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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 ).
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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.
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