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					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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