On the String Equation of the BKP Hierarchy

On the String Equation of the BKP Hierarc h y Hsin-F u Shen 1 and Ming-Hsien T u 2 ∗ 1 Dep artment of Me chanic al Engine ering , WuF eng Institute of T e chnolo gy, Chiayi 621 , T aiwan , 2 Dep artment of Physics, National Chung Ch eng Univers ity, Chiayi 621 , T aiwan No ve m b er 3, 2018 Abstract The Adler-Shio ta-v an Mo erb eke formula is employ ed to derive the W -constraints for the p -reduced BK P hierar ch y constrained by the s tring e quation. W e a lso provide the Grassmannia n description of the s tring eq uation in terms of the s pe c tr al pa rameter. P ACS: 02 .3 0.Ik Keywords: BKP hierarch y , additional symmetries, v ertex op erator s, string equation, W - constraints, Sato Grassma nnia n. ∗ E-mail: ph y mhtu@ccu.edu.t w 1 1 In tro duction String equation is an imp ortan t constrained condition connecting in tegrable hierarc hies with solv able string theories and in tersection theory (see e. g. [8, 28] and references therein). The most famous example is the KdV hierarc hy constrained b y a string equation wh ose solutions corresp onds to p artition function of 2d quan tum gra vit y or generating fun ction of inte rsection n u m b er[29, 30 , 15](see also [13]). Since th e solutions of KdV h ierarc hy can b e c haracterized b y a sin gle fun ction called τ -function, it turns out th at the constrained Kd V hierarch y can b e written as Virasoro constraint s on the τ -function, in w h ic h the lo west one corresp onds to th e string equation[9, 11, 16, 17]. F or higher order KdV hierarc hy (i.e. p -reduced KP h ierarc hy with p > 2) the asso ciated τ -function satisfies the W -constrain ts so that the τ -fun ction is a n u ll-v ector of a set of W + p -algebra w hic h con tains the Virasoro alge bra as a subalgebra with cen tral c h arge c = p [10, 1, 28]. It was pointed out[16, 17, 5] that the string equation asso ciated with the KP/KdV hierarc h y can b e traced b ac k to additional s y m metries (or non-isosp ectral flo ws )of the hierarc hies prop osed b y Orlov and Sc hulman[18](see also [7]). F urthermore, Adler-Shiota-v an Mo erb ek e [2, 3 ] and Dic k ey[6] show ed that additional symmetries acting on w av e function are connected with Sato’s B¨ ac klund symmetries acting on tau function. Th ese prop erties provide a u seful to ol to stud y th e solutions to the string equation of the p -reduced KP hierarc hy w ith p > 2[1] that generalize Kon tsevich’s result for the K dV case. In the presen t wo r k, w e like to stu d y th e string equation asso ciated with the BKP hierarc hy[4, 24]. The BKP h ierarc hy p ossesses many integ rable s tr uctures as the KP hierarc hy suc h as Lax form u lation, tau fun ction, Hirota bilinear equation, f ermion represen tation, soliton and quasi- p erio d ic solutions, etc . In [26] v an de Leur p ro vided the Adler-Shiota-v an Mo erb eke formula for the BKP hierarch y . He also studied the W -constrain t f or the p -reduced BKP hierarch y in terms of the t w isted affine Lie algebra ˆ sl (2 n [27]. In this w ork, we follo w closely the work b y Adler-v an Mo erb ek e[1] a nd Dic k ey[5] to r econsider the strin g equatio n of the BKP hierarc h y from the p oin t of view of additional symm etries in Lax-Orlo v-Sc hulman formulat ion. In par- ticular, we shall sho w that