Instanton Effects in ABJ(M) Theory

Instanton Effects in ABJ(M) Theory Sanefumi Moriyama (NagoyaU KMI) [Fuji+Hirano+M 1106] [Hatsuda+M+Okuyama 1207, 1211, 1301] [HMO+Marino 1306] [HMO+Honda 1306] [Matsumoto+M 1310]

0. Introduction

Physics "Special" Background Highly Symmetric, Simple OR EVEN Solvable "General" Backgrounds By Perturbation Theory or Other Methods (Example) e βh with H=p 2 /2+q 2 /2+g q 4 /4

Perturbative String Theory Perturbative String Theories on 10D Flat Spacetime Perturbation Theory Amplitudes (Small Fluctuation) β Functions (Background EOM)...

Non Perturbatively Situation Changes Drastically! In Strong Coupling Limit, M Theory!! Most Symmetric Cases M Theory on 11D Flat Spacetime M Theory on AdS 7 4 x S M Theory on AdS 7 x S 4 Still Very Mysterious = Unsatisfied

M2 branes [Aharony Bergman Jafferis Maldacena] N=6 N6 Super Chern Simons Theory = M2 on C 4 /Z k [Witten, Pestun,...] Supersymmetric Observables : Matrix Integral Results: ABJM Matrix Model with Interesting Structure Extract As Much Physics As Possible

Contents 1. Quick Review of M Theory What did we learn two decades ago? 2. ABJ(M) Wilson Loop Fractional Branes 3. Exact Instanton Expansion String Democracy

1. Quick Review of M Theory

M is for Mother 5 Consistent tstring Theories in 10D Het E8xE8 IIA Het SO(32) IIB I

M is for Mother 5 Consistent tstring Theories in 10D 5 Vacua of A Unique String Theory Het E8xE8 IIA Het SO(32) String Duality IIB D brane I

M is for Mother Het E8xE8 M (11D) Strong Coupling Limit IIA 10D Het SO(32) IIB I

M is for Membrane Fundamental Solitonic i M2 brane M5 brane Lessons String D2 brane String Theory NOT Just "a theory of strings" Only Consistent withbranes But, Branes also as "Composites of Strings" (from SFT viewpoints)

M is for Mystery DOF N 2 for N D branes Described by Matrix

M is for Mystery M2 brane M2 brane DOF N 3/2 /N 3 for N M2 /M5 branes

To Summarize we only knew little on To Summarize, we only knew little on "What M Theory Is" so far!

2. ABJ(M) Wilson Loop

ABJ(M) Theory [Aharony Bergman Jefferis (Maldacena)] N=6 N6Chern Simons matter matter Theory adjoint U(N 1 ) U(N 2 ) bifundamental adjoint Min(N 1,N 2 )x M2 & N 2 N 1 x fractional M2 on R 8 / Z k

Fractional brane & Wilson loop s Y k (N 1,N 2 ) Partition Function OR VEV of Half BPS Wilson Loop in Rep Y on Min(N 1,NN 2 ) x M2 & N 2 NN 1 x Fractional lm2 s Y k (N 1,N 2 ) = DA exp( S ABJ(M) [A, ]) Tr P exp A μ dx μ + = After Localization Techniques,...

ABJ(M) Matrix Model Physically,

ABJ(M) Matrix Model [Kapustin Willet Yaakov, Drukker Trancanelli, Marino Putrov] (Fresnel Integral) Character of U(N 1 N 2 )in Rep Y (Classified by Young Diagram Y) = Supersymmetric Schur Poly.

Invariant Measure

Invariant Measure Chern Simons (q deform) U(N 1 ) Inv

Invariant Measure Gaussian Matrix Model H: N x N Hermitian Matrix H: N 1 x N 1 Hermitian Matrix Gauge Sym = U(N 1 )

Invariant Measure Chern Simons (q deform) U(N 1 ) Inv

Invariant Measure Chern Simons (q deform) U(N 1 N 2 ) Inv Supersymmetry U(N 1 ) Inv

Invariant Measure Chern Simons (q deform) ABJ(M) Matrix Model GaussianMatrix Model + Supersymmetry Deformation + Chern Simons Deformation U(N 1 N 2 ) Inv Supersymmetry U(N 1 ) Inv

Message ABJ(M) Matrix Model, as Fundamental as Gaussian Model or Chern Simons Model.

