d=4, - PowerPoint Presentation

Slide1

d=4, N=2, Field Theoryand Physical Mathematics

Gregory Moore

ICMP, Aalborg, Denmark, August 8, 2012

Rutgers UniversitySlide2

This is a review talk For my recent research results see my talk at

Strings-Math, Bonn, 2012: http://www.physics.rutgers.edu/~gmoore/Slide3

Introduction

3Wall Crossing 101

1

Conclusion

Review: d=4,

N

=2 field theory

2

3

4

5

6

7

8

9

Defects in Quantum Field Theory

Wall Crossing 102

3D Reduction &

Hyperk

ähler

geometry

Theories of Class S

Spectral NetworksSlide4

Two Important Problems In Mathematical Physics

1. Given a QFT what is the spectrum of the Hamiltonian?

and how do we compute forces, scattering amplitudes, operator vev’s

?

2. Find solutions of Einstein’s equations,

and how can we solve Yang-Mills equations on Einstein manifolds? Slide5

in the restricted case of d=4 quantum field theories with ``N=2

supersymmetry.’’

(Twice as much supersymmetry as in potentially realistic

supersymmetric

extensions of the standard model.)

Today, I will have something to say about each of these problems…Slide6

What we can say about Problem 1In the past 5 years there has been much progress in understanding a portion of the spectrum – the ``BPS spectrum’’ –

of these theories.

A corollary of this progress: many exact results have been obtained for ``line operator’’ and ``surface operator’’ vacuum expectation values. Slide7

What we can say about Problem 2It turns out that understanding the BPS spectrum allows one to give very explicit constructions of ``

hyperkähler metrics’’ on certain manifolds associated to these

d=4, N

=2 field theories.

Hyperk

ähler

(HK)

manifolds are Ricci flat, and hence are solutions to Einstein’s equations. Slide8

Moreover, the results on ``surface operators’’ lead to a construction of solutions to natural generalizations of the Yang-Mills equations on HK manifolds.(Hyperholomorphic

connections.)

(On a 4-dimensional HK manifold a hyperholomorphic connection is the same thing as a self-dual Yang-Mills

instanton

.) Slide9

New Interrelations,

Directions & ProblemsHitchin

systems, integrable systems,

moduli

spaces of flat connections on surfaces, cluster algebras,

Teichm

ű

ller

theory and the ``higher

Teichm

ű

ller theory’’ of

Fock & Goncharov, ….

A good development should open up new questions and directions of research and provide interesting links to other lines of enquiry.

It turns out that solving the above problems leads to interesting relations to … Slide10

Introduction

10Wall Crossing 101

1

Conclusion

Review: d=4,

N

=2 field theory

2

3

4

5

6

7

8

9

Defects in Quantum Field Theory

Wall Crossing 102

3D Reduction &

Hyperk

ähler

geometry

Theories of Class S

Spectral NetworksSlide11

d=4,N=2 Superalgebra

Poincare superalgebraSlide12

Constraints on the TheoryRepresentation theory: Field and particle

multiplets

Lagrangians: Typically depend on very few parameters for a given field content.

BPS Spectrum: Special subspace in the Hilbert space of statesSlide13

Example: N=2 Super-Yang-Mills

Gauge fields:

Doublet of

gluinos

:

Complex

adjoint

scalars:Slide14

Hamiltonian & Classical Vacua

The renormalizable Lagrangian is completely determined up to a choice of Yang-Mills coupling g

2.

Classical

Vacua

: Slide15

Quantum

Moduli Space of Vacua

Claim: The continuous vacuum degeneracy is

an exact property of the quantum theory:



Physical properties depend on

the vacuum

Slide16

Low Energy: Abelian Gauge Theory

Unbroken gauge symmetry:

Low energy theory is described by an

N

=2 extension of Maxwell’s theory:

(r= Rank = K-1)

Maxwell fields F

I

, I=1,…, r. i.e.

