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Wall-crossing and holomorphicanomaly
Hamburg, December 2, 2010
Albrecht Klemm
M. Alim , B. Hagighat, M. Hecht, A.K, M. Rauch, T. Wotschke Dec 2010
2
∙ Objective of the work
– Generalize holomorphic anomaly from string- to gauge
theory,...
– Topological String as an example
∙ M5 brane(s) on a divisor in a CY 3-fold
– Elliptic genus of the MSW string
– N = 4 SYM theory perspective
∙ Wall-crossing for D4-D2-D0 branes
– The Kontsevich-Soibelmann formula
– Gottsche’s Wall-crossing formula
3
∙ The holomorphic anomaly equation
– Zwerger’s anholomorphic regularisation of indefinite Θ
-fcts
– Proof of the holomophic anomaly for r = 2
∙ Conclusions
4
Objective of the work:
∙ Generalize the holomorphic anomaly equation capturing
background dependence of topological string theory.
∙ Extend the relation between modularity and
holomorphicity → Mock modular objects .
∙ Interprete the failure of holomorphicity in terms of Wall
crossing.
∙ Proof a holomorphic anomaly equation for the partition
function of N = 4 SYM on del Pezzo surfaces.
5
Topological String on CY M as example:
Key object of interest in any topological theory:
Topological partition function
Z(t, t) = exp
⎛⎝ ∞∑g=0
�2g−2Fg(t, t)
⎞⎠
6
∙ Fg(t, t) invariant under space-time duality group Γ ∈SP (ℎ3(M),ℤ)
∙ Fg(t, t) yields generating function for M2/D2+D0 BPS
invariants n�g ∈ ℤ in holomorphic limit∑g �
2g−2Fg(t) =∑g
�2g−2 limt→t∞
Fg(t, t)
=∑
m,�∈H2(Mℤ)
n�gm
(2 sin
(k�
2
))2g−2
emt⋅�
t∞ reflects the background dependence of the BPS
numbers n�g ∈ ℤ
7
∙ Fg(t, t) fullfill an holomorphic anomaly equation
∂tkFg =
∫ℳ(g)
∂∂�g =1
2Cij
k
⎛⎝DiDjFg−1 +
g−1∑r=1
DiFrDjFg−r
⎞⎠ .
a b
dc
=
c
=
a
d
b
c
j
φi
jφ
Σ ij η
ba
dΣij
φ φ ji η= φ
φ j
i
Σij
(−1)F
= φφ φi jΣij
η
ηij
ijij
ij
∙ The anholomorphicity comes from the boundary of the
moduli space ℳ(g)
8
Solution
∙ Fg(t, t) are generated by finite polynomials rings of
quasimodular forms Γ ∈ SP (ℎ3(M),ℤ) of weight 3g−3
→ finite numbers of unknows.
∙ Anholomorphic generators are analogs of the
anholomorphic second Eisenstein Series E2(�, �) =
E2(�)− 3�Im(�).
∙ Boundary conditions for limt→tcrit fix the unknows.
9
M5 branes on a divisor
Wrap M5-brane(s) r-times on a divisor P in a Calabi-Yau
3-fold M and extended in S1 × ℝ3,1
Two effective descriptions of the effective action of the
M5-brane
∙ Dimensional reduction of the effective action on M5
on small P leads to the MSW string with (0,4) CFT
worldsheet theory. �-model whose target space are the
def. of M5 on P . Maldacena, Strominger, Witten 97 r = 1.
∙ Reduction of the effective action on M5 on P × T2
10
with small T2 leads to N = 4 U(r) SYM on P .
SL(2,ℤ) invariance of SYM is geometrized on the
complex modulus of the torus � = 4�ig2
+ �2�. Minahan,
Nemeschansky, Warner, Vafa 98
BPS states of the MSW string can be computed by
compatifying M5 on P × T2 as elliptic genus of the
MSW string partition function obeying SL(2,ℤ)
invariance and compared with the topological partition
function of N = 4 U(r) and after splitting of U(1)
SU(r) gauge theory.
11
Elliptic genus of the MSW string
Degrees of freedom (scalars):
∙ R/L scalars �A from reduction of selfdual 3-form ℎ(3) =
d�AR/L ∧ �±A, �±A ∈ H2
±(P,ℤ),
∙ ℎ0(M,ℒP) − 1 right scalars, describing movement of
M5 in M. In this work we consider rigid divisors. It
implies P a del Pezzo surface (b+2 = 1).
