Negative Generators of the Virasoro Constraints for the BKP Hierarchy

NEGA TIVE GENERA TORS OF THE VIRASOR O CONSTRAINTS F OR THE BKP HIERA R CHY JINGSONG HE †‡ , KELEI TIAN † , ANGELA F O ERSTER ‡ † Dep artment of Mathematics, USTC, Hefei, 230026 Anhui, P. R. Chi na ‡ Institu to de F ´ ısic a da UFRGS, Av. Bento Gon¸ calve s 9500, Porto Ale gr e, RS - Br azil Abstract. W e g ive a str aightforw ard deriv a tion of the string equation and Viras oro con- straints on the τ function of the BKP hierarch y b y means of some s pe c ial additional sy mmetr y flows. The explicit forms of the ac tio ns of these additiona l symmetry flows on the wav e function and then the negative Viras o ro genera tors L − k are given, where k is a p o sitive in teger. Keyw ords: BKP hierarch y , a dditio nal symmetries, Virasoro constraints Mathematics Sub ject Class ification(2000) : 17B80, 3 7 K05, 37K10 P A CS(2003) : 02.30.Ik 1. Introduction Since its intro duction in a ve ry con v enien t form in 198 6 [1] a nd stim ulated b y the imp orta nce of the string equation [2], muc h atten tion has b e en paid to the study of the additional sym- metries and the Virasoro constrain ts [3– 5] for the Kadom tsev-P etviash vili(KP) hierarch y [6, 7]. F or this hierarc hy , it is no w w ell established that t here are t w o represen tations of the additional symmetries , i.e. Sa to v ertex op erato r form [6] and Orlov -Sc h ulmann(OS) M-op erato r form [1]. In 1994, they w ere pro v ed to b e equiv alent by the action on the w av e functions of the KP hierarc hy in the t w o differen t f orms [8, 9]. In this pro cess, Adler- Shio t a-v an Mo erb ek e(ASvM) form ula plays a crucial role. Almost at the same time Dick ey presen t ed a ve ry elegant and compact pro of of ASvM f o rm ula [10] based on the Lax op e rator L a nd OS’s M op erator. In addition he also [1 1] deriv ed the string equation, the action of the additional symmetries on the τ function and the Virasoro constraints of t he KP hierarch y . The BKP hierarc hy [6, 12] is a reductional sub-hierarch y o f the KP with a restriction on the Lax op erator L ∗ = − ∂ L∂ − 1 (here ∗ stands for a fo rmal adjoint op eration, L is a Lax op erator of the BKP hierarc h y). Therefore, it w as natural t o expect intensiv e inv estigations on the additional sym metry and its asso ciated structures of the BKP hierarc h y after the disco v ery of this kind of symmetries on the KP hierarch y . In this con text, Johan [13, 14] has obtained the Virasoro constraints on the τ function a nd ASvM form ula of the BKP hierarch y b y an a lge- braic metho d. T ak asaki [15] found a ppropriate restrictions on the generators of the additional symmetries for the BKP hierarch y . V ery recen tly , by using T ak asaki’s result [15] and Dic ke y’s metho d [11], T u [16] has giv en an explicit form of the generators of the additional symmetries, and then a n alternativ e pro of of the ASvM form ula for the BKP hierarch y . This new pro of is more simpler a nd transparen t in comparison with the a lg ebraic metho d presen ted in [14]. Here it is imp orta n t to men tion that due to the reductional conditions of the BKP hierarc h y man y differences in relation to the KP hierarc h y may emerge, turning this in ve stigation highly non-trivial. F or example, the generator s of the additional symmetries of the BKP hierarch y are 1 2 JINGSONG HE †‡ , KELEI TIAN † , ANGELA FOERSTER ‡ also correspondingly restricted [15](sp ecifically ,see eq.