Wall Crossing, Quivers and Dimers

Mina AganagicUC Berkeley

Based on work with Kevin Schaeffer, arXiv:1006.2113

1

Consider IIB string theory on a Calabi-Yau Y. BPS states are D3 branes, labeled by

Δ ∈Η3(Y ,Z )

The number of BPS states in this class,

Ω(Δ)

can be computed, near singularities of Y, in terms of a quiver gauge theory.

For example,

3

Quiver :

+ Superpotential W

Nodes:

Edges:α β

>

α Δα = S3⎡⎣ ⎤⎦

Δα Δβ

P(x, y) = x + et

x+ y + e

s

y+1 Hanany et al

Cachazo-Fiol-Intriligator-Vafa

uv = w,P(x, y) = w,Y :

<<

VV

<<

VV

1

2 3

4

ΩQ (Δ) = TrQ,Δ (−1)F

α =1

r

∏ U(Nα ) quiver gauge theory, andcorrespond to

the Witten index of this theory counts the BPS states:

4

Nα ≥ 0

Δ = Nαα =1

r

∑ Δα ,

States written in terms of the nodesΔ

with positive ranks ,

U(Nα )

α

5

Quiver description is “good” when central charges of the nodes are nearly aligned, and then all the BPS states can

be described in this way.

Nα

θ

1g2

Z(Δα )

This splits the spectrum into the branes and the antibranes, which are bound states with positive and negative

Z(Δ) = Ω(3,0)

Δ∫

Z(Δα )

6

a) Central charges can align, and then BPS degeneracies jump - this is a wall of marginal stability.

Two phenomena affect the spectrum:

Z(Δ1)

Z(Δ2 )

Z(Δ2 )Z(Δ1)

Z(Δ1 + Δ2 )

7

The space of BPS states changes, in that what we call elementary states and their bound states changes.

This is Seiberg duality.

b) A node can leave the upper half plane, and enter it. The split into the branes and anti-branes, and hence also the quiver description, changes.

Δ* −Δ*Z −

−Δ*

Δ*

−Δ*

Q

Δ*

Q’

Cachazo-Fiol-Intriligator-Vafa,Berenstein-Douglas

8

and so do the quivers:

<<

VV

<<

VV

1

2 3

4

<<VV

<<VV

1

2 3

4

Q Q’

For example, in the context of Y, varying the complex structure moduli, the basis of ‘s changesS3

Cachazo-Fiol-Intriligator-Vafa

Kontsevich and Soibelman predict how the spectrum changes:

Δ = Nαα∑ Δα eΔ = Π

αeαNα

eΔ ,eΔ '[ ] = Δ Δ 'eΔ+Δ '

The product

taken in clockwise order, is constant.

Z(Δ)

AΔ = exp( enΔn2)

n=1∑

and

10

In the context of quivers Q and Q’ related by Seiberg duality,

this implies that and are not equal, but are related by

AQ 'AQ

KS “wall of the second kind”

Δ*

−Δ*

Δ*

−Δ*Q Q’

11

There is a simple proof that framed BPS statesof toric quivers, satisfy this.

Framed BPS states = extend the quiver Q by one node, and consider states with ranks .Δ,1( )

no bound states at threshold

Toric quiver Q = quiver which is a periodic graph on T 2

in this case, the quiver gauge theory has a symmetry, can compute the degeneracies by localization.

T 3

12

Toric quiver Q = quiver which is a periodic graph on T 2

The graph on encodes not just the quiver, but also the superpotential.

T 2

Each face gives a superpotential term corresponding to product of its edges, with the sign depending on the orientation. Every edge enters two superpotential terms with opposite

signs.

<<

VV

<<

VV < <

V

VV

<

<

VV

<

<

V

+

+

−

−

T 2

=Hanany et al

13

For a toric quiver, framed BPS degeneracies

are counted by melting crystal configurations.

ΩQ (Δ) = TrQ,Δ (−1)F

<<VV

<<VV

3

41

2<<

VV

<<VV

3

41

20

Mozogovy-Reineke, Ooguri-Yamazaki

Nodes of the quiver - atoms in the crystal Edges - bonds between neighboring atoms

14

T 3

Idea:

Any two paths on the framed toric quiver Q, starting at the framing node 0, and with the same charge turn out to be equivalent modulo F-terms.

T 3

For each such path we get a site in the crystal lattice, with labeled by the charge.

From the crystal, we can explicitly read off the solutions to F-term equations fixed by the torus action. The fixed points in the quiver moduli space correspond to “meltings” of the crystal. These are ideals, such that if a path p (a node) is in the melted crystal, so is pa, for

any path a on Q.

For a specific choice of central charges, all the F-term solutions constructed also solve D-term constraints.

Δ0

Δα >0

15

ZQ (q) = ΩQ (Δ) qΔ

Δ∑

ZQ (q) = (−1)d (Δ )qΔ

crystals∑

The framed BPS degeneracies are computed

by counting melting crystals, with the weight related to the rank of gauge group, corresponding to the number of atoms removed

qΔ = ΠαqαNα

1 q3 2q3q4 q3q42

16

The problem of counting melting crystal configurations is the same as

counting perfect matchings of the bipartite graph on the plane

OkounkovOkounkov-Reshetikhin-Vafa

Mozogovy-Reineke

17

T 2

< <

V

VV

VV

<

<

V

+

+

−

−12 2

3 3

4 33

<

T 2The graph dual to the periodic quiver is a bipartite

graph on

<<

VV

<<

VV

12

3 4

+12 2

3 3

33 4

4

4

nodes faces

+ , - superpotential terms

edges ,edges ,

,nodes

<

18

<<

VV<<

VV

12

3 4

As another example:

T 2

2

3

3

4

1 1

44

4

Consider the lift of this to the plane.