the in v ariance of the p -reduced BKP h ierarc hy w ith resp ect to a particular additional symmetry is cru cial for obtaining the strin g equation whic h together with the Adler-Shiota-v an Mo erb ek e form ula implies the W -constrain ts of the τ function. I n the end of the work w e also giv e a Grassmannian d escription of the s tr ing equation in terms of the sp ectral parameter. The p ap er is organized as follo ws. In section 2, we r ecall some basic notions for th e BKP hierarc hy suc h as Lax formulation, (adjoin t-) wa v e f unction, τ function and (differen tial) F ay iden tity . In section 3, w e introdu ce the Orlo v-Sc hulman op erator for additional sym metries 2 and ve rtex op erators for B¨ ac klund symmetries of th e BKP hierarch y . W e giv e a simple ex- pression of th e generators of v ertex op erator using the F a` a di Br u no p olynomials and p ro vid e the Ad ler-Shiota-v an Mo erb eke formula that conn ects additional symmetries acting on wa v e function with those Sato’s B¨ ac klun d sym metries acting on tau f u nction. In section 4, we s h o w that the solution of the p -reduced BKP hierarc hy constrained by the string equatio n can b e c haracterized by a v acuum condition so that the asso ciated tau function is ann ihilated b y a set of differen tial op erators. In section 5, W e sho w that these different ial op erators constitute a W B + p algebra whic h conta ins a Virasoro algebra as a sub algebra with cen tral c harge c = p . In section 6, w e pr o vide the Grassmannian d escrip tion of the string equation in terms of the sp ectral parameter. Sect ion 7 is d ev oted to the concluding remarks. 2 BKP hierarc hy In this section we recall some b asic prop erties f or th e BKP h ierarc hy [4] . W e shall follo w the notations used in th e p revious w ork[25]. Th e BKP hierarc hycan b e formulated in Lax form as ∂ 2 n +1 L = [ B 2 n +1 , L ] , B 2 n +1 = ( L 2 n +1 ) + , n = 0 , 1 , 2 , · · · (1) where the Lax op erator is d efined by L = ∂ + u 1 ∂ − 1 + u 2 ∂ − 2 + · · · , (2) with co efficien t functions u i dep end ing on the time v ariables t = ( t 1 = x, t 3 , t 5 , · · · ) and satisfies the constrain t L ∗ = − ∂ L∂ − 1 . (3) Here and the r est of the pap er we w ill u se the notatio ns: ( P i a i ∂ i ) + = P i ≥ 0 a i ∂ i , ( P i a i ∂ i ) − = P i< 0 a i ∂ i , ( P i a i ∂ i ) [ k ] = a k , r es( P i a i ∂ i ) = a − 1 and ( P i a i ∂ i ) ∗ = P i ( − ∂ ) i a i . It can b e sho w n [4] th at the constrain t (3) is equiv alen t to the condition ( B 2 n +1 ) [0] = 0. The Lax equation (1) is equiv alen t to the compatibilit y condition of th e lin ear s y s tem Lw ( t, z ) = z w ( t, z ) , ∂ 2 n +1 w ( t, z ) = B 2 n +1 w ( t, z ) , (4) where w ( t, z ) is called w av e function (or Bak er function) of the system and z is the sp ectral parameter. The wh ole hierarch y can b e expressed in terms of a dressing op erator, the so-called Sato’s op erator W , so that L = W ∂ W − 1 , W = 1 + X j =1 w j ∂ − j , 3 and the Lax equation is equiv alen t to the Sato’s equ ation ∂ 2 n +1 W = − ( L 2 n +1 ) − W , (5) with constrain t[23, 24] W ∗ ∂ W = ∂ . (6) Let the solutions of th e linear system (4) b e the form w ( t, z ) = W e ξ ( t,z ) = ˆ w ( t, z ) e ξ ( t,z ) , (7) where ξ ( t, z ) = P i =0 t 2 i +1 z 2 i +1 and ˆ w ( t, z ) = 1 + w 1 /z + w 2 /z 2 + · · · . Then w ( t, z ) is a wa v e function of the BKP hierarch y if and only if it satisfies th e bilinear id en tity[4] res z ( z − 1 w ( t, z ) w ( t ′ , − z )) = 1 , ∀ t , t ′ (8) where we d enote the symb ol res z ( P i a i z i ) = a − 1 . In fact, fr om the bilinear iden tit y (8), solutions of the BKP hierarc h y can b e charac terized by a single function τ ( t ) called τ -fu nction suc h that[4] ˆ w ( t, z ) = τ ( t 1 − 2 z , t 3 − 2 3 z 3 , t 5 − 2 5 z 5 , · · · ) τ ( t ) . (9) F r om (7) and (9 ) the wa v e function w ( t, λ ) can b e expr essed in terms of τ -function as w ( t, z ) = X B ( t, z ) τ ( t ) τ ( t ) , where X B ( t, z ) is the so-called vertex op erator, defi n ed by [4] X B ( t, z ) = e ξ ( t,z ) e − 2 D ( t,z ) ≡ e ξ ( t,z ) G ( z ) , with D ( t, z ) = P n =0 z − 2 n − 1 ∂ 2 n +1 / (2 n + 1). In the follo wing we pro vide tw o u seful identit ies asso ciated with the tau fun ction of the BKP h ierarc hy . Prop osition 1. [25] (F ay identity) The tau function of the BKP hier ar chy satisfies the F ay quadrise c ant identity: X ( s 1 ,s 2 ,s 3 ) ( s 1 − s 0 )( s 1 + s 2 )( s 1 + s 3 ) ( s 1 + s 0 )( s 1 − s 2 )( s 1 − s 3 ) τ ( t + 2[ s 2 ] + 2[ s 3 ]) τ ( t + 2[ s 0 ] + 2[ s 1 ]) + ( s 0 − s 1 )( s 0 − s 2 )( s 0 − s 3 ) ( s 0 + s 1 )( s 0 + s 2 )( s 0 + s 3 ) τ ( t + 2[ s 0 ] + 2[ s 1 ] + 2[ s 2 ] + 2[ s 3 ]) τ ( t ) = 0 wher e ( s 1 , s 2 , s 3 ) stands for cyclic p ermutations of s 1 , s 2 and s 3 . 4 Prop osition 2. [25 ] (Diffe r ential F ay identity ) The f ol lowing e quation holds.  1 s 2 2 − 1 s 2 1  { τ ( t + 2[ s 1 ] + 2[ s 2 ]) τ ( t + 2[ s 2 ]) − τ ( t + 2[ s 1 ] + 2[ s 2 ]) τ ( t ) } =  1 s 2 + 1 s 1  { ∂ τ ( t + 2[ s 2 ]) τ ( t + 2[ s 1 ]) − ∂ τ ( t + 2[ s 1 ]) τ ( t + 2[ s 2 ]) } +  1 s 2 − 1 s 1  { τ ( t + 2[ s 1 ] + 2[ s 2 ]) ∂ τ ( t ) − ∂ τ ( t + 2[ s 1 ] + 2[ s 2 ]) τ ( t ) } . 3 Additional symmetries and v ertex op erators Based on the work of Orlo v and Sc h ulman [18], the Lax equation can be exte nded b y in tro- ducing the Or lo v-Sch ulman op er ator M defined by M = W Γ W − 1 , Γ = X n =0 (2 n + 1) t 2 n +1 ∂ 2 n , whic h satisfies ∂ 2 n +1 M = [ B 2 n +1 , M ] , [ L, M ] = 1 . Th us the linear sy s tem (4) sh ould b e extended to Lw = z w, M w = ∂ z w, ∂ 2 n +1 w = B 2 n +1 w. Note that o n the space o f wa v e f unction w ( t, z ), ( L, M ) is an ti-isomorphic to ( z , ∂ z ) since [ z , ∂ z ] = − 1. Mo r e general, one has M m L l w = z l ∂ m z w, L l M m w = ∂ m z z l w. In f act, one can defin e the adjoint w av e function w ∗ ( t, z ) = ( W ∗ ) − 1 e − ξ ( t,z ) = − z − 1 w x ( t, − z ) and M ∗ = ( L ∗ ) − 1 ∂ M ∂ − 1 L ∗ where we ha v e u se the fact that Γ ∗ = Γ and (6). Then [ L ∗ , M ∗ ] = [ M , L ] ∗ = − 1, and L ∗ w ∗ = z w ∗ , M ∗ w ∗ = − ∂ z w ∗ , ∂ 2 n +1 w ∗ = − B ∗ 2 n +1 w ∗ . Motiv ated by the KP hierarc hy , one can in tro duce a new set of parameters ˆ t ml so that add itional symmetries of the BKP h ierarc hy can b e expressed as ˆ ∂ ml W = − ( A ml ( L, M )) − W , (10) where the generator A ml ( L, M ), d ue to (6 ), has th e form [26 , 25] A ml ( L, M ) = M m L l − ( − 1) l L l − 1 M m L. (11) 5 Let us in tro duce another generator Y B ( λ, µ ) of additional symmetries as[26] Y B ( λ, µ ) = ∞ X m =0 ( µ − λ ) m m ! ∞ X l = −∞ λ − l − m − 1 ( A m,m + l ( L, M )) − , (12) = ∞ X m =0 ( µ − λ ) m m ! ∞ X l = −∞ λ − l − m − 1 ( M m L m + l − ( − 1) m + l L m + l − 1 M m L ) − . W e ment ion that for λ = µ , the generator Y B ( λ, λ ) = 2 P l = odd λ − l − 1 L l − corresp onds to the resolv ent op erator of th e BKP hierarch y . On the other hand, r ecalling the v ertex op erator X B ( λ, µ ) defined b y X B ( λ, µ ) = e − ξ ( t,λ ) e ξ ( t,µ ) G ( − λ ) G ( µ ) whic h provides the infin itesimal Sato’s B¨ ac klu nd transform ations[4] on the space of tau fun c- tion, namely , if τ ( t ) is a solution then τ ( t ) + ǫX B ( λ, µ ) τ ( t ) is a solution as well. In f act, one can T a ylor expand the v ertex op erator X B ( λ, µ ) around µ = λ for large λ as X B ( λ, µ ) = X m =0 ( µ − λ ) m m ! W ( m ) ( λ ) = X m =0 ( µ − λ ) m m ! ∞ X l = −∞ λ − m − l W ( m ) l , where W ( m ) ( λ ) = ∂ m µ X B ( λ, µ ) | µ = λ . I ntro ducing the sy mb ol α ( z ) = P n = odd α n z − n /n w ith α n =    2 ∂ /∂ t n n ∈ Z + odd | n | t | n | n ∈ Z − odd where Z odd = Z + odd ⊕ Z − odd . Then α n ’s satisfy th e comm utation r elations [ α n , α m ] = 2 nδ n, − m , n ∈ Z odd . The v ertex op erator X B ( λ, µ ) can b e expressed as X B ( λ, µ ) = : e α ( λ ) − α ( µ ) : where the normal ord er in g :: demands that α n> 0 m u st b e placed to the righ t of α n< 0 . Therefore, W ( m ) ( λ ) = ∂ m µ X B ( λ, µ ) | µ = λ =: ∂ m λ e − α ( λ ) · e α ( λ ) :=: F m ( − ∂ λ α ( λ )) : where F m ( u ( z )) is th e so-called F a` a d i Bruno p olynomials (see e.g. [7]) defined by the recurren ce relations F m +1 ( u ) = ( ∂ z + u ) F m ( u ). F or in s tance, F 0 = 1 , F 1 = u, F 2 = u ′ + u 2 , F 3 = u ′′ + 3 uu ′ + u 3 . Lemma 3. [6] The fol lowing formula W ( m ) ( λ ) = X m 1 +2 m 2 + ··· = m m ! m 1 ! m 2 ! · · · :  − ∂ λ α 1!  m 1  − ∂ 2 λ α 2!  m 2 · · · : (13) holds, wher e m i ≥ 0 . 6 Pr o of. Introd ucing a generating fun ction of F m as g ( λ, z ) = P m =0 F m z m /m !, then g ( λ, z ) = X m =0 ( d m λ 1) m ! z m , d λ = e α ( λ ) · ∂ λ · e − α ( λ ) = e α ( λ ) · e z ∂ λ e − α ( λ ) = e α ( λ ) e − α ( λ + z ) = Y n =1 e − ∂ n λ αz n /n ! = X m 1 =0  − ∂ λ α 1!  m 1 z m 1 m 1 ! X m 2 =0  − ∂ 2 λ α 2!  m 2 z 2 m 2 m 2 ! · · · = X m =0   X m 1 +2 m 2 + ··· = m m !  − ∂ λ α 1!  m 1 m 1 !  − ∂ 2 λ α 2!  m 2 m 2 ! · · ·   z m m ! . The v ertex op erator generators W ( m ) l can b e easily computed as W (0) n = δ n, 0 , W (1) n =    α n n ∈ Z odd 0 n ∈ Z eve n , W (2) n =    − ( n + 1) α n n ∈ Z odd P i + j = n : α i α j : n ∈ Z eve n , W (3) n =    P i + j + k = n : α i α j α k : + ( n + 1)( n + 2) α n n ∈ Z odd − 3 2 ( n + 2) P i + j = n : α i α j : n ∈ Z eve n , W (4) n =        − 2( n + 3) P i + j + k = n : α i α j α k : − ( n + 1)( n + 2)( n + 3) α n n ∈ Z odd P i + j + k + l = n : α i α j α k α l : − P i + j = n ij : α i α j : − (2 n 2 + 9 n + 11) P i + j = n : α i α j : n ∈ Z eve n , etc. A remark able formula describ ed b elo w p r o vides a b r idge b et wee n additional symmetries acting on wa v e function and Sat o’s B¨ ac klund symmetries a cting on τ fu n ction. This k in d of form u la w as first deriv ed for the KP hierarc h y b y Adler-Shiota-v an Mo erb eke[ 2, 3] and Dic k ey[6], and later for BKP b y v an d e Leur [26] (see also [25]). Theorem 4. [26, 25] The fol lowing formula X B ( λ, µ ) w ( t, z ) = 2 λ  λ − µ λ + µ  Y B ( λ, µ ) w ( t, z ) , (14) 7 holds for