A note on the extended dToda hierarchy
aa r X i v : . [ n li n . S I] D ec A note on the extended dToda hierarchy
Niann-Chern Lee and Ming-Hsien Tu ∗ General Education Center, National Chin-Yi University of Technology , Taichung 411, Taiwan Department of Physics, National Chung Cheng University , Chiayi 621, Taiwan
July 3, 2018
Abstract
We give a derivation of dispersionless Hirota equations for the extended dispersionlessToda hierarchy. We show that the dispersionless Hirota equations are nothing but a directconsequence of the genus-zero topological recursion relation for the topological CP model.Using the dispersionless Hirota equations we compute the two point functions and express theresult in terms of Catalan number. Keywords: extended dToda hierarchy, dispersionless Hirota equation, Catalan number, topologicalfield theory. ∗ [email protected] Introduction
Recently, Kodama and Pierce[12] gave a combinatorial description of the one-dimensional disper-sionless Toda(dToda) hierarchy to solve the two-vertex problem on a sphere. The main strategy isto characterize the free energy F ( t , t ) ( t = ( t , t , · · · )) of the dToda hierarchy by the correspond-ing dispersionless Hirota equations. Then the second derivatives of the free energy ∂ t n ∂ t m F ≡ F n,m satisfy a set of algebraic relations. Surprisingly they found a closed form for the rational numbers F n,m under the conditions F = F = 0 for general n and m . In particular, the formulas of F n,m can be expressed in terms of the Catalan number which is commonly used in the context ofenumerative combinatorics (see e.g. [18]). Their result for F n,m provides a combinatorial meaningof a counting problem of connected ribbon graphs with two vertices of degree n and m on a sphereand is a generalization of the previous works where the problem has been solved only in the caseof the same degree (that is F nn ) [10, 15].In this work, motivated by the aforementioned result, we like to generalize the computationof the two point functions F n,m to the extended dToda hierarchy[8, 7, 5, 6] which is an extensionof the one-dimensional dToda hierarchy by adding logarithmic type conserved densities. Sinceextended dToda hierarchy is the dispersionless limit of the extended Toda hierarchy [23, 2] whichhas been used to govern the Gromov-Witten(GW) invariants(see e.g. [9] and references therein)for the CP manifold. Thus the extended dToda hierarchy becomes the master equation of thegenus zero GW invariants whose generating function is characterized by the free energy of theextended dToda hierarchy. Based on the twistor theoretical method [20, 11] the extended dTodahierarchy can be constructed by adding logarithmic-flow to the one-dimensional dToda hierarchy.The corresponding Orlov-Schulman operator is conjugated with the Lax operator under the Pois-son bracket which imposes an extra condition (the so-called string equation) on the free energyof the extended dToda hierarchy. We will show that the full hierarchy flows can be expressed interms of second derivatives of its associated free energy F and thus can be viewed as the corre-sponding dispersionless Hirota(dHirota) equations. We then investigate the two point functions ofthe extended dToda hierarchy based on the associated dHirota equations and express the result interms of the Catalan number. To make a connection with the topological field theory, we rewritethe dHirota equations in CP time parameters and show that they are indeed a direct consequenceof the genu-zero topological recursion relation[21] of the topological CP model.This paper is organized as follows. In section 2, we recall the Lax formalism of the extendeddToda hierarchy. In section 3, we derive the dHirota equation of the extended dToda hierarchywhich can be expressed as a set of equations in terms of second derivatives of the free energy. Theinitial values of two-point functions of the extended dToda hierarchy are computed in Section 4. Insection 5, we reinterpret the dHirota equations from topological field theory point of view. Section6 is devoted to the concluding remarks. 2 The extended dispersionless Toda hierarchy
