On the Frobenius Manifold

Frobenius manifolds and Frobenius algebra-valued integrable systems

Dafeng Zuo (USTC)

Russian-Chinese Conference on Integrable Systems and Geometry 19–25 August 2018

The Euler International Mathematical Institute

1 Outline:

Part A. Motivations

Part B. Some Known Facts about FM

Part C. The Frobenius algebra-valued KP hierarchy

Part D. Frobenius manifolds and Frobenius algebra-valued In- tegrable systems

2 Part A. Motivations

3 KdV equation : 4ut − 12uux − uxxx = 0

3 2 2 Lax pair: Lt = [L+,L],L = ∂ + 2 u tau function: u = (log τ)xx

Bihamiltonian structure(BH): ({·, ·}2 =⇒ Virasoro algebra)

{ } { } ( ) ∫ ∫ 3 f,˜ ˜g = 2 δf ∂ δg dx, f,˜ ˜g = −1 δf ∂ + 2u ∂ + 2 ∂ u δg dx. 1 δu ∂x δu 2 2 δu ∂x3 ∂x ∂x δu

Dispersionless =⇒ Hydrodynamics-type BH =⇒ Frobenius manifold

······

4 Natural generalization

Lax pair/tau function/BH: KdV =⇒ GDn / DS =⇒ KP (other types)

(N) {·, ·}2: Virasoro =⇒ Wn algbra =⇒ W∞ algebra

Lax pair/tau function/BH: dKdV =⇒ dGDn =⇒ dKP

(N) {·, ·}2: Witt algebra =⇒ wn algbra =⇒ w∞ algebra

FM: A1 =⇒ An−1 =⇒ Inﬁnite-dimensional analogue (?)

5 Other generalizations: the coupled KdV

4vt − 12vvx − vxxx = 0, 4wt − 12(vw)x − wxxx = 0.

τ1 tau function: v = (log τ0)xx, w = ( )xx. τ0 R. Hirota, X.B.Hu and X.Y.Tang, J.Math.Anal.Appl.288(2003)326

BH: A.P.Fordy, A.G.Reyman and M.A.Semenov-Tian-Shansky,Classical r-matrices and compatible Poisson brackets for coupled KdV systems, Lett. Math. Phys. 17

(1989) 25–29.

W.X.Ma and B.Fuchssteiner, The bihamiltonian structure of the perturbation equations of KdV Hierarchy. Phys. Lett. A 213 (1996) 49–55.

6 P.Casati and G.Ortenzi, New integrable hierarchies from vertex operator representations of polynomial Lie algebras. J.Geom.Phys. 56(2006) 418–449.

Johan van de Leur,J.Geom.Phys., h57(2007)435–447.   {( )}  a0 0 0  0 a0 0 0   0 F2 = , F3 = a1 a0 0  , ··· , Fm, ··· a1 a0   a2 a1 a0 -valued Lax pair

When m = 2, it gives the Coupled KdV equation

4vt − 12vvx − vxxx = 0, 4wt − 12(vw)x − wxxx = 0.

7 Question: To construct BH for the coupled KP hierarchy and study the related W-type algebras and Frobenius manifolds.

Method: Adler-Gelfand-Dickey scheme

A conjectural construction :   1 1 ··· 1  m m−1   1 . 1   0 ..  0 tr (A) = the trace of  m 2  A, A ∈ F  . ... .  m ··· 1 0 0 m

0 Observation:(Fm, tr ,Im) is a Frobenius algebra.

8 Part B. Some Known Facts about FM

9 Deﬁnition 1. A Frobenius algebra {A, ◦, e, tr } over K (=R or C) sat- isﬁes the following conditions:

(i). ◦ : A × A → A is a commutative, associative algebra with unity e;

(ii). tr ∈ A⋆ deﬁnes a non-degenerate inner product ⟨a, b⟩ = tr (a ◦ b) .