Melting crystal configurations are in one to one correspondence with perfect matchings of the bipartite graph on the plane, with suitable boundary conditions

4

4

3

4

4

3

4

4

3

4

4

3

1 q3 2q3q4 q3q42

removing atom=rotating dimers around the corresponding face

20

Seiberg duality has a geometric interpretation on the dual graph.

Hanany et al

39

<<

VV

<<

VV

1

2 3

4

<<VV

<<VV

1

2 3

4

Dualizing a node reverses the orientation of arrows to the node = replace the face with the dual face, a copy of the original, with coloring

reversed.

+32

4 4

44 1

1

2

1 4

3 3

44

4

3

1

21

From perfect matchings of one graph, one can get perfect matchings of the dual.

1

<<

VV

<<

VV

3

4

20

<<VV

<<VV

3

41

20

dualize node 2

22

Outside of the dualized face, the map is on to one.

(1+ q*)

(1+ q*)−1

This implies that the original and the dual partition functions, are related by

q 'ΔMQ '

∑ = qΔ

MQ

∑ (1+ q*)#1−# 2

One can show,

#1− #2 = Δ Δ*

independently of the details of the perfect matching, so

Ω

Q'Δ∑ Δ( )q 'Δ ≈ Ω

Q'Δ∑ Δ( )q 'Δ 1+ q*( )Δ* Δ

24

This is exactly what KS formula

implies, in this context.

ZQ = ΩQ (Δ)eΔ+Δ0Δ∑

−Δ*

Δ0

Δ*

Δ0

Δ*

−Δ*

= ΩQ (Δ)(1− eΔ* )

Δ* Δ eΔ+Δ0Δ∑

ZQ ' = AΔ*ZQAΔ*

−1

semi-primitive Denef -Moore w.c.f.

25

The crystals that count the framed degeneracies of toric quivers based on Calabi-Yau

uv = w,P(x, y) = w,Y :

have geometric interpretation in terms of its mirror Calabi-Yau X:

The crystal is a discretization of the mirror geometry.

The mirror Calabi-Yau X is toric

The base of the Calabi-Yau is obtained by forgetting the phases of , and using as the coordinates.zi zi

2

27

The lattice sites corresponding to the framed node are integral points in the base

of the singular Calabi Yau.

28

This follows as paths on the quiver beginning at node 0 and ending at the

framed node = holomorphic functions on X

Holomorphic functions on X are given in terms of monomials

which are gauge invariant:

Hanany et al

29

The macroscopic shape of the crystal is set by the geometry of Calabi-Yau,

while the microscopic details depend on the quiver

<<

VV

<<

VV

3

4

20 <<

VV

<<VV

3

41

20

30

In the mirror, the quiver Q has interpretation as counting BPS bound states of D4, D2 and D0 branes, wrapping compact

submanifolds of X,

<<

VV<<

VV

nodes of Q = a exceptional collection of spherical sheaves on the

surface S in X

The framed quiver is counting bound states of one D6 brane on X, with these branes.

framing node = line bundle on X, dual to the

sheaf on the framed node

Cachazo-Fiol-Intriligator-Vafa

Aspinwall

<<

VV<<

VV

OS (−2,−2)OS (−2,−1)[−1]

OS (−1,−2)[−1]OS (−1,−1)[−2]

OX[−1]

31

The BPS bound states of this with lower dimensional branes, in the chamber corresponding to quiver Q, are

computed by the crystal, with the shape of the Calabi-Yau with Kahler class

J = F

Conjecture:

Consider a D6 brane , with flux

F ∈H 2 (X,Z ).

OX (F)

and with microscopic structure of the crystal set by Q.

32

The shape of the crystal is set by X and the choice of F,

FCa∫ = na

and the details by Q.

Various aspects of this have been checked

for

P1 × P1,P2

33

Removing faces, edges and vertices in the Calabi-Yau crystal

corresponds to adding D4 on divisors, D2 on corresponding curves and D0

branes on the vertices

34

In the limit

F→∞

X and the crystal degenerate, and the microscopic excitations, involving individual nodes of the quiver freeze out. Only the “large” excitations are left.

The degenerate crystal is the same as the degenerate Calabi-Yau crystal of Iqbal-Nekrasov-Okounkov-Vafa,

counting BPS degeneracies of one D6 brane on X, as the path integral of a 6 dimensional, non-comutative, U(1) gauge

theory on X

35

The degenerate crystal is also the same as the Calabi-Yau crystals of Okounkov-Reshetikhin-Vafa, that arize in computing

Gromov-Witten invariants of X.

This re-derives Donaldson-Thomas/Gromov-Witten correspondence

36

This also answers the question (raised by A. Okounkov), about the DT interpretation of finite Calabi-Yau crystals:

They are counting BPS bound states of D6 brane with finite background D4 brane charge F.