Massive ABJM theory on three sphere! and large N phase transition

Transcription

1 Massive ABJM theory on three sphere! and large N phase transition Tomoi Nosaa (KIAS) Based on: [TN-Shimizu-Terashima, ] December 6, KIAS Worshop Current Topics in String Theory

2 Introduction gauge%theory%with%massive%ma0er%fields%can%have% non6trivial%phase%structure%in%large%n N =2 ex.)%%4d%%%%%%%%%%%%%%%u(n)%sym%on S 4 [Russo6arembo] = N =2 S 4 N =2 vectormulaplet%vev vector%mulaplet%+%adj.%ma0er%fields%with%mass%±m localizaaon = diag(a,a 2,,a N ) d e N t Tr m 2 -loop.%.%.% are%distributed%as a i a j. t (0,0) t infinitely%many%phase%transiaons%in% t%hood%coupling t =%vanishing%of%effecave%mass a i a j ± m %of%some%ma0er%components

3 Mass deformation in 3d ass%deformed%abjm%theory N =6 [Hosomichi6Lee%%6Par][Gomis6Rodriguez_Gomez6Van%Raamsdon6Verlinde] 3 =%3d%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%superconformal%CS6ma0er%theory U(N) U(N) +%mass%term%for%ma0er%mulaplets mabjm S 3 = d dee i(tr 2 Tre 2) ( ) t%hood%coupling: t = N Infinitely%many%phase%transiAon%in% t if 2 ir For%%%%%%%%%%%%%%,%no%phase%transiAon 2 R m [Anderson6Russo][Anderson6arembo] Q:%How%about%in%mass%parameter? (0,0) t

4 M-theory interpretation of massive ABJM ABJM%theory%=%N%M26branes mass%deformaaon%=%bacground%flux%normal%to%m2s M26branes%can%expand%in%extra%(fuzzy)%direcAons mabjm Expect%%%%%%%%%%%%%%%%%%%to%have%non6trivial%transiAon S 3 %in%dimensionless%mass%parameter% mr (mr) critical =0 (including%the%possibility%%%%%%%%%%%%%%%%%%%%%%%%%%%%or%%%%%%%) Myers%effect M2s M5 (r:%radius%of%%%%%) S 3 Indeed,%in%two%extremes%in%M6theory%limit%(%%%%%%%%%%%%%%,%:%fixed), (mr = 0) = ABJM = e ( p 2/3)N 3 2 [Druer6Marino6Putrov] log? This%tal N! mr (mr = ) Y (r 2 + m (2) #massive fields ) e mrn 2

5 ABJM theory (in 3d N =2 notation) ABJM%theory%=%2%vector%mulAplets%+%2%bifund%hypermulAplets [Aharony6Bergman6Jafferis6Maldacena][Hosomichi6Lee%%6Par] 3 ev =( e A µ, e, e, e D) < (X I, I,F I ) V =(A µ,,,d) < (Y I, 0 I,F 0 I) I =, 2 (SU(2) SU(2) SO(6) R ) AcAon%on%S 3 (radius=r): S = S CS (V ) S CS ( V e )+ (ma0er%ineac%terms) +S int " i Tr AdA 2 4 S 3 A3 + 3 S 3 p gtr(2d ) # apple p gtr X I X I e 2 + i(x I DX I X IDX e I ) S 3 + Y I Y I e 2 + i(y I e DY I Y I DY I ) + 3 4r 2 ( X I 2 + Y I 2 )

6 Supersymmetric mass term mass%of (X I,Y I ) are%induced%by%fi%term i 2 S 3 p gtr D r + e D e r! S = S CS (V ) S CS ( e V )+ (ma0er%ineac%terms) " i Tr AdA 2 4 S 3 A3 + 3 S 3 p gtr(2d ) # +S int + S FI apple p gtr X I X I e 2 + i(x I DX I X IDX e I ) S 3 + Y I Y I e 2 + i(y I e DY I Y I DY I ) + 3 4r 2 ( X I 2 + Y I 2 ) indeed DD D ed 2 + X IX I Y I Y I + 2 =0 2 e X I X I + Y I Y I + 2 =0 = mx I +[X, X; X]+[X, Y ; Y ] 2 = my I +[Y,Y ; Y ]+[Y,X; X] 2 with%mass m =2/ [Gomis6Rodriguez_Gomez6Van%Raamsdon6Verlinde]

7 We%want%to%evaluate%this%2N%integrals%for%%%%%%%%%%%%%%%%%,%%%%%%%%%:%finite N!, Partition function by Localization mabjm = DFDYe S mabjm e d(y,dy) δ(susy)6exact% damping%factor [Nerasov][Pestun] path%integral%=%6loop%correcaons%around% δ(ψ,χ)=0 %configs 0 = D = 2 0 e... C A, e = D e e = 2 N... e N C A mabjm = (N!) 2 d N e S CS d N e e i P i ( 2 i e2 i )+2 i P i ( i+ e i) Q i<j (2 sinh ( i j)) 2 Q i<j (2 sinh (e i Q i,j (2 cosh ( e i j )) 2 e S FI e j )) 2 [KapusAn6Wille06Yaaov][Hama6Hosomichi6Lee][Jafferis] (vector) -loop (hyper) -loop