& their

superpartners

Slide17

Low-Energy Effective Action

N

=2

susy

constrains the low energy effective

action of the Maxwell theory to be of the form

is a symmetric,

holomorphic

matrix function of the vacuum parameters u.Slide18

Electro-magnetic Charges(Magnetic, Electric) Charges:

The theory will also contain ``

dyonic particles’’ – particles with electric and magnetic charges for the various Maxwell fields FI

, I = 1,…, r.

On general principles they are

in a

symplectic

lattice



u

. Slide19

Dirac Quantization: Slide20

BPS States

Superselection

sectors:

Taking the square of suitable

Hermitian

combinations of

susy

generators and using the algebra shows that in sector

H

Slide21

The Central Charge Function

The central charge function is a linear function

This linear function depends

holomorphically

on the vacuum manifold

B

. Denote it by Z(u).

On

Knowing Z



(u) is equivalent to knowing 

IJ

(u).Slide22

General d=4, N=2 Theories

1. A moduli space

B of quantum vacua,

(a.k.a. the ``Coulomb branch’’).

The low energy dynamics are described by an effective

N

=2

abelian

gauge theory.

The Hilbert space is graded by an integral lattice of charges,

,

with integral

anti-symmetric form. There is a BPS subsector with masses given exactly by |Z

(u)|. Slide23

So far, everything I’ve said follows fairly straightforwardly from general principles.

But how do we compute Z(u) and 

IJ(u) as functions of u ? Slide24

Seiberg-Witten Curve

Seiberg & Witten showed (for SU(2) SYM) that (u) can be computed in terms of the periods of a

meromorphic differential form  on a Riemann surface  both of which depend on u. Slide25

The Promise of Seiberg-Witten Theory

So Seiberg & Witten showed how to determine the LEEA

exactly as a function of u, at least for G=SU(2) SYM.

They also gave cogent arguments for the exact BPS spectrum of this theory.

So it was natural to try to find the LEEA and the BPS spectrum for other d=4

N

=2 theories. Slide26

Extensive subsequent work showed that this picture indeed generalizes to all known solutions for the LEEA of N=2 field theory:Slide27

u

The family of Riemann surfaces is usually called the ``

Seiberg

-Witten curve’’ and the

meromorphic

differential thereupon is the ``

Seiberg

-Witten differential.’’

But, to this day, there is no general algorithm for computing the

Seiberg

-Witten curve and differential for a given

N

=2 field theory. Slide28

Singular Locus On a special complex

codimension one sublocus

Bsingular

the curve degenerates

new

massless

degrees of freedom enhance the Maxwell theorySlide29

In the 1990’s the BPS spectrum was only determined in a handful of cases…

( SU(2) with (N=2 supersymmetric) quarks flavors:

Nf = 1,2,3,4, for special masses: Bilal

& Ferrari)

In the past 5 years there has been a great deal of progress in understanding the BPS spectra in these and infinitely many other

N=2

theories.

One key element of this progress has been a much-improved understanding of the ``wall-crossing phenomenon.’’

But what about the BPS spectrum? Slide30

Introduction

30Wall Crossing 101

1

Conclusion

Review: d=4,

N

=2 field theory

2

3

4

5

6

7

8

9

Defects in Quantum Field Theory

Wall Crossing 102

3D Reduction &

Hyperk

ähler

geometry

Theories of Class S

Spectral NetworksSlide31

Recall the space of BPS states is:

It depends on u, since Z



(u) depends on u.

But even the dimension can depend on u !

It is finite dimensional.

It is a representation of so(3)



su

(2)

RSlide32

BPS IndexAs in the index theory of Atiyah

& Singer, HBPS

is Z

2

graded by (-1)

F

so there is an index, in this case a Witten index, which behaves much better (piecewise constant in u):

J

3

is any generator of so(3)Slide33

The Wall-Crossing Phenomenon

BPS particles can form bound states which are themselves BPS!

