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”

In#grand#poten,al#J(μ), An#mysterious#rela,on#between#instanton#effects#and# refined&topological&string#on#local#P###x#P####was#found:

1#######1

J(µ) = C 3 µ3

eff + Bµeff + A + O(e−µeff)

with Par,,on#func,on#of#ABJ(M)#theory#is#corrected#by#nonDperturba,ve# effect#in#1/N,#which#correspond#to#closed#M2s#winding#on#S##/Z

7 k

=instantons

AdS4 S7 M2#on#S##/Z

3 k

AdS4 S7 M2#on#RP3

Introduction

[HatsudaDMarinoDMoriyamaDOkuyama][HondaDOkuyama]

gs = 2 k

Jnp(µeff) =

∞

X

n,d ∞

X

g=0

(−1)nd n nd

g (2 sin πgs)2g−2e−nd·Teff + ∂

∂gs gs X

n,d,jL,jR

N d

jL,jR

−2 sin πnsL

gs 2 sin πnsR gs

2πn2(2 sin πn

gs )3

e−

nd·Teff gs

ZU(N)k×U(N+M)−k = Z i∞

−i∞

dµ 2πieJ−µN

:#Kahler#parameters

Teff = 4µeff k ± 2πi ⇣1 2 − M k ⌘

Does relation exist for more general backgrounds?

Q:#How#can#we#extend&this&structure#for#general#theories?

· · ·

N##D3s NS5 (1,k)5 AdS4 × S7/Zk

U(N%%)###xU(N%%)

AdS4 × S7/(Zq, Zp, Zk)

O3#plane

AdS4 × S7/ b Dk

(ABJ(M)): new&35cycles&=&new&instantons

1##k############2#Dk

1

N##D3s

2 [ImamuraDKimura][TerashimaDYagi] [ABJ,HLLLP] [AharonyDBergmanDJafferisDMaldacena]# [AharonyDBergmanDJafferis]# [HosomichiDLeeDLeeDLeeDPark]

Jnp(µeff) =

∞

X

n,d ∞

X

g=0

(−1)nd n nd

g (2 sin πgs)2g−2e−nd·Teff + ∂

∂gs gs X

n,d,jL,jR

N d

jL,jR

−2 sin πnsL

gs 2 sin πnsR gs

2πn2(2 sin πn

gs )3

e−

nd·Teff gs

Par,,on#func,on#of#these#theories#allow#Fermi#gas#formalism :#density#operator#of#some#1d#QM

To do: exactly solve “next-simplest model”

:#natural#generalisa,on#of# ○##Solved#only#for#(i)#q=p=2,#N##=N#and#(ii)#orbifold#ABJ(M)

J(µ) ∼ Tr log(1 + eµb ρ)

b ρ = e− b

H

b ρ = 1 (2 cosh

b Q 2 )q

1 (2 cosh

b P 2 )p

b ρABJM ⇣ b ρABJM = 1 2 cosh

b Q 2

1 2 cosh

b P 2

⌘

[MoriyamaDTN]# [HatsudaDHondaDOkuyama]

U(N%%)###xU(N%%)###xU(N%%%%%)####xU(N%%)

1##k%############i##0###########q+1#Dk#############j##0 qD1###################################pD1

○##N##=N##=…=N→

1######2

a

N 1 N 2 N 1 N 2 N 1 N 2 N N N N [HondaDMoriyama]

powerful,#but#not#enough#for#finite#k ○##General#rank##{N##}##is#difficult…

a

○##Next#largest#SUSY looks#different#from This#talk:####O(N%%)###xUSp(2N%%)

1##k###################2##Dk/2

in#localiza,on#computa,on) (due#to#different

(N = 5) b ρ Z1-loop b ρABJM

○ …

To do: exactly solve “next-simplest model”

Par,,on#func,on#of#these#theories#allow#Fermi#gas#formalism :#density#operator#of#some#1d#QM

J(µ) ∼ Tr log(1 + eµb ρ)

b ρ = e− b

H

⇣ b ρABJM = 1 2 cosh

b Q 2

1 2 cosh

b P 2

⌘

powerful,#but#not#enough#for#finite#k O3#plane

○##Instantons

b ρO×USp = b ρU×U · 1 ± b R 2

b ρOSp can#be#rewriken#as#chiral&projec9on#of

( b R : |Qi ! | Qi)

[Honda,1512][Okuyama,1601][MoriyamaDSuyama,1601][MoriyamaDTN,1603]

O(e−2µ) O(e−4µ/k) ,

are#generated#from#those#in#ABJM# ○##Instantons#on#new#cycle O(e−µ) were#determined#for k ∈ N ○

[MoriyamaDTN,1603] [Okuyama,1601][MoriyamaDSuyama,1601]

Instantons#in#O(N%%)###xUSp(2N%%)########were#completely#determined#!