The one-dimensional dToda hierarchy[20, 12] is defined by the Lax equation ∂L∂t n = { B n , L } , B n = ( L n ) ≥ . where L ia a two-variable Lax operator of the form L = p + u + u p − (1)with u and u are functions of the time variables t = ( t , t , . . . ) along with a spatial variable t .Here ( A ) ≥ denotes the polynomial part of A , ( A ) ≤− = A − ( A ) ≥ , and the Poisson bracket { , } is defined by { A ( p, t ) , B ( p, t ) } = p ∂A ( p, t ) ∂p ∂B ( p, t ) ∂t − p ∂A ( p, t ) ∂t ∂B ( p, t ) ∂p . In particular, the fundamental variable u and u can be expressed in terms of second derivativesof F as u = F , u = F = e F where the second equation is just the one-dimensional reduction of the dToda field equation.Following the twistor theoretical construction [20, 11], the extended dToda hierarchy can be con-structed from the one-dimensional dToda hierarchy by adding the ˆ t n -flows as ∂L∂ ˆ t n = { ˆ B n , L } , ˆ B n = ( L n (log L − d n )) ≥ (2)where d n = P nj =1 /j with d ≡ L is defined by the prescriptionlog L = 12 log u + 12 log(1 + u p − + u p − ) + 12 log (cid:18) u u p + 1 u p (cid:19) (3)with the proviso that we shall Taylor expand the second term in p − , whereas in p for the lastterm. Moreover, the associated Orlov-Schulman is given by N ( t , t, ˆ t ) = X n =1 nt n L n + t + X n =1 n ˆ t n L n (log L − d n − ) + X n =1 F n L − n , which satisfies ∂ t n N = { B n , N } , ∂ ˆ t n N = { ˆ B n , N } , { L, N } = L. The symplectic two-form of the extended dToda hierarchy can be written as ω ≡ dpp ∧ dt + ∞ X n =1 dB n ∧ dt n + ∞ X n =1 d ˆ B n ∧ d ˆ t n = dL ∧ dNL which implies the existence of a S function such that dS ( t , t, ˆ t ) = N d log L + log pdt + ∞ X n =1 B n dt n + ∞ X n =1 ˆ B n d ˆ t n N = ∂S∂ log L , log p = ∂S∂t , B n = ∂S∂t n , ˆ B n = ∂S∂ ˆ t n . It is not hard to show that the S function has the form S = X n =1 t n L n + t log L + X n =1 ˆ t n L n (log L − d n ) − X n =1 F n n L − n . Setting ˆ t n = 0 for n ≥
1, it recovers the S function of the one-dimensional dToda hierarchy. Finally,the twistor construction[11] enables us to extract the string equation − ∞ X n =2 nt n ∂L∂t n − + ∞ X n =1 n ˆ t n ∂L∂ ˆ t n − . (4)for the extended dToda hierarchy without referring to the CP matrix model[8]. Proposition 1.
The following relations hold. F n = ( B n ) [0] , F n = res( L n ) , n ≥ F b n = ( b B n ) [0] , F b n = res( L n (log L − d n )) , n ≥ where ( P k a k p k ) [ j ] = a j .Proof. From log p = ∂S/∂t we havelog p = log L − X n =1 F n n L − n (7)or L = pe P n =1 F n L − n = p + F + (cid:18) − u F + 12 F + 12 ( F ) (cid:19) p − + O ( p − )which yields u = F and u = F − ( F ) . Therefore, from the p -term of the Lax equation(2) we have F n = ( B n ) [0] . On the other hand, from B n = ∂S/∂t n we have B n = L n + = L n − X m =1 F nm m L − m . For n = 1, we have u p − = P m =1 F m L − m /m which together with (7) implies F m = mu P m − ( F j /j )where P m ( t ) are Schur polynomials defined by e P j =1 t j z j = P j =0 P j ( t ) z j . In particular, u = e F = F . Also, for the p − -term of the Lax equation (2) we have F n = res( L n ) = u ( B n ) [1] .Furthermore, from ˆ B n = ∂S/∂ ˆ t n we haveˆ B n = [ L n (log L − d n )] + = L n (log L − d n ) − X m =1 F ˆ nm m L − m . The p − -term gives F ˆ n = res( L n (log L − d n )) = u ( L n (log L − d n )) [1] where the last equality isdue to the the identity res( L n (log L − d n ) dL ) = 0. Finally, from the p -term of the Lax equation(2) we have F ˆ n = ( ˆ B n ) [0] . 4 roposition 2. The two point functions F n , F n , F ˆ n , and F ˆ n can be expressed in terms of F and F as follows F n = [ n ] X s =0 n ! s ! s !( n − s )! F n − s e sF (8) F n +1 , = [ n ] X s =0 ( n + 1)! s !( s + 1)!( n − s )! F n − s e ( s +1) F (9) F b n = 12 [ n ] X s =0 n ! s ! s !( n − s )! F n − s e sF ( F − d s ) (10) F [ n +1 , = 12 [ n ] X s =0 ( n + 1)! s !( s + 1)!( n − s )! F n − s e ( s +1) F (cid:18) F − d s − s + 1 (cid:19) . (11) Proof.
