Two constructions of the isomonodromic τ -function Ian Strachan September 24th 2005 Two constructions of the isomonodromic τ -function Ian Strachan September 24th 2005 based on: IMRN, 2003 MRL, 2005 (joint work with A.Kotokov) 1 The isomonodromic τ -function Monodromy Fuchsian System: z ∈ C , Ai ∈ GL(N, C) n dy Ai = y dz i=1 z − ui Monodromy measures the multivaluedness of the solution. Let Y (z) be a fundamental solu- tion: Y (zγ ) = Y (z) Mγ The matrix Mγ is invertible, depending on the homotopy class of γ ρ : π1(C\{u1 , . . . , un}) −→ GL(N, C) The map ρ is the monodromy representation, its image, the monodromy group. 2 Isomonodromy Deform: A → A(u) , Y (z) → Y (z, u) in such a way that the monodromy is invariant: Scheslinger Equations (1912): n ∂Y (z, u) Ai(u) = Y (z, u) , ∂z i=1 z − ui ∂Y (z, u) A (u) = − i Y (z, u) ∂ui z − ui Equivalently: ∂Aj [Ai, Aj ] = , ∂ui ui − uj ∂Ai [Ai, Aj ] = − ∂ui u − uj j=i i 3 The Isomonodromic τ -function Deﬁne n ω= Hm dum m=1 where 1 tr(AiAj ) Hi = 2 j=i ui − uj Theorem dω = 0 so ω = d log τI . e Theorem (Painlev´ property) The singularities of the solutions to Schlesinger’s equations are poles except for ﬁxed singulari- ties The τI function is holomorphic except at the ﬁxed singularities 4 Frobenius Manifold Ingredients • Manifold M , ﬂat metric <, > , • a commutative, associative multiplication on each tangent space ◦ : T M ×T M → T M , with unity vector ﬁeld e • Euler vector ﬁeld E with < X ◦ Y, Z >=< X, Y ◦ Z > and ◦ totally symmetric, e = 0 and every- thing (quasi)-homogeneous LE (◦) = const. ◦ 5 Semisimple Frobenius Manifold: Frobenius algebra on (generic) tangent space semi-simple: Idempotents: ∂ ∂ ∂ ∂ : ◦ = δij i ∂ui ∂ui ∂uj ∂u Metric diagonal: ∂ ∂ ∂ ∂ < , j > = <e◦ , j > ∂ui ∂u ∂ui ∂u ∂ ∂ = < e, i ◦ j > ∂u ∂u ∂ = δij < e, i > ∂u Two natural coordinate systems: ﬂat and canonical 6 <, >-ﬂat; diagonal in canonical coordinates ui <, >= 2 Hi (dui)2 rotation coeﬃcients: γij = ∂j Hi/Hj . Γ = ||γij || U = diag (u1 , . . . , uN ) V = [Γ, U ] In these coordinates 2 Vij ω= dui i=j ui − u j Frobenius Manifolds ↔ Schlesinger Equations Dubrovin connection: ˜ uv = u v + zu ◦ v Extend to M × C d ˜u = 0 dz d ˜d = 0 dz dz 1 ˜ d v = ∂z v + E ◦ v − µv dz z where µ = diag(µ1 . . . µn) Frobenius manifold ≡ Flatness of extended con- nection 7 Solutions of ξ = 0 exist, i.e. ∂ξ = (zEi + Vi) ξ ∂ui ∂ξ V = U+ ξ dz z where (Ei)ab = δiaδib , [U, Vi] = [Ei, V ] . Schlesinger Equations, but with an essential singularity at inﬁnity Problem: Find τI for classes of Frobenius man- ifolds 8 Construction I Orbit spaces and Coxeter Groups Example: W = I2(N ) z → e2πi/N . z z → z Invariant objects: g = 2dzdz t1 = z N + z N t2 = zz New objects on C2/I2(N ) : (t2)N −1 t1 g −1 = t1 t2 E = 2t 1 ∂ + N t2 ∂ ∂t1 ∂t2 ∂ e = ∂t1 <, >−1 = Le g −1 N.B. g deﬁned on C2\∆ while <, > is deﬁned on the whole of C2 9 Generalize to Cn/W Theorem (Saito,K.,Dubrovin, B.) From {g , <, > , E , e} one can construct a Frobe- nius manifold M (in particular ◦) with <, > be- ing a ﬂat, non-degenerate metric Caustics: The set of points where the multiplication is not semisimple is known as the caustic: #Ki K= Ki . i=1 10 Example: W = An λ = pn+1 + a1pn−1 + . . . + an Canonical coordinates: ui = λ(αi) , where λ (αi) = 0 . Caustics: coincident critical points αi = αj , R(λ , λ ) = 0 Maxwell Strata: coincident critical values αi = αj , u i = uj Discriminant: (∆ = 0) ui = 0 , R(λ, λ ) = 0 11 Theorem (IABS) The isomonodromic τ -function for the Frobe- nius manifold CN /W where W is a Coxeter group, is given by #Ki − (Ni−2)2 16Ni τI = κi i=1 where κ−1(0) = Ki are the irreducible compo- i nents on the caustic. Example: W = F4 N1 = 4 , N 2 = N3 = 3 12 Virasoro Constraints Deﬁne En = E ◦ E ◦ . . . ◦ E E0 = e n these satisfy Virasoro algebra: [E n, E m] = (m − n)E m+n−1 and τI satisﬁes a set of constraint equations LE n log τI = γn n = 0,... ,∞ (γ0 = 0 , γ1 constant) Idea: Solve constraint equations rather than deﬁning equations 13 Behaviour near a caustic: I At a generic point, multiplication decomposes into n-one-dimensional algebras: An 1 What happens on/near a caustic (a non-generic point)? The simplest case case is where the multiplica- n−2 tion on the caustic Ki is of the type A1 I2(Ni) , the multiplication decomposes into n − 2 one- dimensional algebras and a single two-dimensional algebra based on the Coxeter group I2(N ) . 