KdV integrability in GUE correlators

KD V INTEGRABILITY IN GUE CORRELA TORS DI Y ANG Abstract. Ok ounko v [36] prov ed a remark able form ula relating n -p oin t GUE (Gaussian unitary ensem ble) correlators of a fixed genus to Witten’s in tersection n um b ers of the same gen us. The partition function of GUE correlators is a tau-function for the T o da lattice hierarch y . In this note, based on the knowledge of these tw o statements w e giv e a new proof of the Witten–Kon tsevich theorem, that relates Witten’s in tersection n umbers to the KdV (Kortew eg–de V ries) integrable hierarc h y . Contents 1. In tro duction 1 2. Review of T o da lattice hierarc h y and GUE partition function 4 3. A new pro of of Witten’s conjecture 6 References 11 1. Introduction The Korteweg–de V ries (KdV) equation (1) ∂ u ∂ t 1 = u ∂ u ∂ t 0 + 1 12 ∂ 3 u ∂ t 3 0 is a nonlinear ev olutionary partial differential equation (PDE). Since the late 1960s, it has b een kno wn from the theory of integrable systems that the KdV equation (1) can b e extended to an infinite system of pairwise commuting ev olutionary PDEs, called the KdV hier ar chy , which can b e written compactly as follows: (2) ∂ u ∂ t d = 1 (2 d + 1)!!  L 2 d +1 2  + , L  , d ≥ 1 . Here, L := ∂ 2 t 0 + 2 u is called the Lax op erator, [ , ] denotes the comm utator, and for a pseudo differen tial op erator P the notation P + means taking the nonnegativ e part of P (see e.g. [12] for more details). The d = 1 equation in (2) coincides with (1). Note that the normalization of KdV flo ws (2) differs from [12] b y rescalings. Let g, n be nonnegativ e in tegers satisfying the stability condition 2 g − 2 + n > 0, and M g ,n the Deligne–Mumford mo duli space of stable algebraic curves of genus g with n distinct marked p oin ts. Denote b y ψ a the first Chern class of the a th tautological line bundle on M g ,n , a = 1 , . . . , n . They are often called psi-classes . The in tegrals (3) Z M g,n ψ d 1 1 · · · ψ d n n =: ⟨ τ d 1 . . . τ d n ⟩ g 1 2 DI Y ANG are called psi-class interse ction numb ers or Witten ’s interse ction numb ers , whic h v anish unless the degree and the dimension matc h: d 1 + · · · + d n = 3 g − 3 + n . Here, d 1 , . . . , d n ≥ 0. The integrals (3) will b e understo o d as 0 if 2 g − 2 + n ≤ 0. Let t = ( t d ) d ≥ 0 b e indetermi- nates. The Witten fr e e ener gy F = F ( t ) is the p ow er series of t defined by (4) F ( t ) = X g ≥ 0 X n ≥ 0 X d 1 ,...,d n ≥ 0 ⟨ τ d 1 . . . τ d n ⟩ g n ! t d 1 · · · t d n . In [39] Witten prop osed a striking conjecture: Witten’s conjecture ([39]). The function u := ∂ 2 F /∂ t 2 0 ob eys the KdV hier ar chy (2) . Witten’s conjecture was first prov ed by Kon tsevic h [29], and is now kno wn as the Witten–Kontsevich the or em . W e refer to [2, 8, 9, 20, 27, 28, 34, 36, 37, 38] ab out other pro ofs. Note that, to pro v e the Witten–Kon tsevic h theorem, it suffices [27, 