Sheaves on Surfaces and Generalized Appell Functions

Sheaves on Surfaces and Generalized Appell Functions

Sheaves on Surfaces and Generalized Appell
Functions
Jan Manschot
GEOQUANT
September 18th, 2015
Outline
• Motivation: Yang-Mills path integral and S-duality
• Sheaves on rational surfaces
• Explicit expressions for generating functions
• Generalized Appell functions
Based on:
1407.7785
1109.4861
Classical Yang-Mills theory with gauge group G on 4-manifold
S:
- Gauge potential: A ∈ Ω1 (S, g)
- Field strength: F = dA + A ∧ A ∈ Ω2 (S, g)
R
R
iθ
- Action: S[A] = − g12 S Tr F ∧ ∗F + 8π
2 S Tr F ∧ F
For quantitative results beyond the classical theory, consider
topologically twisted N = 4 U(r ) Yang-Mills on S:
Vafa, Witten (1994)
Z
hr (τ ) = DADψDX e −S[A,ψ,X ]
with τ =
θ
2π
+
4πi
g2
Path integral localizes on Bogomolny’i-Prasad-Sommerfield
solutions:
- minimize S[A], e.g. ASD-instantons: F = − ∗ F
R
- instanton number: κ(F ) = − 8π1 2 S Tr F ∧ F ∈ Z
- e −S[A] = q κ with q = e 2πiτ
Generating function:
⇒ hr (τ ) =
X
Ω(κ) q κ + R(τ, τ̄ )
κ
- Ω(κ): Euler number of inst. moduli space ∼ BPS-invariant
- R(τ, τ̄ ): arises due to reducible
connections:
A1
0
e.g. r = 2 : A =
0 −A1
S-duality
Electric-magnetic duality:
Montonen, Olive (1977)

 (F , ∗F ) → (∗F , −F )
g → gL
S:

