact (all-genera) 1-point correlators in the Gaussian Hermitianmatrix model [22, 23], C i ( i − x i λ N in λ − λ − λ ) − (1 + λ ) x , (23)where i is even. By using Toda integrability of the model, Morozov and Shakirov derived the2-point generalization of the Harer-Zagier 1-point function [26], C (2 k +1)(2 m +1) (2 k + 1)!!(2 m + 1)!! = coefficient of x k +1 y m +1 λ N in λ ( λ − / arctan ( xy √ λ − √ λ − λ +1)( x + y ) ) p λ − λ + 1)( x + y ) ,C (2 k )(2 m ) (2 k − m − x k y m λ N in λ ( λ + 1) x y (1 − λ ) ( λ − λ )( x + y )) − ( λ − λ )( x + y ) + ( λ − x y ) − − λ ( λ + 1) xy (1 − λ ) / ( λ − λ + 1)( x + y )) − / arctan ( xy √ λ − p λ − λ + 1)( x + y ) ) . (24)However, it should be noted that it is difficult to give the higher correlators in this way. RecentlyMironov and Morozov presented a compact formula for correlators by finite sums over Youngdiagrams of a given size [27], C i i ··· i l ( N ) ≡ O Λ = X R ⊢| Λ | d R χ R { t n = 12 δ n, } · D R ( N ) · ψ R (Λ) , (25)where Λ = { i > i > · · · > i l > } and R are the Young diagrams of the given size P k i k ,and D R ( N ), χ R { t } , ψ R (Λ) and d R are respectively the dimension of representation R for thelinear group GL ( N ), the linear character (Schur polynomial), the symmetric group character andthe dimension of representation R of the symmetric group S | R | divided by | R | !. Furthermore, arepresentation of the correlators in terms of permutations is given by [28] O σ = X R ⊢ m ϕ R ([2 m ]) · D R ( N ) · ψ R ( σ ) , (26)5here ϕ R ([2 m ]) are the symmetric group characters.Let us turn to consider the Virasoro constraints in (22)˜ W l Z G = 0 , l ∈ N . (27)The constraint operators yield the Witt algebra[ ˜ W l , ˜ W l ] = ( l − l ) ˜ W l + l , (28)and null 3-algebra [ ˜ W l , ˜ W l , ˜ W l ] = 0 . (29)When l = 0, by using the expression (21) to calculate left-hand side of (27), we obtain ∞ X i =1 it i ( C · · · | {z } l i ( N ) + ∞ X i =1 C · · · | {z } l ii ( N ) t i + 12! ∞ X i ,i =1 C · · · | {z } l ii i ( N ) t i t i + · · · )+( N + 2 l )( C · · · | {z } l ( N ) + ∞ X i =1 C · · · | {z } l i ( N ) t i + 12! ∞ X i ,i =1 C · · · | {z } l i i ( N ) t i t i + · · · ) − ( C · · · | {z } l +1 ( N ) + ∞ X i =1 C · · · | {z } l +1 i ( N ) t i + 12! ∞ X i ,i =1 C · · · | {z } l +1 i i ( N ) t i t i + · · · ) = 0 . (30)From the fact that the constant term in the left-hand side of (30) should be zero, we have C · · · | {z } l +1 ( N ) = ( N + 2 l ) C · · · | {z } l ( N ) . (31)Taking the special constraint operator ˜ W in (27), it is easy to obtain C ( N ) = N . (32)Thus from (31), we obtain C · · · | {z } l ( N ) = l − Y j =0 ( N + 2 j ) . (33)By collecting the coefficients of t i t i · · · t i k in (30) and setting to zero, we have C · · · | {z } l +1 i · · · i k ( N ) = ( N + i + · · · + i k + 2 l ) C · · · | {z } l i · · · i k ( N )= l Y j =0 ( N + i + · · · + i k + 2 j ) C i ··· i k ( N ) , l ∈ N . (34)6et us take the constraint operator W in (15), i.e.,( ˆ W − − ˆ D ) Z G = 0 . (35)After a straightforward calculation of the left-hand side of (35), we obtain ∞ X j ,j =1 ( j + j + 2) t j + j +2 ( C j j ( N ) + ∞ X i =1 C j j i ( N ) t i + 12! ∞ X i ,i =1 C j j i i ( N ) t i t i + · · · )+ t N (1 + ∞ X i =1 C i ( N ) t i + 12! ∞ X i ,i =1 C i i ( N ) t i t i + 13! ∞ X i ,i ,i =1 C i i i ( N ) t i t i t i + · · · )+2 t N (1 + ∞ X i =1 C i ( N ) t i + 12! ∞ X i ,i =1 C i i ( N ) t i t i + 13! ∞ X i ,i ,i =1 C i i i ( N ) t i t i t i + · · · )+ ∞ X j ,j =1 j + j > j j t j t j ( C j + j − ( N ) + ∞ X i =1 C j + j − ,i ( N ) t i + 12! ∞ X i ,i =1 C j + j − ,i ,i ( N ) t i t i + · · · ) + 2 ∞ X j =1 ( j + 2) t j +2 N ( C j ( N ) + ∞ X i =1 C j i ( N ) t i + 12! ∞ X i ,i =1 C j i i ( N ) t i t i + · · · ) − ∞ X j =1 j t j ( C j ( N ) + ∞ X i =1 C j i ( N ) t i + 12! ∞ X i ,i =1 C j i i ( N ) t i t i + · · · ) = 0 . (36)By collecting the coefficients of t and setting to zero, we obtain C , ( N ) = N. (37)Similarly, for the case of the coefficients of t l with l even, we have C , · · · , | {z } l ( N ) = ( l − N C , · · · , | {z } l − ( N ) . (38)Substituting (37) into the recursive relation (38), we obtain C , · · · , | {z } l ( N ) = ( l − N l , f or l even. (39)Motivated by the exact l -point correlators C , · · · , | {z } l ( N ) and C , · · · , | {z } l ( N ), we now proceed toderive the general l -point correlators C i ··· i l ( N ). Let us consider the Virasoro constraints in (15) W m Z G = 0 . (40)The constraint operators W m also yield the Witt algebra (28) and null 3-algebra (29). By meansof (9) and (10), we may rewrite (40) as( ˆ W − ) m +1 Z G = m Y j =0 ( ˆ D − j ) Z G . (41)7et us focus on the coefficients of t i t i · · · t i l with l P j =1 i j = 2( m + 1) on the both sides of (41).Note that the form of ( ˆ W − ) m +1 appears to become more complicated very rapidly as one proceedsto higher power. We may formally express the ( m + 1)-th power of ˆ W − as( ˆ W − ) m +1 = m +1) X k,l =1 ∞ X j ,j , ··· ,j k =0 X i + i + ··· + i l = ρi ,i , ··· ,i l > P i ,i , ··· ,i l j ,j , ··· ,j k t i · · · t i l ∂∂t j · · · ∂∂t j k , (42)where ρ = k P n =1 j n + 2( m + 1) and P i ,i , ··· ,i l j ,j , ··· ,j k are polynomials in i α , α = 1 , · · · , l and j β , β = 1 , · · · , k . When j β = 0 for β = 1 , · · · , k in (42), the corresponding terms acting on Z G give the coefficientsof t i t i · · · t i l with l P j =1 i j = 2( m + 1) on the left-hand side of (41) m +1) X k =1 X σ P σ ( i ) ,σ ( i ) , ··· ,σ ( i l ) , · · · , | {z } k N k e Nt , (43)where σ denotes all distinct permutations of ( i , i , · · · , i l ). By means of (8), the right-hand sideof (41) becomes m Y j =0 ( ˆ D − j ) Z G = ∞ X s =0 m Y j =0 ( s − j ) Z ( s ) G = e Nt ∞ X s,l =0 X i + ··· + i l = si , ··· ,i l > l ! m Y j =0 ( s − j ) C i ··· i l ( N ) t i · · · t i l . (44)From (44), we obtain that the coefficients of t i t i · · · t i l with l P j =1 i j = 2( m + 1) on the right-handside of (41) are e Nt X σ m +1 ( m + 1)! l ! C σ ( i ) , ··· ,σ ( i l ) ( N ) = 2 m +1 ( m + 1)! λ ( i ··· i l ) l ! e Nt C i ··· i l ( N ) , (45)where we denote by λ ( i ··· i l ) the number of distinct permutations of ( i , i , · · · , i l ).By equating (43) and (45), we obtain the l -point correlators C i ··· i l ( N ) C i ··· i l ( N ) = l !2 m +1 ( m + 1)! λ ( i ··· i l ) 2( m +1) X k =1 X σ P σ ( i ) ,σ ( i ) , ··· ,σ ( i l ) , · · · , | {z } k N k , (46)where l P j =1 i j is even and m = l P j =1 i j − 1. 