30] to pro v e the d = 1 case of (2), i.e., the KdV equation for psi-class in tersection n um b ers. Indeed, Witten prov ed [39] that psi-class in tersection num b ers ob ey the following tw o equations ⟨ τ 0 τ d 1 . . . τ d n ⟩ g = X 1 ≤ a ≤ n, d a > 0 ⟨ τ d a − 1 Y b  = a τ d b ⟩ g + δ n, 2 δ g , 0 , (5) ⟨ τ 1 τ d 1 . . . τ d n ⟩ g = (2 g − 2 + n ) ⟨ τ d 1 . . . τ d n ⟩ g + 1 24 δ g , 1 δ n, 0 , (6) called string e quation and dilaton e quation , respectively , and, according to [27, 30], v alidit y of (5), (6) and the KdV equation for in tersection n um b ers implies v alidity of (2). In Kon tsevic h’s pro of [29] of Witten’s conjecture, a remark able iden tit y w as established: (7) X d 1 ,...,d n ≥ 0 ⟨ τ d 1 . . . τ d n ⟩ g n Y a =1 (2 d a − 1)!! z 2 d a +1 a = X G ∈ G 3 g,n 2 2 g − 2+ n | Aut( G ) | Y e ∈ E ( G ) 1 ˜ z ( e ) (see also [38]). Here, g , n ≥ 0 satisfy 2 g − 2 + n > 0, G 3 g ,n denotes the set of triv alen t maps on an orien ted closed surface of genus g with n cells mark ed b y 1 , . . . , n , Aut(G) is the finite group of symmetries generated b y orien tation preserving homeomorphisms of the surface that map G to G and resp ect the markings, E ( G ) denotes the edge set of G , and the notation ˜ z ( e ) is defined as follo ws: let the v ariables z 1 , . . . , z n corresp ond to the markings of G ∈ G 3 g ,n ; each edge e ∈ E b orders tw o cells; let i and j b e the lab els assigned to these cells (if b oth sides of e b order the same cell then i = j ); ˜ z ( e ) := z i + z j . W e call (7) the Kontsevich main identity . This identit y connects psi-class in tersection n um b ers to com binatorics of triv alent maps. In order to obtain the KdV in tegrability , Kon tsevic h [29] constructed the “matrix Airy function” (the Kon tsevic h matrix mo del) and completed his pro of of Witten’s conjecture. Another pro of of the KdV in tegrabilit y , again after the Kontsevic h main identit y (7) is established, is given by Ok ounko v [37] using a limit form ula (see [36, (2.7) and Prop osition 1], [38] or (11) b elow) and the edge- of-the-sp ectrum mo del [36, 37, 38]. En umeration of maps, or sa y ribb on graphs, appeared naturally in the GUE (Gaussian unitary ensem ble) random matrix mo del [5, 11, 25, 38, 39], which, b y the theory of orthogonal p olynomials, is well kno wn to b e related to the T o da lattice hierarch y ( aka the 1D T o da lattice or the T o da chain), and more sp ecifically to the V olterra lattice hierarch y ( aka the discrete KdV flows) when restricted to even couplings; see e.g. [10, 11, 15, 16, KD V INTEGRABILITY IN GUE CORRELA TORS 3 17, 23, 33, 39]. The goal of this pap er is to prov e Witten’s KdV equation from T o da in tegrabilit y based on the ab ov e-men tioned limit form ula of [36]. W e note that the KdV in tegrabilit y was disco v ered by taking double-scaling/contin uum limit in matrix gra vit y (cf. [3, 7, 11, 