Large N Non-Perturbative Effects in ABJM Theory

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1 Strings Large N Non-Perturbative Effects in ABJM Theory Yasuyuki Hatsuda (DESY) Collaborators: A. Grassi, M. Honda, M. Marino, S. Moriyama & K. Okuyama

2 Basic Flow in Localization Path Integrals Localization Matrix Integrals Not easy... Large N Expansions Typically, leading large N behavior is investigated Remarkably, in ABJM theory, the complete large N expansion has been found! Here, I will talk about a powerful approach to explore the large N expansion in M-theoretic limit Topological string plays a crucial role 2

3 Plan 1. Fermi-Gas Approach and Large N Expansion [YH, Marino, Moriyama & Okuyama, arxiv: ] 2. Quantum Spectral Problem and Quantization Condition [Grassi, YH & Marino, arxiv: , ] 3

4 ABJM Theory What is ABJM? [Aharony, Bergman, Jafferis, Maldacena 08] 3d superconformal Chern-Simons-matter theory with gauge group U(N) k U(N) k Two independent parameters N N Rank of gauge group N & Chern-Simons level k t Hooft coupling = N/k & String coupling g s =2 /k 4

5 Why important? Low energy effective theory on multiple M2 branes has gravity dual at large N M-theory on AdS 4 S 7 /Z k (k N 1/5 ) Type IIA on AdS 4 CP 3 ( t Hooft limit) Using AdS/CFT, the ABJM theory probes nonperturbative aspects of the dual string/m-theory Analogy: Non-critical strings/matrix models 5

6 ABJM Matrix Model Localization: path integral matrix integral [Pestun 07] Z ABJM (N) = 1 N! 2 Z d N µ (2 ) N d N (2 ) N Qi<j [2 sinh( µ i µ j )] 2 [2 sinh( i j )] Qi,j [2 cosh( µ i j exp " ik 4 2 )] 2 # NX (µ 2 i 2 i ) i=1 The reduction is exact [Kapustin, Willett & Yaakov 09] All information is encoded in this matrix integral Still non-trivial to extract the large N result from this integral 6

7 Fermi-Gas Approach Crucial fact: The ABJM partition function can be regarded as the partition function of an ideal Fermigas [Marino & Putrov 11] Generating function (µ, k) =1+ 1X N=1 Z ABJM (N, k)e Nµ 1Y = (1 + e µ E n ) n=0 Grand partition function of ideal Fermi-gas 7

8 The Hamiltonian is quite unconventional e Ĥ = 1 1 (2 cosh ˆx 2 )1/2 2 cosh ˆp 2 1 (2 cosh ˆx 2 )1/2 Eigenvalue problem: Fredholm integral equation Important: The Chern-Simons level k plays the role of the Planck constant! [ˆx, ˆp] =i~, ~ =2 k (Semi-classical limit) (Strong coupling limit) ~! 0 g s = 2 k!1 8

9 One can develop the semi-classical analysis 1X [Marino & Putrov 11] J WKB (µ, k) = 1 k n=0 k 2n J n (µ) Large µ Large N 9

10 One can develop the semi-classical analysis 1X [Marino & Putrov 11] J WKB (µ, k) = 1 k n=0 k 2n J n (µ) Large µ Large N = C 3 µ3 + Bµ + A + 1X (a`(k)µ 2 + b`(k)µ + c`(k))e 2`µ `=1 N 3/2 -behavior What are these corrections? 9

11 One can develop the semi-classical analysis 1X [Marino & Putrov 11] J WKB (µ, k) = 1 k n=0 k 2n J n (µ) Large µ Large N = C 3 µ3 + Bµ + A + 1X (a`(k)µ 2 + b`(k)µ + c`(k))e 2`µ `=1 N 3/2 -behavior What are these corrections? O(e 2µ ) O(e p 2kN ) O(e 2 2p 2 /g s ) Non-perturbative corrections in g s Membrane instantons!? 9