the BKP hier ar chy, wher e it sho uld b e understo o d that the vertex op er ator X B ( λ, µ ) acting on w ( t, z ) is gener ate d by its action on the τ function. W e remark that the pro of of (14) in [25] is based on a simple expression for the generator Y B ( λ, µ ) and the d ifferen tial F ay identit y of the BKP hierarc hy . Through th e f er m ion-b oson corresp onden ce in th e BKP hierarc hy , a realization of Lie algebra g o ( ∞ ) on C [ t 1 , t 3 , t 5 , · · · ] is giv en by[4] Z B ( λ, µ ) = 1 2 µ + λ µ − λ ( X B ( λ, µ ) − 1) , whic h after T aylo r expand ing around µ = λ for large λ has the form Z B ( λ, µ ) = X m =0 ( µ − λ ) m m ! ∞ X l = −∞ λ − m − l Z ( m +1) l . It is easy to show that differenti al op erators Z ( m ) l are related to W ( m ) l as Z (1) l = W (1) l , Z ( m +1) l = W ( m +1) l m + 1 + 1 2 W ( m ) l , m ≥ 1 , and constitute an infinite-dimensional Lie algebra called W B 1+ ∞ -algebra whic h is a subalgebra of W 1+ ∞ asso ciated w ith the KP hierarc hy . 4 p -reduced B KP and string equation The so-calle d p -r ed uced BKP h ierarc hy[4] is defin ed by the Lax op erator (2) such that L p = ( L p ) + , p = o dd in teger . (15) This r eduction must b e compatible with the cond ition (3). F or example, ( L p ) [0] = 0 for o dd p and th u s ( L p ) ∗ + = ( L ∗ ) p = ( − 1) p ∂ L p ∂ − 1 = ( − 1) p ∂ ( L p ) + ∂ − 1 is a d ifferen tial op erator as well. Th er efore, from the Lax equation (1), we h a ve ∂ 2 n +1 L p =  ( L p ) 2 n +1 p + , L p  (16) whic h implies th at L p is indep enden t of the parameters t j p for j = 1 , 3 , 5 , · · · . F or p = 1 case, w e hav e L = L + = ∂ wh ic h is a tr ivial reduction ( u i = 0 for ∀ i ). T he next case is p = 3, wh ich con tains the sim p lest non trivial equ ation called Sa wada-Kote ra equation[20]: u t + 15( u 3 + uu xx ) x + u xxxxx = 0 where x = t 1 , t = t 5 and u = 2(log τ ) xx with ∂ τ /∂ t 3 j = 0 ( j = 1 , 3 , 5 , · · · ). 8 In th e f ollo wing , we w ould like to c haracterize the solutions of the p -reduced BKP hierarch y constrained by the strin g equation [ L p , P ] = 1 (17) where P is a differential op erator. F rom (11), we hav e A 1 , 1 − k =    k L − k k ∈ Z odd 2 M L 1 − k − k L − k k ∈ Z eve n (18) Guided by the K P hierarc hy[5], if we th ink the string equation (17) as the consequence of the addition flo w equation ˆ ∂ 1 , 1 − p W = 0 for o dd p , then (10) an d (18) sh o w that ( A 1 , 1 − p ) − = pL − p = 0 w hic h p ro du ces a con tradictory result. In[27] v an de Luer p oin ted out that one ma y consider the additional flo w equation ˆ ∂ 1 , 1 − 2 p W = − ( A 1 , 1 − 2 p ) − W = 0 from whic h one gets ( M L 1 − 2 p ) − = pL − 2 p (19) and hence the op erator Q ≡ ( M L 1 − 2 p − pL − 2 p ) / 2 p is pu rely d ifferen tial. Then [ L 2 p , Q ] = 1 . (20) Ho we v er, in view of the fact that L p = ( L p ) + , we hav e ( M L 1 − p ) − = ( M L 1 − 2 p L p ) − = pL − p , (21) whic h pr o vides the d ifferen tial op erator P ≡ ( M L 1 − p − pL − p ) /p for the string equation (17). Therefore, (19) can b e regarded as th e symmetry origin of the string equation (17), and thus w e r efer (19) to the pr e-string equation. T aking the resid u e of (19) we obtain X n = p (2 n + 1) t 2 n +1 res( L 2 n − 2 p +1 ) + (2 p − 1) t 2 p − 1 = 0 . (22) F r om the S ato’s equation (5) an d th e f orm u la (9), we hav e res( L 2 n − 2 p +1 ) = 2 ∂ 1 ∂ 2 n − 2 p +1 log τ . Substituting ab ov e b ac k to (22) and int egrating it o v er t 1 , yields X n = p (2 n + 1) t 2 n +1 ∂ ∂ t 2 n +1 − 2 p + (2 p − 1) 2 t 1 t 2 p − 1 + c ! τ ( t ) = 0 . (23) In fact, from (19) and the p -r ed uced BKP hierarc hy flo ws, w e can p ro ve a more general r esult of constrain ts on τ -fun ction. 