But even the

index

can depend on u !Slide34

Denef’s

Boundstate Radius Formula

So the

moduli

space of

vacua

B

is divided into two regions:

The Z’s are functions of the

moduli u B

ORSlide35

R

12 > 0

R12 < 0 Slide36

Wall of Marginal Stability

u

-

u

+

u

ms

Exact binding energy:

Consider a path of

vacua

crossing the wall: Slide37

The Primitive Wall-Crossing Formula

Crossing the wall:

(

Denef

& Moore, 2007)Slide38

Non-Primitive Bound StatesBut this is not the full story, since the same marginal stability wall holds for charges

N1 

1 and N2



2

The full wall-crossing formula, which describes all possible bound states which can form is the ``

Kontsevich-Soibelman

wall-crossing formula’’ Slide39

Line DefectsThese are nonlocal objects associated with dimension one subsets of

spacetime.

There are now several physical derivations of this formula, but – in my view -- the best derivation uses ``line operators’’ – or more properly - ``line defects.’’ Slide40

Introduction

40Wall Crossing 101

1

Conclusion

Review: d=4,

N

=2 field theory

2

3

4

5

6

7

8

9

Defects in Quantum Field Theory

Wall Crossing 102

3D Reduction &

Hyperk

ähler

geometry

Theories of Class S

Spectral NetworksSlide41

Interlude: Defects in Local QFTPseudo-definition: Defects are local disturbances supported on positive

codimension submanifolds of

spacetime.

Extended ``operators’’ or ``defects’’ have been playing an increasingly important role in recent years in quantum field theory. Slide42

Examples of DefectsExample 1

: d=0: Local Operators

Example 2: d=1: ``Line operators’’

Example 3

: Surface defects: Couple a 2-dimensional field theory to an ambient theory. These 2d4d systems play an important role later.

Gauge theory Wilson line:

4d Gauge theory

‘t

Hooft

loop: Slide43

Extended QFT and N-CategoriesThe inclusion of these extended objects enriches the notion of quantum field theory.

Even in the case of topological field theory, the usual formulation of

Atiyah and Segal is enhanced to ``extended TQFT’s’’ leading to beautiful relations to N-categories and the ``

cobordism

hypothesis’’ …

D. Freed; D.

Kazhdan

; N.

Reshetikhin

; V.

Turaev

; L. Crane; Yetter; M. Kapranov

; Voevodsky; R. Lawrence; J. Baez + J. Dolan ; G. Segal; M. Hopkins, J. Lurie, C. Teleman,L. Rozansky, K. Walker, A. Kapustin, N. Saulina

,…Slide44

N

44Slide45

Introduction

45Wall Crossing 101

1

Conclusion

Review: d=4,

N

=2 field theory

2

3

4

5

6

7

8

9

Defects in Quantum Field Theory

Wall Crossing 102

3D Reduction &

Hyperk

ähler

geometry

Theories of Class S

Spectral NetworksSlide46

We will now use these line defects to produce a physical derivation of the Kontsevich-Soibelman

wall-crossing formula. Gaiotto

, Moore, Neitzke; Andriyash, Denef

,

Jafferis

, MooreSlide47

Supersymmetric Line Defects A

line defect L is of type 

=ei



if it preserves:

Example:

47

Physical picture for charge sector

: As if we inserted an infinitely heavy BPS particle of charge 

Our line defects will be at

R

t

x { 0 }



R

1,3Slide48

Framed BPS Index

Framed

BPS States are states in

H

L,



which saturate

the bound.Slide49

Piecewise constant in  and u, but has wall-crossingacross ``

BPS walls’’ (only defined for () 0):

Framed BPS Wall-Crossing

49

BPS particle of charge



binds to the defect states in charge sector



c

to make a new framed BPS state: Slide50

But, particles of charge , and indeed n  for any n>0 can bind in

arbitrary numbers: they feel no relative force, and hence there is an entire

Fock space of boundstates with halo particles of charges n

.