13, 14, 23, 24, 35, 39]) and our pro of resembles this original idea. Let N b e a p ositiv e integer and H ( N ) the space of hermitian matrices of the finite size N . By GUE c orr elators w e mean the normalized Gaussian integrals (cf. [5, 6, 11, 25]) (8) R H ( N ) tr( M i 1 ) . . . tr( M i n ) e − 1 2 tr( M 2 ) dM R H ( N ) e − 1 2 tr( M 2 ) dM =: ⟨ tr M i 1 . . . tr M i n ⟩ , where n ≥ 0, i 1 , . . . , i n ≥ 1, and dM := Q 1 ≤ i ≤ N dM ii Q 1 ≤ i 0, by comparing with the Kontsevic h main iden tit y (7), the follo wing remark able form ula w as pro v ed in [36] (cf. [38]): (11) 2 2 g − 3+ 3 n 2 π n 2 √ x 1 · · · x n Map g ( i 1 , . . . , i n ) 2 | i | κ 3 g − 3+ 3 n 2 → Q g ( x 1 , . . . , x n ) , as (10), for | i | b eing ev en , where Q g ( x 1 , . . . , x n ) denotes the Witten n -p oin t function in genus g , i.e., (12) Q g ( x 1 , . . . , x n ) := X d 1 ,...,d n ≥ 0 ⟨ τ d 1 . . . τ d n ⟩ g x d 1 1 · · · x d n n . By using (11) and T o da in tegrabilit y w e will give a new pro of of Witten’s conjecture. Organization of the pap er. In Section 2 w e review T oda lattice and the GUE partition function. In Section 3 we give a short pro of of Witten’s conjecture based on (11). Ac kno wledgemen ts. I w ould lik e to thank Alessandro Giacchetto for stim ulating lec- tures giv en at MA TRIX, Australia. P art of the work w as done during m y visit at ICTP , Italy; I thank ICTP for warm hospitality . I also thank Don Zagier for several helpful suggestions that improv e a lot the presentation of the paper. The work is supp orted b y NSF C 12371254 and CAS YSBR-032. 4 DI Y ANG 2. Review of Tod a la ttice hierar chy and GUE p ar tition function In this section we review the T o da lattice hierarch y and its tau-functions b y means of the matrix-resolven t metho d (cf. [4, 16, 19, 40]), and review the T o da integrabilit y for the partition function of GUE correlators. 2.1. T au-function for the T o da lattice hierarc h y. Let A b e the p olynomial ring: (13) A := Z [ v 0 , w 0 , v ± 1 , w ± 1 , v ± 2 , w ± 2 , · · · ] . Define the shift op erator Λ : A → A as the linear op erator satisfying (14) Λ( v i ) = v i +1 , Λ( w i ) = w i +1 , Λ( f g ) = Λ( f )Λ( g ) , ∀ i ∈ Z , f , g ∈ A . F or an op erator of the form P = P m ∈ Z P m Λ m , with P m ∈ A , denote P + := P m ≥ 0 P m Λ m and P − := P m< 0 P m Λ m . Let L := Λ + v 0 + w 0 Λ − 1 , where Λ − 1 denotes the inv erse op erator of Λ. The abstr act T o da lattic e hier ar chy is a sequence of deriv ations D i : A → A , i ≥ 1, defined by (15) D i ( L ) = [( L i ) + , L ] and b y requiring D i comm ute with Λ. The op erator L is called the L ax op er ator . It was pro v ed in [22, 32] that the deriv ations D i , i ≥ 1, all comm ute. The Lax op erator L can b e written in the matrix form (16) L = Λ + U ( λ ) , U ( λ ) =  v 0 − λ w 0 − 1 0  . Lemma 1 ([16, 40]) . Ther e exists a unique 2 × 2 matrix series (17) R ( λ ) =  