Orientifold ABJM Matrix Model: ! Chiral Projections and Worldsheet - PowerPoint PPT Presentation

Orientifold ABJM Matrix Model: ! Chiral Projections and Worldsheet Instantons Tomoki Nosaka (KIAS) Based on: [Moriyama-TN, 1603.00615] August 8, YITP Workshop “Strings and Fields 2016”

Introduction Par,,on#func,on#of#ABJ(M)#theory#is#corrected#by#nonDperturba,ve# 7 effect#in#1/ N ,#which#correspond#to#closed#M2s#winding#on#S##/Z k =instantons S 7 S 7 3 M2#on#RP 3 M2#on#S##/Z AdS 4 k AdS 4 An#mysterious#rela,on#between#instanton#effects#and# 1#######1 refined&topological&string #on#local#P###x#P####was#found: [HatsudaDMarinoDMoriyamaDOkuyama][HondaDOkuyama] Z i ∞ In#grand#poten,al# J ( μ ), dµ 2 π ie J − µN Z U ( N ) k × U ( N + M ) − k = − i ∞ J ( µ ) = C 3 µ 3 e ff + Bµ e ff + A + O ( e − µ eff ) J np ( µ e ff ) − 2 sin π ns L g s 2 sin π ns R ∞ ∞ ( − 1) nd g (2 sin π g s ) 2 g − 2 e − n d · T eff + ∂ n d · T eff g s X X X n d N d = e − g s gs j L ,j R 2 π n 2 (2 sin π n g s ) 3 ∂ g s n n, d g =0 n, d ,j L ,j R g s = 2 T e ff = 4 µ e ff ⇣ 1 2 − M with :#Kahler#parameters ⌘ ± 2 π i k k k

Does relation exist for more general backgrounds? N ##D3s 1 U( N%% )###xU( N%% ) (ABJ(M)): 1## k ############2#D k AdS 4 × S 7 / Z k N ##D3s [AharonyDBergmanDJafferisDMaldacena]# (1, k )5 NS5 2 [AharonyDBergmanDJafferis]# [HosomichiDLeeDLeeDLeeDPark] O3#plane · · · AdS 4 × S 7 / ( Z q , Z p , Z k ) AdS 4 × S 7 / b D k [ImamuraDKimura][TerashimaDYagi] [ABJ,HLLLP] new&35cycles&=&new&instantons J np ( µ e ff ) − 2 sin π ns L g s 2 sin π ns R ∞ ∞ ( − 1) nd g (2 sin π g s ) 2 g − 2 e − n d · T eff + ∂ n d · T eff g s X X X n d N d = e − g s gs j L ,j R 2 π n 2 (2 sin π n g s ) 3 ∂ g s n g =0 n, d n, d ,j L ,j R Q:#How#can#we# extend&this&structure #for#general#theories?

To do: exactly solve “next-simplest model” Par,,on#func,on#of#these#theories#allow#Fermi#gas#formalism J ( µ ) ∼ Tr log(1 + e µ b ρ ) ⇣ ⌘ 1 1 :#density#operator#of#some#1d#QM ρ = e − b H b b ρ ABJM = b b Q P 2 cosh 2 cosh 2 2 powerful,#but#not#enough#for#finite# k q D1################################### p D1 U( N%% )###xU( N%% )###xU( N%%%%% )####xU( N%% ) 1## k% ############i##0###########q+1#D k #############j##0 1 1 ○## N ##= N ##=…= N→ :#natural#generalisa,on#of# b 1######2 ρ = b ρ ABJM b b Q P 2 ) p 2 ) q (2 cosh (2 cosh ○##Solved#only#for#(i)# q = p =2,# N ##= N #and#(ii)#orbifold#ABJ(M) a [HondaDMoriyama] [MoriyamaDTN]# [HatsudaDHondaDOkuyama] N N N 2 N 1 N 1 N 2 N 2 N 1 N N ○##General#rank##{ N ##}##is#difficult… a

To do: exactly solve “next-simplest model” Par,,on#func,on#of#these#theories#allow#Fermi#gas#formalism J ( µ ) ∼ Tr log(1 + e µ b ρ ) ⇣ ⌘ 1 1 :#density#operator#of#some#1d#QM ρ = e − b H b b ρ ABJM = b b Q P 2 cosh 2 cosh 2 2 powerful,#but#not#enough#for#finite# k This#talk:####O( N%% )###xUSp(2 N%% ) 1## k ###################2##D k /2 ○##Next#largest#SUSY O3#plane ( N = 5) ○ looks#different#from … b b ρ ρ ABJM (due#to#different in#localiza,on#computa,on) Z 1-loop