Using the binomial expansion of powers of L in (5) and the Taylor expansion in (6) withthe prescription (3) for log L .We come now to the main result of the work; that is to derive the dHirota equation for theextended dToda hierarchy from the Lax formulation. The result will be expressed in terms ofsecond derivatives of the free energy F ( t , t, ˆ t ). Theorem 3.
The free energy F ( t , t, ˆ t ) of the extended dToda hierarchy satisfies the followingequations F n +1 ,m n + 1 + F n,m +1 m + 1 = F m, F n, + F m, F n, , ( n ≥ , m ≥
1) (12) F [ n +1 ,m n + 1 + F ˆ n,m +1 m + 1 = F m, F ˆ n, + F m, F ˆ n, , ( m ≥ , n ≥
0) (13) F [ n +1 , b m n + 1 + F ˆ n, [ m +1 m + 1 = F b m, F ˆ n, + F b m, F ˆ n, , ( m, n ≥ . (14) Proof.
To prove (12), we note that F m,n +1 , = ∂F n +1 , ∂t m = ∂ ( L n +1 ) [0] ∂t m = ( n + 1) (cid:18) L n ∂L∂t m (cid:19) [0] = ( n + 1) (cid:18) ( B n ) [0] ∂u ∂t m + ( B n ) [1] ∂u ∂t m (cid:19) = ( n + 1)( F n, F m, , + F m, , F n, )where u = F and u = e F have been used to reach the last equality. Similarly, we have F n,m +1 , = ( m + 1)( F m, F n, , + F n, , F m, ) . Hence F n +1 ,m n + 1 + F n,m +1 m + 1 = F m, F n, + F m, F n, , ( n ≥ , m ≥ . Equations (13) and (14) can be verified in a similar manner.5 orollary 4.
The two point functions F mn , F b mn , and F b m b n are all determined by the fundamentalvariables F and F .Proof. This is just an immediate consequence of Proposition 2 and Theorem 3.We shall show later on that the expression of (12)-(14) has a simple interpretation from topo-logical field theory.
From dispersionless Hirota equations (12)-(14), we see that the building blocks are the two-pointfunctions (5) and (6). Motivated by the work of Kodama and Pierce [12] we like to consider thetwo-point functions F mn , F b mn , and F b m b n in the case with F = F = 0. Proposition 5. F k, = ( k + 1) C k , F k +1 , = 0 (15) F k +1 , = (2 k + 1) C k , F k, = 0 (16) F c k, = − ( k + 1) d k C k , F \ k +1 , = 0 (17) F \ k +1 , = − (2 k + 1) (cid:18) d k + 12( k + 1) (cid:19) C k , F c k, = 0 (18) where C k is the k -th Catalan number defined by C k = 1 k + 1 (cid:18) kk (cid:19) . Proof.
