Developments in BPS Wall-Crossing

Work done with Davide Gaiotto and Andy Neitzke

arXiv:0807.4723

Strings 2008, Cern, August 22, 2008

And, to appear…

Outline1. Review BPS Wall-Crossing

2. The Kontsevich-Soibelman formula

3. N=2,D=4 Field Theory on

4. Twistor Space

5. One-particle corrections

6. Multi-particle corrections: Riemann-Hilbert

7. Differential Equations

8. Summary & Concluding Remarks

R3 £ S1

Introduction

This talk is about the BPS spectrum of theories with d=4,N=2

Recently there has been some progress in understanding Recently there has been some progress in understanding how the BPS spectrum depends on the vacuum. how the BPS spectrum depends on the vacuum.

These are called Wall-Crossing Formulae (WCF)

Last year: WCF derived with Frederik Denef

This talk will give a physical explanation & derivation of the KS formula

Review of BPS Wall-Crossing-I

Consider a theory on R4 with N = 2 SUSY

Low energy theory: an unbroken rank r abelian gauge theory

For u 2 M v, themoduli spaceof vacua,

¡ : Symplectic latticeof electric & magnetic charges °

Hu =©°2¡ H° ;u

let Hu be theone-particleHilbert space.

BPS -II

Some BPS states are boundstates of other BPS states.

à will decay as u crosses a wall of marginal stabilitywhereZ°1 (u) and Z°2 (u) areparallel.

à 2 HB P S° ;u , a boundstateof BPS states: ° = °1+°2,

(Cecotti,Fendley,Intriligator,Vafa; Seiberg&Witten)

N = 2 central chargeoperator Z = Z° (u) on H° ;u

HB P Su;° : Subspaceof H° ;u with E = jZ° (u)j

­ (°;u) := ¡12TrH B P S

° ;u(2J 3)2(¡ 1)2J 3

Semi-Primitive Wall-Crossing

Marginal Stability Wall:

u-u+

ums

Denef & Moore gave formulae for ¢ ­

for decays of the form ° ! °1+N°2 N ¸ 1

Based on Denef’s multi-centered solutions of sugra, and quiver quantum mechanics.

MS(°1;°2) := fujZ°1 (u)Z°2 (u)

2 R+g

Do not easily generalize to ° ! N1°1+N2°2 N i ¸ 1

BPS Rays

For each 2 associate a ray in the plane:

As u crosses an MS wall some BPS rays will coalesce:

u+ u-ums

° := Z° (u)R¡

Symplectic transformations

KS assign a group element S° to each BPS ray °

K ° : Symplectic transformations of T

T := ¡ ¤ ­ C¤ »= (C¤)2r

S° :=!Y

° 0k°

K ­ (° 0;u)° 0

KS WCF

Main statement: The product is INDEPENDENT OF u

This is a wall-crossing formula !!

For generic u, and convex coneV in the ³-plane

A(V) :=!Y

`° 2V

S° =!Y

¡ Z° (u)2V

K ­ (° ;u)°

KS Transformations

Example for r=1:

²i j =h° i ;° j i ) $ T :=12²i jdX i

X i

dX j

X j

T = C¤ £ C¤ $ T =dxxdyy

T = ¡ ¤ ­ C¤ ) Fourier modes: X ° : T ! C¤

K ° : X ° 0 ! X ° 0eh° ;°0i log(1¡ X ° )

Ka;b : (x;y) ! (x(1¡ xayb)b;y(1¡ xayb)¡ a)

Seiberg-Witten Theory

Locally, we may choose a duality frame

Special coordinates aI =Z®I (u)

M v»= Cr (u2 = hTr©2i , u3 = hTr©3i ;¢¢¢)

¡ »= ¡ el ©¡ mg

Z° (u) = a¢°el +aD ¢°mg

is now a local system¡ ! M v

Consider a d= 4;N = 2 ¯eld theory with asemisimple gaugegroup of rank r.

Low Energy Theory on R4

Choosing a duality frame, I = 1,…r :

L = ¡14¼Im¿I J (daI ¤d¹aJ +F I ¤F J ) +

14¼Re¿I J F I F J +¢¢¢

J u

M vu

Generic ¯bers J u: Abelian varieties

J :

Torus ¯bration: J u := ¡ ¤u ­ (R=2¼Z)

Example of G=SU(2)

§ u : y+¤4

y= x2 ¡ 2u

It’s true!!!

u

¡ ¤2 ¤2

K2;¡ 1K0;1 =K0;1K2;1K4;1¢¢¢K ¡ 22;0¢¢¢K4;¡ 1K2;¡ 1

Low Energy theory on R3 £ S1

3D sigma model with target space

Periodic coordinates

Susy is hyperkähler

for J u

(Seiberg & Witten)

J

' Ie =I

S1A I4dx

4

' m;I =I

S1(AD ;4)I dx4

J

Semiflat Metric

R = radius of S1 R ! 1

KK reduce and dualize the 3D gauge field:

dzI =d' m;I ¡ ¿I J d' Je

gsf =RIm¿jdaj2+(RIm¿)¡ 1jdzj2

The Main Idea

• gsf is quantum-corrected by BPS states(instanton = worldline of BPS particle on S1)

• So, quantum corrections depend on the BPS spectrum

• The spectrum jumps, but the true metric g must be smooth across MS walls.

