Instanton Effects in Orbifold ABJM Theory
Sanefumi Moriyama (NagoyaU/KMI) [Honda+M 1404] [M+Nosaka 1407] Contents
1. Introduc on 2. ABJM 3. Generaliza ons Message
The Exact Instanton Expansion of The ABJM Par on Func on Has Been Found!
Cau on: Not Solved Completely Logically (Par al Assist From Numerical Method) Why ABJM Interes ng?
ABJM Theory Describes Mul ple M2-branes on 4 1 3 C /Zk → R>0 x S x CP (k→∞) • log (Par on Func on) = M2 DOF = N3/2 + ... [Drukker-Marino-Putrov] • Understanding Membrane Interac ons • A prototype of AdS/CFT Why ABJM Exact PF Interes ng?
• A comple on of N3/2 DOF (Understanding Membrane Interac on) • Perturba ve Correc ons 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] Ques on
So Far So Good for ABJM
• How General, Proper es of ABJM?
- More Instantons, More Cancella ons?
- Exact Expansion for General Theories?
Contents
1. Introduc on 2. ABJM 3. Generaliza ons ABJM Matrix Model
• Par on Func on of ABJM Theory • Due to SUSY, Localized to Matrix Model
N1=N2=N Z(N) =
gs=2πi/k [Pestun, Kapus n-Wille -Yaakov] 't Hoo 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 Func on! 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 Hoo Limit But Not Exactly N M-theory Regime k: Fixed 't Hoo Regime k λ=N/k Fixed but Large Fermi Gas Formalism [Marino-Putrov]
• Rewri ng Par on Func on • 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 Poten al 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 Func on 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 Hoo Perturba ve WKB Expansion Sum Expansion [Drukker-Marino-Putrov] [Marino-Putrov]
Worldsheet Membrane Instanton Instanton Furthermore
[Fuji-Hirano-M] 't Hoo Perturba ve WKB Expansion Sum Expansion [Drukker-Marino-Putrov] [Marino-Putrov]
Worldsheet Membrane Instanton Instanton
• Numerical Analysis • Cancella on 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 Poten al Redef
μ → μeff -2μ -4μ μeff = μ + # e + # e + ...
Cancela on 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 Cancela on 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-Wi en invariants) Looking Back, What Makes ABJM Solvable?
• Max SUSY (N=6) !? • Rela ons to Topological Strings?
In Fact, ABJ (N=6 & Topological Strings) Solved Similarly. [Awata-Hirano-Shigemori, Honda, Matsumoto-M, Honda- Okuyama, Kallen] Generaliza ons
• ABJM only Two Types of Instantons → More Non-Trivial Cancella ons To Understand Membrane Moduli Space • What If Rela on To Topological Strings Is Unclear?
Contents
1. Introduc on 2. ABJM 3. Generaliza ons Generaliza ons
• 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) Generaliza ons
Especially, [Gaio o-Wi en, 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 Interpreta on: 4 Orbifold C / (Zkr x Zr) [Benna- Klebanov-Klose-Smedback, Imamura-Kimura, Terashima-Yagi] In Fact, More Generally
As Long As Mul ple Repe on
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 Poten al for Mul ple Repe on
Jr(μ) = ... in terms of J1(μ) • Especially, Perturba ve Coefficients 2 2 -2 Cr = C1 / r Br = B1 - π C1(1-r )/3 Ar = r A1
Results [Honda-M]
• Orbifold ABJM Par on Func on Solved - A New Instanton Effect Originates From Pert exp(-2π2Cμ/k) = exp(-2μ/r) - Interpreta on: 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 Proper es Similar to ABJM Airy Func ons, Cancella on Mechanism, ... [M-Nosaka, see Nosaka's poster]
Summary
• ABJM Par on Func on Solved • Generaliza ons to General N=3 Superconformal Circular Quiver Theories? • Start with N=4 - Orbifold Case: Solved AND Beyond - General Cases: Just Star ng • Hope: Solve All Superconformal Theories & Understand Membrane Moduli Space Thank you for your a en on.