Halo PictureSlide51

Framed BPS Generating Function

When crossing a BPS wall W



the charge sector



c

gains or loses a

Fock

space factor

(The sign takes account of the fact that some halo particles are

bosonic

or

fermionic

.) Slide52

Description via Differential OperatorsSo the change of F(L) across a BPS wall W

 is given by the action of a differential operator

: Slide53

Derivation of the wall-crossing formulaSlide54

The Kontsevich-Soibelman Formula

=

= Slide55

Example 1: The Pentagon Identity

Related to consistency of simple

superconformal

field theories (“

Argyres

-Douglas theories”) coherence theorems in category theory &

associahedra

, 5-term

dilogarithm

identity, …Slide56

Example 2Slide57

The SU(2) Spectrum

uSlide58

(No) Wild Wall ConjectureFor other values of <

1, 2

> rearranging K1 K

2

produces exponentially growing BPS

degeneracies

.

This is in conflict with basic thermodynamics of QFT, and hence for physical reasons we expect that there are never any such ``wild wall crossings’’

This seems very nontrivial from the mathematical viewpoint. Slide59

The wall crossing formula only describes the CHANGE of the BPS spectrum across a wall of marginal stability.

Only half the battle…

It does NOT determine the BPS spectrum!

We’ll return to that in Part 8, for theories of class S. Slide60

Political Advertisement

The first wall-crossing formula was found by

Cecotti

&

Vafa

in the context of d=2 N = (2,2) QFT’s in 1992

The first quantitative WCF (“

semiprimitive

”) for d=4 was written by

Denef

& Moore in 2007. After that the full WCF was announced by

Kontsevich

&

Soibelman

, there are related results by Joyce, and Joyce & Song.

There are other physical derivations of the KSWCF due to

Cecotti

&

Vafa

and

Manschot

,

Pioline

, & Sen. Slide61

Introduction

61Wall Crossing 101

1

Conclusion

Review: d=4,

N

=2 field theory

2

3

4

5

6

7

8

9

Defects in Quantum Field Theory

Wall Crossing 102

3D Reduction &

Hyperk

ähler

geometry

Theories of Class S

Spectral NetworksSlide62
Slide63

Compactification on a circle of radius R leads to a 3-dimensional sigma model with target space

M, a hyperk

ähler manifold.

Strategy

In the large R limit the metric can be solved for easily. At finite R there are mysterious

instanton

corrections.

Finding the HK metric is equivalent to finding a suitable set of functions on the

twistor

space of

M.

The required functions are solutions of an explicit integral equation (resembling

Zamolodchikov’s

TBA). Slide64

Low Energy theory on

3D sigma model with target space

(Seiberg & Witten)

4D scalars reduce to 3d scalars:

Periodic Wilson

scalarsSlide65

Seiberg-Witten Moduli Space M

Relation to

integrable

systems

(

)Slide66

Semiflat Metric

Singular on

B

sing

The leading approximation in the R

  limit is straightforward to compute: Slide67

Twistor SpaceHitchin Theorem: A HK

metric g is equivalent to a fiberwise holomorphic

symplectic form

Fiber above

 is

M

in complex structure 

Slide68

Local ChartsM

has a coordinate atlas {U

} with charts of the form

Contraction with

 defines c

anonical ``

Darboux

functions’’ Y



Canonical

holomorphic

symplectic

form: Slide69

The ``Darboux functions’’ So we seek a ``suitable’’

holomorphic maps

solves the problem.

such that Slide70

Darboux Functions for the Semiflat Metric

For the semiflat metric one can solve for the Darboux

functions in a straightforward way:

Strategy: Find the quantum corrections to the metric from the quantum corrections to the

Darboux

functions:

(

Neitzke

,

Pioline

,

Vandoren

)Slide71

The desired properties of the exact functions

lead to a list of conditions which correspond to a Riemann-Hilbert problem for

Y



on the

-plane.