This is an immediate consequence by setting u = F = 0 and u = F = e F = 1 in theequations (8)-(11).Let us derive the two point functions F n,m from the dHirota equation (12). Theorem 6. (Kodama and Pierce[12]) The two point function F nm for the extended dToda hier-archy with F = 0 and F = 0 are given by F k, = ( k + 1) C k , k = 1 , , · · · F k +1 , l +1 = (2 l + 1)(2 k + 1)( l + 1)( k + 1) l + k + 1 C k C l , k, l = 0 , , , · · · F k, l = lk ( l + 1)( k + 1) l + k C k C l k, l = 1 , , · · · F nm = 0 , otherwise where C k is the k -th Catalan number. roof. Here we present a derivation of F k, l from the dHirota equation (12). Writing F k, l in theexpression F k, l = ( F k, l + 2 l k + 1 F k +1 , l − ) − l k + 1 ( F k +1 , l − + 2 l − k + 2 F k +2 , l − )+ 2 l (2 l − k + 1)(2 k + 2) ( F k +2 , l − + 2 l − k + 3 F k +3 , l − ) + · · · + 2 l (2 l − · · · k + 1)(2 k + 2) · · · (2 k + 2 l −
2) ( F k +2 l − , + 22 k + 2 l − F k +2 l − , ) − l !(2 k + 1)(2 k + 2) · · · (2 k + 2 l − F k +2 l − , . Then, using the dHirota equation (12), we have F k, l = 2 l ( F k, F l − , + F k, F l − , ) − l (2 l − k + 1 ( F k +1 , F l − , + F k +1 , F l − , )+ 2 l (2 l − l − k + 1)(2 k + 2) ( F k +2 , F l − , + F k +2 , F l − , ) + · · ·− l !(2 k + 1)(2 k + 2) · · · (2 k + 2 l − F k +2 l − , . Taking into account (15) and (16) we get F k, l = 2 l (( k + 1) C k (2 l − C l − ) − l (2 l − k + 1 ((2 k + 1) C k lC l − )+ 2 l (2 l − l − k + 1)(2 k + 2) (( k + 2) C k +1 (2 l − C l − ) + · · · + (2 l )!(2 k + 1)(2 k + 2) · · · (2 k + 2 l −
2) (( k + l ) C k + l − ) − (2 l )!(2 k + 1)(2 k + 2) · · · (2 k + 2 l −
1) ((2 k + 2 l − C k + l − )= l − X i =0 (2 l )!(2 k )!(2 l − i − k + 2 i )! [( k + i + 1) − ( l − i )] C k + i C l − i − = l ! k !( l + 1)!( k + 1)! C k C l ( k + l )!( k + l − l − X i =0 (cid:18) k + l − i (cid:19) (cid:20)(cid:18) k + li + 1 (cid:19) − (cid:18) k + li (cid:19)(cid:21) where the formula C k + p = 2 p (2 k + 2 p − k + 1)!(2 k − k + p + 1)! C k for the Catalan numbers has been used. Since for any p >
1, we have p − X i =0 (cid:18) k + l − i (cid:19) (cid:20)(cid:18) k + li + 1 (cid:19) − (cid:18) k + li (cid:19)(cid:21) = p X i =1 (cid:18) k + l − i − (cid:19) (cid:20)(cid:18) k + li (cid:19) − (cid:18) k + li − (cid:19)(cid:21) = p X i =2 (cid:18) k + l − i − (cid:19) (cid:20)(cid:18) k + l − i (cid:19) + (cid:18) k + l − i − (cid:19) − (cid:18) k + l − i − (cid:19) − (cid:18) k + l − i − (cid:19)(cid:21) + ( k + l − (cid:18) k + l − p − (cid:19)(cid:18) k + l − p (cid:19) (cid:0) ab (cid:1) = (cid:0) a − b (cid:1) + (cid:0) a − b − (cid:1) has been used to reach the second equality.Hence, F k, l = l ! k !( l + 1)!( k + 1)! C k C l ( k + l )!( k + l − (cid:18) k + l − l − (cid:19)(cid:18) k + l − l (cid:19) = lk ( l + 1)( k + 1) k + l C k C l . Substituting F k, l into (12) for n = 2 k and m = 2 l + 1, we obtain F k +1 , l +1 = (2 l + 1)(2 k + 1)( l + 1)( k + 1) k + l + 1 C k C l . This is just the result obtained by Kodama and Pierce in [12].
Corollary 7.
The two point functions F mn for m, n ≥ are positive-defined, i.e. F mn ≥ . Next we deal with the two point function F b nm . Theorem 8.