(40)). This fact implies that the genera- tors of the additional symmetries for the BKP hierarc hy m ust b e differen t compared to their coun terpart s on the KP hierarc h y . In this scenario, it w ould b e relev an t to deriv e the Virasoro constrain ts of the BKP hierarc h y using also the p oten tialities of the Dick ey’s method. In this w ork, applications of the additional symmetrie s of the BKP hierarc hy a re studied in detail. In particular, w e find the string equation and negativ e generators of Virasoro constrain ts on the τ function for the BKP hierarc hy b y means of the additional symmetry flows . W e also giv e the explicit forms of the negativ e Virasoro generators by calculating t he action o f the additional symmetry flows on the τ function, which is induced b y the action of the additional symmetry flows on the w av e function of the BKP hierarc hy . The organization of this pap er is as follo ws. In section 2 w e presen t a brief summary of the BKP and its additional symmetry , whic h is follow ed b y string equation and some sp ecial additional symmetry flo w equations in section 3. In section 4 w e deriv e the negativ e generators of the Virasoro constrain ts. Section 5 is devoted to conclusions and disc ussions. 2. B KP Hierarchy and its additional symmetries Let L b e the pseudo-differen tia l op e rator, L = ∂ + u 1 ∂ − 1 + u 2 ∂ − 2 + u 3 ∂ − 3 + · · · , (2.1) and t hen the KP hierarc h y is defined b y the set of partial differen tial equations u i with resp ect to indep enden t v ariables t j ∂ L ∂ t n = [ B n , L ] , n = 1 , 2 , 3 , · · · . (2.2) Here B n = ( L n ) + = n P k =0 a k ∂ k denotes the no n-negativ e p ow ers of ∂ in L n , ∂ = ∂ /∂ x , u i = u i ( x = t 1 , t 2 , t 3 , · · · , ). The other notation L n − = L n − L n + will b e needed b y the sequen t text. L is called the Lax op erator and eq.(2.2) is called the Lax equation of the KP hierarch y . In order to define the BKP hierarch y , we need a fo rmal adjoint op eration ∗ fo r an arbitrary pseudo- differen tial op erator P = P i p i ∂ i , P ∗ = P i ( − 1) i ∂ i p i . F or example, ∂ ∗ = − ∂ , ( ∂ − 1 ) ∗ = − ∂ − 1 , and ( AB ) ∗ = B ∗ A ∗ for t wo op erators. The BKP hierarc h y [6, 12] is a reduction of the KP hierarc hy b y the constrain t L ∗ = − ∂ L∂ − 1 , (2.3) whic h compresses all ev en flows of the KP hierarch y , i.e. the Lax equation of the BKP hierarc hy has only o dd flo ws , ∂ L ∂ t 2 n +1 = [ B 2 n +1 , L ] , n = 0 , 1 , 2 , · · · . (2.4) Th us u i = u i ( t 1 , t 3 , t 5 , · · · ) for the BKP hierarc h y . The La x equation of the BKP hierarch y can b e giv en by the consisten t conditions of the follo wing set of linear par t ial differen tial equations Lw ( t, λ ) = λw ( t, λ ) , ∂ w ( t, λ ) ∂ t 2 n +1 = B 2 n +1 w ( t, λ ) , t = ( t 1 , t 3 , t 5 , · · · ) . (2.5) Here w ( t, λ ) is iden tified as a w av e function. Let φ b e the w av e op e rator(or Sato op erator) of the BKP hierarc hy φ = 1 + P ∞ i =1 w i ∂ − i , then the La x op erator and the wa v e function admit the following represen tation L = φ∂ φ − 1 , w ( t, λ ) = φ ( t ) e ξ ( t,λ ) = ˆ w e ξ ( t,λ ) , (2.6) ADDITION AL SYMMETRIES AND STR ING EQUA TION 3 in which ξ ( t, λ ) = λt 1 + λ 3 t 3 + · · · + λ 2 n +1 t 2 n +1 + · · · , ˆ w = 1 + w 1 λ + w 2 λ 2 + w 3 λ 3 + · · · . Similar to the KP hierarc hy , the BKP hierarc hy also has a sole function, τ function τ ( t ) = τ ( t 1 , t 3 , t 5 , · · · , t 2 n − 1 · · · ) ( n is a p ositiv e integer), suc h that all of the dynamical co ordinates u i can b e expressed, and further the w av e f unction is w ( t, λ ) = ˆ w ( t, λ ) e ξ ( t,λ ) = τ ( t 1 − 2 λ , t 3 − 2 3 λ 3 , t 5 − 2 5 λ 5 , · · · ) τ ( t ) e ξ ( t,λ ) ≡ ˜ τ ( t, λ ) τ ( t ) e ξ ( t,λ ) (2.7) It is easy to sho w that the Lax equation is equiv alen t to Sato equation ∂ φ ∂ t 2 n +1 = − L 2 n +1 − φ, (2.8) and the constrain t on L in eq.(2.3) is transformed to the constrain t on t he w a v e operat o r φ ∗ = ∂ φ − 1 ∂ − 1 . (2.9) Eq.(2.9) is a crucial conditio n to construct the additiona l symmetries of the BKP hierarch y , whic h will affect the action of the additional symm etry on the op e rator φ . It leads t o a distinct explicit fo rm of the generators of the additional symmetry in comparison to the cases o f the KP hierarch y [1, 7] and the CKP hierarch y [1 7], a s w e shall see latter. No w w e recall the additional symmetries giv en b y T u [16] of the BKP hierarc h y . Let the OS’s op e rator M b e given b y M = φ Γ φ − 1 , Γ = ∞ X i =1 (2 i − 1) t 2 i − 1 ∂ 2 i − 2 = t 1 + 3 t 3 ∂ 2 + 5 t 5 ∂ 4 + · · · , (2.10) then they satisfy the useful tec hnical iden tit ies [ M , L l ] = − l L l − 1 , l ∈ Z , (2.11) [ M m , L ] = − mM m − 1 , m ∈ Z + . (2.12) Define the additional flows ∂ φ ∂ t ∗ m,l = − ( A m,l ) − φ, (2.13) or equiv alen tly ∂ L ∂ t ∗ m,l = − [( A m,l ) − , L ] , (2.14) where A m,l = A m,l ( L, M ) are monomials in L and M . As p ointe d in t he last paragraph, constrain ts on L in eq.(2.3), or equiv alen tly on φ in eq.(2.9) imply restrictions on the generators, and then one distinct for m of A m,l [16] is A m,l = M m L l − ( − 1) l L l − 1 M m L. (2.15) Indeed, this generator is different compared to results A m,l = M m L l [1, 7] for the KP hierarc hy and A m,l = M m L l − ( − 1) l L l M m [17] for the CKP hierarc h y . Prop osition 1. ([16])1) The additional flows ar e symmetries of the BKP hier a r ch y. 2)They form a c enterless W B 1+ ∞ -algebr a understanding their actions on φ as e q.(2.13). 