Grand Canonical Ensemble Without Loss of Generality, M=N 2 N 1 0, k>0 [s ( ) Y ] GC k,m(z) = N=0 s Y k (N,N+M) z N Regarding ABJ(M) Partition Function as PF of N Particle Fermi Gas System [Marino Putrov] s GC M(z) = GC k GC Y k,m [s Y ] k,m(z) / [1] k,0(z)

where and H p,q (z) = Theorem [Hatsuda Honda M Okuyama, Honda Matsumoto M] M] s GC Y k,m M(z) ( ) = det (M+r)x(M+r) H p,q( (z) E (1 + z 1 l p QP) 1 E M+q 1 z E (1 + z QP) 1 Q E lp aq M (Q) =... (P) =... (E j ) =... l p :... a q :... (M = N 2 N 1 ) (1 q M) (1 q M r)

where and H p,q (z) = Theorem [Hatsuda Honda M Okuyama, Honda Matsumoto M] M] s GC Y k,m M(z) ( ) = det (M+r)x(M+r) H p,q( (z) E (1 + z 1 l p QP) 1 E M+q 1 z E (1 + z QP) 1 Q E lp aq M (Q) ν,μ = [2cosh(ν μ)/2] 1 (P) μ,ν = [2cosh(μ ν)/2] 1 (E = e (j+1/2)ν j ) ν (Q) ν,μ, (P) μ,ν as Matrix & (E j ) ν as Vector (M = N 2 N 1 ) (1 q M) (1 q M r) with Continuous o Indices μ, ν Matrix Multiplication = Integration with Dμ, Dν

where H p,q (z) = Theorem [Hatsuda Honda M Okuyama, Honda Matsumoto M] M] s GC Y k,m M(z) ( ) = det (M+r)x(M+r) H p,q( (z) E (1 + z 1 l p QP) 1 E M+q 1 z E (1 + z QP) 1 Q E lp aq M (M = N 2 N 1 ) (1 q M) (1 q M r) l p : p th leg length a q : q th arm length

Frobenius Symbol (a 1 a 2 a r l 1 l 2 l r+m ) U(N) or U(N N) U(N N+3) (320 975421) (3,2,0 9,7,5,4,2,1) or (6532 6421) (6,5,3,2 6,4,2,1) ( 1, 2, 3,3,2,0 9,7,5,4,2,1) 3320 975421)

where H p,q (z) = Theorem [Hatsuda Honda M Okuyama, Honda Matsumoto M] M] s GC Y k,m M(z) ( ) = det (M+r)x(M+r) H p,q( (z) E (1 + z 1 l p QP) 1 E M+q 1 z E (1 + z QP) 1 Q E lp aq M (M = N 2 N 1 ) (1 q M) (1 q M r) leg # arm if not distinguishing between (1+zQP) 1 or z(1+zqp) 1 Q

Example GC k,m=3 det 1 # 9 1 # 7 1 # 5 1 # 4 1 # 2 1 # 1 2 # 9 2 # 7 2 # 5 2 # 4 2 # 2 2 # 1 3 # 9 3 # 7 3 # 5 3 # 4 3 # 2 3 # 1 3 # 9 3 # 7 3 # 5 3 # 4 3 # 2 3 # 1 2 # 9 2 # 7 2 # 5 2 # 4 2 # 2 2 # 1 0 # 9 0 # 7 0 # 5 0 # 4 0 # 2 0 # 1

Especially, ABJM Wilson loop det " General Representation = det Hook Representations "

Especially, ABJM Wilson loop Without VEVs,... det "(General Representation) = det (Hook Representations)" Giambelli Formula Giambelli Formula A Well Known Classical Formula in Mathematics

Especially, ABJM Wilson loop What we have proved... Possible to Put... GC k,m In Giambelli Formula "Giambelli Compatibility" for ABJM & "(Generalized) Giambelli Compatibility" for ABJ