The two point function F b nm for the extended dToda hierarchy with F = 0 and F = 0 are given by F c k, = − ( k + 1) d k C k , k = 1 , , · · · F \ k +1 , l +1 = − (2 l + 1)(2 k + 1)( l + 1)( k + 1) l + k + 1 (cid:18) d k + 12( l + k + 1) (cid:19) C k C l , k, l = 0 , , , · · · F c k, l = − lk ( l + 1)( k + 1) l + k (cid:18) d k − l k ( l + k ) (cid:19) C k C j k, l = 1 , , · · · F b nm = 0 , otherwise . Proof.
Following the same procedure by using the dHirota equation (13) and (15)-(18), we have F c k, l = − l − X i =0 (2 l )!(2 k )! C k + i C l − i − (2 l − i − k + 2 i )! (cid:20) ( k + i + 1) d k + i − ( l − i ) (cid:18) d k + i + 12( k + i + 1) (cid:19)(cid:21) = ( I ) + ( II )where ( I ) = − k ! l !( k + 1)!( l + 1)! C k C l l − X i =0 ( k − l + 2 i + 1) d k ( l − i )!( l − i − k + i + 1)!( k + i )!( II ) = − k ! l !( k + 1)!( l + 1)! C k C l l − X i =1 ( k − l + 2 i + 1)( k + i + · · · + k +1 ) − ( l − i )2( k + i +1) ( l − i )!( l − i − k + i + 1)!( k + i )! . Part (I) can be computed as before and it gives( I ) = − lk ( l + 1)( k + 1) k + l d k C k C l . − lk ( l + 1)( k + 1) k + l C k C l k + l ) P l − i =1 (cid:0) k + l − i − (cid:1) [ (cid:0) k + li (cid:1) − (cid:0) k + li − (cid:1) ]( k + l − i + · · · + k +1 ) − P l − i =0 (cid:0) k + li (cid:1) k + l ) (cid:0) k + l − l (cid:1)(cid:0) k + l − k (cid:1) where the first summation of the numerator in the bracket can be simplified as l − X i =1 (cid:18) k + l − i − (cid:19) (cid:20)(cid:18) k + li (cid:19) − (cid:18) k + li − (cid:19)(cid:21) (cid:18) k + l − i + · · · + 1 k + 1 (cid:19) = 1 k + 1 l − X i =1 (cid:18) k + l − i − (cid:19) (cid:20)(cid:18) k + li (cid:19) − (cid:18) k + li − (cid:19)(cid:21) + 1 k + 2 l − X i =1 (cid:18) k + l − i − (cid:19) (cid:20)(cid:18) k + li (cid:19) − (cid:18) k + li − (cid:19)(cid:21) + · · · + 1 k + l − X i =1 (cid:18) k + l − i − (cid:19) (cid:20)(cid:18) k + li (cid:19) − (cid:18) k + li − (cid:19)(cid:21) = 1 k + 1 (cid:18) k + l − l − (cid:19)(cid:18) k + l − l − (cid:19) + 1 k + 2 (cid:18) k + l − l − (cid:19)(cid:18) k + l − l − (cid:19) + · · · + 1 k + l − (cid:18) k + l − (cid:19)(cid:18) k + l − (cid:19) = l − X i =1 k + l − i (cid:18) k + l − i − (cid:19)(cid:18) k + l − i (cid:19) . Hence the numerator in the bracket is given by2( k + l ) l − X i =1 k + l − i (cid:18) k + l − i − (cid:19)(cid:18) k + l − i (cid:19) − l − X i =0 (cid:18) k + li (cid:19) = l − X i =1 " (cid:18) k + l − i − (cid:19)(cid:18) k + li (cid:19) − (cid:18) k + li (cid:19) − (cid:18) k + l (cid:19) = l − X i =1 (cid:18) k + li (cid:19) (cid:20) (cid:18) k + l − i − (cid:19) − (cid:18) k + l − i (cid:19) − (cid:18) k + l − i − (cid:19)(cid:21) + 1= l − X i =1 (cid:20)(cid:18) k + l − i (cid:19) + (cid:18) k + l − i − (cid:19)(cid:21) (cid:20)(cid:18) k + l − i − (cid:19) − (cid:18) k + l − i (cid:19)(cid:21) + 1= − (cid:18) k + l − l − (cid:19) which implies F c k, l = − lk ( l + 1)( k + 1) l + k (cid:18) d k − l k ( l + k ) (cid:19) C k C j . Substituting F c k, l into (13) for n = 2 k and m = 2 l + 1, we obtain F \ k +1 , l +1 = − (2 l + 1)(2 k + 1)( l + 1)( k + 1) k + l + 1 (cid:18) d k + 12( l + k + 1) (cid:19) C k C l . orollary 9. The two point functions F b mn for mn = 0 are negative-defined, i.e. F b mn < .Proof. The only case to be considered is F c k, l in which d k − l k ( l + k ) = (cid:18) d k − k (cid:19) + 12( k + l ) > . Finally, we compute the two point function F b n b m . Theorem 10.