4 JINGSONG HE †‡ , KELEI TIAN † , ANGELA FOERSTER ‡ 3. Some spec ial additional s ymmetr y flow equa tions W e further concen trate on some sp ecial additional symmetry flo ws in order to find suitable additional flows implying t he Virasoro constrain ts on the τ function of the BKP hierarc h y . So t w o examples a r e calculated in the follow ing. Prop osition 2. The action on L of the additional flows asso cia te d with A 1 ,l = − ( l − 1) L l − 1 is in the form of ∂ t ∗ 1 ,l L = ( l − 1)[( L l − 1 ) − , L ] =  0 , f or l = 0 , − 2 , − 4 , − 6 , · · · . − ( l − 1)( ∂ t 1 − l L ) , f or l = 2 , 4 , 6 , · · · . (3.1) Although this r esult is different with its coun t erpar t in the KP hierarc hy , this case is no t in teresting enough b ecaus e this additional symmetry flow s are almost equiv a len t to the CKP flo ws acing on the space of the Lax op erators L . The reason is that l is an ev en integer. Therefore w e consider A 1 , − ( l − 1) , and calculate its action o n L l . F or this end, from now on assume that l = 2 k and k is a p ositive integer. By using eq.(2.11), the A 1 , − ( l − 1) can b e expresse d as A 1 , − ( l − 1) = 2 M L − ( l − 1) − l L − l , (3.2) and then ∂ t ∗ 1 , − ( l − 1) L l = − [( A 1 , − ( l − 1) ) − , L l ] = [( A 1 , − ( l − 1) ) + , L l ] + [ − ( A 1 , − ( l − 1) ) , L l ] = [( A 1 , − ( l − 1) ) + , L l ] + 2 l . (3.3) Th us w e get t he following prop os ition based on the actions of the additiona l symmetry A 1 , − ( l − 1) on the L l . Prop osition 3. L et l = 2 m (2 n + 1) , m, n = 1 , 2 , 3 , · · · , and L l is indep end e nt of t ∗ 1 , − ( l − 1) , then the string e quation of the BKP hier ar chy is [ L l , 1 2 l ( A 1 , − ( l − 1) ) + ] = 1 . (3.4) F urthermor e, this e quation c an b e written in a mor e explici t form as fol lows, [ L 2 k , 1 2 k M L − (2 k − 1) − 1 2 L − 2 k ] = 1 , k = m (2 n + 1) . (3.5) Pro of The eq.(3.3) and ∂ t ∗ 1 , − ( l − 1) L l = 0 deduce directly eq.(3.4). Moreo ve r ∂ t ∗ 1 , − ( l − 1) L l = 0 infers ( A 1 , − ( l − 1) ) − = 0, and then ( M L − (2 k − 1) ) − = k L − 2 k and ( A 1 , − (2 k − 1) ) + = 2 M L − (2 k − 1) − 2 k L − 2 k . T aking this bac k into eq.(3.5), t hen eq.(3.5) is pro v ed, whic h completes the pro of.  Note that eq.(3.5) was also obtained by Joha n [13 ] from the Virasoro constrain ts o n the τ function of the BK P hierarch y . Ho w ev er, his equation is not the string equation without the restrictions of l . In other words, L l can not equal ( L l ) + with an arbitrary po sitive eve n integer l . Corollary 1. I f L l satisfy the e q.(3.4),then − 1 2 k X n ≥ k +1 (2 n − 1) t 2 n − 1 ( ∂ t 2 n − (2 k +1) L 2 k ) = 1 . (3.6) Let k = 1, the zero order terms of ab ov e equation tell us 1 2 X n ≥ 2 (2 n − 1) t 2 n − 1 ( ∂ t 2 n − 3 τ ) + 1 8 x 2 τ = 0 . (3.7) ADDITION AL SYMMETRIES AND STR ING EQUA TION 5 This result is indeed distinct with the case of K P hierarc h y giv en by corollary 1.2 of Ref. [18 ]. Pro of By a direct calculation, the left hand side of eq.(3.4) b ecomes 1 =  L 2 k , 1 2 k ( M L − (2 k − 1) ) +  = " L 2 k , 1 2 k φ ∞ X n =1 (2 n − 1) t 2 n − 1 ∂ 2 n − 2 k − 1 φ − 1 ! + # =   L 2 k , 1 2 k φ ∞ X n ≥ k +1 (2 n − 1) t 2 n − 1 ∂ 2 n − 2 k − 1 φ − 1 ! +   . Note that the c ha ng e in index of summation is due to the iden t it y ( φ∂ − m φ − 1 ) + = 0 , here m is a p ositive in teger. W e a lso should no te L k + = ( φ∂ k φ − 1 ) + with k ≥ 0, and then get 1 = " L 2 k , 1 2 k ∞ X n ≥ k +1 (2 n − 1) t 2 n − 1 ( L 2 n − 2 k − 1 ) + # = − 1 2 k ∞ X n ≥ k +1 (2 n − 1) t 2 n − 1 ( ∂ t 2 n − 2 k − 1 L 2 k ) , whic h is eq.