Recent developments in model

1##k####################2##Dk/2

O × USp b ρABJM

theories

O × USp

r : (z1, z2, z3, z4) → (iz∗

2, −iz∗ 1, iz∗ 4, −iz∗ 3)

Zk : (z1, z2, z3, z4) → (e

2πi k z1, e 2πi k z2, e 2πi k z3, e 2πi k z4)

b Dk = (Zk, r)

O3D 1/2#NS5 (1,k)/2#5 O3+

O(2N)###xUSp(2N+2M)

k###############################Dk/2

O(2N+1)###xUSp(2N+2M)

k%%%%%%%%%%#####################Dk/2 1/2#NS5 (1,k)/2#5 O3D ～ O3+ ～

O(2N+2M)###xUSp(2N)

k#####################Dk/2

O(2N+2M+1)###xUSp(2N)

k%%%%%%%%%%###########Dk/2

AdS4 × S7/ b Dk

with

O3+ O3D O3D ～ O3+ ～ D3#charge (NSNS,RR) G (0,0) (1/2,0) (0,1/2) (1/2,1/2) O(2N) O(2N+1) USp(2N) D1/4 +1/4 (O3D#=#1/2#D3#on#O3D) ～

Goal:#to#determine#perturba,ve#part#and#instanton#coefficients (leading#in#4d#SUGRA)

|Z(N)| = Z i∞

−i∞

dµ 2πieJ(µ)−µN

for

J(µ) O × USp

:#closed#M2s#winding#on#3Dcycle

J(µ) = µ3 3π2k + · · · + O(e−µ) log Z ∼ −2π √ kN 3/2 3 α`(µ)e− 4`µ

k

β`(µ)e−2`µ γ`(µ)e−`µ

S3/Zk ⊂ S7/ b Dk RP3 ⊂ S7/ b Dk RP30/r ⊂ S7/ b Dk

r ∈ b Dk : (z1, z2, z3, z4) → (iz∗

2, −iz∗ 1, iz∗ 4, −iz∗ 3)

(instanton) ○##Large#N#expansion#of#Z(N)

J(µ)

µ

○

⊂ S7/ b Dk

Large######expansion#of

(µ ∼ √ kN ∼ R3

AdS)

Large μ contributions to

Hint 1: O3 = chiral projection for

b ρO×USp = b ρU×U · 1 ± b R 2

○##Consistent#with#HananyDWiken’s#sDrules#and#duality ○##Recently#we#found#simple#rela,on#between b

ρ

U(N)xU(N+2M) O(2N+1)xUSp(2N+2M) O(2N)xUSp(2N+2M) O(2N+2M)xUSp(2N) U(N)xU(N+2M+1) U(N+2MD1)xU(N)

ー ＋ ー

U(N+2M)xU(N) O(2N+2M+1)xUSp(2N)

ー

[Honda][Moriyama-Suyama] [Moriyama-TN]

b R : |Qi ! | Qi

○##Let#us#consider#

• b

ρU(N)×U(N+M)

• ± and#denote J±(µ) = Σ(µ) ± ∆(µ)

2

○##For#large#N#expansion#it#is#more#convenient#to#rewrite ○##Modified#grand#poten,al###########differs#from##########by#“oscilla,ons”

Hint 2: Total vs Modified grand potential

|Z(N)| = Z πi

−πi

dµ 2πie

e J(µ)−µN

|Z(N)| = Z i∞

−i∞

dµ 2πieJ(µ)−µN

e J(µ) J(µ)

e

e J(µ) =

X

n∈Z

eJ(µ+2πin) e

e J(µ) =

X

N≥0

eµN|Z(N)| e

e J(µ) = det(1 + eµb

ρ)

○##Total#grand#poten,al###########is#given#by

e J(µ)

e J(µ) = J(µ) + log h 1 + X

n6=0

eJ(µ+2πin)J(µ)i

J+ + J− = JU×U + (oscillations)

Josc = log h 1 + X

n6=0

eJ(µ+2πi)J(µ)i J = µ3 k + · · · = O(e−µ/k)

Perturbative & O(e−µ) in Σ(µ)

det(1 + eµb ρ+) det(1 + eµb ρ−) = det(1 + eµb ρ) e

e J+(µ)

e

e J−(µ)

e

e JU×U(µ)

In#modified#grand#poten,al, = = =

ΣMB(µ) = JMB

U×U =

∂ ∂gs gs X

n,d,jL,jR

N d

jL,jR

−2 sin πnsL

gs 2 sin πnsR gs

2πn2(2 sin πn

gs )3

e−

nd·Teff gs

Σpert(µ) = Jpert = C 3 µ3

eff + Bµeff + A

○##No#direct#rela,on#to#unprojected#grand#poten,al ex.#M=0:

[Moriyama-Suyama] [Okuyama]

∆(µ)

e JU×U(µ), JU×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

∆(µ) = µ 2 + A0 +

1

X

`=1

γ`e`µ

X

`

γ`e−`µ = 8 > > > > > > > > > < > > > > > > > > > : 1 4 log 1 + 2 √ 2e−µ + 4e−2µ 1 − 2 √ 2e−µ + 4e−2µ (k ≡ 1, 7 mod 8) −1 4 log 1 + 2 √ 2e−µ + 4e−2µ 1 − 2 √ 2e−µ + 4e−2µ (k ≡ 3, 5 mod 8) 1 4 log(1 + 16e−2µ) (k ≡ 2, 6 mod 8) 1 2 log(1 + 4e−µ) (k ≡ 0 mod 8)

= perturbative + half instantons

in

O(e−µ/k) Σ(µ)

n− n+ m n

n± = n ± m

X

n

eJU×U(µ+2πin) = X

n+

eJ+(µ+2πin+) X

n−

eJ−(µ+2πin−)

X

n+,n−

= X

m,n

+((n+ − n−): odd)

e∆(µ+2πin+)−∆(µ+2πin−) = in+−n−

eJU×U(µ) = X

m

(−1)me

Σ(µ+2πim) 2

+ Σ(µ−2πim)

2

−Σ(µ)

○##Special#simplifica,on#for :#Direct#rela,on#between#worldsheet#instanton#coefficients no#contribu,on

Worldsheet instantons

[Moriyama-TN]

⇣ J±(µ) = Σ(µ) ± ∆(µ) 2 ⌘

New “Gopakumar-Vafa invariants”

:#GV#invariants#for#local#P##x#P

1######1

polynomial#(nonDlinear)#of# nd

g :#?

n0d

g =

ΣWS(µ) = 2

1

X

n,d 1

X

g=0

(−1)nd n n0d

g

⇣ 2 sin 2π k ⌘2g2 e 4nd·µ

k

JWS

U(N)×U(N)(µ) = ∞

X

n,d ∞

X

g=0

(−1)nd n nd

g

⇣ 2 sin 2π k ⌘2g−2 e− 4nd·µ

k

d (0,1) (1,1) (1,2) (1,3) (2,2) (1,4) (2,3) n0d −1 −2 −3 −4 −16 −5 −55 n0d

1

1 4 10 53 20 318 n0d

2

−1 −6 −64 −21 −757 n0d

3

1 37 8 1002 n0d

4

−10 −1 −792 n0d

5

1 378 n0d

6

−106 n0d

7

16 n0d

8

−1 d (0,1) (1,1) (1,2) (1,3) (2,2) (1,4) (2,3) nd −2 −4 −6 −8 −32 −10 −110 nd

1

9 68 nd

2

−12 nd

3

JU(N)×U(N)(µ) = Σ(µ) + log " 1 + 2

∞

X

m=1

(−1)m exp hΣ(µ + 2πim) + Σ(µ − 2πim) 2 − Σ(µ) i#

=#GV#invariants#of#new#CY###?

n0d

g

Can#we#interpret#OSp#results#as#topological#string#quan,,es? ○##How#to#unify#ws#&#mb#with#different&topological&invariants?# does#ambigui,es#of ○ ○##Should#half#instantons#be#unified#with#ws#&#mb?

3

N d

jL,jR

○##not#determined#as#func,on#of#g#～#1/k#yet

s

Toward whole structure of O x USp

○##Interpreta,on#for#“reshuffling”? helps? Ini,al#Ques,on:#Extension#of#HMMO#formula

This#induces#orien,fold#ac,on#on#instantons#in#unprojected#theory

Summary and future work

We#found#“orien,fold#ac,on”#on#Fermi#gas#system:

b ρO×USp = b ρU×U · 1 ± b R 2

transforms#among#themselves ○##worldsheet#instantons ○##membrane#instantons O(e−2µ)

O(e−4µ/k)

kept#unchanged#(×1/2#in#J%%%) ○##In#ABJM#case nonDlinearly#reshuffles#GV#invariants ± O3Dplane#nicely#acts,#does#not#to#spoil#the#beauty#of#instantons (c.f.#repe,,on#of#quiver [Honda-Moriyama] :#evidence#for#generalisa,on? Can#we#extend#the#class#of#such#opera,ons? )