...a journey from Physics to Combinatorics
Renzo Cavalieri
Colorado State University
SACNAS 2009
Renzo Cavalieri
Oh The Places You’ll Go
Physics
Physicists tell us that matter consists of tiny little strings that
wiggle their way through Space Time. In doing so they trace
surfaces, which represent the evolution of a physical system.
These surfaces come with a complex structure.
SpaceTime
Complex Surface
Strings
C|
Time
It seems reasonable that physicists want to know as much as
possible about analytic functions between surfaces.
Renzo Cavalieri
Oh The Places You’ll Go
Geometry
Geometers have been studying such maps for over a century,
and know they are ramified covers: away from a finite number
of points (branch points), there are exactly d preimages.
X
|
C,z
w=z 2
C,w
|
|
C,z
w=z 3
F
|
C,w
Y
Above a branch point, the local expression of the function is
z 7→ z n , and the collection of the n’s above a branch is called
the ramification profile.
Renzo Cavalieri
Oh The Places You’ll Go
The Quest
Q UESTION : how many covers of degree d from a genus g
surface to a sphere, with specified ramification profile over two
points and generic ramification over r other points?
Torus
1
3
3
2
H1((3), (2,1))
F
Sphere
(2,1)
(2,1)
(3)
Such number is called a Hurwitz number, denoted Hgr (α, β).
Renzo Cavalieri
Oh The Places You’ll Go
Algebra
Algebra gives us a way to compute Hurwitz number via the
following construction:
|
C,z
X
3
|
C,z
2
1
w=z 2
|
C,w
w=z 3
F
|
C,w
Y
(12)
(123)
In order to count covers, we instead count (r + 2)-tuples
σ0 , τ1 , . . . , τr , σ∞ of permutations of d points such that:
1
(σ0 , σ∞ ) have cycle type (α, β);
2
τ ’s are simple transpositions;
3
σ0 τ1 . . . τr σ∞ = Id
4
hσ0 , τ1 , . . . , τr , σ∞ i acts transitively on the d points
Renzo Cavalieri
Oh The Places You’ll Go
Combinatorics
The cut and join equations tell us how a permutation can
change when you compose it with a simple transposition:
2
CUT
4
σ = (1234)(56) τ1 = (13)
2
⇒ τ1 σ = (12)(34)(56)
2
2
4
⇒ τ1 σ = (123564)
σ = (1234)(56) τ1 = (35)
JOIN
6
2
This gives us a way to compute Hurwitz number by instead
counting weighted trivalent graphs.
-2
2
H02 ((2, 1), (2, 1))
=
4
=
-1
1
+
1
Renzo Cavalieri
2
3
-1
Oh The Places You’ll Go
1
-2
Sketch of further coolness...
x1 + y2 >0
y
x1
x1 +y1 <0
x1 +y1 =0
x1 +y1 >0
y
x1
1
x1+x2
1
x1+x2
x2
y2
x2
y2
x1
y1
x1
y1
-x1-y
x1+y
1
1
x
2
y
x
y2
x1
2
x1
x1+y2
y
2
x
1
2x 1
2
y2
x1+y2
x
y
2
2(x1+y1 )
Renzo Cavalieri
y
2
1
-2y1
Oh The Places You’ll Go