Download Counting curls and hunting monomials: Tropical mirror symmetry for

Counting curls and hunting monomials: Tropical
mirror symmetry for elliptic curves
Hannah Markwig joint with Janko Böhm, Kathrin Bringmann, Arne
Buchholz
Eberhard Karls Universität Tübingen
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Tropical geometry is...
... geometry over the max-plus semifield
(→ optimization, complexity theory)
... a finite shadow of non-archimedean analytic geometry
... an efficient combinatorial tool to study degenerations in algebraic
geometry
ΘΕΤ Σϑ 6ΜΙΘΕΡΡ ΩΨςϑΕΓΙΩ
ΓΣΘΦΜΡΕΞΣςΜΕΠ ΩΛΕΗΣ[
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Tropical covers of an elliptic curve
Fix a tropical elliptic curve E, i.e. a circle, e.g. of length 6.
9
C
1
3
P
3
2
1
2
π
E
6
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No contrast
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Tropical geometry in enumerative geometry
Theorem (Correspondence Theorem (Cavalieri-Johnson-M,
Bertrand-Brugallé-Mikhalkin))
trop
Nd,g = Nd,g
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Gross-Siebert mirror symmetry
Counts
X∨
X
Correspondence
Theorems
Integrals
?
B
Tropical counts
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Gross-Siebert mirror symmetry
Counts
X∨
X
Correspondence
Theorems
Integrals
?
B
Tropical counts
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Gross-Siebert mirror symmetry
Counts
X∨
X
Correspondence
Theorems
Integrals
?
B
Tropical counts
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Tropical mirror symmetry of elliptic curves
2013: Böhm, Bringmann, Buchholz, M.
IΓ = Feynmanintegral for the graph Γ
P
d
Nd,g q d
P
Dijkgraaf
Correspondence
theorem
Γ
IΓ
IΓ
?
P
d
trop d
Nd,g
q
generating series
of tropical
Hurwitz numbers
of type Γ
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Feynman graphs
Feynman graphs: Let Γ be a 3-valent connected graph of genus g. Γ has
2g − 2 vertices and 3g − 3 edges.
Fix a labeling z1 , . . . , z2g−2 for the vertices and q1 , . . . , q3g−3 for the edges.
z1
q2
q1
q3
z4
z2
q5
q4
q6
z3
Notation
For the edge qk , we say zk1 and zk2 are its adjacent vertices (no matter which
order).
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Feynman integrals
Definition (The Propagator)
P (z, q) :=
1
1
℘(z, q) + E2 (q 2 ).
4π 2
12
Integration paths:
γ2g−2
...
γ1
Definition
IΓ (q) =
Z
...
γ2g−2
Z
γ1
3g−3
Y
k=1
!
P (zk1 − zk2 , q) dz1 . . . dz2g−2 .
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Example
z1
q2
q1
q3
z2
q5
z4
q4
q6
z3
Definition
IΓ (q) =
Z
...
γ2g−2
Z
γ1
3g−3
Y
k=1
!
P (zk1 − zk2 , q) dz1 . . . dz2g−2 .
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Labeled tropical covers
1
z1
q1
q2
3
z2
q3
4
2
q6
z4
3
q5
q4
z3
1
p1
p2
p0
p4
p3
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Numbers of labeled tropical covers
Let a = (a1 , . . . , a3g−3 ) be a tuple of natural numbers.
Fix a Feynman graph Γ. Fix a base point p0 ∈ E.
Definition
trop
= weighted number of labeled tropical covers π : C → E
Na,Γ
where C is of combinatorial type Γ,
where the weighted sum of points in π −1 (p0 ) ∩ qk is ak
with ramifications (3-valent vertices) at the fixed points.
Q
Each such cover is counted with weights.
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Tropical mirror symmetry
Definition (Refined integrals)
IΓ (q1 , . . . , q3g−3 ) =
R
γ2g−2
...
R
γ1
Q3g−3
k=1
!
P (zk1 − zk2 , qk ) dz1 . . . dz2g−2 .
Theorem (BBBM, 13)
trop
Na,Γ
= the coefficient of q a in IΓ (q1 , . . . , q3g−3 ).
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Tropical mirror symmetry
Theorem (BBBM, 13)
trop
= the coefficient of q a in IΓ (q1 , . . . , q3g−3 ).
Na,Γ
Corollary (Right arrow in the triangle)
X trop
X
Nd,g q d =
IΓ .
d
Γ
Proof: Set qk = q for all k and count correctly.
Corollary (Top arrow)
Mirror symmetry for elliptic curves.
Theorem (Goujard-Möller, 16)
New quasimodularity results for generating function for fixed graph.
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Computing the integrals
Coordinate change xk = eiπzk .
Produces circles from integration paths.
Yields a factor of
function).
1
xk
in the integral for every k (derivative of the inverse
Computing the integral= computing residues, resp. the constant term of
the product of propagators after coordinate change.
Theorem (Propagator after coord. change)
P (x, q) =
∞
X
w=1
For the integral, plug in
xk1
xk2
wxw +
∞ X
X
w(xw + x−w )q a .
a=1 w|a
= eiπ(zk1 −zk2 ) .
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Hunting monomials
For a, the coefficient of q a in IΓ (qi ) is the in all xi constant coefficient of
Y
k|ak =0
Y
k|ak 6=0
i.e.
PQ
k
∞
X
wk ·
wk =1


X
wk
wk |ak
xk1
xk2
wk !
xk1
xk2
wk
·
+
xk2
xk1
wk , where the sum goes over all tuples
!
x wk k1
a, wk
xk2
k
such that the product of the wk
ak > 0.
xk1 wk
xk2
wk 
,
is constant in all xi , and wk |ak if
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Curls from monomials
Draw “cut” E with the pi ,
draw xi above pi ,
if ak = 0 draw edge from xk1 to xk2 of weight wk directly,
if ak > 0, draw edge from xk1 to xk2 , but “curled”
ak
wk
times
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