Integrable Systems, Monopoles and Modifications of Bundles over Elliptic Curves

Transcription

1 Integrable Systems, Monopoles and Modifications of Bundles over Elliptic Curves p. 1/23 Integrable Systems, Monopoles and Modifications of Bundles over Elliptic Curves Andrei Zotov, Conformal Field Theory, Integrable Models and Liouville gravity, Chernogolovka, 2009 Imperial College, London 2009 (in collaboration with M.Olshanetsky, A.Levin, A.Smirnov) Institute of Theoretical and Experimental Physics, Moscow

2 Integrable Systems, Monopoles and Modifications of Bundles over Elliptic Curves p. 2/23 Integrable system Integrable systems related to elliptic curve. Hitchin type systems: Elliptic Calogero-Moser systems, Elliptic Gaudin systems, Elliptic tops, Painleve VI, XYZ-model, Landau-Lifshitz equation, Calogero-Moser field theory; Their trigonometric, Whittaker-Inozemtsev and rational degenerations - XXZ, XXX-models, sin-gordon equation, Toda models,...

3 Integrable Systems, Monopoles and Modifications of Bundles over Elliptic Curves p. 3/23 PLAN 1. Integrable systems 2. Modifications of Bundles: Relations between Integrable systems 3. Characteristic classes and new systems 4. Modifications of Bundles: Bogomolny equations and Monopoles

4 Integrable Systems, Monopoles and Modifications of Bundles over Elliptic Curves p. 4/23 Integrable system LAX EQUATION L = [L,M] L - Lax operator L = L(v,u,S;z) (v,u,s) - dynamical variables (with Poisson brackets), z - spectral parameter z Σ τ - elliptic curve (Σ τ = C/(τZ + Z)). L takes values in g=lie(g), where G is a gauge group. Commuting integrals: {trl k (z), trl n (w)} = 0 tr(l k (z)) - generating function of integrals of motion.

5 Integrable Systems, Monopoles and Modifications of Bundles over Elliptic Curves p. 5/23 Integrable system V - holomorphic bundle over Σ τ. For s Γ(V ): s(z + 1) = Q(z)s(z), s(z + τ) = Λ(z)s(z), Q(z + τ) Λ(z) = Λ(z + 1)Q(z) L(z) is a section of End(V ) with simple poles. Typical example: Gaudin model. Rational case: L = n a=1 S a z x a ; Elliptic case ({T α } - is a basis in g=lie(g)): L = n S a αϕ α (z x a,ω α )T α a=1 α

6 Integrable Systems, Monopoles and Modifications of Bundles over Elliptic Curves p. 6/23 Lax operator 1. L(z) is a meromorphic map L(z) : (Σ τ = C/(τZ+Z)) g a simple complex Lie algebra 2.ResL(z) z=xa = S a O a coadjoint orbits of G 3.L(z + 1) = Q(z)L(z)Q 1 (z), L(z + τ) = Λ(z)L(z)Λ 1 (z). τ Q(z + τ) Λ(z) = Λ(z + 1)Q(z) 0 Λ x 1 x 2 x 3 Q x 4 1

7 Integrable Systems, Monopoles and Modifications of Bundles over Elliptic Curves p. 7/23 Examples 1. Q = Id, Λ = exp2πidiag(u 1,...,u N ) QΛQ 1 Λ 1 = Id L(z) - Lax operator of the Elliptic Calogero-Moser System. L ij (z) = δ ij v i + 1νΦ(z,u i u j ), Φ(z,u) = ϑ (o)ϑ(z + u) ϑ(z)ϑ(u) Canonical Poisson brackets {v i,u j } = δ ij Hamiltonian: H = N vi 2 + ν 2 (u i u j ) i j i=1

8 Examples 2.Q = diag(1,ω,...,ω N 1 ) ω = exp(2πi 1 ), detq = 1 N Λ = , detλ = L(z) - Lax operator of Elliptic Top. L(z) = S α1 α 2 exp( 2πi α 2z N )Φ( α 1 + α 2 τ,z)t α1 α N 2 α=(α 1,α 2 ) Z N Z N but for Λ = exp( 2πi z+ τ 2 N )Λ: QΛQ 1 Λ 1 = ωid Q Λ(z) = Λ(z + 1)Q. Integrable Systems, Monopoles and Modifications of Bundles over Elliptic Curves p. 8/23