Recent developments in model O × USp ○ ρ OSp can#be#rewriken#as# chiral&projec9on #of b b ρ ABJM ρ U × U · 1 ± b R ( b R : | Q i ! | � Q i ) ρ O × USp = b b 2 [Honda,1512][Okuyama,1601][MoriyamaDSuyama,1601][MoriyamaDTN,1603] ○##Instantons O ( e − 4 µ/k ) , O ( e − 2 µ ) are# generated #from#those#in#ABJM# [MoriyamaDTN,1603] ○##Instantons#on#new#cycle O ( e − µ ) were#determined#for k ∈ N [Okuyama,1601][MoriyamaDSuyama,1601] Instantons#in#O( N%% )###xUSp(2 N%% )########were#completely#determined#! 1## k ####################2##D k /2

～ ～ ～ ～ ～ O × USp theories (NSNS,RR) D3#charge G (0,0) O(2 N ) D1/4 O3D O(2 N +1) (0,1/2) O3D (1/2,0) +1/4 O3+ USp(2 N ) (1/2,1/2) (O3D#=#1/2#D3#on#O3D) O3+ (1, k )/2#5 O3D 1/2#NS5 (1, k )/2#5 1/2#NS5 O3D O3+ O3+ O(2 N )###xUSp(2 N +2 M ) O(2 N +1)###xUSp(2 N +2 M ) k ###############################D k /2 k%%%%%%%%%% #####################D k /2 O(2 N +2 M )###xUSp(2 N ) O(2 N +2 M +1)###xUSp(2 N ) k #####################D k /2 k%%%%%%%%%% ###########D k /2 AdS 4 × S 7 / b D k with 2 π i 2 π i 2 π i 2 π i b Z k : ( z 1 , z 2 , z 3 , z 4 ) → ( e k z 4 ) D k = ( Z k , r ) k z 1 , e k z 2 , e k z 3 , e r : ( z 1 , z 2 , z 3 , z 4 ) → ( iz ∗ 2 , − iz ∗ 1 , iz ∗ 4 , − iz ∗ 3 )

J ( µ ) Large μ contributions to for O × USp Z i ∞ dµ 2 π ie J ( µ ) − µN | Z ( N ) | = − i ∞ ○##Large# N #expansion#of#Z( N ) Large######expansion#of J ( µ ) µ µ 3 √ ○ kN ∼ R 3 J ( µ ) = 3 π 2 k + · · · + O ( e − µ ) ( µ ∼ AdS ) ⊂ S 7 / b :#closed#M2s#winding#on#3Dcycle D k (instanton) √ kN 3 / 2 log Z ∼ − 2 π α ` ( µ ) e − 4 ` µ S 3 / Z k ⊂ S 7 / b 3 D k k (leading#in#4d#SUGRA) β ` ( µ ) e − 2 ` µ RP 3 ⊂ S 7 / b D k γ ` ( µ ) e − ` µ RP 3 0 /r ⊂ S 7 / b D k r ∈ b D k : ( z 1 , z 2 , z 3 , z 4 ) → ( iz ∗ 2 , − iz ∗ 1 , iz ∗ 4 , − iz ∗ 3 ) Goal:#to#determine#perturba,ve#part#and#instanton#coefficients

＋ ー ー ー b R : | Q i ! | � Q i Hint 1: O3 = chiral projection for ○##Recently#we#found#simple#rela,on#between b ρ ρ U × U · 1 ± b R ρ O × USp = b b 2 O(2N+1)xUSp(2N+2M) U(N)xU(N+2M) [Honda][Moriyama-Suyama] O(2N+2M+1)xUSp(2N) U(N+2M)xU(N) O(2N)xUSp(2N+2M) U(N)xU(N+2M+1) [Moriyama-TN] O(2N+2M)xUSp(2N) U(N+2MD1)xU(N) ○##Consistent#with#HananyDWiken’s# s Drules#and#duality � � ± and#denote J ± ( µ ) = Σ ( µ ) ± ∆ ( µ ) ○##Let#us#consider# b ρ U ( N ) × U ( N + M ) 2

Hint 2: Total vs Modified grand potential Z π i dµ J ( µ ) = e e X J ( µ ) − µN e µN | Z ( N ) | | Z ( N ) | = e 2 π ie − π i N ≥ 0 ○##Total#grand#poten,al###########is#given#by e J ( µ ) = det(1 + e µ b e J ( µ ) ρ ) e ○##For#large# N #expansion#it#is#more#convenient#to#rewrite Z i ∞ dµ 2 π ie J ( µ ) − µN | Z ( N ) | = − i ∞ ○##Modified#grand#poten,al###########differs#from##########by#“oscilla,ons” e J ( µ ) J ( µ ) h e J ( µ +2 π in ) � J ( µ ) i X e e J ( µ ) = X e J ( µ +2 π in ) J ( µ ) = J ( µ ) + log 1 + e n 6 =0 n ∈ Z