The two point function F b n b m for the extended dToda hierarchy with F = 0 and F = 0 are given by F c k, b = − k + 12 d k C k , k = 1 , , · · · F \ k +1 , [ l +1 = (2 l + 1)(2 k + 1)( l + 1)( k + 1) l + k + 1 × (cid:20)(cid:18) d k + 12( l + k + 1) (cid:19) (cid:18) d l + 12( l + k + 1) (cid:19) + 14( k + l + 1) (cid:21) C k C l , k, l = 0 , , , · · · F c k, b l = lk ( l + 1)( k + 1) l + k (cid:20)(cid:18) d k − l k ( l + k ) (cid:19) (cid:18) d l − k l ( l + k ) (cid:19) + 14( k + l ) (cid:21) C k C l , k, l = 1 , , · · · F b n b m = 0 , otherwise . Proof.
Using the dHirota equation (14) and taking into account (17)-(18) we have F c k, b l = l − X i =0 (2 l )!(2 k )! C k + i C l − i − (2 l − i − k + 2 i )! × (cid:20) ( k + i + 1) d k + i ( d l − i − + 12( l − i ) ) − ( d k + i + 12( k + i + 1) )( l − i ) d l − i − (cid:21) − l !2 k !2(2 k + 2 l )! ( k + l + 1) d k + l C k + l = l ! k !( l + 1)!( k + 1)! C k C l k + l )!) [( I ) + ( II ) + ( III ) + ( IV ) + ( V )]where ( I ) = 2(( k + l )!) l − X i =0 h ( k − l + 2 i + 1)( d k + ( k + i + · · · + k +1 )) − ( l − i )2( k + i +1) i d l ( l − i )!( l − i − k + i + 1)!( k + i )!( II ) = 2(( k + l )!) l − X i =0 h − ( k − l + 2 i + 1)( l − i + · · · + l ) + ( k + i +1)2( l − i ) i d k ( l − i )!( l − i − k + i + 1)!( k + i )!( III ) = 2(( k + l )!) l − X i =0 h − ( k − l + 2 i + 1)( k + i + · · · + k +1 )( l − i + · · · + l ) i ( l − i )!( l − i − k + i + 1)!( k + i )!( IV ) = 2(( k + l )!) l − X i =0 h ( k + i +1)2( l − i ) ( k + i + · · · + k +1 ) + ( l − i )2( k + i +1) ( l − i + · · · + l ) i ( l − i )!( l − i − k + i + 1)!( k + i )!10 V ) = − d k + l . Each term can be calculated as follows:( I ) = 2( k + l ) (cid:18) k + l − k (cid:19)(cid:18) k + l − l (cid:19) (cid:20) d k − l k ( l + k ) (cid:21) d l ( II ) = d k − k + l ) (cid:18) k + l − k (cid:19)(cid:18) k + l − l (cid:19) (cid:20) d k k l ( k + l ) (cid:21) ( III ) = − l − X j =0 j + 1 j X i =0 (cid:18) k + li (cid:19) − l − X j =1 k + l − j j X i =1 (cid:18) k + li (cid:19) + 1 k + l (cid:18) k + l − l − (cid:19)(cid:18) k + l − l (cid:19) + l − X i =0 k + l − i ( IV ) = l − X j =1 k + l − j j X i =1 (cid:18) k + li (cid:19) + l − X j =0 j + 1 j X i =0 (cid:18) k + li (cid:19) ( V ) = − d k − l − X i =0 k + l − i . It turns out that F c k, b l = lk ( l + 1)( k + 1) l + k (cid:20)(cid:18) d k − l k ( l + k ) (cid:19) (cid:18) d l − k l ( l + k ) (cid:19) + 14( k + l ) (cid:21) C k C l . Substituting F c k, b l into (14) for n = 2 k and m = 2 l + 1, we obtain F \ k +1 , [ l +1 = (2 l + 1)(2 k + 1)( l + 1)( k + 1) l + k + 1 × (cid:20)(cid:18) d k + 12( l + k + 1) (cid:19) (cid:18) d l + 12( l + k + 1) (cid:19) + 14( k + l + 1) (cid:21) C k C l . Corollary 11.