Riemann-Hilbert ProblemSlide72

Solution Via Integral Equation

(

Gaiotto

, Moore,

Neitzke

: 2008)Slide73

Remarks1. Solving by iteration converges for large R for sufficiently tame BPS spectrum.

3. The coordinates

Y

are cluster coordinates.

(A typical field theory spectrum will be tame; a typical black hole spectrum will NOT be tame!)

2. The HK metric carries an ``imprint’’ of the BPS spectrum, and indeed the metric is smooth

iff

the KSWCF holds!Slide74

Other Applications of the Darboux Functions

The same functions allow us to write explicit formulae for the vev’s of line defects:

Exact results on line defect

vevs

. (Example below).

Deformation quantization of the algebra of

holomorphic

functions on

MSlide75

These functions also satisfy an integral equation strongly reminiscent of those used in inverse scattering theory.

Generalized Darboux Functions & Generalized Yang-Mills Equations

In a similar way, surface defects lead to a generalization of Darboux

functions.

Geometrically, these functions can be used to construct

hyperholomorphic

connections on

M

(A

hyperholomorphic

connection is one whose

fieldstrength

is of type (1,1) in all complex structures. )Slide76

Introduction

76Wall Crossing 101

1

Conclusion

Review: d=4,

N

=2 field theory

2

3

4

5

6

7

8

9

Defects in Quantum Field Theory

Wall Crossing 102

3D Reduction &

Hyperk

ähler

geometry

Theories of Class S

Spectral NetworksSlide77

We now turn to a rich set of examples of d=4, N=2 theories,

In these theories many physical quantitieshave elegant descriptions in terms of Riemann surfaces and flat connections.

the theories of class S.

(‘’S’’ is for six ) Slide78

The six-dimensional theories

Claim, based on string theory constructions:

There is a family of stable interacting field theories, S[

g

] ,

with six-dimensional (2,0)

superconformal

symmetry.

(Witten;

Strominger

;

Seiberg

).

These theories have not been constructed – even by physical standards - but some characteristic properties of these hypothetical theories can be deduced from their relation to string theory and M-theory.

These properties will be treated as axiomatic. Later they should be theorems. Slide79

Theories of Class SConsider

nonabelian (2,0) theory S[g] for ``gauge algebra’’

g

The theory has half-BPS

codimension

two defects D

Compactify

on a Riemann surface C with

D

a

inserted at punctures

z

a

Twist to preserve d=4,N=2

Witten, 1997

GMN, 2009

Gaiotto

, 2009

79

Type II duals via ``geometric engineering’’

KLMVW 1996Slide80

Most ``natural’’ theories are of class S:

For example, SU(K) N=2 SYM coupled to ``quark flavors’’.

But there are also (infinitely many) theories of class S with no (known) Lagrangian

, e.g.

Argyres

-Douglas theories, or the

trinion

theories of (

Gaiotto

, 2009). Slide81

Relation to Hitchin systems

5D

g

SYM

-Model:

81Slide82

Effects of Defects

Physics depends on choice of

&

Physics of these defects is still being understood: (

Gaiotto

, Moore,

Tachikawa

;

Chacaltana

,

Distler

,

Tachikawa

)Slide83

Relation to Flat Complex Gauge Fields

is flat:

solves the

Hitchin

equations then

If

a

moduli

space of flat SL(K,

C

) connections. Slide84

We will now show how

Seiberg-Witten curve & differential 

Charge lattice & Coulomb branch

B

BPS states

Line & surface defects

can all be formulated geometrically in terms of the geometry and topology of the UV curve C and its associated flat connection

A.Slide85

SW differential

For

g

=

su

(K)

is a K-fold branched cover

Seiberg

-Witten Curve

85

UV CurveSlide86

Coulomb Branch & Charge LatticeCoulomb branch

Local system of charges

(Actually,

 is a

subquotient

. Ignore that for this talk. )

{

Meromorphic

differential with prescribed singularities at

z

a

}Slide87

BPS States: Geometrical Picture Label the sheets of the covering

  C by i,j,=

1,…, K.