1 0 0 0  + O  λ − 1  ∈ Mat  2 , A [[ λ − 1 ]]  satisfying the e quation (18) Λ( R ( λ )) U ( λ ) − U ( λ ) R ( λ ) = 0 along with the normalization c onditions (19) tr R ( λ ) = 1 , det R ( λ ) = 0 . The unique series R ( λ ) in the ab o ve lemma is called the b asic matrix r esolvent . If we think of v 0 , w 0 as tw o functions v = v ( x ; ϵ ), w = w ( x ; ϵ ), the op erator Λ as e ϵ∂ x , and v m , w m as v ( x + mϵ ; ϵ ), w ( x + mϵ ; ϵ ), with ϵ b eing a parameter, then the deriv ations D i lead to a hierarc hy of differen tial-difference equations: ϵ ∂ v ∂ s i = D i ( v 0 ) , ϵ ∂ w ∂ s i = D i ( w 0 ) , i ≥ 1 , (20) whic h are the T o da lattic e hier ar chy . The i = 1 equation reads ϵ ∂ v ∂ s 1 = (Λ − 1)( w ) , ϵ ∂ w ∂ s 1 = w · ( v − Λ − 1 ( v )) , (21) KD V INTEGRABILITY IN GUE CORRELA TORS 5 whic h is the T o da lattice equation. The second flo w of (20) reads ϵ ∂ v ∂ s 2 = (Λ − 1)  w · ( v + Λ − 1 ( v ))  , (22) ϵ ∂ w ∂ s 2 = w · (Λ( w ) − Λ − 1 ( w ) + v 2 − (Λ − 1 ( v )) 2 ) . (23) Lemma 2 ([16, 40]) . F or an arbitr ary solution ( v ( x, s ; ϵ ) , w ( x, s ; ϵ )) to the T o da lattic e hier ar chy (20) , ther e exists a function τ ( x, s ; ϵ ) such that ϵ 2 X i,j ≥ 1 1 λ i +1 µ j +1 ∂ 2 log τ ( x, s ; ϵ ) ∂ s i ∂ s j = tr ( R ( λ ; x, s ; ϵ ) R ( µ ; x, s ; ϵ )) ( λ − µ ) 2 − 1 ( λ − µ ) 2 , (24) ϵ X i ≥ 1 1 λ i +1 ∂ ∂ s i  log τ ( x + ϵ, s ; ϵ ) τ ( x, s ; ϵ )  = ( R ( λ ; x, s ; ϵ )) 21 , (25) τ ( x + ϵ, s ; ϵ ) τ ( x − ϵ, s ; ϵ ) τ ( x, s ; ϵ ) 2 = w ( x, s ; ϵ ) . (26) Her e, R ( λ ; x, s ; ϵ ) = R ( λ ) | v m 7→ v ( x + mϵ, s ; ϵ ) , w m 7→ w ( x + mϵ, s ; ϵ ) , m ∈ Z . The function τ ( x, s ; ϵ ) in the ab ov e lemma is called a tau-function of the solution ( v ( x, s ; ϵ ) , w ( x, s ; ϵ )) to the T o da lattice hierarc hy [16]. 2.2. The GUE partition function. Lik e in e.g. [16], it is conv enien t to in tro duce the ’t Ho oft coupling constan t x = N ϵ and define the GUE fr e e ener gy F G ( x, s ; ϵ ) as follows: F G ( x, s ; ϵ ) = X g ≥ 0 X n ≥ 1 X i 1 ,...,i n ≥ 1 2 − 2 g − n + | i | 2 ≥ 1 Map g ( i 1 , . . . , i n ) n ! s i 1 · · · s i n ϵ 2 g − 2 x 2 − 2 g − n + | i | 2 (27) + x 2 2 ϵ 2  log x − 3 2  − log x 12 + ζ ′ ( − 1) + X g ≥ 2 ϵ 2 g − 2 B 2 g 4 g ( g − 1) x 2 g − 2 , where ζ ( s ) denotes the Riemann zeta function and B m the m th Bernoulli num ber. W e call the exp onen tial exp F G ( x, s ; ϵ ) =: Z G ( x, s ; ϵ ) the p artition function of GUE c orr elators or the GUE p artition function , whic h is well known to satisfy the follo wing equations: X i ≥ 1 i  s i − 1 2 δ i, 2  ∂ Z G ( x, s ; ϵ ) ∂ s i − 1 + xs 1 ϵ 2 Z G ( x, s ; ϵ ) = 0 , (28) X i ≥ 1 i  s i − 1 2 δ i, 2  ∂ Z G ( x, s ; ϵ ) ∂ s i + x 2 ϵ 2 Z G ( x, s ; ϵ ) = 0 . (29) Equation (28) is called the string e quation for Z G ( x, s ; ϵ ). Equation (29) implies that (30) Map g (2 , i 1 , . . . , i n ) = | i | · Map g ( i 1 , . . . , i n ) + δ g , 0 δ n, 0 . 