8hen particularized to the 1-point correlators in (46), we have C i ( N ) = 1 √ i ( i )! i X k =1 P i , · · · , | {z } k N k . (47)Comparing (46) with (25) and (26), we see that (46) is different from the other two expressions.Hence (46) is a new formula for correlators, where the operators ( ˆ W − ) m +1 play an crucial role todeterminate the polynomials in i α , α = 1 , · · · , l in the correlators.For clarity of calculation, let us consider the m = 1 case in (46), i.e., l P j =1 i j = 4. From theexpression( ˆ W − ) = ∞ X i ,i =0 X i + i = i + i +4 ( i + i + 2)˜ t i ˜ t i ∂∂t i ∂∂t i + 2 ∞ X i ,i =0 i i ˜ t i + i +2 ∂∂t i + i − +2 ∞ X i ,i ,i =0 ( i + i − t i ˜ t i ˜ t i ∂∂t i + i + i − + 2 ∞ X i ,i ,i ,i =0 ˜ t i ˜ t i ˜ t i + i +2 ∂∂t i + i − ∂∂t i ∂∂t i + ∞ X i ,i ,i ,i =0 ˜ t i ˜ t i ˜ t i ˜ t i ∂∂t i + i − ∂∂t i + i − + ∞ X i ,i ,i ,i =0 ˜ t i + i +2 ˜ t i + i +2 ∂∂t i ∂∂t i ∂∂t i ∂∂t i +4 ∞ X i ,i ,i =0 i ˜ t i + i +2 ˜ t i ∂∂t i ∂∂t i + i − + 2 ∞ X i ,i ,i =0 ( i + i + 2)˜ t i + i + i +4 ∂∂t i ∂∂t i ∂∂t i , (48)where ˜ t j = jt j , we have P = 8 , P , , = 16 , P , , = 6 , P , , = 18 , P , , = 8 ,P , , , , = 4 , P , , , , = 1 , P , , = P , , = P , , , , = 4 . (49)Substituting (49) into (46), we obtain C ( N ) = 12 · · λ (4) ( P N + P , , N ) = 2 N + N,C , ( N ) = 2!2 · · λ (1 , ( P , , + P , , ) N = 3 N ,C , ( N ) = 2!2 · · λ (2 , ( P , , N + P , , , , N ) = N + 2 N ,C , , ( N ) = 3!2 · · λ (1 , , [( P , , + P , , ) N + P , , , , N ] = N + 2 N,C , , , ( N ) = 4!2 · · λ (1 , , , P , , , , N = 3 N , (50)where λ (4) = 1, λ (1 , = 2, λ (2 , = 1, λ (1 , , = 3 and λ (1 , , , = 1.9 Summary It is known that the partition function of the Gaussian Hermitian matrix model can be obtainedby acting on an elementary function with exponent of the operator ˆ W − . This operator increasesthe grading in the sense (6). Based on the operators ˆ W − and ˆ D preserving the grading, we haveconstructed the W ∞ constraints (15) for the Gaussian model, where the constraint operators yieldnot only the W ∞ algebra, but also the closed W ∞ n -algebra. In contrast with the operatorˆ W − , we observed that the operator ∂∂t decreases the grading in the sense (13). Another W ∞ constraints (22) for the Gaussian model have been presented in terms of the operators ∂∂t and ˆ D ,where the constraint operators also constitute the closed W ∞ ( n -)algebras. When particularizedto the Virasoro constraints in (15) and (22), respectively, the corresponding constraint operatorsgive the null 3-algebra.Based on the Virasoro constraints (27), we have presented the exact correlators C ··· ( N ) (33).However, it appears to be impossible to obtain arbitrary correlators from (27). With the helpof another Virasoro constraints (40), we have derived a new formula (46) for correlators in theGaussian Hermitian matrix model. Our results confirm that the constraint operators which leadto the higher algebraic structures provide new insight into the matrix models. Acknowledgments We would like to thank the referee for his/her helpful comments. This work is supported by theNational Natural Science Foundation of China (Nos. 11875194, 11871350 and 11605096).