Since tr (a) = ⟨e, a⟩ the inner product determines the form tr and visa- versa. This linear form tr is often called a trace form (or Frobenius form).

10 ∈ R Fε Example 2. Given ε , let N be an N-dimensional commutative associative algebra over C with a basis e1, e2, ··· , eN satisfying { ei+j−1, i + j ≤ N + 1; ei ◦ ej = εei+j−1−N , i + j > N + 1. Fε → C We introduce N “basic” trace forms tr : N deﬁned by ∑N − N ε ··· tr ( ajej) = ak + aN (1 δk )δ0, k = 1, ,N. j=1 {Fε ◦} Obviously, all N , tr , e1, are Frobenius algebras of the rank N. The Fε algebra N has a matrix representation as follows

0 j−1 e1 7→ Θ = IdN, ej 7→ Θ , j = 2, ··· ,N, j ∈ C j j+1 1 j where Θ = (Θi ) gl(N, ) with the element Θi = δi + εδi δN and 0 th Θ = IdN is the N -order identity matrix.

11 Deﬁnition 3. (B.Dubrovin) The set {M, ◦, e, ⟨ , ⟩,E} is a Frobenius manifold if each tangent space TtM carries a smoothly varying Frobe- nius algebra with the properties:

(i). ⟨ , ⟩ is a ﬂat metric on M;

(ii). ∇e = 0, where ∇ is the Levi-Civita connection of ⟨ , ⟩;

(iii). the tensors c(u, v, w) := ⟨u ◦ v, w⟩ and ∇zc(u, v, w) are totally symmetric;

(iv). A vector field E exists, linear in the flat-variables, such that the corresponding group of diffeomorphisms acts by conformal transforma- tion on the metric and by rescalings on the algebra on TtM.

12 These axioms imply the existence of the prepotential F which satisfies the WDVV-equations of associativity in the flat-coordinates of the metric (strictly speaking only a complex, non-degenerate bilinear form) on M. The multiplication is then defined by the third derivatives of the prepotential: ∂ ∂ ∂ ◦ = cγ (t) ∂tα ∂tβ αβ ∂tγ where ∂3F c = αβγ ∂tα∂tβ∂tγ ∂ ∂ and indices are raised and lowered using the metric η = ⟨ , ⟩. αβ ∂tα ∂tβ

13 Witten-Dijgraﬀ-Verlinde-Verlinde equations

2-D TFT: ﬁnd free energy F = F (t1, ··· , tn) satisfying WDVV equations of associativity (1991):

∂3F ∂3F ∂3F ∂3F ηλµ = ηλµ , ∂tα∂tβ∂tλ ∂tµ∂tδ∂tγ ∂tδ∂tβ∂tλ ∂tµ∂tα∂tγ with a quasihomogeneity condition

LEF = (3 − d)F + quadratic polynomial in t, and ∂3F = ηαβ ∂tα∂tβ∂t1

(ηαβ): constant nondegenerate.

14 k Example 4. Suppose cij are the structure constants for the Frobenius A ◦ k ⟨ ⟩ algebra , so ei ej = cijek and ηij = ei, ej . For such an algebra one obtains a cubic prepotential 1 F = c titjtk , 6 ijk 1 = tr (t ◦ t ◦ t) , t = tie . 6 i ∑ ∂ The Euler vector ﬁeld takes the form E = ti and E(F ) = 3F. i i ∂t The notation A will be used for both the algebra and the corresponding Frobenius manifold.

15 ∗ On the Frobenius manifold M, its cotangent space Tt M also carries a Frobenius algebra structure, with an invariant bilinear form ⟨dtα, dtβ⟩∗ = ηαβ and a product given by

α β αβ γ αβ αϵ β dt · dt = cγ dt , cγ = η cϵγ.

16 Let

αβ α β g = iE(dt · dt ), then (dtα, dtβ)∗ = gαβ deﬁnes a symmetric bilinear form, called the ∗ intersection form, on Tt M.