τ → −1/τ
Translations:
1
8π 2
⇒
Z
Tr F ∧ F ∈ Z
T : τ → τ + 1 is a symmetry of the theory
S + T generate SL2 (Z) S-duality group
Modularity
Instanton equations are invariant under S-duality
⇒ modular transformation properties are expected for hr (τ ):
hr
aτ + b
cτ + d
∼ (cτ + d)
−χ(S)/2
hr (τ ),
a b
c d
∈ SL2 (Z)
Verification of EM-duality for U(r = 1, 2) YM and complex
surfaces using algebraic-geometric techniques
Vafa, Witten (1994)
From YM theory to vector bundles
Computation of hr (τ ) for rational surfaces is possible using
methods for complex surfaces Gottsche, Nakajima, Yoshioka,. . .
Donaldson-Uhlenbeck-Yau theorem:
ASD connections ⇔
- stable holomorphic vector bundles
- reducible connections
Characterized by their Chern classes ci ∈ H 2i (S, Z):
c1 =
i
Tr F ,
2π
1
1
c2 − c12 = 2 Tr F ∧ F
2
8π
Gieseker stability
Let C (S) ∈ H 2 (S, R) be the ample cone of S.
Choose polarization (Kähler modulus) J ∈ C (S), and define the
slope:
c1 (E ) · J
µJ (E ) :=
r (E )
Definition:
A sheaf E is stable if for every subsheaf E 0 ( E , µJ (E 0 ) < µJ (E ) (
semi-stable ⇒ µJ (E 0 ) ≤ µJ (E ) )
Agrees with stability based on central charge in large volume limit:
limJ→∞ Z (E , B + iJ)
Douglas (2000), Bridgeland (2002)
Wall-crossing
On varying J, a sheaf E can cease to be stable or reversely become
stable.
This happens accross walls of marginal stability (WMS)
Definition:
Let 0 ⊂ E 0 ⊂ E , and G = E /E 0 and. A wall W (E 0 , E ) ⊂ C (S) is a
codimension 1 subspace of C (S) such that for Jw ∈ W (E , E 0 )
µJw (E 0 ) − µJw (E ) = 0
but for J ∈
/ W (E 0 , E )
µJ (E 0 ) − µJ (E ) 6= 0
Invariants
Algebraic-geometric approach can determine more refined
invariants than the Euler number
Let MJ (γ) (MJ (γ)) be the moduli space (stack) of semi-stable
sheaves with γ = (r , c1 , c2 )
Let γ be relatively prime, J · KS < 0 and not on a wall ⇒ MJ (γ)
is smooth, compact and of expected dimension
Let I(γ, w ; J) be the virtual Poincaré function of MJ (γ)
Joyce (2004)
If γ is relatively prime, then
I(γ, w ; J) :=
where p(X , s) =
w − dimC MJ (γ)
p(MJ (γ), w ),
w − w −1
P2 dimC (X )
i=0
bi s i with bi = dim H i (X , Z)
Generating function
Generating function:
Hr ,c1 (z, τ ; S, J) :=
X
I(γ, w ; J) q r ∆−
c2
−1 2
c1 )
- discriminant: ∆ = 1r (c2 − r 2r
aτ +b
- modular parameter: q = e 2πiτ , τ → cτ
+d
- elliptic parameter: w = e 2πiz , z → cτz+d
r χ(S)
24
Rational ruled surfaces
Σl
Rational ruled surface
π : Σ` → P1
- H2 (Σ` , Z): generators C , f
- Intersection numbers: C 2 = −`, f 2 = 0, C · f = 1
- Restriction to the rational ruled surface Σ1 in this talk
See the refs for other ruled surfaces & their blow-ups/downs
- Ample cone: C (Σ1 ) = {m(C + f ) + nf ; m, n > 0}
Outline of computations
n
O
J0,1
n
m
1. Determine invariants
for boundary polarization
W1 W2
O
2. Wall-crossing
W3
m
Suitable polarization
Conjecture:
JM (2011)
Hr ,c1 (z, τ ; Σ` , J0,1 ) =
0
if c1 · f =
6 0 mod r
Hr (z, τ ) if c1 · f = 0 mod r
where Hr (z, τ ) is the infinite product:
Hr (z, τ ) :=
i (−1)r −1 η(τ )2r −3
,
θ1 (2z, τ )2 θ1 (4z, τ )2 . . . θ1 ((2r − 2)z, τ )2 θ1 (2rz, τ )
1
with η(τ ) = q 24
Q∞
n=1 (1
− q n ),
2
θ1 (z, τ ) =
r r2
r ∈Z+ 12 (−1) q
P
wr
∼ P1 are equivalent to direct sums ⇒
Reason: bundles on f =
unstable for c1 · f 6= 0 mod r
Proofs: r = 1: Gottsche (1990), r = 2: Yoshioka (1995), general r : Mozgovoy (2013)
Wall-crossing
Determine I(γ, w ; J 0 ) by adding (substracting) HN-filtrations for J
(J 0 ) which correspond to semi-stable sheaves for J 0 (J)
Wall-crossing formula:
I(γ, w ; J 0 ) =
X
S({γi }, J, J 0 ) w −
P
i<j (ri cj −rj ci )·KS
γ=γ1 +···+γ`
ri ≥1 ∀i
`
Y
I(γi ; w , J)
i=1
For all i = 1, . . . , ` − 1, we have either
P
P
1. µJ (γi ) ≤ µJ (γi+1 ) and µJ 0 ( ij=1 γj ) > µJ 0 ( `j=i+1 γj ),
or
P
P
2. µJ (γi ) > µJ (γi+1 ) and µJ 0 ( ij=1 γj ) ≤ µJ 0 ( `j=i+1 γj ),
then S = (−1)k with k # 1. is true, else S = 0.
Joyce (2004)
Explicit expressions for r = 1, 2 & Σ1
Rank 1:
H1,c1 (z, τ ; Jm,n ) = H1 (z, τ )
Gottsche (1990)
Rank 2: using wall-crossing formula
H2,βC −αf (z, τ ; Jm,n )
=
1 < |w | < |q
δβ,0 H2 (z, τ )
+ H1 (z, τ )
2
X
1 ( sgn(b
2
−1
4
|
1 b 2 + 1 ab
2
− ε) − sgn(bn − am − ε) ) w −b+2a q 4
(a,b)=−(α,β) mod 2
n
+
J
Indefinite theta function:
O
m
+
D ab=
 