6 DI Y ANG Define tw o p o w er series v G ( x, s ; ϵ ) and w G ( x, s ; ϵ ) of s b y v G ( x, s ; ϵ ) = ϵ (Λ − 1) ∂ F G ( x, s ; ϵ ) ∂ s 1 , (31) w G ( x, s ; ϵ ) = e F G ( x + ϵ, s ; ϵ )+ F G ( x − ϵ, s ; ϵ ) − 2 F G ( x, s ; ϵ ) . (32) It is well kno wn (see e.g. [1, 16, 23, 35]) that ( v G ( x, s ; ϵ ) , w G ( x, s ; ϵ )) is a solution to the T o da lattice hierarch y (20) and that Z G ( x, s ; ϵ ) is a tau-function of this solution. W e recall that (see e.g. [16]) the functions v G ( x, s ; ϵ ) , w G ( x, s ; ϵ ) can b e uniquely determined b y the T o da lattice hierarc h y (20) together with the initial data (33) v G ( x, 0 ; ϵ ) = 0 , w G ( x, 0 ; ϵ ) = x. Recall that the even GUE fr e e ener gy F eG ( x, s even ; ϵ ) is defined by (34) F eG ( x, s even ; ϵ ) = F G ( x, s ; ϵ ) | s odd = 0 , where s even = ( s 2 , s 4 , s 6 , . . . ), s odd = ( s 1 , s 3 , s 5 , . . . ). The exponential e F eG ( x, s even ; ϵ ) =: Z eG ( x, s even ; ϵ ) is called the even GUE p artition function , which plays an imp ortan t role in quantum gravit y [23, 39]. By definition w e kno w that (35) v G ( x, s ; ϵ ) | s odd = 0 ≡ 0 . Denote w eG ( x, s even ; ϵ ) = w G ( x, s ; ϵ ) | s odd = 0 , which by definition (see (32)) satisfies (36) w eG ( x, s even ; ϵ ) = e F eG ( x + ϵ, s even ; ϵ )+ F eG ( x − ϵ, s even ; ϵ ) − 2 F eG ( x, s even ; ϵ ) . Putting s odd = 0 on b oth sides of (23) and using (35), one finds that w eG ( x, s even ; ϵ ) satisfies the V olterra lattice equation (also called the discrete or difference KdV equation): (37) ϵ ∂ w eG ( x, s even ; ϵ ) ∂ s 2 = w eG ( x, s even ; ϵ )( w eG ( x + ϵ, s even ; ϵ ) − w eG ( x − ϵ, s even ; ϵ )) . This was known in [17, 23, 39]. Similarly , it is clear (see e.g. [15, 17, 23, 39]) that w eG ( x, s even ; ϵ ) satisfies (38) ϵ ∂ w eG ( x, s even ; ϵ ) ∂ s 2 j = [( L 2 j e ) + , L e ] , j ≥ 1 , whic h is the V olterr a lattic e hier ar chy , aka the discr ete KdV hier ar chy . Here, the Lax op erator has the expression L e = Λ + w eG ( x, s even ; ϵ )Λ − 1 . 3. A new pr oof of Witten’s conjecture In this section w e giv e a new pro of of Witten’s conjecture. Let us do some preparations. In Section 1, for 2 g − 2 + n > 0 the Witten n -p oint function in gen us g , i.e., Q g ( x 1 , . . . , x n ), is defined. F or the cases with 2 g − 2 + n ≤ 0, like in e.g. [2, 30, 37], it is con venien t to extend the definition of Q g ( x 1 , . . . , x n ) as follows: Q 0 ( ∅ ) = Q 1 ( ∅ ) = 0 , Q 0 ( x 1 ) = 1 x 2 1 , Q 0 , 2 ( x 1 , x 2 ) = 1 x 1 + x 2 . (39) Then, as shown b y Liu–Xu [30], equation (5) implies that for an y g ≥ 0 and n ≥ 1, (40) Q g ( x 1 , . . . , x n , x n +1 = 0) = ( x 1 + · · · + x n ) Q g ( x 1 , . . . , x n ) . KD V INTEGRABILITY IN GUE CORRELA TORS 7 By induction we kno w that for g ≥ 0, n ≥ 1, I = { 1 , . . . , n } and for any s ≥ 0, Q g ( x I , x n +1 = 0 , . . . , x n + s = 0) = | x I | s Q g ( x I ) . (41) Here