Toda tau functions with quantum torus symmetries

T o da tau functions with quan tum torus symmetries Kanehisa T ak asaki ∗ Graduate Sc ho ol of Human a nd Environmen tal Studies, Ky oto Univ ersit y Y oshida, S a ky o, Kyo to, 606-8501 , Japan Abstract The quan tum torus algebra pla ys an imp ortan t role in a sp ecial class of solutio ns of the T o da hierarc h y . T ypical examples are the so- lutions related to the melting crystal mo del of top olog ical strings and 5D SUSY gauge theories. The quantum torus algebra is r ea lized b y a 2D complex free f e rmion sys t em that underlies th e T o da hierarch y , and exhibits m y ste rious “shift symmetries”. This article is based on collaboration with T oshio Nak atsu. Key words: T o da hierarch y , me lting crystal model, quan tum torus algebra 1 In tro duction This pap er is a review of our recen t work [1, 2] o n an integrable structure of the melting crys tal mo del o f top ological strings [3] and 5D gaug e theories [4]. It is sho wn here that the partition function of this mo del, on b eing suitably deformed b y sp ecial external p oten tials, is essen tially a tau f unction of the T o da hierarc h y [5 ]. A tec hnical clue to this observ atio n is a kind of symmetries (referred to as “shift symmetries”) in the underlying quan tum torus algebra. These symmetries enable us, firstly , to conv ert the deformed partition function to t he tau f un ction and, se condly , to show the existence of hidden symmetries of the ta u function. These results can b e exn tended to some other T o da tau f un ctions that are related to the top ological v ertex [6] and the double Hurw itz n um b ers of the Riemann sp here [7]. ∗ E-mail: tak a saki@math.h.kyoto-u.ac.jp 1 2 Quan tum torus algebra Throughout this pap er, q denotes a constant with | q | < 1, and Λ and ∆ denote the Z × Z matrices Λ = X i ∈ Z E i,i +1 = ( δ i +1 ,j ) , ∆ = X i ∈ Z iE ii = ( iδ ij ) . Their combinations v ( k ) m = q − k m / 2 Λ m q k ∆ ( k , m ∈ Z ) (1) satisfy the comm utation relations [ v ( k ) m , v ( l ) n ] = ( q ( lm − k n ) / 2 − q ( kn − lm ) / 2 ) v ( k + l ) m + n (2) of the quan tum torus algebra. This Lie algebra can th us b e em b edded into the Lie algebra gl( ∞ ) of Z × Z matrices A = ( a ij ) for whic h ∃ N suc h that a ij = 0 for | i − j | > N . T o form ulate a fermionic realization of this Lie a lgebra, w e intro duc e the creation/annihilat io n op erators ψ i , ψ ∗ i ( i ∈ Z ) with an ti-comm utation relations ψ i ψ ∗ j + ψ ∗ j ψ i = δ i + j, 0 , ψ i ψ j + ψ j ψ i = 0 , ψ ∗ i ψ ∗ j + ψ ∗ j ψ ∗ i = 0 and the 2D free fermion fields ψ ( z ) = X i ∈ Z ψ i z − i − 1 , ψ ∗ ( z ) = X i ∈ Z ψ ∗ i z − i . The v acuum states h 0 | , | 0 i of the F o c k space and its dual space are c harac- terized b y the v acuum conditions ψ i | 0 i = 0 ( i ≥ 0) , ψ ∗ i | 0 i = 0 ( i ≥ 1) , h 0 | ψ i = 0 ( i ≤ − 1) , h 0 | ψ ∗ i = 0 ( i ≤ 0) . T o any elemen t A = ( a ij ) of gl( ∞ ), one can associat e the fermion bilinear b A = X i,j ∈ Z a ij : ψ − i ψ ∗ j : , : ψ − i ψ ∗ j : = ψ − i ψ ∗ j − h 0 | ψ − i ψ ∗ j | 0 i . 2 These fermion bilinears fo r m a one- dimensional cen tral extension \ gl( ∞ ) of gl( ∞ ). The sp ecial fermion bilinears [1, 2 ] V ( k ) m = d v ( k ) m = q k / 2 I dz 2 π i z m : ψ ( q k / 2 z ) ψ ∗ ( q − k / 2 z ): (3) satisfy the comm utation relations [ V ( k ) m , V ( l ) n ] = ( q ( lm − k n ) / 2 − q ( kn − lm ) / 2 )  V ( k + l ) m + n − q k + l 1 − q k + l δ m + n, 0  (4) for k and l with k + l 6 = 0 and [ V ( k ) m , V ( − k ) n ] = ( q − k ( m + n ) / 2 − q k ( m + n ) / 2 ) V (0) m + n + mδ m + n, 0 . (5) Th us \ gl( ∞ ) contains a cen tra l extension of the quan t um torus algebra, in whic h the d u (1) algebra is r e alized b y J m = V (0) m = c Λ m ( m ∈ Z ) . (6) 3 Shift s ym metries Let us in tro duce the opera t ors G ± = exp ∞ X k =1 q k / 2 k (1 − q k ) J ± k ! , W 0 = X n ∈ Z n 2 : ψ − n ψ ∗ n : . (7) G ± ’s pla y the role of “tra ns fer matrices” in the melting crystal mo del[3, 4]. W 0 is a fermionic form of the so called “cut-and-join” op erator for Hurwitz n um b ers [8]. G ± and q W 0 / 2 induce the follo wing tw o t yp es of “shift symm etries” [1, 2] in the (cen trally extende d) quan tum torus alg ebra. • First shift symmetry