Representation as Fermion Representation Y as Fermion Excitation (Example) Hook Representation Hook Representation Fundamental Excitation

Representation as Fermion Hook Representation = Fundamental Excitation General Representation = Solitonic Excitaion " General Representation = det Hook Representations " " Solitonic Excitation = det Fundamental Excitation "

Fermion as String String Fluctuation Wilson loop in Fundamental Representation Fermion as String

Especially, Fractional brane ABJ Without Wilson Loop Insertion GC 1 # 2 1 # 1 1 # 0 det 2 # 2 2 # 1 2 # 0 k,m=3 3 # 2 3 # 1 3 # 0 ABJ Partition Function In terms of "Hook" Fractional Branes from Fundamental Strings?

Break Summary So far Giambelli Compatibility for ABJM Wilson loop Generalized Giambelli Compatibility for ABJ Fractional Brane as String Composite Hereafter Exact Instanton Expansion

3. Exact Instanton Expansion

ABJM Theory [Aharony Bergman Jefferis Maldacena] N=6 N6Chern Simons matter matter Theory adjoint U(N) bifundamental U(N) adjoint N x M2 on R 8 / Z k (= R + x S 7 / Z k )

Pictorially S 7 / Z k S 7 / Z k k CP 3 x S 1

Shorthand Notation ABJM Partition Function ABJM Grand Potential Z(N) = 1 k (N,N) N) exp J(log z)= [1] GC k,0(z)

Developments Free Energy F(N)= Log Z(N)in large N Limit Perturbative ti Sum F(N) N 3/2 Z(N) ( ) = Ai[N] ( exp N 3/2 ) [Drukker Marino Putrov] [Fuji Hirano M, KEK, Marino Putrov]

Developments (Cont'd) Worldsheet Instanton (F1 wrapping CP 1 CP 3 ) [Drukker Marino Putrov, Hatsuda M Okuyama] Membrane Instanton (D2 wrapping RP 3 CP 3 ) Bound dstatet [Drukker Marino Putrov, Marino Putrov, Hatsuda M Okuyama, Calvo Marino] (Basically From Numerical Studies) [Hatsuda M Okuyama]

All Explicitly In Topological Strings [Hatsuda M Okuyama, Hatsuda M Marino Okuyama] Marino J(μ)=J pert (μ eff )+J WS (μ eff )+J MB (μ eff ) J pert (μ)=cμ 3 /3+Bμ+A J WS (μ eff )=F T top (T eff 1,T eff 2,λ) J MB (μ eff )=(2πi) 1 λ [λf NS (T 1 eff /λ,t 2 eff /λ,1/λ)] F top (T 1,T 2,τ) =... T eff 1 =4μ eff /k iπ F NS (T 1,TT 2,τ) =... T eff 2 =4μ eff /k+iπ C=2/π 2 k, B=..., A=... λ=2/k μ ( 1) k/2 2e 2μ 4F (1,1,3/2,3/2;2,2,2;( 1) k/2 16e 2μ ) k=even μ eff 4 3 = μ+e 4μ 4F 3 (1,1,3/2,3/2;2,2,2; 16e 4μ ) k=odd

All Explicitly In Topological Strings [Hatsuda M Okuyama, Hatsuda M Marino Okuyama] Marino J(μ)=J pert (μ eff )+J WS (μ eff )+J MB (μ eff ) J pert (μ)=cμ 3 /3+Bμ+A J WS (μ eff )=F T top (T eff 1,T eff 2,λ) J MB (μ eff )=(2πi) 1 λ [λf NS (T 1 eff /λ,t 2 eff /λ,1/λ)] F(T 1,T 2,τ 1,τ 2 ): Free Energy of Refined Top Strings T 1,T 2 : Kahler Moduli τ 1,τ 2 : Coupling Constants Topological Limit F top (T 1,T 2,τ) = lim τ1 τ,τ 2 τ F(T 1,T 2,τ 1,τ 2 ) NS Limit F NS (T 1,T 2,τ) = lim τ1 τ,τ 2 0 2πiτ 2 F(T 1,T 2,τ 1,τ 2 )