• This implies a WCF!

Twistor Space

A HK metric g is equivalent to a fiberwise holomorphic symplectic form

$ 2 ­ 2Z =CP 1 ­ O(2)

$ ³ = ³ ¡ 1! + +! 3+³! ¡ , ³ 2 C¤

Z := J £ CP 1 p! CP 1

J ³ has complex structure ³ 2 CP 1

Holomorphic Fourier Modes

9 a basis of holomorphic functions X° (³), ° 2 ¡

$ ³ =X(³)¤($ T )

J ³ ! M v is a torus ¯bration

Restriction to torus (J ³ )u ) X° (³) = exp[iµ° +¢¢¢]

J u = ¡ ¤u ­ (R=2¼Z) Tu = ¡ ¤u ­ C¤

X° (³) = X (³)¤(X ° )X (³) : J ³ ! T

Semiflat holomorphic Fourier modes

X sf° = exp

£¼R³ ¡ 1Z° +iµ° +¼R³ ¹Z°

¤

(Neitzke, Pioline, & Vandoren)

Strategy: Compute quantum corrections to X sf°

X° =X sf° X

inst°

Recover the metric from $ ³ =X(³)¤($ T )

One Particle Corrections

• Work near a point u* where one HM, H becomes massless

• Dominant QC’s from instantons of these BPS particles

• Choose a duality frame where H has electric charge q>0

• Do an effective field theory computation

Periodic Taub-NUT

(S&W, Ooguri & Vafa, Seiberg & Shenker)

r = 1: Gibbons-Hawkingmetric on U(1) ¯bration over R2 £ S1

gPT N =V ¡ 1(d' m +A)2+V£jdaj2+(d' e)2

¤

V sf = ¡ q2R³log

a¤+ log

¹a¹¤

´V = V sf +V inst

V inst =q2RX

n

einq' eK 0(Rjnqaj)

Twistor coordinates for PTN

From themetric gPT N wecompute$ P T N

Differential equation for twistor coordinates

Xe(³) = exp[i' e+¢¢¢]

Xm(³) = exp[i' m +¢¢¢]

$ P T N =X ¤($ T ) =dXeXe

dXmXm

Explicit PTN twistor coordinates

Xm(³) = X sfmX

instm

X instm (³) » exp

(

qZ

`° e

[d³0]log[1¡ Xe(³0)q]

³0¡ ³

)

Xe(³) = X sfe =exp[R³ ¡ 1a+ i' e+R³¹a]

X sfm = exp[R³ ¡ 1aD +i' m +R³¹aD ]

Key features of the coordinates

The discontinuity is given by a KS transformation!

1. As a function of ³ , Xm is discontinuousacross theelectric BPS ray °e .

2. Xm(³) » ³ ! 0;1 X sfm (³)

As befits instanton corrections.

(Xe;Xm)ccw =K °e (Xe;Xm)cw

Multi-Particle Contributions

• To take into account the instanton corrections from ALL the BPS particles we cannot use an effective field theory computation.

• Mutually nonlocal fields in Leff are illegal!

• We propose to circumvent this problem by reformulating the instanton corrections as a Riemann-Hilbert problem in the plane.

Riemann-Hilbert problem

Piecewise holomorphic family

1. Across each BPS ray °

Exponentially fast for

2. X (³) ! X sf(³)

³ ! 0;1

X (³)ccw = S°X (³)cw

X (³) : J ³ ! T

Solution to the RH problem

X° (³) = X sf° (³)X

inst° (³)

Explicit instanton expansion as a sum over trees

X° (³) = X sf° (³)¢

¢exp

8<

:

X

° 0­ (°0;u)h°;°0i

Z

`° 0[d³0]

log(1¡ X° 0(³0))³0¡ ³

9=

;

KS WCF = Continuity of the metric

As u crosses an MS wall, BPS rays pile up

³+

³¡

³+

³¡

³+

³¡

KS WCF Discontinuity from ³+ to ³¡ is unchanged

$ and hence g is continuous across a wall !

Differential Equations

The Riemann-Hilbert problem is equivalent to a flat system of differential equations:

@uX = (³ ¡ 1A¡u +A0

u)X

R@RX = (³ ¡ 1A¡R +A0

R +³A+R )X

³@³X = (³ ¡ 1A¡³ +A0

³ +³A+³ )X

holomorphy

scale symmetry

U(1)R symmetry

Stokes factors are independent of u,R

Compute at large R: Stokes factors = KS factors S

Summary

• We constructed the HK metric for circle compactification of SW theories.

• Quantum corrections to the dim. red. metric gsf encode the BPS spectrum.

• Continuity of the metric across walls of MS is equivalent to the KS WCF.

• Use the twistor transform to include quantum corrections of mutually nonlocal particles

• The are Wilson-’tHooft-Maldacena loop operators, and generate the chiral ring of a 3D TFT

• Analogies to tt* geometry of Cecotti & Vafa

• Relations to Hitchin systems and D4/NS5 branes following Cherkis & Kapustin.

Other Things We Have Studied

X°

Open Problems

• Singularities at superconformal points

• Relations to integrable systems?

• Meaning of KS ``motivic WCF formula’’?

• Relation to the work of Joyce & Bridgeland/Toledano Laredo

• Generalization to SUGRA

• QC’s to hypermultiplet moduli spaces