Summing up all genus free energy of ABJM matrix model

Summing up all genus free energy of ABJM matrix model

Seminar @ National Taiwan Univ.

Shinji Hirano (Nagoya Univ.)

Based on

H. Fuji, SH, S. Moriyama (1106.4631), O. Bergman, SH (0902.1743)

Introduction

Sum up all-loop all-genus free energy of strongly coupled 3d CFT!

3d CFT (ABJM theory) = U (N )_k × U (N )_−k Chern-Simons-Matter theory

• ’t Hooft coupling λ ≡ N/k  1

• Genus expansion g_s ≡ 2πi/k = 2πiλ/N

ABJM theory (on S

)

U (N )_k × U (N )_−k theory with 4 bifundamental matters:

S_ABJM = S_CSk + S_CS−k + S_M where

S_CSk = k Z

d³xTr



A ∧ dA + 2i

3 A³ − ¯λλ + 2Dσ



with DR of 4d N = 1 vector muliplet (A_µ, σ, λ, D)

S_M = Z

d³x

4

X

I=1



D_µφ¯^ID^µφ^I − i ¯ψ^Iγ^µD_µψ^I + 3 4

φ¯^Iφ^I + · · · + Z

d²θW + h.c.



with bifundamental chiral multiplets (φ^I, ψ^I, F^I)

ABJM Conjecture

Type IIB brane configuration of (UV completed) ABJ/M:

5 (1,k)

D3 N

D3 N+l

NS5

. &

. Gravity Gauge theory &

IR limit of M-theory lift IR limit of SYM + CSM

AdS₄ × S⁷/Z_k ABJM theory

AdS₄/CFT₃ ⊂ AdS/CFT ⊂ Gravity/Gauge duality:

M-theory on AdS₄ × S⁷/Z_k = U (N )_k × U (N )_−k Chern-Simons-Matter

When k  1, dimensional reduction 11d → 10d

Type IIA on AdS₄ × CP³ = U (N )_k × U (N )_−k Chern-Simons-Matter

Result

F^ABJM (λ, N ) = log



2πC₁Ai





"

√π 2

 N λ

²

λ^3/2_ren

#2/3





 + O 

e^−2π

√

λ−₂₄¹ 

up to worldsheet instantons O 

e^−2π

√

λ−₂₄¹ 

λ_ren = λ − 1

24 − λ² 3N²

Note:

(1) Stringy corrections : 1

√λ − expansion = α⁰ − expansion (2) QG loop corrections : 1

N² − expansion = G_N − expansion

PROSPECT:

Quantum Gravity test of AdS/CFT !

Technique: LOCALIZATION

Strong coupling λ  1 non-planar computation on gauge side is now possible!

Note: Integrability in AdS₅ – exact computation from λ  1 to λ  1 BUT only at N = ∞

Gravity prediction of non-planar corrections (Bergman, S.H.)

M2-brane charge N shifted to

N → N − 1

24k + 1 24k

In terms of ’t Hooft coupling (IIA description)

λ → λ − 1

24 + λ²

24N² (large N, k with λ = N/k)

Implying renormalization of AdS radius

R_AdS = (32π²kN)^1/6`_p →



32π²k



N − 1

24k + 1 24k

^1/6

`_p

(1) M-theory (11d SUGRA action):

S₁₁ = 1 2κ²₁₁

Z

d¹¹x √

−G



R − 1

2|G₄|²



− 1 6

Z

C₃ ∧ G₄ ∧ G₄+ (2π)² Z

C₃ ∧ I₈



+ S_{M 2}

where I₈ = _23·4!(2π)¹ ₄ 

TrR⁴ − ¹₄ TrR^2
2

(2) M2 Maxwell equation:

d ∗ G₄ = (2π)⁴(kN )δ⁸(x) − 1

2G₄ ∧ G₄ + (2π)²I₈

(I) (localized) M2 brane source

(II) flux (discrete torsion – H₃(S⁷/Z_k, Z) = Z_k) (III) higher curvature (Duff-Liu-Minasian)

(3) Membrane charge (M₈ = C⁴/Z_k and ∂M₈ = S⁷/Z_k):

Q_{M 2} ≡ 1 (2π)⁴

Z

∂M₈

∗ G₄ = N − 1 2(2π)⁴

Z

M₈

G₄ ∧ G₄ − χ_blk 24 where χ_blk bulk contribution to C⁴/Z_k Euler characteristic

χ_blk(C⁴/Z_k) = k − 1 k

Flow chart of free energy computation

3d N = 6 U (N )_k × U (N )_−k Chern-Simons-Matter conformal field theory

⇓

Localization

⇓

U (N ) × U (N ) ABJM matrix model

matrix potential = Gaussian + 2d Coulomb repulsion

⇓

Analytic continuation to Lens space matrix model (N, N ) → (N₁, −N₂)

⇓

Large N technique to find planar solution

⇓

Chain of dualities

Lens MM ^{geom trans}−→ Top A ^mirror−→ Top B

⇓

Holomorphic Anomaly Equation = recursion for higher genus free energies

⇓

Solve HAE neglecting O  e⁻

√λ

⇓

All genus free energy up to worldsheet instantons

(1) Once genus zero free energy F₀(λ) is given, HAE can be explicitly found (2) Together with genus one data, HAE can be solved (at least) recursively

We managed to sum it up !