The above two bilinear forms on T ∗M compose a pencil gαβ + µ ηαβ of ﬂat metrics with parameter µ, hence they induces a bi-hamiltonian structure { , } + µ{ , } of hydrodynamic type on the loop space { } 2 1 S1 → M .

17 Furthermore, on the loop space one can choose a family of functions

θα,p(t) with α = 1, 2, . . . , n and p ≥ 0 such that 2 β ∂F ∂ θα,p ϵ ∂θα,p−1 θ 0 = η t , θ 1 = , = c for p > 1. α, αβ α, ∂tα ∂tλ∂tµ λµ ∂tϵ The principal hierarchy associated to M is the following system of Hamiltonian equations γ { ∫ } ∂t γ = t (x), θα,p dx , α, γ = 1, 2, . . . , n; p ≥ 0, ∂T α,p 1 in which x is the coordinate of S1. This hierarchy can be written in a bi-hamiltonian recursion form if certain nonresonant condition is fulﬁlled. (See ref.Dubrovin-Zhang 2001)

18 1 3 ∂ ∂ 11 Example 5. F (u) = 6u ,E = u∂u, e = ∂u, η = 1. Denote := ! α,n := , then = 1 n+1. So the principal hier- Tn n T , θn θα,n θn (n+1)!u archy is given by

∂u n = uxu , n = 1, 2, ··· dispersionless KdV ∂Tn with two compatible Poisson structure ∂ ∂ ∂ P = , P = u + u 1 ∂x 2 ∂x ∂x which gives a ﬂat pencil η11 + µg11 on the Frobenius manifold C.

Conversely, under certain conditions one can obtain the prepotential F from the two compatible hydrodynamical Poisson structure.

19 Part C. The Frobenius algebra-valued KP hierarchy

20 Problem: To study the Frobenius algebra-valued KP hierarchy.

∑ i For an A-valued operator P = Pi∂ , P+ is the pure diﬀerential of i the operator P and ∑ ∗ i i ∂ P− = P − P+, res(P ) = P−1,P = (−1) ∂ Pi, ∂ = . i ∂x

21 Let

−1 −2 L = Im∂ + U1∂ + U2∂ + ··· (1) be an A-valued ΨDO with coeﬃcients U1, U2, ··· being smooth A- valued functions of an inﬁnitely many variables t = (t1, t2, ··· ) and t1 = x.

Deﬁnition. The A-KP hierarchy is the set of equations ∂L r ··· = [Br,L],Br = L+, r = 1, 2, (2) ∂tr or equivalently,

∂Bl ∂Br − + [Bl,Br] = 0. (3) ∂tr ∂tl

22 2 3 Example 3.1.[r = 2, l = 3] Using B2 = Im∂ + 2U1 and B3 = Im∂ + 3U1∂ + 3U2 + 3U1,x, the system (3) becomes

U1,t2 = U1,xx + 2U2,x, 2U1,t3 = 2U1,xxx + 3 U2,xx + 3 U2,t2 + 6 U1U1,x.

If we eliminate U2 and rename t2 = y, t3 = t and U = U1, we obtain

(4Ut − 12UUx − Uxxx)x − 3Uyy = 0. (4) ( ) F 0 u0 0 When we choose a 2 -valued smooth function U = , the system (4) reads u1 u0 the coupled KP equation (e.g., P.Casati and G.Ortenzi 2006) { (4u0t − 12u0u0x − u0xxx)x − 3u0yy = 0, (4u1t − 12u0u1x − 12u0xu1 − u1xxx)x − 3u1yy = 0.

Especially if u0y = u1y = 0, the coupled KP equation reduces to the coupled KdV equation (e.g.,A.P.Fordy et al 1989; Ma W.X et al,1996; Hirota-Hu-Tang 2003)

4u0t − 12u0u0x − u0xxx = 0, 4u1t − 12u0u1x − 12u0xu1 − u1xxx = 0.