The two point functions F b m b n for mn = 0 are positive-defined, i.e. F b m b n ≥ . C P model The relationship between integrable systems and topological field theories has dramatic advancesin the past two decades(see e.g. [1, 3, 4, 13, 14, 21, 22]). For the extended dToda hierarchythe corresponding topological field is described by two primary fields (or observables) {O = 1 ∈ H ( CP ) , O = ω ∈ H ( CP ) } with coupling parameters T α, , α = 1 ,
2. When the theory couplesto topological gravity, a set of new variables emerge as gravitational descendants { σ n ( O α ) } withnew coupling constants { T α,n } . The identity operator now becomes the puncture operator O = P and we also denote O = Q . The space spanned by { T α,n , n = 0 , , , · · · } is called the full phase11pace and the subspace parametrized by T α, the small phase space. The generating function ofcorrelation function is the full free energy defined by F ( T ) = X g =0 F g = ∞ X g =0 h e P α,n T α,n σ n ( O α ) i g . Since the free energy F ( t , t, ˆ t ) of the extended dToda hierarchy corresponds to the genu-zerogenerating function F of CP under the identification t n +1 = T ,n ( n + 1)! , ˆ t n = 2 T ,n n ! , n ≥ . where ˆ t = 2 t = 2 T , = 2 x . Hence a generic genus-zero m -point correlation function can becalculated as follows h σ n ( O α ) σ n ( O α ) · · · σ n m ( O α m ) i = ∂ m F∂T α ,n ∂T α ,n · · · ∂T α m ,n m . In particular, the metric on the space of primary fields is defined by three-point correlation function η αβ = h P O α O β i with η = η = 0 and η = η = 1, and hence O = O and O = O .The Lax equations of the extended dToda hierarchy can be written as ∂L∂T α,n = { B α,n , L } , α = 1 , n = 0 , , , · · · where B ,n = 2 n ! ( L n (log L − d n )) ≥ , B ,n = 1( n + 1)! ( L n +1 ) ≥ and the string equation (4) becomes0 = 1 + ∞ X n =1 T ,n ∂L∂T ,n − + ∞ X n =1 T ,n ∂L∂T ,n − . Shifting the variable T , → T , − ∂L∂T , = 1 + ∞ X n =1 T ,n ∂L∂T ,n − + ∞ X n =1 T ,n ∂L∂T ,n − which, after extracting the p term, yields t ( T ) = T , + X α ∞ X n =1 T α,n h σ n − ( O α ) Q i ,t ( T ) = T , + X α ∞ X n =1 T α,n h σ n − ( O α ) P i where we identify the flat coordinate t α = h P O α i as t = u = h P Q i , t = log u = h P P i . t = T , and t = T , . The condition F = F = 0 then correspondsto T α,n = 0 , ∀ α, n . In the Landau-Ginzburg formulation of the topological CP model, it can beshown [21] that the following genus-zero topological recursion relation holds. h σ n ( O α ) XY i = X β h σ n − ( O α ) O β ihO β XY i . (19) Proposition 12.