9 Prop osition 5. F or the p -r e duc e d L ax op e r ator L p c onstr aine d by the string e quation, the fol lowing formulas hold f or m ≥ 0 . ( M m L j p + m ) − =        Q m − 1 r =0 ( p − r ) L − 2 p j = − 2 Q m − 1 r =0 ( p − r ) L − p j = − 1 0 j = 0 , 1 , 2 , ... (24) ( L j p + m − 1 M m L ) − = ( − 1) m ( M m L j p + m ) − (25) with the pr oviso that the factor Q m − 1 r =0 ( p − r ) should b e set to 1 for m = 0 . Pr o of. W e shall follo w [1] to p ro ve it by induction on m and j . F or m = 0, the pro of is ob vious. F or m = 1 , j = − 2, this is j u st the equation (19). Assu me that the formula (24) holds up to some in teger m > 0 f or j = − 2, then for j ≥ − 1, one sees that ( M m L j p + m ) − = (( M m L m − 2 p ) − L ( j +2) p ) − = m − 1 Y r =0 ( p − r ) L − 2 p L ( j +2) p ! − =    Q m − 1 r =0 ( p − r ) L − p j = − 1 0 j = 0 , 1 , 2 , ... . Next, f or the j = − 2 , we ha ve ( M m +1 L m +1 − 2 p ) − = ( M m M L m L 1 − 2 p ) − = ( M m L m M L 1 − 2 p ) − − m ( M m L m − 2 p ) − = ( M m L m ( M L 1 − 2 p ) − ) − − m ( M m L m − 2 p ) − = p ( M m L m − 2 p ) − − m ( M m L m − 2 p ) − = ( p − m ) m − 1 Y r =0 ( p − r ) L − 2 p = m Y r =0 ( p − r ) L − 2 p where the third equalit y is du e to the f act that M m L m is a differen tial op erator. The formula (25) can b e p ro ved in a similar wa y . W e lik e to m en tion that a similar result as (24) for the CKP h ierarc hy h as b een d eriv ed in [12] to discuss the asso ciated additional symmetries and strin g equation. Prop osition 6. L et L p b e the L ax op er ator of the p -r e duc e d BKP hier ar chy c onstr aine d by the string e qu ation (17 ). Then f or m ≥ 0 and j ≥ − 2 , Z ( m +1) j p τ ( t ) = c · τ ( t ) . (26) 10 wher e c is a c onstant. Pr o of. Using (12) and the Adler-Shiota-v an Moer b ek e form u la (14 ), w e ha v e ( A m,m + j p ) − ω ( t, z ) = r es λ  λ j p + m ∂ m µ | µ = λ Y B ( λ, µ ) ω ( t, z )  = res λ  λ j p + m − 1 ∂ m µ | µ = λ 1 2 λ + µ λ − µ X B ( λ, µ ) ω ( t, z )  = − ω ( t, z )( G ( z ) − 1)res λ  λ j p + m − 1 ∂ m µ | µ = λ Z B ( λ, µ ) τ ( t ) τ ( t )  = − ω ( t, z )( G ( z ) − 1) Z ( m +1) j p τ ( t ) τ ( t ) ! . On the other hand, fr om (24) and (25) we hav e ( A m,m + j p ) − ω ( t, z ) = (( M m L j p + m ) − − ( − 1) j p + m ( L j p + m − 1 M m L ) − ) ω ( t, z ) = (1 − ( − 1) j p )( M m L j p + m ) − ω ( t, z ) = 2 m − 1 Y r =0 ( p − r ) L − p ω ( t, z ) δ j, − 1 = 2 m − 1 Y r =0 ( p − r ) z − p ω ( t, z ) δ j, − 1 . Noticing that 2 z − p = − ( p ( t p − 2 z − p /p ) − pt p ) = − ( G ( z ) − 1) Z (1) − p τ ( t ) τ ( t ) , and hence ( G ( z ) − 1) Z ( m +1) j p τ ( t ) τ ( t ) − m − 1 Y r =0 ( p − r ) δ j, − 1 Z (1) − p τ ( t ) τ ( t ) ! = 0 . Since ( G ( z ) − 1) f ( t ) = 0 imp lies f ( t ) is a constant, we hav e Z ( m +1) j p − m − 1 Y r =0 ( p − r ) δ j, − 1 Z (1) − p ! τ ( t ) = c · τ ( t ) . No w we can drop th e term inv olving Z (1) − p = pt p without h arm b ecause the p -redu ced BKP hierarc hy d o es not dep end on the v ariables t p . 