All Explicitly In Topological Strings [Hatsuda M Okuyama, Hatsuda M Marino Okuyama] Marino J(μ)=J pert (μ eff )+J WS (μ eff )+J MB (μ eff ) J pert (μ)=cμ 3 /3+Bμ+A J WS (μ eff )=F T top (T eff 1,T eff 2,λ) J MB (μ eff )=(2πi) 1 λ [λf NS (T 1 eff /λ,t 2 eff /λ,1/λ)] F(T 1,T 2,τ 1,τ 2 ) = jl,j R n d1,d 2 N jl,j R d 1,d 2 χ n(d T +d T ) jl (q L ) χ jr (q R ) e 1 1 2 2 /[n(q 1 n/2 q 1 n/2 )(q 2 n/2 q 2 n/2 )] N jl j R d 1,d 2 :BPS Index of local P 1 x P 1 q 1 =e 2πiτ 1 q 2 =e 2πiτ 2 q L =e πi(τ 1 τ 2 ) q R =e πi(τ 1 +τ 2 ) j L,j R (Gopakumar Vafa or Gromov Witten invariants)

Methods 't Hooft Expansion [Drukker Marino Putrov] Perturbative WKB Expansion [Marino Putrov] Worldsheet Instanton Membrane Instanton Numerical Studies Cancellation Mechanism [Hatsuda M Okuyama]

Numerical Studies for k = 1, 2, 3, 4, 6 Compute Exact Values of Z(N) up to N max Read off Exact Values of J(μ) from Z(N) Numerically, After Subtracting Perturbative and Major Instanton Contribution Compare with Worldsheet Instanton

Instanton Effects Non Perturbative Part of Grand Potential J(μ) J k=1 (μ) = [#μ 2 +#μ+#]e 4μ + [#μ 2 +#μ+#]e 8μ + [#μ 2 +#μ+#]e 12μ +... J k=2 (μ) = [#μ 2 +#μ+#]e 2μ + [#μ 2 +#μ+#]e 4μ + [#μ 2 +#μ+#]e 6μ +... J k=3 (μ) = [#]e [] 4μ/3 + [#]e [] 8μ/3 + [#μ μ 2 +#μ+#]eμ 4μ +... J k=4 (μ) = [#]e μ + [#μ 2 +#μ+#]e 2μ + [#]e 3μ +...... J k=6 (μ) = [#]e 2μ/3 + [#]e 4μ/3 + [#μ 2 +#μ+#]e 2μ +...

Instanton Effects Non Perturbative Part of Grand Potential J(μ) J k=1 (μ) = [#μ 2 +#μ+#]e 4μ + [#μ 2 +#μ+#]e 8μ + [#μ 2 +#μ+#]e 12μ +... J k=2 (μ) = [#μ 2 +#μ+#]e 2μ + [#μ 2 +#μ+#]e 4μ + [#μ 2 +#μ+#]e 6μ +... J k=3 (μ) = [#]e [] 4μ/3 + [#]e [] 8μ/3 + [#μ μ 2 +#μ+#]eμ 4μ +... J k=4 (μ) = [#]e μ + [#μ 2 +#μ+#]e 2μ + [#]e 3μ +...... J k=6 (μ) = [#]e 2μ/3 + [#]e 4μ/3 + [#μ 2 +#μ+#]e 2μ +... WS(1) WS(2) WS(3)

Instanton Effects Worldsheet Instanton J k=1 (μ) = [#μ 2 +#μ+#]e 4μ + [#μ 2 +#μ+#]e 8μ + [#μ 2 +#μ+#]e 12μ +... J k=2 (μ) = [#μ 2 +#μ+#]e 2μ + [#μ 2 +#μ+#]e 4μ + [#μ 2 +#μ+#]e 6μ +... J k=3 (μ) = [#]e [] 4μ/3 + [#]e [] 8μ/3 + [#μ μ 2 +#μ+#]eμ 4μ +... J k=4 (μ) = [#]e μ + [#μ 2 +#μ+#]e 2μ + [#]e 3μ +... Match well with Topological String... J k=6 (μ) = [#]e 2μ/3 + [#]e 4μ/3 + [#μ 2 +#μ+#]e 2μ +... Prediction of WS WS(1) WS(2) WS(3)