∙ right scalars in the center of mass multiplett, describing
movement in ℝ3,1.
12
Symmetries and charges (type IIA picture):
∙ rpA D4 brane charge
∙ M2 brane gives rise to
– qA: D2 brane charges in Λ = H2±(X,ℤ)∣P .
– q0 momentum of the M2 around the S1 ∈ T2
The charges are
Γ = (Q6, Q4, Q2, Q0) = r(0, pA, qA, q0),
13
One defines the modified elliptic genus
Z ′(r)X,P(�, z) = TrℋRR
(−1)FR F 2R q
L′0−cL24 qL
′0−
cR24e2�iz⋅Q2,
which is expected (proven for r = 1) to be a (0, 2)
((−32,
12) after seperating the CMM) SL(2,ℤ) Jacobiform,
with modular parameter � and elliptic parameter z.
It receives contributions only from states annihilated by(L0 −
cR
24− 1
2q2R
)∣q⟩ = 0 .
14
Using the spectral flow symmetry with k ∈ Λ
− q0 7→ −q0 + k ⋅ q +1
2k ⋅ k,
q 7→ q + k,
one can show using the symmetries of d(r,Q,Q0)
Z(r)X,P(�, z) =
∑Q0;QA
d(r,Q,Q0) e−2�i�Q0 e2�iz⋅Q2
=∑
q0;q∈Λ∗+[P ]2
d(r, q,−q0) e−2�i�rq0 e2�irz⋅q,
15
for arbitrary rank the decomposition with q0 = −q0 − 12q
2
Z(r)X,P(�, z) =
∑�∈Λ∗/Λ
f (r)� (�)�(r)
� (�, z), (1)
f (r)� (�) =
∑q0≥−
cL24
d(r)� (q0)e
2�i�rq0,
�(r)� (�, z) =
∑k∈Λ+
[P ]2
(−1)rp⋅(k+�)e2�i�r(k+�)2
2 e2�irz⋅(k+�).
Note the shift [P ]2 due to the Freed-Witten anomaly,
whenever P is not a spin manifold.
16
N = 4 gauge theory perspective
The generating function
f(r)�,J(�) =
∑d≥ 0
(−1)rp⋅� Ω(Γ; J) qd−r�(P )24
of the BPS invariants Ω(Γ; J), defined through
Ω(Γ; J) =∑m∣Γ
Ω(Γ/m; J)
m2
in terms of Euler# of moduli sp. of coherent sheaf on P
Ω(Γ, J) = (−1)dimℂℳJ(Γ)�(ℳ(Γ), J)
17
Coherent sheaves represent the gauge theory instantons
configurations of N = 4 SYM theory Vafa, Witten 94.
They depend in general through Wall-Crossing behaviour
on the Kahlerclass J in the Kahlercone C(P ) on P .
Also the Θ-functions depend on the Kahlerparameter J
through a choice of polarization:
k2+ =
(k ⋅ J)2
J ⋅ J, k2
− = k2 − k2+.
18
�(r)�,J(�, z) =
∑k∈Λ+
[P ]2
(−1)rp⋅(k+�)e2�i�r(k+�)2+
2 e2�i�r(k+�)2−
2 e2�irz⋅(k+�),
rank=1: L. Gottsche 97∑q0
d(1, �, q0) e2�i� q0 =
1
��(P ).
For a single M5-brane Ω and Ω become identical and
19
independent of J . This is reflected by the holomorphicity(∂� +
1
4�i∂2z+
)Z
(1)X,P(�, z) = 0.
For multiple M5-branes and if the branes cannot seperate
(rigid case) one expects boundstates at threshold and a
holomorphic anomaly on the rhs. Minahan, Nemeschansky, Warner,
Vafa 98. A famous example for the not rigid case is P is a
K3. Here we get the elliptic genus of the heterotic string
which is holomophic.
20
Wall-crossing for D4-D2-D0 branes
In the large volume limit the even D-branme charges are
given by the topological data of the sheaf ℰ on P
Γ = r
(0, [P ], i∗F (ℰ),
�(P )
24+
∫P
1
2F (ℰ)2 −Δ(ℰ)
),
with
Δ(ℰ) :=1
r(ℰ)
(c2(ℰ)− r(ℰ)− 1
2r(ℰ)c1(ℰ)2
), �(ℰ) :=
c1(ℰ)
r(ℰ), F (ℰ) := �(ℰ)+
[P ]
2.