(3.6). F urthermore, let k = 1, taking L 2 k = L 2 = ∂ 2 + 2 u 1 +lo w er or der terms and u 1 = 2(ln τ ) xx bac k in to eq. (3.6), we get the zero order terms in b oth sides, − 1 2 X n ≥ 2 (2 n − 1) t 2 n − 1 (4 ∂ t 2 n − 3 (ln τ ) xx ) = 1 . By exc ha ng ing the order of the deriv ativ e with resp ect to x and t 2 n − 3 , then − 2 X n ≥ 2 (2 n − 1) t 2 n − 1  1 τ ( ∂ t 2 n − 3 τ )  xx = 1 . In tegra ting the ab o v e form ula t wo times on x and choo sing suitable constan ts, then eq.(3.7) is reac hed, and t hus completes t he pro of.  4. Viras or o gene ra tors It is easy to find tha t eq.(3.4) is equiv alen t to ∂ t ∗ 1 , − ( l − 1) φ = 0 , with l = 2 k . T o get the Virasoro constrain ts on the τ function and the Virasoro generators, firstly we shall pa ss the action of the flo ws ∂ t ∗ 1 , − ( l − 1) on the w a v e op erator φ t o the action on the w a v e function w , and then on the τ function of the BKP hierarc hy . In this con text, ˆ w ( t, z ) plays the role of a bridge connecting actions on the w av e op erator φ and on the τ function. The follow ing lemmas are necess ary to do this. Lemma 1. F or l = 2 k , k = 1 , 2 , 3 , 4 , · · · , ( A 1 , − ( l − 1) ) − = 2 φ k X n =1 (2 n − 1) t 2 n − 1 ∂ 2( n − k ) − 1 ! φ − 1 + 2 ∞ X n = k +1 (2 n − 1) t 2 n − 1 L 2( n − k ) − 1 − − lL − l (4.1) Pro of According to the definitions of M and L , ( M L − ( l − 1) ) − = ( φ Γ φ − 1 φ∂ − ( l − 1) φ − 1 ) − = ( φ Γ ∂ − ( l − 1) φ − 1 ) − = φ  k X n =1 (2 n − 1) t 2 n − 1 ∂ 2 n − 2 k − 1 + ∞ X n = k +1 (2 n − 1) t 2 n − 1 ∂ 2 n − 2 k − 1  φ − 1 ! − 6 JINGSONG HE †‡ , KELEI TIAN † , ANGELA FOERSTER ‡ = φ k X n =1 (2 n − 1) t 2 n − 1 ∂ 2 n − 2 k − 1 ! φ − 1 + ∞ X n = k +1 (2 n − 1) t 2 n − 1 φ∂ 2 n − 2 k − 1 φ − 1 ! − = φ k X n =1 (2 n − 1) t 2 n − 1 ∂ 2 n − 2 k − 1 ! φ − 1 + ∞ X n = k +1 (2 n − 1) t 2 n − 1 L 2( n − k ) − 1 − In the second term of the last second equalit y , φ pass the t 2 n − 1 b ecause φ is inv olv ed only with ∂ t 1 , but there 2 n − 1 > 1. Th us, taking this represen ta tion of ( M L − ( l − 1) ) − in to the generator A 1 , − ( l − 1) = 2 M L − ( l − 1) − lL − 1 , and then the lemma is prov ed.  Prop osition 4. L et l = 2 k as b efor e, and ˆ w ( t, z ) is given by e q.(2.7), then ∂ t ∗ 1 , − ( l − 1) ˆ w ( t, z ) = − 2 z − 2 k + 1 ( ∂ ∂ z ˆ w ) + k X n =1 (2 n − 1) t 2 n − 1 z 2 n − 2 k − 1 ˆ w ! + 2 ∞ X n = k +1 (2 n − 1) t 2 n − 1 ∂ ˆ w ∂ t 2 n − 2 k − 1 + 2 k z − 2 k ˆ w . (4.2) Pro of First of all, b y using lemma 1, the additional symmetry flow acts o n the w av e o p erator φ as ∂ t ∗ 1 , − ( l − 1) φ = − ( A 1 , − ( l − 1) ) − φ = − 2 φ k X n =1 (2 n − 1) t 2 n − 1 ∂ 2( n − k ) − 1 +2 ∞ X n = k +1 (2 n − 1) t 2 n − 1 ( ∂ t 2( n − k ) − 1 φ ) + l φ∂ − l . Note that this is an op erator equation, thus w e can apply the function e xz to b ot h side sim ul- taneously . Therefore, b y applying to b oth sides of the last fo rm ula e xz and using tw o iden tities: [ φ, x ] e xz = ( ∂ ∂ z ˆ w ) e xz and φ∂ − l e xz = φz − l e xz = z − l ˆ w e xz , we ac hiev e that ( ∂ t ∗ 1 , − ( l − 1) ˆ w ) e xz = − (( A 1 , − ( l − 1) ) − φ ) e xz = − 2 φ ( xz − 2 k + 1 e xz + k X n =2 (2 n − 1) t 2 n − 1 z 2 n − 2 k − 1 e xz ) +2 ∞ X n = k +1 (2 n − 1) t 2 n − 1 ( ∂ t 2( n − k ) − 1 ˆ w ) e xz + 2 k z − 2 k ˆ w e xz = − 2( z − 2 k + 1 ( ∂ ∂ z ˆ w ) + k X n =2 (2 n − 1) t 2 n − 1 z 2 n − 2 k − 1 ˆ w ) e xz +2 ∞ X n = k +1 (2 n − 1) t 2 n − 1 ( ∂ t 2( n − k ) − 1 ˆ w ) e xz + 2 k z − 2 k ˆ w e xz . Dividing from ab ov e equalit y the factor e xz , the result of the prop osition is obtained, and thus completes the pro of.  