Perturbative & O ( e − µ ) in Σ ( µ ) det(1 + e µ b ρ + ) det(1 + e µ b ρ − ) = det(1 + e µ b ρ ) = = = e e e J + ( µ ) J U × U ( µ ) J − ( µ ) e e e In#modified#grand#poten,al, J + + J − = J U × U + ( oscillations ) J = � µ 3 + · · · k h e J ( µ +2 π i ) � J ( µ ) i X = O ( e − µ/k ) J osc = log 1 + n 6 =0 Σ pert ( µ ) = J pert = C 3 µ 3 e ff + Bµ e ff + A − 2 sin π ns L g s 2 sin π ns R ∂ n d · T eff g s X N d Σ MB ( µ ) = J MB e − U × U = g s gs j L ,j R 2 π n 2 (2 sin π n g s ) 3 ∂ g s n, d ,j L ,j R

∆ ( µ ) = perturbative + half instantons e ○##No#direct#rela,on#to#unprojected#grand#poten,al J U × U ( µ ) , J U × U ( µ ) ○##S,ll#we#can#compute#small#k#expansion# #####and#exact#values#of#leading#instanton#coefficients#for#a#few#k’s Whole#structure#for#general#( k , M )#was#guessed#as 1 ∆ ( µ ) = µ 2 + A 0 + X γ ` e � ` µ ` =1 ex.# M =0: √ 2 e − µ + 4 e − 2 µ 8 1 4 log 1 + 2 ( k ≡ 1 , 7 mod 8) > √ > 2 e − µ + 4 e − 2 µ > 1 − 2 > > √ 2 e − µ + 4 e − 2 µ > − 1 4 log 1 + 2 > > > ( k ≡ 3 , 5 mod 8) γ ` e − ` µ = < √ X 2 e − µ + 4 e − 2 µ 1 − 2 [Moriyama-Suyama] 1 [Okuyama] > ` 4 log(1 + 16 e − 2 µ ) ( k ≡ 2 , 6 mod 8) > > > > > 1 > > 2 log(1 + 4 e − µ ) ( k ≡ 0 mod 8) > :

O ( e − µ/k ) Σ ( µ ) Worldsheet instantons in X X e J + ( µ +2 π in + ) X e J U × U ( µ +2 π in ) = e J − ( µ +2 π in − ) n n + n − J ± ( µ ) = Σ ( µ ) ± ∆ ( µ ) ⇣ ⌘ 2 ○##Special#simplifica,on#for e ∆ ( µ +2 π in + ) − ∆ ( µ +2 π in − ) = i n + − n − n − m n X X = +(( n + − n − ): odd) n ± = n ± m n + ,n − m,n no#contribu,on n + e J U × U ( µ ) = Σ ( µ +2 π im ) + Σ ( µ − 2 π im ) X − Σ ( µ ) ( − 1) m e 2 2 m [Moriyama-TN] :#Direct#rela,on#between#worldsheet#instanton#coefficients

New “Gopakumar-Vafa invariants” " i# ∞ h Σ ( µ + 2 π im ) + Σ ( µ − 2 π im ) ( − 1) m exp X J U ( N ) × U ( N ) ( µ ) = Σ ( µ ) + log 1 + 2 − Σ ( µ ) 2 m =1 ∞ ∞ ( − 1) nd 2 sin 2 π ⌘ 2 g − 2 ⇣ e − 4 n d · µ X X J WS n d U ( N ) × U ( N ) ( µ ) = k g n k g =0 n, d 1 1 ( − 1) nd 2 sin 2 π ⌘ 2 g � 2 ⇣ e � 4 n d · µ X X Σ WS ( µ ) = 2 n 0 d k g n k g =0 n, d (0,1) (1,1) (1,2) (1,3) (2,2) (1,4) (2,3) d (0,1) (1,1) (1,2) (1,3) (2,2) (1,4) (2,3) d n 0 d − 1 − 2 − 3 − 4 − 16 − 5 − 55 n d − 2 − 4 − 6 − 8 − 32 − 10 − 110 0 0 n 0 d 0 1 4 10 53 20 318 n d 0 0 0 0 9 0 68 1 1 n 0 d 0 0 − 1 − 6 − 64 − 21 − 757 n d 0 0 0 0 0 0 − 12 2 2 n 0 d 0 0 0 1 37 8 1002 n d 0 0 0 0 0 0 0 3 3 n 0 d 0 0 0 0 − 10 − 1 − 792 4 n 0 d 0 0 0 0 1 0 378 1######1 5 :#GV#invariants#for#local#P##x#P n 0 d 0 0 0 0 0 0 − 106 6 n 0 d 0 0 0 0 0 0 16 7 n 0 d 0 0 0 0 0 0 − 1 8 polynomial#(nonDlinear)#of# n d g :#? n 0 d g =