9 Integrable Systems, Monopoles and Modifications of Bundles over Elliptic Curves p. 9/23 Relations between systems Modification Symplectic Hecke Correspondence L(z) ΞL(z)Ξ 1 Ξ is a singular gauge transformation. Example: L top = ΞL Cal Ξ 1, ( Ξ(z) = Ξ(z) diag ( 1) l j<k;j,k l ) 1 ϑ(u k u j,τ), where Ξ ij (z,u 1,...,u N,τ) = θ i N 1 2 N 2 (z Nu j,nτ), det Ξ ϑ(z) i>j ϑ(u i u j )

10 Integrable Systems, Monopoles and Modifications of Bundles over Elliptic Curves p. 10/23 Example g = A 1 SL(2, C) - Calogero-Moser system for two particles: SL(2, C)-Euler top: H SL(2,C) = 1 2 v2 + ν 2 (2u), {v,u} = 1 H SL(2,C) = 1 2 (S2 1 (τ/2) + S 2 2 ( 1 + τ 2 ) + S2 3 ( 1 2 )), {S j,s k } = ǫ ijk S i Ξ = ( θ00 (z 2u, 2τ) θ 00 (z + 2u, 2τ) θ 10 (z 2u, 2τ) θ 10 (z + 2u, 2τ) )

11 Integrable Systems, Monopoles and Modifications of Bundles over Elliptic Curves p. 11/23 Example S 1 = v θ 10(0) θ 11(0) θ 10 (2u) θ 11 (2u) ν θ 2 10(0) θ 00 (0)θ 01 (0) θ 00 (2u)θ 01 (2u) θ 2 11(2u), S 2 = v θ 00(0) iθ 11(0) θ 00 (2u) θ 11 (2u) ν θ00(0) 2 iθ 10 (0)θ 01 (0) θ 10 (2u)θ 01 (2u) θ 2 11(2u), S 3 = v θ 01(0) θ 11(0) θ 01 (2u) θ 11 (2u) ν θ 2 01(0) θ 00 (0)θ 10 (0) θ 00 (2u)θ 10 (2u) θ 2 11(2u). θ ab - theta functions with characteristics. Ξ : (v,u,ν) (S 1,S 2,S 3 ), ν 2 = S S S 2 3

12 Integrable Systems, Monopoles and Modifications of Bundles over Elliptic Curves p. 12/23 Local description Ξ GL(N,C) (z) = z , det Ξ GL(N,C) (z) = z is a modification of GL(N, C)-bundle at point z = 0 in direction (1, 0,...,0) T SL(N, C) case: Ξ SL(N,C) (z) = z 1 N ΞGL(N,C) (z), det Ξ SL(N,C) (z) = 1

13 Integrable Systems, Monopoles and Modifications of Bundles over Elliptic Curves p. 13/23 Relations between systems Dimension of moduli space (in GL(N, C) case) equals g.c.d.(degree, rank) deg = 0 deg = 1... deg = k... deg = N deg = 0 CM Top... Backlund CM [Λ,Q] G = 1 [Λ,Q] G = ω [Λ,Q] G = ω k [Λ,Q] G = ω N = 1 N = pl - Interacting elliptic tops Z N - is a center of SL(N, C). The models are classified by solutions of [Λ,Q] G = ζ, ζ Z(G)

14 Integrable Systems, Monopoles and Modifications of Bundles over Elliptic Curves p. 14/23 Monodromies Ḡ - a simply connected simple complex Lie group. Z(Ḡ) - its center. G ad = Ḡ/Z(Ḡ). Examples: 1. Ḡ = SL(N, C), Z(SL(N, C)) = diag(ω,...,ω), ω N = 1, G ad = PSL(N, C). ζ generator of Z(Ḡ) ΛQΛ 1 Q 1 = ζ Λ and Q are not monodromies of Ḡ-bundle, but transition operators for G ad -bundle. ζ is an obstruction to lift G ad -bundle to Ḡ-bundle. ζ H 2 (Σ τ, Z(Ḡ)) Z(Ḡ) ζ is the characteristic class of the bundle

15 Centers of universal covering groups Integrable Systems, Monopoles and Modifications of Bundles over Elliptic Curves p. 15/23 Ḡ Lie (Ḡ) Z(Ḡ) G ad SL(n, C) A n 1 Z n SL(n, C)/Z n Spin 2n+1 (C) B n Z 2 SO(2n + 1) Sp n (C) C n Z 2 Sp n (C)/Z 2 Spin 4n (C) D 2n Z 2 Z 2 SO(2n)/Z 2 Spin 4n+2 (C) D 2n+1 Z 4 SO(2n)/Z 2 E 6 (C) E 6 Z 3 E 6 (C)/Z 3 E 7 (C) E 7 Z 2 E 7 (C)/Z 2

