Instanton Effects in Orbifold ABJM Theory Sanefumi Moriyama (NagoyaU/KMI) [Honda+M 1404] [M+Nosaka 1407] Contents 1. IntroducOon 2. ABJM 3. Generalizaons Message The Exact Instanton Expansion of The ABJM ParOOon FuncOon Has Been Found! CauOon: Not Solved Completely Logically (ParOal Assist From Numerical Method) Why ABJM InteresOng? ABJM Theory Describes MulOple M2-branes on 4 1 3 C /Zk → R>0 x S x CP (k→∞) • log (ParOOon FuncOon) = M2 DOF = N3/2 + ... [DrukKer-Marino-Putrov] • Understanding Membrane InteracOons • A prototype of AdS/CFT Why ABJM Exact PF InteresOng? • A compleon of N3/2 DOF (Understanding Membrane InteracOon) • Perturbave Correcons Sum Up To Airy Z(N) ~ Ai[N] ~ exp(N3/2) [Fuji-Hirano-M] • Poles from Coefficients of Two Instantons Cancel Between Themselves [Hatsuda-M-OKuyama] • Completely by Refined Topological Strings [Hatsuda-Marino-M-OKuyama] Queson So Far So Good for ABJM • How General, ProperOes of ABJM? - More Instantons, More Cancellaons? - Exact Expansion for General Theories? Contents 1. IntroducOon 2. ABJM 3. Generalizaons ABJM Matrix Model • ParOOon FuncOon of ABJM Theory • Due to SUSY, Localized to Matrix Model N1=N2=N Z(N) = gs=2πi/k [Pestun, KapusOn-Willem-Yaakov] 't Hooq Expansion • N→∞ Expansion with λ=N/k Fixed • Genus Expansion [DrukKer-Marino-Putrov] 2 0 -2 -4 log[Z(N)] = N F0(λ) + N F1(λ) + N F2(λ) + N F3(λ) + ... 2 pert -2π√2λ -4π√2λ = N [F0 (λ) + # e + # e + ...] 0 pert -2π√2λ -4π√2λ + N [F1 (λ) + # e + # e + ...] + ... ... Sum Up To Airy Funcon! Worldsheet Instanton -2π√2λ 1 [Fuji-Hirano-M, Honda et al] e = exp[- TF1 Area(CP )] F-String Wrapping CP1⊂CP3 [Cagnazzo-Sorokin-Wulff, DMP] M-theory Expansion [Herzog-Klebanov-Pufu-Tesileanu] 4 • M-theory BacKground: C /Zk • N→∞ with k Fixed • Close To 't Hooq Limit But Not Exactly N M-theory Regime k: Fixed 't Hooq Regime k λ=N/k Fixed but Large Fermi Gas Formalism [Marino-Putrov] • RewriOng ParOOon FuncOon • Regarding as Fermi Gas with Hamiltonian e-H -1 σ -H Z(N) = (N!) ∑σ∈S(N) (-1) ∫ ∏i dqi 〈qi| e |qσ(i)〉 e-H ~ (2 cosh q/2)-1 (2 cosh p/2)-1 [q,p] = i 2πk • Introducing Grand PotenOal J(μ) ∞ μN μ -H e = ∑N=0 Z(N) e = det (1 + e e ) WKB (small k) expansion -1 1 3 5 J(μ) = k J0(μ) + k J1(μ) + k J2(μ) + k J3(μ) + ... -1 pert 2 -2μ -4μ = k [ J0 (μ) + (#μ +#μ+#)e + (...) e + ... ] pert 2 -2μ -4μ + k [ J1 (μ) + (#μ +#μ+#)e + (...) e + ... ] + ... ... = (#μ3+#μ+#) + (#μ2+#μ+#)e-2μ + (...)e-4μ + ... Airy FuncOon Reproduced [Marino-Putrov Membrane Instanton ] -2μ -π√2Nk 3 e ≈ e = exp[- TD2 Area(RP )] D2-brane Wrapping RP3⊂CP3 [DrukKer-Marino-Putrov] Short Summary [Fuji-Hirano-M] 't Hooq Perturbave WKB Expansion Sum Expansion [DrukKer-Marino-Putrov] [Marino-Putrov] Worldsheet Membrane Instanton Instanton Furthermore [Fuji-Hirano-M] 't Hooq Perturbave WKB Expansion Sum Expansion [DrukKer-Marino-Putrov] [Marino-Putrov] Worldsheet Membrane Instanton Instanton • Numerical Analysis • Cancellaon Mechanism [Hatsuda-M-OKuyama] Bound States [Hatsuda-M-OKuyama 1301] MB WS Bound States e-2lμ-4mμ/k Taken Care Of By WS instantons with Chemical PotenOal Redef μ → μeff -2μ -4μ μeff = μ + # e + # e + ... Cancelaon Mechanism [Hatsuda-M-OKuyama 1211] 2 -4μ 2 -8μ 2 -12μ Jk=1(μ) = [#μ +#μ+#]e + [#μ +#μ+#]e + [#μ +#μ+#]e + ... 2 -2μ 2 -4μ 2 -6μ Jk=2(μ) = [#μ +#μ+#]e + [#μ +#μ+#]e + [#μ +#μ+#]e + ... -4μ/3 -8μ/3 2 -4μ Jk=3(μ) = [#]e + [#]e + [#μ +#μ+#]e + ... -μ 2 -2μ -3μ Jk=4(μ) = [#]e + [#μ +#μ+#]e + [#]e + ... ... -2μ/3 -4μ/3 2 -2μ Jk=6(μ) = [#]e + [#]e + [#μ +#μ+#]e + ... WS(1) WS(2) WS(3) ↑ ↑ ↑ Free Energy of Topological String Cancelaon Mechanism [Hatsuda-M-OKuyama 1211] (Raison d'Etre for M-theory!?) 2 -4μ 2 -8μ 2 -12μ Jk=1(μ) = [#μ +#μ+#]e + [#μ +#μ+#]e + [#μ +#μ+#]e + ... 2 -2μ 2 -4μ 2 -6μ Jk=2(μ) = [#μ +#μ+#]e + [#μ +#μ+#]e + [#μ +#μ+#]e + ... -4μ/3 -8μ/3 2 -4μ Jk=3(μ) = [#]e + [#]e + [#μ +#μ+#]e + ... -μ 2 -2μ -3μ MB(2) Jk=4(μ) = [#]e + [#μ +#μ+#]e + [#]e + ... ... -2μ/3 -4μ/3 2 -2μ Jk=6(μ) = [#]e + [#]e + [#μ +#μ+#]e + ... WS(1) WS(2) WS(3) MB(1) ↑ ↑ ↑ Free Energy of Topological String All Explicitly In Topological Strings [..., Hatsuda-M-Marino-OKuyama] J(μ)=Jpert(μeff)+JWS(μeff)+JMB(μeff) pert 3 J (μ)=Cμ /3+Bμ+A WS eff eff eff J (μ )=Ftop(T1 ,T2 ,λ) MB eff -1 eff eff J (μ )=(2πi) ∂λ[λFNS(T1 /λ,T2 /λ,1/λ)] eff eff Ftop(T1,T2,τ) = ... T1 =4μ /k-iπ eff eff FNS(T1,T2,τ) = ... T2 =4μ /k+iπ λ=2/k C=2/π2k, B=..., A=... k/2 -2μ k/2 -2μ eff μ-(-1) 2e 4F3(1,1,3/2,3/2;2,2,2;(-1) 16e ) k=even μ = -4μ -4μ μ+e 4F3(1,1,3/2,3/2;2,2,2;-16e ) k=odd All Explicitly In Topological Strings [..., Hatsuda-M-Marino-OKuyama] J(μ)=Jpert(μeff)+JWS(μeff)+JMB(μeff) pert 3 J (μ)=Cμ /3+Bμ+A WS eff eff eff J (μ )=Ftop(T1 ,T2 ,λ) MB eff -1 eff eff J (μ )=(2πi) ∂λ[λFNS(T1 /λ,T2 /λ,1/λ)] F(T1,T2,τ1,τ2): Free Energy of Refined Top Strings T1,T2: Kahler Moduli τ1,τ2: Coupling Constants Topological Limit F (T ,T ,τ) = lim F(T ,T ,τ ,τ ) top 1 2 τ1→τ,τ2→-τ 1 2 