F urther, w e know from prop osition 4 that the equiv alent form of eq.(3.4 ), ∂ t ∗ 1 , − ( l − 1) φ = 0, implies the constraints on w a v e function, ∂ t ∗ 1 , − ( l − 1) ˆ w = 0, sp ecifically . So it is v ery natural to express these constrain ts on the w a v e function b y means of the τ function. By pro ce eding in this w a y , the explicit fo rm of the Virasoro generators will b e obtained as follo ws. ADDITION AL SYMMETRIES AND STR ING EQUA TION 7 Lemma 2. The action of additional symmetries on ˆ w c an b e expr ess e d as a sp e cial form of ( ∂ t ∗ m,n ˆ w ) = f ( t ; z ) ˜ τ ( t, z ) τ ( t ) , (4.3) wher e f ( t ; z ) = g 1 ( t 2 n − 1 → t 2 n − 1 − 2 (2 n − 1) z 2 n − 1 ; ˜ τ ( t, z )) − g 1 ( t ; τ ( t )) ≡ ˜ g ( t ; z ) − g ( t ) , which is c al le d a similarity shifte d function. Pro of By a straightforw ard calculation, w e ha ve ( ∂ t ∗ m,n ˆ w ) =  ∂ t ∗ m,n ˜ τ ( t, z ) τ ( t )  = τ ( t ) ∂ t ∗ m,n ˜ τ ( t, z ) − ˜ τ ( t, z ) ∂ t ∗ m,n τ ( t ) τ 2 ( t ) =  ∂ t ∗ m,n ˜ τ ( t, z ) ˜ τ ( t, z ) − ∂ t ∗ m,n τ ( t ) τ ( t )  ˜ τ ( t, z ) τ ( t ) = f ( t ; z ) ˜ τ ( t, z ) τ ( t ) , as required. This is t he end of the pro of .  This lemma reminds us that w e should tra nsform the ( ∂ t ∗ 1 , − ( l − 1) ˆ w ) giv en b y eq.(4.2) to b e the form of eq.(4.3) in order to find ( ∂ t ∗ 1 , − ( l − 1) τ ( t )), and then find Virasoro generators. Ho w ev er, in general, o nly this is not enough to guar a n t ee us to get the correct Virasoro generators. No r ma lly , the for m of ( ∂ t ∗ m,n ˆ w ) is not unique, as p ointed out b y Dic k ey [11] for the KP hierarc hy , whic h is also true for the BKP hierarch y , b ecaus e there exists a freedom of gauge tr a nsformation with constan t co efficien ts. One has to choose a suitable gauge suc h that a go od form of ( ∂ t ∗ m,n ˆ w ) can b e reac hed, and then the latter can lead to the corr ect Vir a soro generators b y means of ( ∂ t ∗ m,n τ ). In part icular, for the case of n ≥ 0, it is very complicated to get a general simple expression of ( ∂ t ∗ 1 ,n +1 ˆ w ) for the KP hierarch y [11] a nd BKP hierarch y . Ho w ev er, it is simpler for the n < 0. This is a reason for us to only study ( ∂ t ∗ 1 , − ( l − 1) ˆ w ) in the last prop osition 4 and t o study ( ∂ t ∗ 1 , − ( l − 1) τ ( t )) in the sequen t prop osition with l ≥ 2. Lemma 3. Supp ose L − k = 1 2 ∞ X n = k +1 (2 n − 1) t 2 n − 1 ∂ ∂ t 2 n − 2 k − 1 + 1 8 X n + m = k +1 (2 n − 1)(2 m − 1) t 2 n − 1 t 2 m − 1 , (4.4) wher e n and m in the se c ond term take value fr om 1 to k under the c ondition of n + m = k + 1 , then the Vir a s or o c ommutation r elations [ L − k , L − l ] = ( − k + l ) L − ( k + l ) (4.5) hold for inte gers k , l ≥ 1 . Prop osition 5. If L l satisfies e q.