Localization

∃ exact nilpotent quantum Gassmann-odd symmetry Q (such as SUSY &

BRST), partition function

Z = Z

Dϕ exp (−S[ϕ]) = Z

Dϕ exp (−S[ϕ] − tQV [ϕ])

(1) With QV ≥ 0 (choosing V cleverly) sending t → ∞

Path integral localizes on the locus {ϕ = ϕ_i | δS[ϕ_i] ≡ QV = 0}

(2) In fortunate situations, localization locus {ϕ_i=1,···,N} is finite dimensional Path integral reduces to finite dimensional integrals (Matrix Model) (3) Partition function one-loop exact (ϕ = ϕ_i + ^√1

tδϕ with t → ∞):

Z =

Z ^N Y

i=1

Dϕ_i v u u u t

det ^δ2∆S

δϕ²_F [ϕⁱ_F] det ^δ2∆S

δϕ²_B [ϕⁱ_B] exp (−S[ϕ_i])

ABJM application: (Kapustin-Willet-Yaakov, Hama-Hosomichi-Lee) (1) Nilpotent Grassmann-odd symmetry δ¯ (one of supercharges)

(2) Localization action ¯δV ≡ ¯δV_gauge + ¯δV_matter δV¯ _gauge = ¯δδ

2

X

A=1

Tr  1 2

λ¯^Aλ^A − 2D^Aσ^A



= S_YM

δV¯ _matter = ¯δδ ¯ψ^Iψ^I − 2i ¯φ^I σ¹ − σ²
φ^I − ¯φ^Iφ^I

= S_m

where

L_YM = Tr 1

4F_µν² + 1

2(D_µσ)² + 1

2(D + σ)² + · · ·



(3) Localization locus

A^A_µ = φ^I = 0 , D^A = −σ^A = const

(4) ABJM Matrix Model:

Z^ABJM = 1 (N !)²

Z ^N Y

i=1

dµ_i 2π

N

Y

a=1

dν_a 2π

Q

i<j



2 sinh_µ

i−µ_j 2

² Q

a<b 2 sinh ^νa−ν₂ <sup>b

2</sup> Q

i,a 2 cosh ^µi−ν₂ <sup>a

2</sup> e^−2gs1 (^Piµ²_i−P

aν_a²)

where g_s = 2πi/k

Luckily, these integrals can be done exactly!

Drukker-Mari˜ no-Putrov

Genus zero free energy of ABJM theory at strong coupling λ  1

F₀(λ) = ⁴

√2π²

3 λ − ₂₄¹
3/2

+ O 

e^−2π

√

λ−₂₄¹ 

Precisely agrees with classical SUGRA result!

−F₀(λ)/g_s² = S_SUGRA[AdS₄ × S⁷/Z_k]

Higher genus – solving recursion

(1) H(olomorphic)A(nomaly)E(quation) (Bershadsky-Cecotti-Ooguri-Vafa):

∂I¯F_g = 1

2CI ¯¯J ¯Ke^2KG^{J ¯J}G^{K ¯K} D_JD_KF_g−1 +

g−1

X

r=1

D_JF_rD_KF_g−r

!

where

G_{I ¯J} = Im∂_I∂_JF₀ C_{IJ K} = ∂_I∂_J∂_KF₀

(2) ABJM case: moduli space coordinate X^I = λ (one-dimensional)

(3) Turns out, neglecting worldsheet instanton, HAE yields

F_g⁰(x) = 1

4x⁴F_g−1⁰⁰ (x) + 12x − 1

12 x²F_g−1⁰ (x) + x⁴ 4

g−2

X

r=2

F_r⁰(x)F_g−r⁰ (x)

where x ≡ 1/√

2π²λ + · · ·

(4) (Modular) weight zero free energy

F_g := F_g^[0]x^3g−3 + O x^{3g−4
t≡−ig}sx^3/2

−→ F (t) :=

∞

X

g=2

F_g^[0]t^2g−2

(5) It sums up to (1st step sum = weight zero sum)

F (t) = logh

2πC₁e^−3t22 t⁻¹³Ai(t^−4/3)i

(6) 2nd step sum = gravity expectation (w/ ∗)

λ → λ_ren = λ − 1

24 − λ²

3N² ⇐⇒ x → y ≡ x

p1 + (g_sx)²/6

⇓

Replace x in F (t) by y

⇓

F^ABJM (λ, N ) = log



2πC₁Ai





"

√π 2

 N λ

²

λ^3/2_ren

#2/3





 + O 

e^−2π

√

λ−₂₄¹ 

This indeed solves HAE!

Conclusions and discussions

(1) Summed up all-loop all genus free energy of ABJM theory! (except for worldsheet instantons)

(2) Expect similar all genus free energy for ABJ theory (unequal rank U (N₁)_k × U (N₂)_−k theory)

w/ λ_ren = λ − k(B² − 1/4)/2 − 1/24− 1/(3k²)

(Aharony-Hashimoto-SH-Ouyang, DMP)

(3) Mismatch between field theory and gravity results at non-planar

λ^FT_ren = λ − 1

24 − λ²

3N² vs. λ^Gravity_ren = λ − 1

24 + λ² 24N²

Is it field theory or gravity?