1 2 NS Limit F (T ,T ,τ) = lim 2πiτ F(T ,T ,τ ,τ ) NS 1 2 τ1→τ,τ2→0 2 1 2 1 2 All Explicitly In Topological Strings [..., Hatsuda-M-Marino-OKuyama] J(μ)=Jpert(μeff)+JWS(μeff)+JMB(μeff) Jpert(μ)=Cμ3/3+Bμ+A WS eff eff eff J (μ )=Ftop(T1 ,T2 ,λ) MB eff -1 eff eff J (μ )=(2πi) ∂λ[λFNS(T1 /λ,T2 /λ,1/λ)] F(T ,T ,τ ,τ ) = ∑ ∑ ∑ N d1,d2 1 2 1 2 jL,jR n d1,d2 jL,jR χ (q ) χ (q ) e-n(d1T1+d2T2) jL L jR R n/2 -n/2 n/2 -n/2 /[n(q1 -q1 )(q2 -q2 )] 2πiτ1 2πiτ2 πi(τ1-τ2) πi(τ1+τ2) q1 =e q2 =e qL=e qR=e N d1,d2 : BPS Index of local P1 x P1 jL,jR (Gopakumar-Vafa or Gromov-Wimen invariants) LooKing BacK, What Makes ABJM Solvable? • Max SUSY (N=6) !? • Relaons to Topological Strings? In Fact, ABJ (N=6 & Topological Strings) Solved Similarly. [Awata-Hirano-Shigemori, Honda, Matsumoto-M, Honda- OKuyama, Kallen] Generalizaons • ABJM only Two Types of Instantons → More Non-Trivial Cancellaons To Understand Membrane Moduli Space • What If Relaon To Topological Strings Is Unclear? Contents 1. IntroducOon 2. ABJM 3. Generalizaons Generalizaons • ABJM in Quiver Diagram k -k • More General N=3 Circular Quivers (Σi ki = 0) k 2 k3 k1 k k5 4 (Expected to be M2 on General BacKground) Generalizaons Especially, [Gaiomo-Wimen, Hosomichi-Lee3-ParK] r [U(N)k x U(N)-k] -k k -k k k -k -k k -k k k Enhanced to N=4 -k Interpretaon: 4 Orbifold C / (Zkr x Zr) [Benna- Klebanov-Klose-SmedbacK, Imamura-Kimura, Terashima-Yagi] In Fact, More Generally As Long As MulOple RepeOOon k k4 5 k1 k3 k2 k2 k k3 2 k3 k1 k1 k4 k5 k4 k5 k5 k4 k k1 3 k2 Results [Honda-M] • Grand PotenOal for MulOple RepeOOon Jr(μ) = ... in terms of J1(μ) • Especially, Perturbave Coefficients 2 2 -2 Cr = C1 / r Br = B1 - π C1(1-r )/3 Ar = r A1 Results [Honda-M] • Orbifold ABJM ParOOon FuncOon Solved - A New Instanton Effect Originates From Pert exp(-2π2Cμ/k) = exp(-2μ/r) - Interpretaon: 3 D2 wrapping A New Lagragian Submfd RP /Zr More N=4 Superconformal Theories N=4 Enhancement, Not Restricted To {ka} = {k, -k, k, -k, ..., k, -k} But [Imamura-Kimura] ka = (k/2) (sa-sa-1) sa = ±1 Many ProperOes Similar to ABJM Airy FuncOons, Cancellaon Mechanism, ... [M-Nosaka, see Nosaka's poster] Summary • ABJM ParOOon FuncOon Solved • Generalizaons to General N=3 Superconformal Circular Quiver Theories? • Start with N=4 - Orbifold Case: Solved AND Beyond - General Cases: Just StarOng • Hope: Solve All Superconformal Theories & Understand Membrane Moduli Space ThanK you for your aenOon. .