8 Saddle point approximation Regard%eigenvalue%distribuAon%as%funcAon%on%interval%(6/2,/2): i! (s), e i! e (s), mabjm! NX i=! N 2 2 ds DlD e le f [l,e l] with%d%acaon f = in ds( 2 e2 ) 2 in ds( + e ) N s<s 2 dsds 0 log[(2 sinh ( (s) (s 0 ))) 2 ] 0 N 2 dsds 0 log[(2 sinh ( e (s) e (s 0 ))) 2 ]+N 2 dsds 0 log[(2 cosh ( (s) e (s 0 ))) 2 ] s<s 0 Since N! is%classical%limit, mabjm e f( saddle, e saddle)

9 Large limit Ansatz: (s) = + in + u(s), e (s) = + in + v(s) u, v : O(N 0 ) log[(2 cosh ( (s) e (s 0 ))) 2 ]= 4 +2 (u(s) v(s0 )) + O(e 4 ) cancel%with%fi f = 4 N 2 + in in dsu 2 N 2 s<s 0 dsds 0 log[(2 sinh (u(s) u(s 0 ))) 2 ] dsv 2 N 2 s<s 0 dsds 0 log[(2 sinh (v(s) v(s 0 ))) 2 ] Du Dv mabjm e 4 N2 pure CS (, N) pure CS (, N) 6loop%effect%of%massive%chiral%mulAplets

10 Large limit Indeed,%saddle%point%equaAons%for%pure%CS%matrix%model 0= f u = iu N 0= f v = iv N s 0 6=s s 0 6=s ds 0 cot[ (u(s) u(s 0 )] ds 0 cot[ (v(s) v(s 0 )] are%solved%with u(s),v(s) =is + O(N ) Hence%restricAon u, v z / is%jusafied%in%saddle%point%approximaaon Also, f 4pzN 2 = 2N logn 4pzN2 log mabjm 4 N 2 for z!

11 Small : log mabjm N 3/2 regime For%saddle%configuraAon,%large%N%scaling%of%each%term%will%balance Ansatz: = N z (s)+z 2 (s), e = N z (s) z 2 (s) ( > 0) N +2 N + N + f = in ds( 2 e2 ) 2 in ds( + e ) N s<s 2 dsds 0 log[(2 sinh ( (s) (s 0 ))) 2 ] 0 N 2 dsds 0 log[(2 sinh ( e (s) e (s 0 ))) 2 ]+N 2 dsds 0 log[(2 cosh ( (s) e (s 0 ))) 2 ] s<s 0 N 2+ N 2 N 2 :%vanishing%trivially = 2 [Herzog6Klebanov6Pufu6Tesileanu](ζ=0)

12 Small : log mabjm N 3/2 regime f =4 N 3 2 h dz ds iz z 2 iz z2 2 ds i + O(N) saddle%point%equaaon%for (z (s),z 2 (s)) %=%EoM%+%boundary%condiAon 2nd%order%diff.%eq. two%constraints%from%s=±/2 unique%soluaon: z (s)= i h r p 2 4m 2 + 2is m 2m 2m i z 2 (s)= i 8 dz 2 ds m = 2z where f = pp 2( + 4m 2 ) 3 N 3 2 [Jafferis6Klebanov6Pufu6Safdi](ζ%6%iR) [TN6Shimizu6Terashima]

13 Critical value: z = /4 z (s)= i h r p 2 4m 2 + 2is m 2m 2m i m = 2z s = 2 s =0 < S=60 S=+0 r 4m 2 + 2is m If z > /4,%soluAon%is%disconAnuous%at%s=0 addiaonal%boundary%constraint%at%s=0,%which%is%not%saasfied s = 2 SoluAon%with%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%exists%only%for log N 3/2 z < /4 log p p z 2 2 N 3 2? 4pzN 2 z 4

14 What s happening in >/4? The%soluAon%of%%%%%%%%%%%%%%%%%%%%%%%%%%%%exists%even%for%finite f (s) f =4 N 2 / =2 in 2 in 2 N 2 s 0 6=s ds 0 coth ( (s) e (s 0 )) + 2 N 2 ds 0 tanh ( (s) e (s 0 )) (s) = N i, + + s e N (s) = + + s i h 2 i ds 0 tanh + i(s s0 ) = for%any > 0 suggests%that%the%coefficient%%%%%%%%%%%%%%%%is%exact%for%finite (4 /) log p p z 2 2 N 3 2 = N 2?? 4pzN 2 z? 4 However,%since%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%is%finite%and%conAnuous%in%%%%%, mabjm (N<) coefficient%of%%%%%%%%%%%must%vanish%when%exponent%jumps N 2 N 3 2! N 2

15 Conclusion log p p z 2 2 N 3 2 unsolved 4pzN 2 z 4 %%(?6th%order)%phase%transiAon%will%occur%to%change%exponent%3/2 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%phase%never%connect%directly%to%%%%%%%%%%%%%%%%%%%%%phase f =4 N 2 / f N 3/2 phase%transiaon%will%occur%at#least#twice%in /4 < apple More%to%be%solved%in%3d%mass%deformaAon

16 %%GeneralizaAon%of%ansatz%is%required%in%SUGRA? Future Problems %How%to%study%%%%%%%%%%%%%%%%%%%%%in%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%? /4 < < %%Transform mabjm S 3 %%saddle%point%eqs.%cannot%be%solved%even%numerically mabjm S 3 = d N d N e e i( 2 e2) ( ) (N. 80) into%some%well6behaved%integrals%(s6dual,%fermi%gas%formalism)%? 2%Gravity%dual%for ABJM%on >/4? S 3 4d%AdS%SUGRA%with%AdS%boundary%(z=0)%=% S 3 mass%deformaaon boundary%value%of%dual%fields%at%z=0 f = p N 3 for%any? 2 [Freedman6Pufu]