1 1
1 0
+ -
Hoppe, Spindler (1980); Ellingsrud, Strømme (1987), Yoshioka (1994/5), Gottsche, Zagier (1996)
Explicit expressions II
Specialize to J = J1,0 , β = 1:
Geometric sum =⇒
1 2
H2,C −αf (z, τ ; Σ1 , J1,0 ) = H1 (z, τ )
2
X
b=1 mod 2Z
Yoshioka (1994/5), Bringmann, JM (2010)
Specialization of Appell function A` (u, v ; τ )
1
w 2α−b q 4 b + 2 αb
1 − w 4qb
Appell functions I
Appell function:
A(u, v ; τ ) = e πiu
X (−1)n e 2πinv q n(n+1)/2
n∈Z
1 − e 2πiu q n
• Appell (1884): building blocks of doubly periodic functions
• Ramanujan (1920): mock theta functions
• Zwegers (2002): non-holomorphic completion Â(u, v ; τ )→
transforms as weight 1 Jacobi form
Completion: Â(u, v ; τ ) = A(u, v ; τ ) + θ1 (z, τ ) R(u − v ; τ )
Z i∞
1
ga,b (u)
1 q a2 e −2πia(b+ 2 )
du
Period integral: R(aτ + b; τ ) = 2i
p
−i(z + τ )
−τ̄
Appel functions II
Higher level/multi-variable generalizations appearing in
mathematical physics:
• Vacuum character of N = 2, 4 superconformal algebra
Eguchi, Taormina (1988)
ˆ
• Characters of admissible sl(m|n)-representations
Kac, Wakimoto
(2000), Semikhatov, Taormina, Tipunin (2003)
• Mirror symmetry of elliptic curves
Polishchuk (1998)
Completion of the generalizations all involve 1-dimensional period
integrals
Hr ,c1 (z, τ ; J1,0 ) for r ≥ 2
Hr ,c1 (z, τ ; J1,0 ) can be determined explicitly for all (r , c1 ) using
wall-crossing formula of D. Joyce (2004)
JM (2014), Toda (2014)
Example:
X w −6k q 3k 2
H3,0 (z, τ ; J1,0 ) = H3 (z, τ ) + 2H1 (z, τ )H2 (z, τ )
1 − w 6 q 3k
k∈Z
+H1 (z, τ )3
X
k1 ,k2 ∈Z
2
2
w −2(k1 +2k2 ) q k1 +k2 +k1 k2
(1 − w 4 q 2k1 +k2 )(1 − w 4 q k2 −k1 )
• Quadratic form for A2 root lattice in numerator
• Two terms in denominator
• Function cannot be written as product of two Appell functions
Generalized Appell functions
Suggests further generalization:
=⇒ Appell functions with signature (n+ , n− )
Let:
• Λ: n+ -dimensional positive definite lattice
• {m j }: set of n− vectors ∈ Λ∗
1
X
q 2 Q(k) e 2πiv ·k
AQ,{m j } (u, v ; τ ) =
Qn −
2πiuj q m j ·k )
j=1 (1 − e
k∈Λ
Modular completion will require higher dimensional period integral
Blow-up formula
Blow-up φ : S̃ → S
Bott & Tu
Easy relation between generating functions:
Hr ,c̃1 (z, τ ; φ∗ J, S̃) = Br ,k (z, τ ) Hr ,c1 (z, τ ; J, S),
with c̃1 = φ∗ c1 − kCe and Br ,k (z, τ ) is a Ar −1 theta function
Identities for Appell functions
Blow-up formula ⇒ relations between Hr ,c̃1 (z, τ ; φ∗ J, S̃)
⇒ relations between Appell functions
Let S = P2 and S̃ = Σ1
Then
Hr ,C +f (z, τ ; J1,0 , Σ1 )
Hr ,f (z, τ ; J1,0 , Σ1 )
=
Br ,0 (z, τ )
Br ,1 (z, τ )
E.g. r = 2:
2
1
2
X q b − 4 w 2b−3
X q b +b w 2b−2
1
1
−
θ3 (2z, 2τ )
1 − w −4 q 2b−1 θ2 (2z, 2τ )
1 − w −4 q 2b
b∈Z
=−
b∈Z
iη(τ )3
θ1 (4z, τ ) θ2 (2z, 2τ )
Familiar quasi-periodicity relation
Zwegers (2002)
For A2 Appell functions
Identities take the following form:
1
2
b3,0 (z) k ,k ∈Z (1 −
1 2
+
=
2iη 3
w 2 q 2k1 +k2 )(1
X w
−
−3k 3k 2
q
θ1 (2z)b3,0 (z) k∈Z 1 − w 3 q 3k
η 6 θ1 (z)
θ1 (2z)2 θ1 (3z) b3,0 (z)
−
2
2
w −k1 −2k2 q k1 +k2 +k1 k2
X
w 2 q k2 −k1 )
−
iη 3

θ1 (2z)b3,1 (z)

.
Proven by Bringman, JM, Rolen (2015)
1
X
2
1
k +k +k k +k +k +
w −k1 −2k2 −1 q 1 2 1 2 1 2 3
b3,1 (z) k ,k ∈Z (1 −
1 2
w 2 q 2k1 +k2 +1 )(1
X
3k 2 −2k+ 1
3
w −3k+1 q
k∈Z
1 − w 3 q 3k−1
+
−
w 2 q k2 −k1 )
2
1
X w −3k−1 q 3k +2k+ 3
k∈Z
1 − w 3 q 3k+1


Conclusions
Generating functions of invariants of moduli spaces can be
determined explicitly in terms of generalized Appell functions with
signature (n+ , n− )
Applications:
• Semi-stable sheaves
• q-series
• Electric-magnetic duality
• Quantum black hole state counting
• Relation to topological strings on elliptic Calabi-Yau manifolds
• ...