14 Behaviour near a caustic: II Deﬁnition (Hertling, Manin): F-manifold LX◦Y (◦) = X ◦ LY (◦) + Y ◦ LX (◦) , LE (◦) = const. ◦ . Theorem (Hertling, C.) Let (M, ◦, e, E, g) be a simply connected semi- simple Frobenius manifold. Suppose that at generic points of the caustic Ki the germ of the n−2 underlying F-manifold is of type I2(Ni)A1 for one ﬁxed number Ni ≥ 3. a) The form d log τI has a logarithmic pole (Ni−2)2 along Ki with residue − 16N along Ki ∩ Kreg . i 15 Useful facts about orbits spaces Consider the Frobenius manifold structure on the orbit space Cn/W where W is a Coxeter group. Then: • Everything polynomial in the ﬂat coordi- nates; • E= n i∂ i=1 dit ∂ti , di > 0 ; • Number of caustics Ki known; • Structure on caustics of type I2(Ni)An−2. 1 16 Proof of Theorem LE n log τI = γn ↓ ∂ log τI i = rational functions ∂t ↓ log τI = rational functions plus log. terms ↓ only singularities logarithmic log τI = polynomial functions plus log. terms ↓ LE log τI = constant log τI = log. terms ↓Hertling/known f acts (Ni − 2)2 log τI = − log κi i 16Ni ↓ #Ki − (Ni−2)2 16Ni τI = κi i=1 17 Construction II Hurwitz spaces Hg,N (k1 , . . . , kl ) Equivalence classes [λ : Σ → P1] of N -fold branched coverings p : Σ → P1, where Σ is a compact Riemann surface of genus g and the holomorphic map λ of degree N is subject to the following conditions: • M simple ramiﬁcation points P1, . . . , PM with distinct ﬁnite images λ1, . . . , λM ∈ C ⊂ P1; • λ−1(∞) = {∞1, . . . , ∞l }, ramiﬁcation index at ∞j is kj (1 ≤ kj ≤ N ). 18 Riemann surface theory Prime-form on a Riemann surface E(P, Q) B(P, Q) = dP dQ log E(P, Q) Singular as P → Q : B(x(P ), x(Q)) = 1 1 + SB (x(P )) + o(1) dx(P )dx(Q) (x(P ) − x(Q))2 6 Bergmann projective connection SB Theorem (Kokotov, Korotkin) 1 Hm = SB (xm)|xm=0 24 19 g = 0 , g = 1 (Kotokov/IABS) Two steps: • Construct Hi and hence τI in terms of Rie- mann surface data; • Express result in terms of ﬂat-coordinates/ geometric data. g ≥ 2 (Kotokov/Korotkin) • Construct Hi and hence τI in terms of Rie- mann surface data; 20 g = 0 : Rational functions k1 −2 l ki c(i,αi) p(z) = z k1 + ar z r − i , r=0 i=2 αi=1 (z − bi)α Calculation of ﬂat-coordinates: pi = bi , i = 2,... ,l, ti = ki c(i,ki)1/ki so ∼ H0,N (k1 , . . . , kl ) = CM \S1 ∪ S2 where ∼ S1 = {pi − pj = 0 , i = j} , ∼ S = {t = 0 , i = 2 . . . , l} . 2 i 21 Theorem The isomonodromic τ -function for the space H0,N (k1 , . . . , kl ) is given by the formula −48 τI = R(f, f ) , i j (ki+1)(kj +1) l (k +1)(ki−2) i=j (p − p ) i=2 ti i where f = numerator of λ . 22 g = 1 : Elliptic functions Theorem The isomonodromic τ -function for the space H1,N (k1 , . . . , kl ) is given by the formula −48 τI = η(t0)48 κ i j (ki+1)(kj +1) l (k +1)(ki−2) i=j (b − b ) i=1 ti i where κ= σ(αr − αs) r=s and αi , i = 1 , . . . , M , are the critical points of the map p . Here η is the Dedekind Discriminant function. 23 Summary Theme: Singular data as building blocks: −48 caustic structures τI = boundary structures [g = 1 : η = 0 degenerate elliptic curve] Properties: • irreducibility properties; • τI ﬁnite and non-zero at semi-simple points; • τI singular only on caustics; • function of critical data only. 24 What next? • boundary of Hurwitz spaces in general; R(f, g) = g(α) {α:f (α)=0} • F -manifold structure on boundaries of Hur- witz spaces; • zeros/poles of τI in general; isomonodromy data and F -manifold structure • Virasoro constraints at higher genus. 25

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