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wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDY STRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van den Bleeken 2008 Sungay Lee+P.Y. 2011 Heeyeon Kim+Jaemo Park+ZhaolongWang+P.Y. 2011 Sen 2011 Bena+Berkooz+de Boer+El-Showk+Van den Bleeken 2012 Seungjoo Lee+ZhaolongWang+P.Y. 2012 Manschot+Pioline+Sen 2010/2011/2012/2013 2008 Kontsevich+Soibelman 2008/2009/2010/2011/2012 Gaiotto+Moore+Neitzke 2007/2009 Derksen+Weyman+Zelevinsky 2011 Keller 2011 Alim+Cecotti+Cordova+Espahbodi+Rastogi+Vafa 2012 Xie 2012 Andriyash+Denef+Jafferis+Moore 2012 Chuang+Diaconescu+Manschot+Moore+Soibelman
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Page 1: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

wall crossing redux

PILJIN YIKOREA INSTITUTE for ADVANCED STUDY

STRINGS 2013

Denef 2002Denef+Moore 2007

de Boer+El-Showk+Messamah+Van den Bleeken 2008Sungay Lee+P.Y. 2011

Heeyeon Kim+Jaemo Park+Zhaolong Wang+P.Y. 2011Sen 2011

Bena+Berkooz+de Boer+El-Showk+Van den Bleeken 2012Seungjoo Lee+Zhaolong Wang+P.Y. 2012

Manschot+Pioline+Sen 2010/2011/2012/2013

2008 Kontsevich+Soibelman2008/2009/2010/2011/2012 Gaiotto+Moore+Neitzke2007/2009 Derksen+Weyman+Zelevinsky2011 Keller 2011 Alim+Cecotti+Cordova+Espahbodi+Rastogi+Vafa2012 Xie2012 Andriyash+Denef+Jafferis+Moore2012 Chuang+Diaconescu+Manschot+Moore+Soibelman

Page 2: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

wall crossing redux

PILJIN YIKOREA INSTITUTE for ADVANCED STUDY

STRINGS 2013

constructive

2008 Kontsevich+Soibelman2008/2009/2010/2011/2012 Gaiotto+Moore+Neitzke2007/2009 Derksen+Weyman+Zelevinsky2011 Keller 2011 Alim+Cecotti+Cordova+Espahbodi+Rastogi+Vafa2012 Xie2012 Andriyash+Denef+Jafferis+Moore2012 Chuang+Diaconescu+Manschot+Moore+Soibelman

Denef 2002Denef+Moore 2007

de Boer+El-Showk+Messamah+Van den Bleeken 2008Sungay Lee+P.Y. 2011

Heeyeon Kim+Jaemo Park+Zhaolong Wang+P.Y. 2011Sen 2011

Bena+Berkooz+de Boer+El-Showk+Van den Bleeken 2012Seungjoo Lee+Zhaolong Wang+P.Y. 2012

Manschot+Pioline+Sen 2010/2011/2012/2013

Page 3: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

Zhaolong Wang Sungjay LeeSeung-Joo LeeHeeyeon Kim

Page 4: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

wall-crossing of BPS states with 4 (or less) supersymmetries

marginal stability wall

Page 5: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

first, some pre-history

Page 6: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

prototype : D=4 N=2 SU(2) U(1) Seiberg-Witten

Gauntlett+Harvey 1995Bilal+Ferrari 1996-1997

Page 7: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

D=4 N=2 SU(r+1) U(1) Seiberg-Witten

Lerche 2000

r

dyons in hyper

Page 8: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

1998 Lee + P.Y.N=4 SU(n) ¼ BPS states via semiclassical multi-center dyon solitons

1997 Bergman¼ BPS states as open string-web, and decay thereof

1997 Dasgupta + Muhki / Senstring junctions

Page 9: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

generic 4 SUSY BPS “particles” are loose bound states of charge centerswall-crossing one or more distances diverge

Page 10: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

generic 4 SUSY BPS “particles” are loose bound states of charge centerswall-crossing ~ supersymmetric Schroedinger problem

Page 11: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

1998 Lee + P.Y.N=4 SU(n) ¼ BPS states via semiclassical multi-center dyon solitons

1999 Bak + Lee + Lee + P.Y. N=4 SU(n) ¼ BPS states via semi-classical multi-center monopole dynamics

1999-2000 Gauntlett + Kim + Park + P.Y. / Gauntlett + Kim + Lee + P.Y. / Stern + P.Y.N=2 SU(n) BPS state counting via semi-classical multi-center monopole dynamics

Page 12: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

1998 Lee + P.Y.N=4 SU(n) ¼ BPS states via semiclassical multi-center dyon solitons

1999 Bak + Lee + Lee + P.Y. N=4 SU(n) ¼ BPS states via semi-classical multi-center monopole dynamics

1999-2000 Gauntlett + Kim + Park + P.Y. / Gauntlett + Kim + Lee + P.Y. / Stern + P.Y.N=2 SU(n) BPS state counting via semi-classical multi-center monopole dynamics

Page 13: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

the BPS supermultiplet of the bound state

in effect, N=2 SUSY multi-particle “primitive wall-crossing formula”

Gauntlett + Kim + Park + P.Y. Gauntlett + Kim + Lee + P.Y.

Stern + P.Y. 1999-2000

Page 14: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

Gauntlett + Kim + Park + P.Y. Gauntlett + Kim + Lee + P.Y.

Stern + P.Y 1999-2000

for Seiberg-Witten theories of rank > 1, arbtrarily high-spin BPS dyons in the weakly coupled regime & the accompanying infinite number of marginal stability walls

qualitative difference of BPS spectra between rank 1 vs. rank > 1

Page 15: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

Gauntlett + Kim + Park + P.Y. Gauntlett + Kim + Lee + P.Y.

Stern + P.Y 1999-2000

Chuang + Diaconescu + Manschot+ Moore + Soibelman 2012

for Seiberg-Witten theories of rank > 1, arbtrarily high-spin BPS dyons in the weakly coupled regime & the accompanying infinite number of marginal stability walls

qualitative difference of BPS spectra between rank 1 vs. rank > 1

Page 16: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

1998 Lee + P.Y.N=4 SU(n) ¼ BPS states via semiclassical multi-center dyon solitons

1999 Bak + Lee + Lee + P.Y. N=4 SU(n) ¼ BPS states via semi-classical multi-center monopole dynamics

1999-2000 Gauntlett + Kim + Park + P.Y. / Gauntlett + Kim + Lee + P.Y. / Stern + P.Y.N=2 SU(n) BPS state counting via semi-classical multi-center monopole dynamics

also,

2001 Argyres + Narayan / Ritz + Shifman + Vainshtein + VoloshinUV-incomplete string-web picture for N=2 BPS dyons in Seiberg-Witten description again, multi-particle picture of BPS states in strongly coupled regime

Page 17: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

SU(2) Seiberg-Witten

vector multiplet

Page 18: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

SU(2) Seiberg-Witten

Page 19: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

1998 Lee + P.Y.N=4 SU(n) ¼ BPS states via semiclassical multi-center dyon solitons

1999 Bak + Lee + Lee + P.Y. N=4 SU(n) ¼ BPS states via semi-classical multi-center monopole dynamics

1999-2000 Gauntlett + Kim + Park + P.Y. / Gauntlett + Kim + Lee + P.Y. / Stern + P.Y.N=2 SU(n) BPS state counting via semi-classical multi-center monopole dynamics

2000 DenefN=2 supergravity via classical multi-center black holes attractor solutions

2001 Argyres + Narayan / Ritz + Shifman + Vainshtein + VoloshinUV-incomplete string-web picture for N=2 BPS dyons

Page 20: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

generic 4 SUSY BPS black hole solutions are loose bound states of many charged singe-center BPS black holes

Page 21: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

1998 Lee + P.Y.N=4 SU(n) ¼ BPS states via semiclassical multi-center dyon solitons

1999 Bak + Lee + Lee + P.Y. N=4 SU(n) ¼ BPS states via semi-classical multi-center monopole dynamics

1999-2000 Gauntlett + Kim + Park + P.Y. / Gauntlett + Kim + Lee + P.Y. / Stern + P.Y.N=2 SU(n) BPS state counting via semi-classical multi-center monopole dynamics

2000 DenefN=2 supergravity via classical multi-center black holes attractor solutions

2001 Argyres + Narayan / Ritz + Shifman + Vainshtein + VoloshinUV-incomplete string-web picture for N=2 BPS dyons

2002 Denef / 2007 Denef + Moorequiver dynamics of BPS states / (semi-)primitive wall-crossing formula ………

Page 22: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

KS, GMN, and …

real-space wall-crossing

quivers & quiver invariants

quiver mutation

Page 23: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

Kontsevich-Soibelman& Gaiotto-Moore-Neitzke

Page 24: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

-+ sideside =

a marginal stability wall

Schwinger product

Kontsevich + Soibelman 2009

Page 25: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

with the 2nd helicity trace for N=2 BPS states

on [ a spin ½ + two spin 0 ] x [ angular momentum multiplet ]l

Page 26: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

=

a marginal stability wall

+ side - side

Page 27: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

is KS true physically ?how to see from BPS state building/counting ?

why rational invariants ?

input data ?

Page 28: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

is KS true physically ? how to see from BPS state building/counting ?

why rational invariants ?

or, more generally

Page 29: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

is KS true for SW ? how to see from BPS state building/counting ?

why rational invariants ?

input data ?

Page 30: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

SW on a circle, Stokes phenomena, (2,0), Hitchin system, Darboux/Fock-Goncharov,framed BPS state, spectrum generator, BPS quiver,

spectral network, …

Gaiotto + Moore + Neiztke 2008, 2009, 2010, 2011, 2012(also, Chen + Dorey + Petunin 2010 for GMN 2008)

Alim+Cecotti+Cordova+Espahbodi+Rastogi+Vafa 2011Xie 2012

Chuang+Diaconescu+Manschot+Moore+Soibelman 2012

……

Page 31: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

(i) proves KS for general Seiberg-Witten theories

=

continuity of Darboux coordinatesacross marginal stability walls

KS operator string = Stokes’ phenomena

Gaiotto + Moore + Neiztke 2008, 2009 also, see Chen + Dorey + Petunin 2010

Page 32: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

(i) proves KS for general Seiberg-Witten theories

discontinuous spectra = continuity of vacuum physics

Gaiotto + Moore + Neiztke 2008, 2009 also, see Chen + Dorey + Petunin 2010

Page 33: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

(ii) spectrum generator

independent computation of the two sets of such holomorphic quantityfrom which the KS operator string can be in principle extracted

Gaiotto + Moore + Neiztke 2009

Page 34: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

(ii) spectrum generator

independent computation of the two sets of such holomorphic quantityfrom which the KS operator string can be in principle extracted

Gaiotto + Moore + Neiztke 2009

Page 35: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

(iii) hypermultiplets as geodesic segments on the Gaiotto curve

explicit construction of SU(2) theory spectra

how to read off BPS quivers from triangulation

also more recently, SU(n>2) partial spectra & higher spin states Gaiotto+Moore+Neitzke 2012

Xie 2012Chuang+Diconescu+Manschot+Moore+Soibelman 2012

Galakhov+Longhi+Mainiero+Moore+Neitzke 2013

Gaiotto+Moore+Neiztke 2009

Gaiotto+Moore+Neitzke 2009Alim+Cecotti+Cordova+Espahbodi+Rastogi+Vafa 2011

Page 36: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

is KS true for all physical wall-crossing ?how to see from BPS state building/counting ?

why rational invariants ?

input data ?

Page 37: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

real space wall-crossing

Denef 2002Denef + Moore 2007

de Boer + El-Showk + Messamah +Van den Bleeken 2008Manschot+Pioline+ Sen 2010/2011

Lee+P.Y. / Kim+Park+Wang+P.Y. 2011Sen 2011

Page 38: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

BPS dyon Schroedinger problem for Seiberg-Witten theories each dyon feels the rest via long-range tails

Page 39: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

for which UV-incomplete long distance solutions will do

Seiberg+Witten 1994Chalmers + Rocek + von Unge 1996

Ritz + Shifman + Vainshtein + Voloshin 2001Argyres + Narayan 2001

Page 40: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

each SW dyon feels the rest via long-range tails

Lee+P.Y. 2011cf) Ritz+Vainshtein 2008

Page 41: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

treat all charge-centers on equal footing & supersymmetrize

Page 42: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

step (1) real space N=4 susy quantum mechanicsfor Seiberg-Witten dyons near marginal stability wall

Lee+P.Y. 2011Kim+Park+Wang+P.Y, 2011

~ ab initio, for SW theories, reproduction of Denef’s Coulomb phase dynamics

Page 43: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van
Page 44: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

deform & localize N=4 3(n-1) dimensional dynamicsN=1 2(n-1) dim nonlinear sigma model with U(1) bundle

Page 45: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

reduces to a N=1 Dirac indexof a nonlinear sigma model on the manifold

3(n-1) bosons + 4(n-1) fermions 2(n-1) bosons + 2(n-1) fermions

Kim+Park+Wang+P.Y. 2011

Page 46: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

step (2) an index theorem before quantum statistics

Manschot+Pioline+ Sen 2010/2011Kim+Park+Wang+P.Y. 2011

Page 47: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

Bose/Fermi statistics an iterative sum overfixed submanifolds under permutation of identical particles

Page 48: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

imposing Bose/Fermi statistics orbifolding of the index

Page 49: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

P.Y. 1997Green + Gutperle 1997

Kim + Park + Wang + P.Y. 2011

for p identical particles & with internal degeneracy rational invariants !!!

cf) Manschot + Pioline + Sen 2010/2011

Page 50: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

Manschot+Pioline+Sen 2011Kim+Park+Wang+P.Y. 2011

step (3) universal wall-crossing formula from ab initio, real space dynamics of charge centers

Page 51: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

Manschot+Pioline+Sen 2011Kim+Park+Wang+P.Y. 2011

= sum over all partition of charges, with rational invariants,with particles in each partition treated as if distinguishable

Page 52: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

step (4) evaluate & compare

Jan’s talk, tomorrow, for equivariant evaluation

and also for how to include quiver invariants to the counting

?

Manschot+Pioline+Sen 2010/2011

Page 53: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

equivariant index on computes

reduction to

Kim+Park+Wang+P.Y. 2011

Page 54: wall crossing redux - Kavli IPMU-カブリ数物連携宇 …...wall crossing redux PILJIN YI KOREA INSTITUTE for ADVANCED STUDYSTRINGS 2013 Denef 2002 Denef+Moore 2007 de Boer+El-Showk+Messamah+Van

protected spin character for field theory BPS states

protected spin characterGaiotto, Moore, Neitzke 2010 / Maldacena

2nd helicity trace

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1) all relevant charges on a single plane of the charge lattice~ always true, anyhow, near marginal stability walls

2) in the absence of scaling regimes ~ in the absence of quiver invariants~ more likely to hold for field theory BPS states

scaling regime is more typical of black hole examples

Ashoke Sen 2011

step (5) equivalence to KS with two assumptions

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is KS true physically ?how to see from BPS state building/counting ?

why rational invariants ?

input data ?

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quivers & quiver invariants

Denef 2002Denef+Moore 2007

Bena+Berkooz+de Boer+El-Showk+Van den Bleeken 2012Lee+Wang+P.Y. 2012

Manschot+Pioline+Sen 2012/2013

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D3 wrapped on 3-cycles in CY3 BPS quiver quantum mechanicsDenef 2002

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Coulomb “phase” multi-center picture of BPS states

small & “positive” FI constants

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thus, index thm above for real-space wall-crossing alsoalso address general BPS states including black holes as well ?