A

WKB path

of phase



is an integral path on C

Generic WKB paths have both ends on singular points

z

a

Separating

WKB paths begin on

branch points, and for generic

,

end on singular points

where

i

, j are two sheets of the covering. Slide88

WKB paths generalize the trajectories of quadratic differentials, of importance in Teichmuller theory:

(Thurston, Jenkins, Strebel,Zorich,….) Slide89

But at critical values of =c

``string webs appear’’: String Webs – 1/4Slide90

String Webs – 2/4

Closed WKB pathSlide91

These webs lift to closed cycles  in  and represent BPS states with

A ``

string web

’’ is a union of WKB paths with endpoints on

branchpoints

or such junctions.

String Webs – 4/4

At higher rank, we get string

junctions

at critical values of

:Slide92

Line defects in S[g,C,D]

6D theory S[g] has supersymmetric

surface defects:

Line defect in 4d

labeled

by a

closed

path

. Slide93

Line Defect VEVsExample: SU(2) SYM Wilson line

Large R limit gives expected terms

Surprising

nonperturbative

correctionSlide94

Canonical Surface Defect in S[g,C,D]

For z  C we have a

canonical surface defect Sz

This is a 2d-4d system. The QFT on the surface

S

z

is a d=2

susy

theory whose massive

vacua

are naturally identified with the points on the SW curve covering z.

There are many exact results for

S

z

. As an example we turn to spectral networks…Slide95

Introduction

95Wall Crossing 101

1

Conclusion

Review: d=4,

N

=2 field theory

2

3

4

5

6

7

8

9

Defects in Quantum Field Theory

Wall Crossing 102

3D Reduction &

Hyperk

ähler

geometry

Theories of Class S

Spectral NetworksSlide96

As we have emphasized, the WCF does not give us the BPS spectrum.

For theories of class S we can solve this problem – at least in principle – with the technique of spectral networks. Slide97

What are Spectral Networks ? Spectral networks are combinatorial

objects associated to a covering of Riemann surfaces   C

C

Spectral network

branch pointSlide98

The combinatorial method for extracting the BPS spectrum in theories of class S is based on the behavior under variation of the phase 

Spectral networks are defined by the physics of two-dimensional

solitons on the surface defect

S

z

Paths in the network are constructed from WKB paths of phase

 according to known local rulesSlide99

Movies: http://www.ma.utexas.edu/users/neitzke/spectral-network-movies

/Slide100

Movies: http://www.ma.utexas.edu/users/neitzke/spectral-network-movies

/Slide101
Slide102

One can write very explicit formulae for the BPS degeneracies

() in terms of the combinatorics of the change of the spectral network.

GMN, Spectral Networks, 1204.4824

Finding the BPS SpectrumSlide103

Mathematical Applications of Spectral Networks

They thereby construct a system of coordinates on moduli spaces of flat connections which generalize the cluster coordinates of Thurston,

Penner, Fock,

Fock

and

Goncharov

.

Spectral networks are the essential data to construct a

symplectic

``

nonabelianization

map’’ Slide104

Application to WKB TheoryThe spectral network can be interpreted as the network of Stokes lines for the

0,  asymptotics of the differential equation.

The equation for the flat sections

is an ODE generalizing the Schrodinger equation (K=2 cover)

The

asymptotics

for

  0 ,  is a problem in WKB theory. K>2 is a nontrivial extension of the K=2 case.Slide105

Introduction

105Wall Crossing 101

1

Conclusion

Review: d=4,

N

=2 field theory

2

3

4

5

6

7

8

9

Defects in Quantum Field Theory

Wall Crossing 102

3D Reduction &

Hyperk

ähler

geometry

Theories of Class S

Spectral NetworksSlide106

Conclusion: Main Results1. A good, physical, understanding of wall crossing. Some understanding of the computation of the BPS spectrum, at least for class S.

2. A new construction of

hyperkähler

metrics and

hyperholomorphic

connections.