5 String equation as the lo w est Virasoro constrain t In this section, we lik e to discuss the algebraic structur e of the equation (26). Giv en a infinite- dimensional algebra W B 1+ ∞ defined by th e v ertex op erator, one can introdu ce t wo sub algebras as follo ws: W B p = { generated by W ( m ) j p , 1 ≤ m ≤ p, j ∈ Z , t p = t 3 p = · · · = 0 } 11 and its truncated sub-algebra: W B + p = { generated by W ( m ) j p , 1 ≤ m ≤ p , j ≥ − 2 , t p = t 3 p = · · · = 0 } W e shall sho w that the τ fu nction satisfying the p -reduced BKP hierarc hy and the string equation is a null-v ector of the W B + p -algebra. S ince the algebra W + p for the K P hierarch y has no cen tral term[10], th us w e exp ect that th e su balgebra W B + p ⊂ W + p has also no cen tral term. T o see this, w e h a ve to pr op erly combine the generators Z ( m ) j p so that eve ry r ed efined element W ( m ) n of W B + p can b e expressed as a comm utator of tw o elemen ts of W B + p . As a consequence, the constan t c in (26) can b e remo ved and W ( m ) n τ ( t ) = 0 , 1 ≤ m ≤ p, n ≥ − m + 1 . (27) W e remark that the condition for the subscript n in (27) is due to the fact that for higher-spin generators W ( m ) n , one has [ W (2) − 1 , W ( m ) n ] = ( − ( m − 1) − n ) W ( m ) n − 1 , and thus, u n der br ack eting with generators W (2) − 1 , one can reac h the low est one W ( m ) − m +1 whic h comm utes with W (2) − 1 . L et u s demonstrate the fir st f ew W -constrain ts. F or m = 0, (26) sho w s that Z (1) j p τ ( t ) = W (1) j p τ ( t ) = 2 ∂ τ ( t ) /∂ t j p = 0 , j = 1 , 3 , 5 , · · · whic h is j ust the condition L p = ( L p ) + for p -reduced BKP h ierarc hy . Th us we hav e W (1) n τ ( t ) = W (1) (2 n +1) p τ ( t ) = 0 , n ≥ 0 F or m = 1, (26) sho w s that Z (2) 2 kp τ = c (2) k ( p ) τ with k ≥ − 1 where Z (2) 2 kp = 2 k p − 1 X n =0 ∂ 2 n +1 ∂ 2 kp − 2 n − 1 + 2 ∞ X n =0 (2 n + 1) t 2 n +1 ∂ 2 kp +2 n +1 for k ≥ 0 and Z (2) 2 kp = 1 2 − k p − 1 X n =0 (2 n + 1)( − 2 k p − 2 n − 1) t 2 n +1 t − 2 k p − 2 n − 1 +2 X n =0 (2 n + 1 − 2 k p ) t 2 n +1 − 2 kp ∂ 2 n +1 for k < 0. Define l k = Z (2) 2 kp / 4 p with k ≥ − 1 then l k satisfy cen terless Virasoro algebra [ l n , l m ] = ( n − m ) l n + m except that [ l 1 , l − 1 ] = 2  l 0 + 1 16 p + p 2 − 1 24 p  . 12 This means that the constan ts c (2) k 6 =0 ( p ) = 0 and c (2) 0 ( p ) = − (1 / 4 + ( p 2 − 1) / 6) . If w e redefin e a new set of op erators as W (2) n ≡ L n = l n + δ n, 0  1 16 p + p 2 − 1 24 p  , n ≥ − 1 then we hav e [ L n , L m ] = ( n − m ) L n + m for m, n ≥ − 1. Therefore, the Virasoro constraint b ecomes W (2) n τ ( t ) = 0 , n ≥ − 1 . In particular, the low est Virasoro constraint L − 1 τ = 0 is giv en by 1 2 p − 1 X n =0 (2 n + 1)(2 p − 2 n − 1) t 2 n +1 t 2 p − 2 n − 1 + 2 X n =0 (2 n + 1 + 2 p ) t 2 n +1+2 p ∂ 2 n +1 ! τ = 0 whic h is ju st the p r e-string equation (23 ). M oreo v er, when n extends to all inte gers, W (2) n indeed constitute t he generators of a standard Viraso ro algebra with cen tr al charge c = p , namely , [ L n , L m ] = ( n − m ) L n + m + δ n + m, 0 n 3 − n 12 p. F or m = 2, w e m a y define the spin-3 generators as W (3) n = Z (3) (2 n +1) p whic h indeed satisfy the comm utation r elation [ L n , W (3) m ] = (2 n − m ) W (3) n + m and the constraint equation W (3) n τ ( t ) = 0 , n ≥ − 2 . The higher-spin generators can b e treated in a similar mann er. 6 Grassmannian description of the string equation In this section w e like to giv e a geometric description of the string equation for the p -reduced BKP hierarc hy . Let H b e a Hilb ert space d efined by formal p o wer series in z that can b e decomp osed in to tw o infin ite-dimensional subspaces as H = H + ⊕ H − where H + = sp an { z 0 , z 1 , z 2 , · · ·} , H − = sp an { z − 1 , z − 2 , z − 3 , · · ·} . The Grassmannian Gr is defined by th e set of all subs paces V ⊂ H with the f ollo wing conditions:[19, 22] Gr = { V | V ⊂ H , p + | V : V → H + (F r ed holm) , p − | V : V → H − (compact) } 13 where p ± are pro jection op erators. If p + | V : V → H + is a bijection, then V is called trans versal to H − , or transversal for sh ort (i.e. V b elongs to the big cell Gr 0 ⊂ Gr ). The K P hierarc hy can b e r egarded as a simp le d ynamical system on Gr (see [19] for the detail). The BKP hierarc hy is d efined by th e subv ariet y Gr B ⊂ Gr (see e.g. [23]) so that an infinite- dimensional plane V 0 ∈ Gr B can b e represen ted as follo w s[19, 22]: V 0 = span { ω ( t, z ) | t =0 , ∂ x ω ( t, z ) | t =0 , ∂ 2 x ω ( t, z ) | t =0 , · · ·} = span { ω ( t, z ) , for all t ∈ C ∞ } . where t = ( t 1 , t 3 , t 5 , · · · ) and ω ( t, z ) satisfies the b ilinear equation (8). F or the p -reduced BKP hierarc hy , ∂ p W = − ( L p ) − W and the wa v e fu nction ω ( t, z ) asso ciated with V 0 satisfies ∂ p ω ( t, z ) = ( L p ) + ω ( t, z ) = L p ω ( t, z ) = z p ω ( t, z ) ∈ V 0 Therefore z p V 0 ⊂ V 0 . On the other hand, the string equat ion (17) can b e traced bac k to the Sato equation of the additional flo w ˆ ∂ 1 , 1 − 2 p W = − ( A 1 , 1 − 2 p ) − W = 0 which follo ws ˆ ∂ 1 , 1 − 2 p ω ( t, z ) = − ( A 1 , 1 − 2 p ) − ω ( t, z ) = 0 and hence Qω ( t, z ) = 1 2 p ( M L 1 − 2 p − pL − 2 p ) ω ( t, z ) = 1 2 p ( z 1 − 2 p ∂ z − pz − 2 p ) ω ( t, z ) ≡ A 2 p ω ( t, z ) ∈ V 0 . Therefore, the plane V 0 ∈ Gr B asso ciated with ω ( t, z ) of p -reduced BKP h ierarc hy constrained b y the string equation [ L p , P ] = 1 is inv ariant under the action of d ifferential op erators L p and Q ≡ 1 2 p ( M L 1 − 2 p − pL − 2 p ). Th ey act as z -op erators: L p 7→ z p , Q 7→ A 2 p ≡ z p d dz 2 p z − p and z p V 0 ⊂ V 0 , A 2 p V 0 ⊂ V 0 , with [ A 2 p , z 2 p ] = 1 . (28) The ab ov e d iscussions enable us to transf orm the original problem for solving the solution of the p -reduced BKP hierarc h y constrained by the str in g equation to that d escrib ed in z -op erators in the con text of Grassmannian. 14 7 Concluding Remarks W e h av e inv estigated the strin g equation of the BKP hierarch y fr om additional symmetries p oint of view. W e sho w that the p -reduced BKP hierarch y constrained by the string equation can b e formulate d in terms of Lax and Orlov- Sc hulman op erators as wh at h as b een d one for the K P hierarch y . In particular, the in v ariance of the additional sym metry with r esp ect to the ˆ t 1 , 1 − 2 p -flo w is cr u cial for ob taining the string equation w h ic h together w ith the Ad ler-S hiota- v an Mo erb ek e form u la implies the W -constrain ts of the τ function. 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