23 We have shown that (Ian and Zuo 2015, JMP)

(1). The A-KP hierarchy has an A-valued τ function

(2). The A-KP hierarchy admits bi-Hamiltonian structures

(3). Local matrix generalizations of the classical W-algebras

24 ( ) τ 0 Example 3.2. [The coupled KdV equation]. Assume that τ = 0 τ1 τ0 F0 is its 2 -valued τ-function, then we have ( )   (log τ0)xx 0 u0 0 ∂ −1   = U = (τxτ ) = τ1 . u1 u0 ∂x ( )xx (log τ0)xx τ0 τ1 So u0 = (log τ0)xx, u1 = ( )xx. Taking a τ-function τ0 = 1+exp(2ax+ τ0 ( ) a 0 2a3t) of the KdV equation, then for A = , the function τ = b a 3 I2 + exp(2Ax + 2A t), i.e., ( ) 1 + exp(2ax + 2a3t) 0 τ = (2bx + 2b3t) exp(2ax + 2a3t) 1 + exp(2ax + 2a3t) is a τ-function of the coupled KdV equation. Consequently, we obtain a solution given by ( ) a2 (2bx + 2b3t) exp(2ax + 2a3t) u = , u = . 0 2 3 1 3 cosh (ax + a t) 1 + exp(2ax + 2a t xx

25 Hamiltonian structures of A-KP : the canonical AGD method and the trace form tr : A −→ K.

{ } ∫ The space of functionals: De = f˜= tr F (V )dx tr F (V ) ∈ D .

∑m δF For V = vqeq, the variational derivative is deﬁned by q=1 δV ∫ ( ) ∫ ( ) δF ∑m δf ˜f(v + δv) − ˜f(v) = tr ◦ δV + o(δV) dx = δvq + o(δv) dx, δV q=1 δvq ∞ ∑m δf ∑ ∂f where f(v) = tr F (V ), δV = δv e ∈ A and = (−∂)j . q q (j) q=1 δvq j=0 ∂vq

26 The ﬁrst and the second Poisson brackets of the A-KP hierarchy 0 n associated with the Fm-valued ΨDO L are given by { } ∫ n(∞) ∞ δf δg f,˜ ˜g = tr res Hn( )( ) dx ∫ ( δL δL ) δf δf δg = tr res [L−, ( )+]− − [L+, ( )−]+ dx δL δL δL and { } ∫ n(0) δf δg f,˜ ˜g = tr res Hn(0)( ) dx ∫ ( δL δL ) δf δf δg = tr res (L )+L − L ( L)+ dx, δL δL δL n∑−1 δf − − δF ∫ ∫ where = ∂ i 1 . and f˜= tr f(V )dx, ˜g = tr g(V )dx. L δ i=−∞ δVi

27 ∂L Theorem. The A-KP hierarchy = [Br,L] admits a bi-Hamiltonian ∂tr representation given by ( ) ( ) ∂L δhr ∞ δgr = Hn(0) = Hn( ) (5) ∂tr δL δL with the Hamiltonians ∫ ∫ n r m n+r ˜hr = tr res L dx and ˜gr = − tr res L dx, r r + n n(0) n(∞) where H (X) = (LX)+L − L(XL)+ and H (X) = [L−,X+]− −

[L+,X−]+.

0 Cor. The coupled KP (Fm-KP) hierarchy deﬁned in [CO2006] has at least m “basic” diﬀerent local bi-Hamiltonian structures.