3. Nontrivial results on line and surface defects in theories of class S:

Vev’s

and associated BPS states.

4. Theories of class S define a ``conformal field theory with values in d=4

N

=2 quantum field theories.’’ Slide107

107

S-Duality and the modular groupoid

Higgs branches

AGT:

Liouville

& Toda theory

-backgrounds,

Nekrasov

partition functions,

Pestun

localization

. Cluster algebras

Z(S3 x S1) Scfml

indx Three dimensions, Chern

-Simons, and mirror symmetryNekrasov-Shatashvili: Quantum Integrable systems

Holographic duals

N=4 scatteringSlide108

Conclusion:

Some Future Directions & Open Problems

1. Make the spectral network technique more effective. Spectrum Generator?

3. Can the method for producing HK metrics give an explicit nontrivial metric on K3 surfaces?

2. Geography problem: How extensive is the class S? Can we classify d=4 N=2 theories?

4. + many, many more. Slide109

Conclusion: 3 Main Messages1.

Seiberg and Witten’s breakthrough in 1994, opened up many interesting problems. Some were quickly solved, but some remained stubbornly open.

But the past five years has witnessed a renaissance of the subject, with a much deeper understanding of the BPS spectrum and the line and surface defects in these theories. Slide110

Conclusions: Main Messages2. This progress has involved nontrivial and surprising connections to other aspects of Physical Mathematics:

Hyperkahler

geometry, cluster algebras, moduli

spaces of flat connections,

Hitchin

systems,

instantons

,

integrable

systems,

Teichm

űller

theory, …Slide111

Conclusions: Main Messages

3. There are nontrivial

superconformal fixed points in 6 dimensions.

(They were predicted many years ago from string theory.)

We have seen that the mere existence of these theories leads to a host of nontrivial results in quantum field theory.

Still, formulating 6-dimensional

superconformal

theories in a mathematically precise way remains an outstanding problem in Physical Mathematics. Slide112

112A Central Unanswered Question

Can we construct S[g]? Slide113

NOT

113Slide114

Some ReferencesSpectral Networks and Snakes, to appear

Spectral Networks, 1204.4824 Wall-crossing in Coupled 2d-4d Systems: 1103.2598

Framed BPS States: 1006.0146Wall-crossing,

Hitchin

Systems, and the WKB Approximation: 0907.3987

Four-dimensional wall-crossing via three-dimensional field theory: 0807.4723

Gaiotto

, Moore, &

Neitzke

:

Andriyash, Denef, Jafferis

& Moore, Wall-crossing from supersymmetric galaxies, 1008.0030Denef and Moore, Split states, entropy

enigmans, holes and halos, hep-th/0702146 Diaconescu

and Moore, Crossing the wall: Branes vs. Bundles, hep-th/0702146 Slide115

Kontsevich & Soibelman, Motivic Donaldson-Thomas Invariants: Summary of Results, 0910.4315

Pioline, Four ways across the wall, 1103.0261

Cecotti

and

Vafa

, 0910.2615

Manschot

,

Pioline

, &

Sen

, 1011.1258Slide116

Generalized Conformal Field Theory

``Conformal field theory valued in d=4 N=2 field theories’’

S[g

,C,D

] only depends on the conformal structure of C.

Twisting

For some C, D there are subtleties in the 4d limit.

Space of coupling constants =



g,n

116

This is the essential fact behind the AGT conjecture, and other connections to 2d conformal field theory.

(Moore &

Tachikawa

) Slide117

Gaiotto Gluing Conjecture -A D.

Gaiotto, ``N=2 Dualities’’

Slogan: Gauging = Gluing

Gauge the diagonal G



G

L

x G

R

symmetry with q = e

2

i

:117Slide118

Gaiotto Gluing Conjecture - B

Nevertheless, there are situations where one gauges just a subgroup – the physics here could be better understood. (Gaiotto, Moore,

Tachikawa)

Glued surface: