YITP wor shop Dvlopmnt of Quantum Fld Thory and trng Thory Crcular Wlson loop oprator and mastr fld hoch Kawamoto OCAMI, Osaa Cty Unvrsty atonal Tawan ormal Unvrsty from August Wth T. Kuro Ryo and A. Mwa Toyo U., Komaba 3 July 8
Introducton BP objcts hav playd mportant rols n study of Ad/CFT corrspondnc strngugra n a waly curvd bacground trongly coupld gaug thory BP objcts wll not gt corrctd by quantum ffcts.g. chral prmars I n n Φ Φ Today, w ar gong to loo at dffrnt typs of BP objcts. 3 July 8, YITP worshop hoch Kawamoto Osaa Cty Unvrsty /4
Introducton contnud In Ad/CFT corrspondnc, Wlson loops corrspond to strng D-bran world volum: W-boson trng world-sht Gravty dual W C strng Wlson loop oprators n 4 supr Yang-Mlls may prsrv som of suprsymmtry. Crcular loop traght ln 3 July 8, YITP worshop hoch Kawamoto Osaa Cty Unvrsty 3/4
Wlson loops n 4 supr Yang-Mlls Wlson loop oprators n YM: [ ] ds A s s Φ y s y W C P p s In gnral, th stnc of th loop bras som of th symmtry of vacuum. s Howvr, f th loop has a partcular shap, t may prsrv som of thm. UY transformaton: δ ε A ΨΓ, δ Φ ΨΓε Γ Γ y δw C P ds Ψ ε δ A ε, AΦ ε ε Γ ε BP condton: Γ Γ ε Poncar UY y ΨΓ Γ ν δ Φ ΨΓ Γν suprconformal ν ν ε y A half of ε locally solvs th quaton. local BP condton. y θ θ 3 July 8, YITP worshop hoch Kawamoto Osaa Cty Unvrsty 4/4
uprsymmtrc Wlson loops Whn θ s constant, w fnd two half-bp solutons: traght ln: s s,,, 5 θ δ Crcular loop: s Rcoss, Rsns,, 5 θ δ Γ Γ5 ε Γ ε ε 5 RΓ5ε Zarmobo, Drur,, Ths loops prsrv L, R O3 O5, symmtry as wll. 3 July 8, YITP worshop hoch Kawamoto Osaa Cty Unvrsty 5/4
Crcular Wlson loops and matr modl ν j A s Φ θ s A s Φ θ s hghr W C ds ds ν j g s s s s O g dsds 4 4π s s O n : Planar Laddr appro. W C Prturbatv calculaton of th pctaton valu of a crcular Wlson loop. g 4 4π ds ds / 4 8 / 4 n! n M An 4 YM n n! Gaussan n 3 July 8, YITP worshop hoch Kawamoto Osaa Cty Unvrsty 6/4 / # of planar graphs A n s Rcos s, Rsn s,, A n can b calculatd by Gaussan matr modl: A M n M n n M Z dm M Ercson-mnoff-Zarmbo O M dm O Z
3 July 8, YITP worshop hoch Kawamoto Osaa Cty Unvrsty 7/4 Multply wound loops and gnvalu ff M d Z d Z Z < j j ff ln δ δ ρ If a crcular loop gos around th sam path tms Consdr th followng partton functon wth a sngl trac oprator π ρ d ln ρ ff 4 6 * κ κ Dfn th gnvalu dstrbuton functon O O ~ O O : sm-crcl O: ffctv acton for * Egnvalu dstrbuton.yamaguch
Rsolvnt To dtrmn th gnvalu dstrbuton wth havng ths loop oprator, consdr th pctaton valu of th rsolvnt: R z z M z M z M M By mployng th loop quaton D q., M z M M cyl z M M cyl z M By Laplac transformaton, M pz dp z M M p M o / corrcton solatd pol coms from a mrgd loop. ot: to ths ordr th rsolvnt s gvn by R z z * z z 3 July 8, YITP worshop hoch Kawamoto Osaa Cty Unvrsty 8/4
Wlson loops as D3-bran Fundamntal strngs may ntract on anothr A bttr dscrpton s gvn by D3-bran Gant Wlson loop D3 TD3 oluton for th crcular loop: Drur-Fol L ds d sn φ dt πα [ ] d cos d dρ snh ρdω L Ω g F snθ snh dρdθ 4 sn ρ T D3 cos P C 4 πα 4 sn F 4 L ρ 5 Ansatz ρ, A ρ snhρ coshρ cosθ cos sn cosh ρ snh ρ cosθ oluton of th quatons of moton: : numbr of strngs flu sn F κ snhρ 8π snh 4 3 July 8, YITP worshop hoch Kawamoto Osaa Cty Unvrsty 9/4 κ ρ L κ Inducd mtrc: ds 3 cos Ω d d D L κ d Ad sn κ sn ρ
Boundary trms ds to add th boundary trms B P A Π Drur-Gross-Oogur ar th boundary, th world-volum tnds nfntly., A P,Π to cancl dvrgnc fundamntal strng At th boundary, nd to f Π P,Π Drchlt b.c. : numbr of fundamntal strng Lgndr transformaton: δ B.o.m. δp A δπ ow th acton bcoms th functonal of P and Π B sol κ κ snh κ W crcl 4 YM sol B holds for larg Drur-Fol M 3 July 8, YITP worshop hoch Kawamoto Osaa Cty Unvrsty /4
Boundary condton for D3-Wlson loop.k.-kuro-mwa W hav addd th boundary trms, B ΦP A Π Φ Φ π α Y L At th boundary, w nd to mpos boundary condtons on P and Π. s,ys ar boundary, D3 world-volum nd to shrn down to b on dmnsonal. Factorzd part B W propos A s Φ y s ds Π boundary s PΦ boundary Rparamtrzaton nvarant Charg balanc condton of Callan- Maldacna strng y s 3 July 8, YITP worshop hoch Kawamoto Osaa Cty Unvrsty /4
Gaug thory rsolvnt from D3-bran.K.-Kuro-Mwa In th gaug thory sd, / corrcton an solatd pol appars as th contrbuton from a mrgd loop: R pol z z M M dp pz p M W would l to valuat th r.h.s by usng D3-bran. D3-bran wth p-flu pz p M pz DBI WZ B p dp dp classcal sol. B P A Π P PΦ π p d ΠA p Π d A Φ Φ Φ P P π Φ L π α y y 3 July 8, YITP worshop hoch Kawamoto Osaa Cty Unvrsty /4
3 July 8, YITP worshop hoch Kawamoto Osaa Cty Unvrsty 3/4 Gaug thory rsolvnt from D3-bran * * classcal sol. pol 3 z z A d z z R D B WZ DBI p P P Φ π Π Π A d p A d snh 4 κ ρ ρ ρ ρ O d F d d A d * Egnvalu dstrbuton It rproduc th corrct poston of th solatd pol 4 6 * κ κ Prform Laplac transformaton w.r.t. p
Concluson Th solatd gnvalu of th matr modl s dntfd wth th lctrc flu of th D3-bran soluton. W propos boundary condtons for D3-bran soluton. Anothr approach: Bubblng Wlson loop Yamaguch, Lunn, Ouda-ancanll, 4 Ad ovr a two-plan,y. y 4 shrns shrns Drur-Fol s soluton 3 July 8, YITP worshop hoch Kawamoto Osaa Cty Unvrsty 4/4
3 July 8, YITP worshop hoch Kawamoto Osaa Cty Unvrsty 5/4
3 July 8, YITP worshop hoch Kawamoto Osaa Cty Unvrsty 6/4
pm M pm M pm M cylndr dp pz pm M z M z M M dp pz pm M Z Φ Φ, X Φ3 Φ4 3 July 8, YITP worshop hoch Kawamoto Osaa Cty Unvrsty 7/4