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large & “positive” FI constants

Higgs phase :

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Higgs phase :

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F. Denef 2002 + A. Sen 2011

small FI constantslarge FI constants

?

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the equality is known to fail for quivers with loops

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a simple but illuminating example

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Denef + Moore 2007

a simple but illuminating example

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what physical & mathematical properties characterize these Higgs-only wall-crossing-safe BPS states ?

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also, some of wall-crossing formulae need input data when

how to isolate and count wall-crossing-safe states in such cases ?

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quiver invariants for cyclic Abelian quivers

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wall-crossing vs. wall-crossing-safe

complete intersection

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wall-crossing vs. wall-crossing-safe

S.J. Lee + Z.L. Wang + P. Y., 2012Bena + Berkooz + de Boer + El-Showk + d. Bleeken, 2012

complete intersection

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general proof & explicit counting !

S.L. Lee + Z.L. Wang + P. Y., 2012Manschot + Pioline + Sen, 2012

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the total equivariant index ~ Hirzebruch character

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which is easily computable here, via Riemann-Roch theorem

embedding map

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and decomposed into two parts

embedding map

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wall-crossing states vs. wall-crossing-safe states

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wall-crossing states vs. wall-crossing-safe states

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wall-crossing states vs. wall-crossing-safe states

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wall-crossing states vs. wall-crossing-safe states

many-body bound stateswall-crossing

angular momentummultiplets

single-center stateswall-crossing-safe

angular momentumsinglets

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wall-crossing states vs. wall-crossing-safe states

many-body bound stateswall-crossing

polynomial degeneracy:most of familiar BPS states in

field theories belong here

single-center stateswall-crossing-safe

exponential degeneracy:single-center BH’s

belong here

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more examples of quiver invariants

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more examples of quiver invariants

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Manschot+Pioline+Sen 2012/2013

the real-space wall-crossing formulae must be revised recursively such that

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quiver mutationsis there a more intelligent way to count BPS states ?

Derksen + Weyman + Zelevinsky 2007/2009Keller 2011

Alim+Cecotti+Cordova+Espahbodi+Rastogi+Vafa 2011

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quiver mutation ~ Seiberg duality for quiver rep. theory

Derksen + Weyman + Zelevinsky 2007/2009Keller 2011

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quiver mutation ~ Seiberg duality

Derksen + Weyman + Zelevinsky 2007/2009Keller 2011

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quiver mutation ~ Seiberg duality

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quiver mutation ~ Seiberg duality

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BPS quiver ~ quiver of 2r+f basis hypermultiplets

Andriyash+Denef+Jafferis+Moore 2012

Alim+Cecotti+Cordova+Espahbodi+Rastogi+Vafa 2011

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BPS quiver ~ quiver of 2r+f basis hypermultiplets

Alim+Cecotti+Cordova+Espahbodi+Rastogi+Vafa 2011

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BPS quiver ~ quiver of 2r+f basis hypermultiplets

for hypers, one can mutate enough to find eventually

which by itself can determine entirehyper content in a given chamber;finite chambers offer ideal test-bed

Alim+Cecotti+Cordova+Espahbodi+Rastogi+Vafa 2011Xie 2012

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quiver mutations in its most general form may prove to be a very efficient particle#-reducing algorithm

if we can unclutter the intricate chamber dependences

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summary

KS & GMN represent giant leaps over what we knew before, bits and pieces, mostly for Seiberg-Witten theories; various technical difficulties for rank > 1 field theories & for BH’s still remain

real-space-based, constructive approach to wall-crossing has grown verycompetitive, last 2~3 years, with partial equivalence to KS shown

the intuitive Coulomb picture with recursive wall-crossing augmented by the comprehesive Higgs picture with the extra wall-crossing-safe states

quiver invariants = wall-crossing-safe states (input data for KS, e.g.) are essential ingredient to the wall-crossing beyond simple examples Jan’s talk, tomorrow, for how to do this……