Given a choice of J , a sheaf ℰ is called �-semi-stable if
21
for every sub-sheaf ℰ ′
�(ℰ ′) ⋅ J ≤ �(ℰ) ⋅ J
Walls of marginal stability in J are given by
�(ℰ ′) ⋅ J = �(ℰ) ⋅ J
22
The Kontsevich-Soibelmann wall-crossing formula
Define a symplectic pairing
⟨Γ1,Γ2⟩ = r1r2(�2 − �1) ⋅ [P ] ,
a Lie algebra
[eΓ1, eΓ2] = (−1)⟨Γ1,Γ2⟩⟨Γ1,Γ2⟩eΓ1+Γ2.
and Lie group elements
UΓ = exp
⎛⎝−∑n≥1
enΓ
n2
⎞⎠ . (2)
23
The Kontsevich-Soibelman wall-crossing formula
↷∏Γ:Z(Γ;J)∈V
UΩ(Γ;J+)Γ =
↷∏Γ:Z(Γ;J)∈V
UΩ(Γ;J−)Γ , (3)
where J+ and J− denote Kahler classes on the two sides
of the wall, V is a region in IR2 bounded by two rays
starting at the origin and ↷ denotes a clockwise ordering
of the central charges.
24
Rank 2: All commutators with rank > 2 vanish and with
the Baker-Campell-Hausdorff one gets the primitive
wall-crossing formula
ΔΩ(Γ) = (−1)⟨Γ1,Γ2⟩−1⟨Γ1,Γ2⟩∑
Q0,1+Q0,2=Q0
Ω(Γ1) Ω(Γ2).
25
Gottsche’s wall crossing formula:
Idea: Count invariants of sheaves on surfaces with
b+2 = 1 using the wall-crossing behaviour:
∙ Provide counting function in one chamber
– by a vanishing theorem
– using blow-up formulas
∙ Use the primitive wall-crossing formula
∙ Sum over (infinitly many) walls to obtain indefinite
Theta function
26
Program completed for rank 2. Let Γ = (2, �, d),
d = d1 + d2 + � ⋅ � where � = �1 − �. A wall is given by
W � = {J ∈ C(P ) ∣ � ⋅ J = 0}. From primitive
wall-crossing formula:
∑d≥ 0
(Ω(Γ; J+)− Ω(Γ; J−))qd−�(P )/12
=∑
d1,d2≥ 0, �
(−1)2�⋅[P ] � ⋅ [P ] Ω(Γ1)Ω(Γ2)qd1+d2+�2−2�(P )
24
= (−1)2�⋅[P ]−1 1
�(�)2�(P )
∑�
� ⋅ [P ] q�2,
27
= (−1)2�⋅[P ]−1 1
�(�)2�(P )Coeff2�iy(Θ
J+,J−Λ,� (2�, [P ]y))
Here
ΘJ,J ′
Λ,� (�, x) :=1
2
∑�∈Λ+�
(sgn(J ⋅ �)− sgn(J ′ ⋅ �)) q12�
2e2�i�⋅x.
These theta functions obey a cocycle condition
ΘF,GΛ,� + ΘG,H
Λ,� = ΘF,HΛ,� ,
28
Let J and J ′ be in arbitrary chambers∑d≥0(Ω(Γ; J)−Ω(Γ; J ′))qd−�(P )/12
=(−1)2�⋅[P ]
�(�)2�(P )Coeff2�iy(Θ
J,J ′
Λ,� (2�, [P ]y)).
29
The holomorphic anomaly equation
Idea: Regularisation of the indefinite Θ-function
ΘJ,J ′
Λ,� (�, x) :=1
2
∑�∈Λ+�
(sgn(J ⋅ �)− sgn(J ′ ⋅ �)) q12�
2e2�i�⋅x.
is incompatible with the SL(2,ℤ) invariance of N = 4
SYM.
One has to use a modular regularisation of the theta
function. Such a regularisation has been provided by
Zwegers treatment of Mock-modular forms Zwegers 09. It is
non-holomorphic an was used to construct an
30
anholomophic extension to make Ramanujan’s Mock
modular forms into modular forms.
This idea appeared in similar context in the work of Jan
Manschot Manschot 08 & 09
Using the an-holmophicity of Zwergers regularistion we
will prove the holomorphic anomaly equation.
Zwerger’s anholomorphic regularisation of indefinite
Θ-fcts
31
f(2)�,J ′(�)− f (2)
�,J(�) =#Λ⊥(�)2
�2�(P )(�)Coeff2�iy(Θ
J,J ′
Λ,� (2�, [P ]y)).
transforms not well under SL(2,ℤ).
Zwegers regularization:
Θc,c′
Λ,�(�, x) = 12
∑� ∈Λ+�
((E
((c⋅(�+Im (x)
�2))√�2√
−Q(c)
)−
E
((c′⋅(�+Im (x)
�2))√�2√
−Q(c′)
))e2�i�⋅xqQ(�),
32
here Q(x) = 12x
2 and E denotes the incomplete error
function
E(x) = 2
∫ x
0
e−�udu.
The non-holomorphic function Θc,c′
Λ,�(�, x) transforms as a
Jacobi form of weight 12r(Λ).
Using
E(x) = sgn(x)(1− �12(x2)),
one splits
Θc,c′
Λ,�(�, x) = Θc,c′
Λ,�(�, x)− Φc�(�, x) + Φc′
�(�, x),
33
with
Φc�(�, x) =
1
2
∑� ∈Λ+�
⎡⎣sgn(� ⋅ c)− E
⎛⎝(c ⋅ (� + Im (x)�2
))√�2√
−Q(c)
⎞⎠⎤⎦ e2�i�⋅xqQ(�).
containing all anholomorphicity. Using ideas review in
Zagier 07 one shows
∂�Coeff2�iyΦc�(2�, [P ]y) = −�
−32
2
8�i
c ⋅ [P ]√−c2
(−1)4�2 �(2)�,c(�, 0),
34
Splitting
f(2)�,J(�) = f
(2)�,J ′(�)− 1
�2�(P )Coeff2�iyΘ
J,J ′� (2�, [P ]y),
into the holomophic ambiguity f(2)�,J ′(�) and writting the
reduced elliptic genus (spitting the center of mass mode
off)
Z(2)X,P(�, z) =
∑�∈Λ∗/Λ
f(2)�,J(�)�
(2)�,J(�, z).
With
Dk = ∂� +i
4�k∂2z+,
35
one proofs the holomorphic anomaly equation at rank
two for general surfaces P
D2Z(2)X,P(�, z) = �
−3/22
1
16�i
J ⋅ [P ]√−J2
(Z
(1)X,P(�, z)
)2
.
36
Conclusions
∙ We generalized the holomorphic anomaly in topological
string theory counting D2-D0 brane BPS invariants to
D4-D2-D0 brane invariants associated to N = 4 gauge
theory instantons..
∙ The form that we proof is compatible with the
conjectured form for the E-string related to gauge
theory on half K3 Minahan, Nemeschansky, Warner, Vafa 98
∂�Z(n)(q1) =
i�−22
16�
n−1∑s=1
s(n− s)Z(s)(q1)Z(n−s)(q1) ,
37
∙ Asuming this one can make a prediction for the general
anomaly of the f
∂f�(r) =∑n=1,�
�(n), �(r−n)
n(r − n)f�(n)f�(r−n)e
(r n (r − n)�
2�2
),
with � ∈ Λ + �(r) − �(n) + �(r−n) + P2 .
∙ Quasimodularity of the topological string gets replaced
by Mock modularity. To every mock modular form ℎ of
weight k there exists a shadow g ∈M2−k such that
ℎ(�) = ℎ(�) + g∗(�)
38
transforms as a wheight k modular from. Let gc(z) =
g(−z), and g∗(�) is defined by
g∗(�) = −(2i)k∫ ∞−�
(z + �)−kgc(z) dz.
Then∂ℎ
∂�=∂g∗
∂�= �−k2 g(�).
E2 indead is a trivial case of a mock modular form with
constant shadow:
E2(�) = E2(�)− 3
��2.
39
From ∂�E2 = �−22
3i2� we get g = 3i
2�
g∗(�) = − (2i)2 ∫∞−�(z + �)−2 3i
2�dz
= −6i�
[ −1z+�
]∞−� = − 3
��2.