16 Integrable Systems, Monopoles and Modifications of Bundles over Elliptic Curves p. 16/23 Bogomolny equation Dirac String S 2 y Σ τ Σ + τ A + z = Ξ 1 z Ξ + Ξ 1 A z Ξ A ± z L± (z)

17 Integrable Systems, Monopoles and Modifications of Bundles over Elliptic Curves p. 17/23 Bogomolny equation Σ τ elliptic curve, (z, z) - complex coordinates. W(y,z, z) = R Σ g Fields: 1)A = (A z,a z,a y ) - connections in the adjoint representation of simple Lie algebra g. F = (F z, z,f z,y,f z,y ). 2)φ- scalar field in the adjoint representation of simple Lie algebra g - (the Higgs field). The Bogomolny equation on W F = Dφ ( the Hodge operator on W)

18 Integrable Systems, Monopoles and Modifications of Bundles over Elliptic Curves p. 18/23 Bogomolny equation x = (z, z,y), x 0 = (0, 0, 0) D Dφ = Mδ( x x 0 ) For metrics ds 2 = g dz 2 + dy 2 and gauge fixation A z = 0, A y = iφ: z A z = ig 2 yφ, y A z 2i z φ + 2i[A z,φ] = 0, In scalar case: yφ g z z φ = cδ( x x 0 )

19 Bogomolny equation Abelian rational case φ im 2 1 y2 + z z A + z (z, z,y) im 2 A z (z, z,y) im 2 ( ( 1 z 1 z ) y 1 y2 +z z z ) y + 1 y2 +z z z + const, + const, A + z = A z + i z log z m, Ξ = z m Σ + τ F = Σ τ F + m, Integrable Systems, Monopoles and Modifications of Bundles over Elliptic Curves p. 19/23

20 Integrable Systems, Monopoles and Modifications of Bundles over Elliptic Curves p. 20/23 Bogomolny equation Abelian elliptic case c γ Γ χ(γ, x) ((παy) 2 + z + γ 2 ) 1 2 = φ(x,z,y) φ(x,z,y) = c 4π Consider a generalization γ Γ 1 γ + x e 2π γ+x y χ(γ + x,z). I(s,x,z,y) = 2cπ s y s 1 2 γ Γ K s 1(2π y γ + x ) 2 χ(γ + x,z). γ + x s 1 2 Here K ν is the Bessel-Macdonald function

21 Integrable Systems, Monopoles and Modifications of Bundles over Elliptic Curves p. 21/23 Bogomolny equation K ν (2πyz) = Γ(ν )(z)ν 2(πy) ν Γ( 1 2 ) + e 2πipy dp (p 2 + z 2 ) ν+1 2. The function I(s,x,z,y) is the Green function for the pseudo-differential operator ( 2 y + 4α 2 π 2 z z ) s on R Σ τ with the boundary conditions The series is a three-dimensional generalization of the Kronecker series K(x,x 0,s) = γ χ(γ,x 0 ) x + γ 2s

22 Integrable Systems, Monopoles and Modifications of Bundles over Elliptic Curves p. 22/23 Bogomolny equation Using the Poisson summation formula Kronecker proved that Γ(s)K(x,x 0,s) = α 1 2s Γ(1 s)k(x 0,x, 1 s)χ(x,x 0 ). Our purpose is to generalize this functional equation for the 3-dimensional case Σ τ R. It takes the following form. The function I(s,x,z,y) satisfies the functional equation: I(s,x,z,y) = χ(x,z)π 1 2 α 2s+1 + dk I( 3 2 s,z,x, k πα )e 2πiky

23 where E 1 (z) = ln ϑ(z) is the so-called first Eisenstein series. Integrable Systems, Monopoles and Modifications of Bundles over Elliptic Curves p. 23/23 Bogomolny equation Gauge fixation condition: A z = 0. Solution: A z (z, z,y,x) = ic 4π 1 π 2 α 2 sgn(y) γ Γ 1 γ + x e 2π γ+x y χ(γ+x,z), Notice that the jump of A (while coming through y = 0, z = 0) is obviously defined by the jump of sgn(y). In order to compare elliptic configuration with the rational case we take x = 0. Then on the line y = 0 the connection is proportional to A z γ 0 1 γ χ(γ,z) = E 1(z) α 1 (z z),