Instanton Effects Worldsheet Instanton, Divergent at Certain k J k=1 (μ) = [#μ 2 +#μ+#]e 4μ + [#μ 2 +#μ+#]e 8μ + [#μ 2 +#μ+#]e 12μ +... J k=2 (μ) = [#μ 2 +#μ+#]e 2μ + [#μ 2 +#μ+#]e 4μ + [#μ 2 +#μ+#]e 6μ +... J k=3 (μ) = [#]e [] 4μ/3 + [#]e [] 8μ/3 + [#μ μ 2 +#μ+#]eμ 4μ +... J k=4 (μ) = [#]e μ + [#μ 2 +#μ+#]e 2μ + [#]e 3μ +... Match well with Topological String... J k=6 (μ) = [#]e 2μ/3 + [#]e 4μ/3 + [#μ 2 +#μ+#]e 2μ +... Prediction of WS WS(1) WS(2) WS(3)

Instanton Effects Worldsheet Instanton, Divergent at Certain k Divergence Cancelled by Membrane Instanton J k=1 (μ) = [#μ 2 +#μ+#]e 4μ + [#μ 2 +#μ+#]e 8μ + [#μ 2 +#μ+#]e 12μ +... J k=2 (μ) = [#μ 2 +#μ+#]e 2μ + [#μ 2 +#μ+#]e 4μ + [#μ 2 +#μ+#]e 6μ +... J k=3 (μ) = [#]e [] 4μ/3 + [#]e [] 8μ/3 + [#μ μ 2 +#μ+#]eμ 4μ +... k=3 MB(2) J k=4 (μ) = [#]e μ + [#μ 2 +#μ+#]e 2μ + [#]e 3μ +... Match well with Topological String... J k=6 (μ) = [#]e 2μ/3 + [#]e 4μ/3 + [#μ 2 +#μ+#]e 2μ +... Prediction of WS WS(1) WS(2) WS(3) MB(1)

Divergence Cancellation Mechanism Aesthetically Reproducing the Lessons String Theory, Not Just 'a theory of strings' Practically Helpful in Determining Membrane Instanton

An Incorrect but Suggestive Interpretation Worldsheet Inst S 7 / Z k 1 Instanton k InstantonI Off Fixed Pt cf: Twisted Sectors in String Orbifold

Cancellation New Branch in WS inst Divergence Cancelled by MB Inst

Compact Moduli Space? Compactified by Membrane Instanton NonPerturbatively!? Perturbative WorldSheet Instanton Moduli

Another Implication J(μ)=J pert (μ eff )+J WS (μ eff )+J MB (μ eff ) J pert (μ)=cμ 3 /3+Bμ+A J WS (μ eff )=F T top (T eff 1,T eff 2,λ) J MB (μ eff )=(2πi) 1 λ [λf NS (T 1 eff /λ,t 2 eff /λ,1/λ)] F(T 1,T 2,τ 1,τ 2 ) = jl,j R n d1,d 2 N jl,j R d 1,d 2 χ n(d T +d T ) jl (q L ) χ jr (q R ) e 1 1 2 2 /[n(q 1 n/2 q 1 n/2 )(q 2 n/2 q 2 n/2 )] NonPerturbative Topological Strings on General Background by Requiring Divergence Cancellation [Hatsuda Marino M Okuyama]

Short Summary Summary Explicit Form of Membrane Instanton Exact Large N Expansion of ABJM Partition Function Divergence Cancellation Moduli Space of Membrane?

Summary & Further Directions ABJM Partition Function Exact Large N Expansion Divergence Cancellation Fractional lmembrane from Wilson Loop Generalization for M2 Orbifolds, Orientifolds, Ellipsoid/Squashed S 3 Implication of Cancellation for M5 Exploring Moduli Space of M theory

Thank you for your attention!