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Developments in BPS Wall-Crossing

This discussion explores the BPS spectrum of d=4, N=2 theories with a focus on Wall-Crossing Formulae (WCF). It delves into the Kontsevich-Soibelman formula, field theory in Twistor Space, one-particle and multi-particle corrections, and differential equations in the context of BPS wall-crossing. The presentation provides insights into the physical explanations and derivations of key formulas.

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Developments in BPS Wall-Crossing

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  1. Developments in BPS Wall-Crossing Strings 2008, Cern, August 22, 2008 Work done with Davide Gaiotto and Andy Neitzke arXiv:0807.4723 And, to appear… TexPoint fonts used in EMF: AAAAAAAAAAAAA

  2. Outline 1. 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

  3. Subsequently, Kontsevich & Soibelman proposed a remarkable wall-crossing formula for generalized Donaldson-Thomas invariants of CY 3-folds Introduction This talk is about the BPS spectrum of theories with d=4,N=2 Recently there has been some progress in understanding 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

  4. Review of BPS Wall-Crossing-I Low energy theory: an unbroken rank rabelian gauge theory

  5. BPS -II Some BPS states are boundstates of other BPS states. (Cecotti,Fendley,Intriligator,Vafa; Seiberg&Witten)

  6. Semi-Primitive Wall-Crossing Marginal Stability Wall: ums u+ u- Denef & Moore gave formulae for for decays of the form Based on Denef’s multi-centered solutions of sugra, and quiver quantum mechanics. Do not easily generalize to

  7. g2 g1 + g2 g1 + g2 g1 g2 g1 BPS Rays For each 2 associate a ray in the z plane: As u crosses an MS wall some BPS rays will coalesce: u+ ums u-

  8. Symplectic transformations

  9. KS WCF Main statement: The product is INDEPENDENT OF u This is a wall-crossing formula !!

  10. KS Transformations Example for r=1:

  11. Seiberg-Witten Theory is now a local system Locally, we may choose a duality frame Special coordinates

  12. Low Energy Theory on R4 Choosing a duality frame, I = 1,…r :

  13. Example of G=SU(2) u It’s true!!!

  14. Low Energy theory on (Seiberg & Witten) 3D sigma model with target space Periodic coordinates for is hyperkähler Susy

  15. Semiflat Metric R = radius of S1 KK reduce and dualize the 3D gauge field:

  16. 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!

  17. Twistor Space A HK metric g is equivalent to a fiberwise holomorphic symplectic form

  18. Holomorphic Fourier Modes

  19. Semiflat holomorphic Fourier modes (Neitzke, Pioline, & Vandoren) Strategy: Compute quantum corrections to Recover the metric from

  20. 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

  21. Periodic Taub-NUT (S&W, Ooguri & Vafa, Seiberg & Shenker)

  22. Differential equation for twistor coordinates Twistor coordinates for PTN

  23. Explicit PTN twistor coordinates

  24. Key features of the coordinates The discontinuity is given by a KS transformation! As befits instanton corrections.

  25. 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 z plane.

  26. Riemann-Hilbert problem Piecewise holomorphic family Exponentially fast for

  27. Solution to the RH problem Explicit instanton expansion as a sum over trees

  28. KS WCF = Continuity of the metric As u crosses an MS wall, BPS rays pile up KS WCF Discontinuity from to is unchanged and hence g is continuous across a wall !

  29. Differential Equations The Riemann-Hilbert problem is equivalent to a flat system of differential equations: U(1)R symmetry scale symmetry holomorphy Stokes factors are independent of u,R Compute at large R: Stokes factors = KS factors Sg

  30. 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

  31. Other Things We Have Studied • 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.

  32. 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

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