28 If we restrict to Vn−1 = 0, the ﬁrst Hamiltonian structure automatically reduces to this submanifold, but the second one is reducible if and only if δf res [L, ] = 0. (6) δL { }n(∞) { }n(0) We denote the corresponding reduced brackets by , and , D . δf Assume that Vn = e and Xi = ∈ A, then the condition (6) is equiv- δVi alent to ( ) ( )  n∑−2 n∑−1 1  −i − 1 (n−i−1) −i − 1 (j−i−1) X − = X + (X V ) . n 1 n − i i j − i i j n i=−∞ j=i+1

29 { } Deﬁnition. In terms of the basis v[i]q , the second Poisson bracket { }n(0) n { }n(0) n , for L and the reduced bracket , D for L with the constraint Vn−1 = 0 will provide two kinds of W-type algebras, we (m,n) (m,n) call them the WKP -algebra and the W∞ -algebra respectively. Un- n der the reduction L− = 0, the corresponding algebras are called the (m,n) WGD -algebra and the W(m,n)-algebra respectively. All of them are local matrix generalizations.

30 Example 3.3. Taking φ(x) = tr Vn−2(x), the reduced Poisson bracket is given by ( ) 3 (0) n − n {φ(x), φ(y)}n = − ∂3 + φ ∂ + ∂ φ δ(x − y). D 12 (m,n) This means that both the W∞ -algebra and the W(m,n)-algebra con- tain the Virasoro algebra as its subalgebra.

31 Example 3.4. Consider the A-KdV hierarchy with the Lax operator 2 2 2 −2 −1 L = e∂ + V , i.e., L− = 0. We denote X = ∂ X1 + ∂ X0 and −2 −1 Y = ∂ Y1 + ∂ Y0, and have

2(∞) 2 ′ H = [X,L ]+ = −2X0 and

′ ′ 1 ′′′ H2(0)(X) = (L2 ◦ X) ◦ L2 − L2 ◦ (X ◦ L2) = 2V ◦ X + X ◦ V + X . + + 0 0 2 0 Thus two compatible Poisson brackets of the A-KdV hierarchy are given by { } ∫ 2(∞) δf ∂ δg f,˜ ˜g = 2 tr ◦ dx δV ∂xδV and ( ) { } ∫ 2(0) 1 δf ∂3 ∂ ∂ δg f,˜ ˜g = − tr ◦ e + 2V + 2 V ◦ dx. D 2 δV ∂x3 ∂x ∂x δV

A F0 In particular, if one chooses the algebra to be the algebra 2 , one 32 F0 obtains the 2 -KdV equation for V = ve1 + we2 given by

4vt − 12vvx − vxxx = 0, 4wt − 12(vw)x − wxxx = 0. (7) Moreover, the system (7) can be written as ( ) ( )   ( )   δH2 δH1 v 0 ∂  δv  0 J0  δv  = δH = δH w ∂ 0 2 J0 J1 1 t δw δw with Hamiltonians ∫ ∫ 3 2 1 H1 = vwdx, H2 = ( v w + vwxx)dx; S1 S1 2 4 and     ( ) ( ) e ( ) e δH2 δH1 v 0 ∂  δv  0 J0  δv  = e = e w ∂ −∂ δH2 J0 J1 − J0 δH1 t δw δw with Hamiltonians ∫ ∫ f 1 2 f 3 2 1 1 3 1 H1 = ( v + vw)dx, H2 = ( v w + vwxx + v + vvxx)dx, S1 2 S1 2 4 2 8 1 where J = ∂3 + v∂ + ∂v and J = w∂ + ∂w. 0 4 1 0 Example 3.5. [The Fm-dBoussinesq hierarchy]. In this case, the Lax operator is given by

3 0 L = Imp + V1p + V0,V0,V1 ∈ Fm. 0 The BH structure of the Fm-dBoussinesq hierarchy is { } ∫ ′ ′ δf δf f,˜ ˜g = 3 tr (X Y + X Y )dx, here X = ,Y = (8) 1 1 0 0 1 k k δVk δVk and { } ∫ ( ) ∫ ( ) 1 ′ ′ ′ ′ f,˜ ˜g = tr X Y − X Y V 2 dx + tr X Y − X Y V dx 2 0 0 0 0 1 1 1 1 1 1 3 ∫ ( ) ′ ′ ′ ′ + tr 2X1Y0 − X1Y0 + X0Y1 − 2X0Y1 V0 dx. (9)