and b elow, x I = ( x i ) i ∈ I and | x I | = P i ∈ I x i . Under the extended definition of Q g ( x 1 , . . . , x n ) we also ha ve the follo wing lemma. Lemma 3. F ormula (11) also holds for ( g , n ) = (0 , 1) and ( g , n ) = (0 , 2) . Pr o of. F or g = 0 and n = 1, it is well kno wn (see e.g. [25]) that (42) Map 0 (2 j 1 ) = 1 j 1 + 1  2 j 1 j 1  , ∀ j 1 ≥ 1 , whic h b y Stirling’s formula implies that 2 − 3 2 π 1 2 √ x 1 Map 0 (2 j 1 ) 2 2 j 1 κ − 3 2 tends to 1 x 2 1 = Q 0 ( x 1 ) as (10). F or g = 0 and n = 2, it is known from [25] that for an y j 1 , j 2 ≥ 1, (43) Map 0 (2 j 1 , 2 j 2 ) =  2 j 1 j 1  2 j 2 j 2  j 1 j 2 j 1 + j 2 , Map 0 (2 j 1 − 1 , 2 j 2 − 1) =  2 j 1 − 1 j 1  2 j 2 − 1 j 2  j 1 j 2 j 1 + j 2 − 1 . Applying Stirling’s formula we see that b oth π √ x 1 x 2 Map 0 (2 j 1 , 2 j 2 ) 2 2 j 1 +2 j 2 and π √ x 1 x 2 Map 0 (2 j 1 − 1 , 2 j 2 − 1) 2 2 j 1 +2 j 2 − 2 tend to 1 x 1 + x 2 = Q 0 ( x 1 , x 2 ) as (10). The lemma is prov ed. □ W e note that for ( g , n ) = (0 , 0), (1 , 0), form ula (11) still holds with the extended definition (39), but trivially . The following lemma giv es an equiv alent description of Witten’s KdV equation. Lemma 4. The d = 1 c ase of Witten ’s c onje ctur e is e quivalent to validity of the fol lowing r elations: for al l g ≥ 0 and n ≥ 1 , setting I = { 1 , . . . , n } , (44) (2 g + n − 1) | x I | 2 Q g ( x I ) = | x I | 5 12 Q g − 1 ( x I ) + X g 1 , g 2 ≥ 0 g 1 + g 2 = g X A  = ∅ ,B  = ∅ A ⊔ B = I | x A | 2 | x B | 3 Q g 1 ( x A ) Q g 2 ( x B ) . Pr o of. In terms of the Witten free energy F ( t ) (see (4)), the d = 1 case of Witten’s conjecture reads as follo ws: (45) ∂ 3 F ( t ) ∂ t 1 ∂ t 2 0 = ∂ 2 F ( t ) ∂ t 2 0 ∂ 3 F ( t ) ∂ t 3 0 + 1 12 ∂ 5 F ( t ) ∂ t 5 0 , whic h, b y comparing co efficien ts of p o wers of t and under degree-dimension matching (or b y dilaton equation), is equiv alent to the follo wing relations: for all g , n, d 1 , . . . , d n ≥ 0, (46) ⟨ τ 1 τ 2 0 τ d I ⟩ g = 1 12 ⟨ τ 5 0 τ d I ⟩ g − 1 + X g 1 , g 2 ≥ 0 g 1 + g 2 = g X A ⊔ B = I ⟨ τ 2 0 τ d A ⟩ g 1 ⟨ τ 3 0 τ d B ⟩ g 2 (see also [21, 30, 39]). Here, τ d A = Q a ∈ A τ d a . F or n = 0, the equality (46) holds trivially . F or n ≥ 1, noticing ⟨ τ 3 0 ⟩ 0 = 1, b y (6) and (41) the equality (46) simplifies to (44). The lemma is prov ed. □ Remark 1. The relation (44) is similar to [30, (ii) of Prop osition 2.1] (see also [31]). W e note that there is another v ersion of Witten’s conjecture, prov ed in [29], whic h says that the function Z ( t ) := exp F ( t ) is a tau-function for the KdV hierarc hy . F or details ab out KdV tau-function see e.g. [4, 12, 19, 26, 29]. It implies the original v ersion of Witten’s 8 DI Y ANG conjecture, whic h, together with (5) and (6), also implies back the tau-function version. A particular case of the tau-function version sa ys (cf. e.g. [4, 19, 31]) (47) ∂ 2 F ( t ) ∂ t 1 ∂ t 0 = 1 2  ∂ 2 F ( t ) ∂ t 2 0  2 + 1 12 ∂ 4 F ( t ) ∂ t 4 0 (differen tiation with resp ect to t 0 giv es (45)). The follo wing relations w ere obtained b y Liu–Xu [31] from (47), (5), (6): (48) (2 g + n − 1) | x I | Q g ( x I ) = | x I | 4 12 Q g − 1 ( x I ) + X g 1 , g 2 ≥ 0 g 1 + g 2 = g X A  = ∅ ,B  = ∅ A ⊔ B = I | x A | 2 | x B | 2 Q g 1 ( x A ) Q g 2 ( x B ) (cf. also [30] for the discov ery of (48) again assuming Witten’s conjecture). The following lemma giv es an iden tit y for the even GUE free energy . Lemma 5. The fol lowing identity holds for F eG = F eG ( x, s even ; ϵ ) : (49) ϵ Λ − 1 Λ + 1  ∂ 2 F eG ∂ x∂ s 2  =  ϵ Λ − 1 Λ + 1  ∂ F eG ∂ s 2  ·  (Λ − 1)(1 − Λ − 1 )  ∂ F eG ∂ x  . Pr o of. It is kno wn from e.g. [16] (cf. (25)) that (50) ϵ (Λ − 1)  F G ( x, s ; ϵ ) ∂ s 2  = ( v G ( x, s ; ϵ )) 2 + (Λ + 1)( w G ( x, s ; ϵ )) . T aking s odd = 0 , using (35), and applying (Λ + 1) − 1 to the resulting iden tity , we get (51) ϵ Λ − 1 Λ + 1  F eG ( x, s even ; ϵ ) ∂ s 2  = w eG ( x, s even ; ϵ ) (this identit y was obtained in [15, (3.32)]). Dividing b oth sides of (37) by w eG ( x, s even ; ϵ ), applying the op erator ∂ x Λ − 1 ◦ Λ Λ+1 , and using (36), (51), w e obtain (49). □ W e note that by (36) and (51) the even GUE free energy F eG satisfies (52) ϵ Λ − 1 Λ + 1  ∂ F eG ∂ s 2  = e (Λ − 1)(1 − Λ − 1 )( F eG ) . Let us also give a deriv ation of (49) just using (52). Similarly to [18], applying ∂ x to b oth sides of (52), we obtain ϵ Λ − 1 Λ + 1  ∂ 2 F eG ∂ x∂ s 2  = e (Λ − 1)(1 − Λ − 1 )( F eG ) ·  (Λ − 1)(1 − Λ − 1 )  ∂ F eG ∂ x  and, replacing the exp onential term of the right-hand side using (52), we get (49). W e are ready to give a new pro of of the Witten–Kon tsevic h theorem. Theorem 1 (Kon tsevic h [29]) . Witten ’s c onje ctur e holds. Pr o of. W e will prov e the d = 1 case, which, as men tioned in Section 1, together with (5), (6) for psi-class in tersection n umbers implies all cases. KD V INTEGRABILITY IN GUE CORRELA TORS 9 Let F norm ( x, s even ; ϵ ) := F eG ( x, s even ; ϵ ) − F eG ( x, 0 ; ϵ ) b e the normalized even GUE free energy . In terms of F norm ( x, s even ; ϵ ), identit y (49) reads (53) ϵ Λ − 1 Λ + 1  ∂ 2 F norm ∂ x∂ s 2  =  ϵ Λ − 1 Λ + 1  ∂ F norm ∂ s 2  ·  1 x + (Λ − 1)(1 − Λ − 1 )  ∂ F norm ∂ x  . Noting that (54) ϵ Λ − 1 Λ + 1 = X g ≥ 0 ϵ 2 g +2 2 2 g +3 − 2 (2 g + 2)! B 2 g +2 ∂ 2 g +1 x , and for fixed h ≥ 0 comparing co efficien ts of ϵ 2 h on b oth sides of (53), we obtain X g , g ′ ≥ 0 g + g ′ = h (2 2 g +3 − 2) B 2 g +2 (2 g + 2)! ∂ 2 g +2 x  ∂ F norm g ′ ∂ s 2  (55) = X g 1 , g 2 , g ′ 1 , g ′ 2 ≥ 0 g 1 + g 2 + g ′ 1 + g ′ 2 = h (2 2 g 1 +3 − 2) B 2 g 1 +2 (2 g 1 + 2)! ∂ 2 g 1 +1 x  ∂ F norm g ′ 1 ∂ s 2  δ g 2 , 0 δ g ′ 2 , 0 x + 2 ∂ 2 g 2 +3 x ( F norm g ′ 2 ) (2 g 2 + 2)!  , where F norm g = F norm g ( x, s even ) := [ ϵ 2 g − 2 ] F norm ( x, s even ; ϵ ), g ≥ 0. Successively taking deriv atives with resp ect to s 2 j 1 , . . . , s 2 j n on b oth sides of (55), we get X g , g ′ ≥ 0 g + g ′ = h (2 2 g +3 − 2) B 2 g +2 (2 g + 2)! ∂ 2 g +2 x  ∂ 1+ n F norm g ′ ∂ s 2 ∂ s 2 j I  = X A ⊔ B = I X g 1 , g 2 , g ′ 1 , g ′ 2 ≥ 0 g 1 + g 2 + g ′ 1 + g ′ 2 = h (2 2 g 1 +3 − 2) B 2 g 1 +2 (2 g 1 + 2)! ∂ 2 g 1 +1 x  ∂ 1+ | A | F norm g ′ 1 ∂ s 2 ∂ s 2 j A  ×  δ B , ∅ δ g 2 , 0 δ g ′ 2 , 0 x + 2 (2 g 2 + 2)! ∂ 2 g 2 +3 x  ∂ | B | F norm g ′ 2 ∂ s 2 j B  , where I = { 1 , . . . , n } , and for a subset A ⊂ I , | A | denotes the cardinality of A and 2 j A = (2 j a ) a ∈ A . T aking s even = 0 giv es X g , g ′ ≥ 0 g + g ′ = h (2 2 g +3 − 2) B 2 g +2 (2 g + 2)! ∂ 2 g +2 x  Map g ′ (2 , 2 j I ) x 2 − 2 g ′ − n + | j |  = X A ⊔ B = I X g 1 , g 2 , g ′ 1 , g ′ 2 ≥ 0 g 1 + g 2 + g ′ 1 + g ′ 2 = h (2 2 g 1 +3 − 2) B 2 g 1 +2 (2 g 1 + 2)! ∂ 2 g 1 +1 x  Map g ′ 1 (2 , 2 j A ) x 2 − 2 g ′ 1 −| A | + | j A |  ×  δ B , ∅ δ g 2 , 0 δ g ′ 2 , 0 x + 2 (2 g 2 + 2)! ∂ 2 g 2 +3 x  Map g ′ 2 (2 j B ) x 2 − 2 g ′ 2 −| B | + | j B |   , 10 DI Y ANG where | j | = j 1 + · · · + j n and | j A | = P a ∈ A j a . By using (30), w e then ha ve, for any fixed h, n ≥ 0, and for all j 1 , . . . , j n ≥ 1, X g , g ′ ≥ 0 g + g ′ = h (2 2 g +3 − 2) B 2 g +2 · (2 | j | Map g ′ (2 j I ) + δ g ′ , 0 δ n, 0 )  2 − 2 g ′ − n + | j | 2 g + 2  (56) = X A ⊔ B = I X g 1 , g 2 , g ′ 1 , g ′ 2 ≥ 0 g 1 + g 2 + g ′ 1 + g ′ 2 = h (2 2 g 1 +3 − 2) B 2 g 1 +2 2 g 1 + 2  2 − 2 g ′ 1 − | A | + | j A | 2 g 1 + 1  × (2 | j A | Map g ′ 1 (2 j A ) + δ g ′ 1 , 0 δ A, ∅ ) ×  δ B , ∅ δ g 2 , 0 δ g ′ 2 , 0 + 2(2 g 2 + 3)  2 − 2 g ′ 2 − | B | + | j B | 2 g 2 + 3  Map g ′ 2 (2 j B )  . F or n ≥ 1, w e hav e, after a relativ ely length y simplification, (2 3 − 2) B 2 · (2 h + n − 1)( | j | + 2 − 2 h − n )( | j | − 2 h − n )Map h (2 j I ) (57) + (2 5 − 2) B 4 · | j | Map h − 1 (2 j I ) ( | j | +4 − 2 h − n )( | j | +3 − 2 h − n )( | j | +2 − 2 h − n )( | j | +1 − 2 h − n ) 12 + h X g =2 (2 2 g +3 − 2) B 2 g +2 · 2 | j | Map h − g (2 j I )  2 − 2( h − g ) − n + | j | 2 g + 2  = X A  = ∅ ,B  = ∅ A ⊔ B = I X g ′ 1 , g ′ 2 ≥ 0 g ′ 1 + g ′ 2 = h (2 3 − 2) B 2 · (2 − 2 g ′ 1 − | A | + | j A | ) | j A | Map g ′ 1 (2 j A ) × ( | j B | + 2 − 2 g ′ 2 − | B | )( | j B | + 1 − 2 g ′ 2 − | B | )( | j B | − 2 g ′ 2 − | B | )Map g ′ 2 (2 j B ) + X A  = ∅ ,B  = ∅ A ⊔ B = I X g 1 , g 2 , g ′ 1 , g ′ 2 ≥ 0 g 1 + g 2 > 0 g 1 + g 2 + g ′ 1 + g ′ 2 = h (2 2 g 1 +3 − 2) B 2 g 1 +2 g 1 + 1  2 − 2 g ′ 1 − | A | + | j A | 2 g 1 + 1  | j A | Map g ′ 1 (2 j A ) × 2(2 g 2 + 3)  2 − 2 g ′ 2 − | B | + | j B | 2 g 2 + 3  Map g ′ 2 (2 j B ) + (2 3 − 2) B 2 · 10  2 − 2( h − 1) − n + | j | 5  Map h − 1 (2 j I ) + h X g 2 =2 (2 3 − 2) B 2 · 2(2 g 2 + 3)  2 − 2( h − g 2 ) − n + | j | 2 g 2 + 3  Map h − g 2 (2 j I ) + h X g 1 =1 (2 2 g 1 +3 − 2) B 2 g 1 +2 2 g 1 + 2  2 − 2( h − g 1 ) − n + | j | 2 g 1 + 1  Map h − g 1 (2 j I ) · 2 | j | . 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