(3.4), i.e., L is indep end ent of t ∗ 1 , − ( l − 1) , l = 2 k , k = 1 , 2 , 3 , · · · , then the Vir a s or o c onstr aints imp ose d on the τ function of the BKP hier ar chy ar e L − k τ = 0 , (4.6) wher e L − k ar e Vir asor o gener ators given by e q.( 4.4). Ob viously , let k = 1 , L − 1 τ = 0 is consisten t with the coro llary 1. The 1 k L − k giv es the same result as that obtained in Ref. [13] using a completely different approac h. In pa r t icularly , the Virasoro generators for the BKP are indeed differen t with ones [19] of the K P hierarch y . Pro of F or conv enience, we mark t he four terms of ( ∂ t ∗ 1 , − ( l − 1) ˆ w ) in eq.(4.2) b y a ) , b ) , c ) , d ), resp ectiv ely . The pro of has the following steps . In this pro of, ˜ τ = ˜ τ ( t, z ). 8 JINGSONG HE †‡ , KELEI TIAN † , ANGELA FOERSTER ‡ 1). First of all, we try to construct the similarity shifted function structure in t w o terms, a ) and c ), because only they ha ve the deriv ative s of τ a nd ˜ τ . A direct calculation sho ws a ) ≡ − 2 z − 2 k + 1 ∂ ˆ w ∂ z = − 2 z − 2 k + 1 1 τ ∂ ˜ τ ∂ z = − 4 τ ∞ X n = k +1 1 z 2 n − 1 ∂ ˜ τ ∂ t 2 n − 2 k − 1 , (4.7) and c ) ≡ 2 X n = k +1 (2 n − 1) t 2 n − 1 ( ∂ ∂ t 2 n − 2 k − 1 ˜ τ τ ) = 2 τ ∞ X n = k +1 (2 n − 1) t 2 n − 1 ∂ ˜ τ ∂ t 2 n − 2 k − 1 − 2 ˜ τ τ 2 ∞ X n = k +1 (2 n − 1) t 2 n − 1 ∂ τ ∂ t 2 n − 2 k − 1 . F or the thir d term, w e try to mak e a similarity shifted function delib erately b y insertion o f one term, and th us c ) = ( 2 ˜ τ ∞ X n = k +1 (2 n − 1)( t 2 n − 1 − 2 (2 n − 1) z 2 n − 1 ) ∂ ˜ τ ∂ t 2 n − 2 k − 1 − 2 τ ∞ X n = k +1 (2 n − 1) t 2 n − 1 ∂ τ ∂ t 2 n − 2 k − 1 ) ˜ τ τ + 4 τ ∞ X n = k +1 1 z 2 n − 1 ∂ ˜ τ ∂ t 2 n − 2 k − 1 . (4.8) F urther, a ) + c ) =  2 ˜ τ ∞ X n = k +1 (2 n − 1)( t 2 n − 1 − 2 (2 n − 1) z 2 n − 1 ) ∂ ˜ τ ∂ t 2 n − 2 k − 1 − 2 τ ∞ X n = k +1 (2 n − 1) t 2 n − 1 ∂ τ ∂ t 2 n − 2 k − 1  ˜ τ τ (4.9) p ossess one similarit y shifted f unction as w e exp ected. 2)F or the b)+d), it can not b e tra nsformed to a similarit y shifted function if w e only mak e a shift t 2 n − 1 → t 2 n − 1 − 2 (2 n − 1) z 2 n − 1 in b), similar to what lik e w e hav e done in c). So w e hav e to try it b y using its pro duct, as the second simplest case. T o do this, b y rewriting b) as a symmetrical form, and then using one identit y , w e get b ) ≡ − 2 k X n =1 z 2 n − 2 k − 1 (2 n − 1) t 2 n − 1 ˆ w = − ( X n + m = k +1 (2 n − 1)(2 m − 1) t 2 n − 1 ˆ w (2 m − 1) z 2 m − 1 + X n + m = k +1 (2 n − 1)(2 m − 1) t 2 m − 1 ˆ w (2 n − 1) z 2 n − 1 ) = − 1 2  − X n + m = k +1 (2 n − 1)(2 m − 1)( t 2 m − 1 − 2 (2 m − 1) z 2 m − 1 )( t 2 n − 1 − 2 (2 n − 1) z 2 n − 1 ) + X n + m = k +1 (2 m − 1)(2 n − 1) t 2 n − 1 t 2 m − 1  ˆ w − 2 X n + m = k +1 ˆ w z 2( n + m ) − 2 = 1 2  X n + m = k +1 (2 m − 1)(2 n − 1)( t 2 m − 1 − 2 (2 m − 1) z 2 m − 1 )( t 2 n − 1 − 2 (2 n − 1) z 2 n − 1 ) ADDITION AL SYMMETRIES AND STR ING EQUA TION 9 − X n + m = k +1 (2 m − 1)(2 n − 1) t 2 n − 1 t 2 m − 1  ˆ w − 2 k z − 2 k ˆ w . (4.10) Therefore, b ) + d ) = 1 2  X n + m = k +1 (2 n − 1)(2 m − 1)( t 2 m − 1 − 2 (2 m − 1) z 2 m − 1 )( t 2 n − 1 − 2 (2 n − 1) z 2 n − 1 ) − X n + m = k +1 (2 m − 1)(2 n − 1 ) t 2 n − 1 t 2 m − 1  ˜ τ τ . (4.11) 3) T aking a ) + c) in eq.