The genus-zero topological recursion relation (19) implies the dHirota equations(12)-(14).Proof.
Using (19) we have ∂∂T , [ h σ n +1 ( Q ) σ m ( Q ) i + h σ n ( Q ) σ m +1 ( Q ) i ]= h σ n +1 ( Q ) σ m ( Q ) P i + h σ n ( Q ) σ m +1 ( Q ) P i = ∂∂T , [ h σ n ( Q ) Q ih σ m ( Q ) P i + h σ n ( Q ) P ih σ m ( Q ) Q i ]which, after integrating over T , , implies h σ n +1 ( Q ) σ m ( Q ) i + h σ n ( Q ) σ m +1 ( Q ) i = h σ n ( Q ) Q ih σ m ( Q ) P i + h σ n ( Q ) P ih σ m ( Q ) Q i . Similarly, we have h σ n +1 ( P ) σ m ( Q ) i + h σ n ( P ) σ m +1 ( Q ) i = h σ n ( P ) Q ih σ m ( Q ) P i + h σ n ( P ) P ih σ m ( Q ) Q i , h σ n +1 ( P ) σ m ( P ) i + h σ n ( P ) σ m +1 ( P ) i = h σ n ( P ) Q ih σ m ( P ) P i + h σ n ( P ) P ih σ m ( P ) Q i . The proof is completed by noting the following identifications: h σ m ( P ) σ n ( P ) i = 4 F ˆ m ˆ n m ! n ! , m, n ≥ h σ m ( P ) σ n − ( Q ) i = 2 F ˆ mn m ! n ! , m ≥ , n ≥ h σ m − ( Q ) σ n − ( Q ) i = F mn m ! n ! , m, n ≥ . We thus show that the integrable structure associated with the genus-zero topological CP model is the extended dToda hierarchy. Furthermore, integrating the two point functions h σ n ( P ) P i and h σ n ( Q ) P i over T , we obtain the one-point functions h σ n ( P ) i = 2( n + 1)! F [ n +1 , , h σ n ( Q ) i = 1( n + 2)! F n +2 , . In particular, their values in the limit of zero couplings ( T α,n = 0 ∀ α, n ) are h σ k − ( P ) i = − d k ( k !) , h σ k − ( Q ) i = 1( k !) . Concluding remarks
We have introduced the extended dToda hierarchy from the one-dimensional dToda hierarchy byadding logarithmic flows. The full hierarchy equations of the extended dToda system can besummarized by a set of dHirota equations which involve second derivatives of the free energy F in time parameters t , t n and ˆ t n . Based on these dHirota equations we computed the two pointfunctions F n,m , F ˆ n,m , and F ˆ n, ˆ m in the case with F = F = 0. Our results extend the previousformula obtained by Kodama and Pierce for the one-dimensional dToda system to those results forthe extended dToda system. Furthermore, we have shown that, in terms of CP time parameters,the dHirota equations are nothing but a direct consequence of the genus-zero topological recursionrelations. This provides another route to realize that the integrable structure associated with thetopological CP model at genus-zero level is the extended dToda hierarchy.There are two remarks in order. First, Milanov [17] has studied the Hirota quadratic equationsassociated with the extended Toda hierarchy by constructing some vertex operators taking valuesin the algebra of differential operators on the affine line. The peculiar properties of these Hirotaequations have been studied in some recent works [16, 19]. It would be interesting to investigatethe dispersionless limit of the Hirota quadratic equations. Second, in [12] a combinatorial meaningof the two point functions F nm has been investigated from large- N expansion of unitary ensembleof random matrices. It is quite natural to ask how to realize the geometric/topological meaningof the rational numbers F ˆ n,m and F ˆ n, ˆ m from the CP matrix integral[8] which contains extralogarithmic terms. We hope to back to all these issues in our future works. Acknowledgments
We like to thank H.F.Shen for useful discussions. This work is partially supported by the NationalScience Council of Taiwan under Grant No. NSC99-2115-M-167-001(NCL) and NSC100-2112-M-194-002-MY3(MHT).
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