G − G +  V ( k ) m − δ m, 0 q k 1 − q k  ( G − G + ) − 1 = ( − 1) k  V ( k ) m + k − δ m + k, 0 q k 1 − q k  (8) • Second shift symmetry q W 0 / 2 V ( k ) m q − W 0 / 2 = V ( k − m ) m (9) 3 4 T o da tau function in meltin g crystal mo del A general tau f un ction of the 2D T o da hierarc h y [5] is giv en b y τ ( s, T , ¯ T ) = h s | exp ∞ X k =1 T k J k ! g exp − ∞ X k =1 ¯ T k J − k ! | s i , (10) where T = ( T 1 , T 2 , · · · ) and ¯ T = ( ¯ T 1 , ¯ T 2 , · · · ) are time v ariables of the T o da hierarc h y , h s | and | s i are the ground states h s | = h − ∞| · · · ψ ∗ s − 1 ψ ∗ s , | s i = ψ − s ψ − s +1 · · · | − ∞i in the c harge- s sector o f the F o c k space, and g is an elemen t o f G L( ∞ ) = exp  gl( ∞ )  . On the other hand, t he partition function Z ( Q, s, t ) of the deformed melt- ing crystal mo del [1, 2] can b e cast into the apparen tly similar (but essen t ially differen t) form Z ( s, t ) = h s | G + e H ( t ) Q L 0 G − | s i , (11) where Q and t = ( t 1 , t 2 , · · · ) are coupling constan ts of the mo del, and H ( t ) and L 0 the f ollo wing op erators: H ( t ) = ∞ X k =1 t k H k , H k = V ( k ) 0 , L 0 = X n ∈ Z n : ψ − n ψ ∗ n : . (12) The sh ift symmetries (8 ) and ( 9) imply the op erator iden tit y G + e H ( t ) G − 1 + = exp ∞ X k =1 t k q k 1 − q k ! G − 1 − q − W 0 / 2 exp ∞ X k =1 ( − 1) k t k J k ! q W 0 / 2 G − . Inserting this iden tit y and using the fa c t that h s | G − 1 − q − W 0 / 2 = q − s ( s +1)(2 s +1) / 12 h s | , q − W 0 / 2 G − 1 + | s i = q − s ( s +1)(2 s +1) / 12 | s i , w e can rewrite Z ( s, t ) as Z ( Q, s, t ) = exp  ∞ X k =1 t k q k 1 − q k  q − s ( s +1)(2 s +1) / 6 τ ( s, T , 0 ) , T k = ( − 1) k t k , ( 1 3) 4 where the GL( ∞ ) elemen t g defining the tau function is giv en b y g = q W 0 / 2 G − G + Q L 0 G − G + q W 0 / 2 . (14) Actually , the s hift symmetries imply the operato r iden tit y G − 1 − e H ( t ) G − = exp ∞ X k =1 t k q k 1 − q k ! G + q W 0 / 2 exp ∞ X k =1 ( − 1) k t k J − k ! q − W 0 / 2 G − 1 + as w ell. This leads to another ex pression of Z ( Q, s, t , ) in whic h τ ( s, T , 0 ) is replaced with τ ( s, 0 , − T ). The existence of differen t expressions can b e explained by the in tertw ining relations J k g = g J − k ( k = 1 , 2 , . . . ) , (15) whic h, to o, are a consequence of the shift symmetries. These in tert wining relations imply the constrain ts  ∂ T k + ∂ ¯ T k  τ ( s, T , ¯ T ) = 0 ( k = 1 , 2 , . . . ) (16) on the tau function. The tau function τ ( s, T , ¯ T ) thereb y b ecomes a function τ ( s, T − ¯ T ) of the difference T − ¯ T . In particular, τ ( s , T , 0 ) and τ ( s, 0 , − T ) coincide. The reduced function τ ( T , s ) may b e thought of as a tau function of the 1D T o da hierarc h y . (15) are a s p ecial case of the more g e neral in tert wining relations ( V ( k ) m − δ m, 0 q k 1 − q k ) g = Q − k g ( V ( − k ) − 2 k − m − δ 2 k + m, 0 q − k 1 − q − k ) . (17) W e can translate these relations to the language of the Lax formalism of the T o da hierarch y . A study on this issue is now in progress. 5 Other mo de ls The follow ing T o da tau functions can b e treated more or less in the same w a y as the foregoing tau function. W e shall discuss this issue elsew here. 1. The generating function of the t w o-leg amplitude W λµ in the top ological v ertex [6] is a T o da ta u function determine d b y g = q W 0 / 2 G + G − q W 0 / 2 . (18) 5 2. The generating function of double Hurwitz n um b ers of the Riemann sphere [7] is a T o da ta u function determined b y g = e − β W 0 Q L 0 . (19) The parameter q is in terpreted as q = e − β . Ac kno wledgeme n ts This work has b een partly supp orted b y the JSPS Gran ts-in-Aid for Scien t ific Researc h No. 19104 0 02, No. 21 5 40218 and No. 22540 186 from t he Japa n So ciet y for the Promotion of Science. References [1] T. Nak a ts u and K. T ak asaki, Melting crystal, quan tum torus and T o da hierarc h y , Comm. Math. Phy s. 285 (2 009), 445–46 8 [2] T. Nak atsu and K. T a k asaki, In t egr a ble structure of melting crystal mo del with external p oten tia l, Adv anced Studies in Pure. Math. v ol. 59 (Math. Soc. Japan, 2010 ), pp. 20 1–223 [3] A. Ok ounk o v, N. R e shetikhin a nd C. 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