(4.9) and b) + d) in eq.(4.11) in to ( ∂ t ∗ 1 , − ( l − 1) ˆ w ) in eq.(4.2), then ( ∂ t ∗ 1 , − ( l − 1) ˆ w ) =  2 ˜ τ ∞ X n = k +1 (2 n − 1)( t 2 n − 1 − 2 (2 n − 1) z 2 n − 1 ) ∂ ˜ τ ∂ t 2 n − 2 k − 1 + 1 2 X n + m = k +1 (2 n − 1)(2 m − 1)( t 2 n − 1 − 2 (2 n − 1) z 2 n − 1 )( t 2 m − 1 − 2 (2 m − 1) z 2 m − 1 )  ˜ τ τ −  2 τ ∞ X n = k +1 (2 n − 1) t 2 n − 1 ∂ τ ∂ t 2 n − 2 k − 1 + 1 2 X n + m = k +1 (2 n − 1)(2 m − 1) t 2 n − 1 t 2 m − 1  ˜ τ τ (4.12) On the one side, taking into accoun t an equiv alen t f orm of the eq.(3.4), i.e., ∂ t ∗ 1 , − l − 1 φ = 0, a nd the lemma 2, w e hav e ∂ t ∗ 1 , − ( l − 1) ˆ w =  ∂ t ∗ 1 , − l − 1 ˜ τ ˜ τ − ∂ t ∗ 1 , − l − 1 τ τ  ˜ τ τ = 0 , (4.13) and then deduce ( ∂ t ∗ 1 , − l − 1 τ ) = c 1 τ . (4.14) where c 1 is a constant. O n the other side, comparing eq.(4.12 ) and eq.(4.13) infers ( ∂ t ∗ 1 , − ( l − 1) τ ) =  2 ∞ X n = k +1 (2 n − 1) t 2 n − 1 ∂ τ ∂ t 2 n − 2 k − 1 + 1 2 X n + m = k +1 (2 n − 1)(2 m − 1) t 2 n − 1 t 2 m − 1 τ  + c 2 τ =4 L − k τ + c 2 τ (4.15) with an arbitrary constan t c 2 , and L − k as w e expected. Therefore, the Virasoro constrain ts on the τ function L − k τ = 0 is obtained from eqs.(4.14) and (4.15) with c 1 = c 2 . This is the end of the pro of.  5. Conclusions and Discus sions W e hav e studied the applications of the additional symmetry flows of the BKP hierarch y , and thus pro vided the following main results: • 1) the action of the special additional symmetry flows on L l and the string equation in prop osition 3; • 2) the explicit forms of t he actions of the additional symmetry flo ws on the wa v e function ( ∂ t ∗ 1 , − ( l − 1) ˆ w ) in prop osition 4; 10 JINGSONG HE †‡ , KELEI TIAN † , ANGELA FOERSTER ‡ • 3) t he explicit forms of the negative Virasoro generator s and the Virasoro constrain ts on the τ function of the BKP hierarc hy in pro p osition 5. In addition, the similarit y shifted function f ( t ; z ) in ( ∂ t ∗ m,n ˆ w ) giv en b y lemma 2 is also crucial to get the action of the additional symmetry flo ws on the τ function. Our route is a dditional symmetry → ad ditional symmetry flow e quations e q.(3.4) asso ciate d with A 1 , − ( l − 1) → actions of the addi tion al symm etry flows on the wave function → Vir asor o c onstr aints on the τ function → Vir asor o gene r a tors . F or the further researc h related to this topic, the extension of the additional symmetry a nd its asso ciated structures for the m ulti-comp onent BKP hierarc h y [20] would b e very interesting and relev a n t although very complicated. Moreo v er, it is also an in teresting pro blem to calculate out the whole set o f L k for the Vira soro constraints and W n for the W -constrain ts f o r the BKP hierarc hy . Ac kno w ledgmen ts Th is work is su p p orted by the NS F of China u nder Grant No.103 01030 and No.106 71187, and S R FDP of Chin a under Grant No.2004 035800 1. Supp ort of th e join t p ost- do c f ello w- ship of TW AS(Italy) and CNPq(Brazil)at UFRGS is gratefully ac kno wledged. 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