33 When m = 1, it is easy to get a Frobenius manifold as follows ∂ ∂ ∂ 1 2 − 1 4 2 F(V) = V0V1 V1, e = , E = V0 + V1 . 2 72 ∂V0 ∂V0 3 ∂V1 When m = 2, we write ( ) ( ) v1 0 v3 0 0 V0 = ,V1 = ∈ F2 . v2 v1 v4 v3 { } By using (8) and (9), we could get explicit formulas of v (x), v (y) , i j k k = 1, 2. We can write { } ′ v (x), v (y) = 3ηij(v)δ (x − y) + h (v; v )δ(x − y) i j 1 1 x and { } ′ v (x), v (y) = 3gij(v)δ (x − y) + h (v; v )δ(x − y) i j 2 2 x

34 ij ij for known functions hk(v; vx), where η (v) and g (v) form a ﬂat pencil of metrics given by   0 0 0 1    0 0 1 −1  (ηij(v)) =    0 1 0 0  1 −1 0 0 and   0 −2v2 0 v  9 3 1   −2 2 2 2 − 4 −  ij  9v3 9v3 9v3v4 v1 v2 v1  (g (v)) =  2  .  0 v1 0 3v3  − 2 2 − v1 v2 v1 3v3 3(v4 v3)

35 Observation: A direct calculation gives (F (v), e, E) as follows ∂ e = ∂v1 and ∂ ∂ 2 ∂ 2 ∂ E = v1 + v2 + v3 + v4 ∂v1 ∂v2 3 ∂v3 3 ∂v4 and 1 1 1 1 F (v) = v2v + v2v + v v v − v3v − v4 2 1 4 2 1 3 1 2 3 18 3 4 72 3 1 1 = tr ( V2V − V4). 2 0 1 72 1

Here tr (a1e1 + a2e2) = a1 + a2.

36 Part D. Frobenius manifolds and Frobenius algebra-valued Integrable systems

37 Question: { } { } Frobenius manifold L9999K Principal hierarchy       y y

{ } { } ???? L9999K A − valued principal hierarchy

39 Deﬁnition.[Lifting map] Let {A, ◦, e, tr } be a Frobenius algebra over

K with the basis e = e1, e2, ··· , em and f an analytic function on M (that is, analytic in the flat coordinates for M). The A-valued function fb is defined to be: b f = f| α (αi) t 7→t ei c A with fg = fˆ◦ ˆg and 1ˆ = e1 . The evaluation f of fˆ is defined by ( ) A f = tr fˆ , where tr ∈ A⋆ is the Frobenius form. Theorem.([Ian-Zuo 2017]) Let

α α α ut = K (u, ux, ··· ), u = {u (x, t)} (10) be a Hamiltonian system with the Hamiltonian H[u], then the corresponding A-valued system

cα α \ ··· ut = K (u, ux, ) (11) ( ) d is also Hamiltonian with the Hamiltonian H[ub] = tr H[u] .

Cor. The A-KP hierarchy admits local bi-Hamiltonian structures.

40 Theorem. ([Ian-Zuo 2017]) Let F be the prepotential of a Frobenius manifold M and let A be a Frobenius algebra with 1-form tr . Then the function ( ) FA = tr Fb deﬁnes a Frobenius manifold, namely the manifold M ⊗ A .

(The tensor product was due to Kaufmann, Kontsevich and Manin.)

Remark. This construction could be generalized to TQFT and F- manifold.

41 References:

[1]. Ian Strachan and D.Zuo, Integrability of the Frobenius algebra- valued Kadomtsev-Petviashvili hierarchy. J.Math.Phys. 56 (2015), no. 11, 113509, 13 pp.

[2]. Ian Strachan and D.Zuo, Frobenius manifolds and Frobenius algebra-valued integrable systems, Lett.Math.Phys. 107(2017)996- 1027.

42 THANKS