12 One can develop the semi-classical analysis 1X [Marino & Putrov 11] J WKB (µ, k) = 1 k n=0 k 2n J n (µ) Large µ Large N = C 3 µ3 + Bµ + A + 1X (a`(k)µ 2 + b`(k)µ + c`(k))e 2`µ `=1 N 3/2 -behavior What are these corrections? O(e 2µ ) O(e p 2kN ) O(e 2 2p 2 /g s ) Non-perturbative corrections in g s Semi-classical analysis provides non-perturbative corrections in g s Membrane instantons!? 9

13 Remarkable Connection In principle, one can compute the WKB expansions of the membrane instanton corrections order by order There is a remarkable connetion with topological string 2 cosh ˆx 2 2 cosh ˆp 2 10 i =e E i Canonical transform (eû + z 1 e û +eˆv + z 2 e ˆv 1) i =0 Quantization of the mirror curve of local P 1 P 1! [Aganagic, Cheng, Dijkgraaf, Krefl & Vafa 11]

14 The quantized mirror curve describes the refined topological string in Nekrasov-Shatashvili limit! ( 1, 2 )! (~, 0) [Aganagic et al. 11] cf. Unrefined slice: 1 = 2 The quantum parameter just corresponds to the Chern-Simons coupling k [YH, Marino, Moriyama & Okuyama 13] B a`(k) b`(k) = (Quantum corrected A-period) = (Quantum corrected B-period) A T (p) +U(x) =E c`(k) = (Combination of a and b) Membrane instantons are computed as quantum periods! 11 Figure from [Kallen & Marino 13]

15 Non-Perturbative Corrections The membrane instanton corrections have poles... b 1 (k) = 2 cos k 2 cot k 2 This divergence is cured by non-perturbative corrections in the Planck constant! Surprisingly, this correction can be computed by unrefined topological string free energy J np (µ, k) = X g,n,d n d g Gopakumar-Vafa invariants for local P 1 P 1 12 diverges at k =2, 4,... [YH, Moriyama & Okuyama 12] ( 1 = 2 ) ( 1) dn 2 sin 2 n 2g 2 e 4dnµ k n k has poles!

16 From Large N to Finite N Combining all corrections, we obtain the complete large µ expansion for any k J(µ, 1) = µ µ + log 2 4 Z FG (N, k) = Z C Order dµ 2 i ej(µ,k) Nµ e 4µ e 8µ Worldsheet + Membrane (3) µ2 +4µ e 4µ + O(e 8µ ) Z FG (N =1,k = 1) Z(N =1,k = 1) = e 24µ 96-digit precision! The large N expansion reproduces the finite N result! 13

17 2. Quantum Spectral Problem and Quantization Condition [Grassi, YH & Marino, arxiv: , ]

18 Quantum Spectral Problem The eigenvalue problem of the Fermi-gas is written as a Fredholm-type integral equation e Ĥ ni =e E n ni Z 1 1 dx 0 (x, x 0 ) n(x 0 )=e E n n(x) (x, x 0 )= 1 2 k 1 1 (2 cosh x 2 )1/2 2 cosh x x0 2k 1 (2 cosh x0 2 )1/2 Recall: k is the Planck constant 15

19 How to solve this eigenvalue problem? We already know the solution! 16

20 How to solve this eigenvalue problem? We already know the solution! We have constructed this function with the help of the topological string 1Y (µ, k) = (1 + e µ E n ) n=0 The energy spectrum can be read off as zeros µ = E n + i(+2 im) 16

21 Quantization Condition The vanishing condition for leads to an exact quantization condition Captured by semi-classical analysis [Grassi, YH & Marino, 14] Special cases: [Kallen & Marino 13] (E n,k)= WKB (E n,k)+ np (E n,k)=n Non-perturbative in k E 7 E 2 The exact quantization 6 5 E 1 condition is valid for any 4 finite k E 0 k

22 Test Spectrum at k = 3 Energy levels for k =3,M =0 Order E 0 E 1 e 4E/ e 12E/ e 24E/ e 32E/ e 40E/ e 52E/ Numerical value Spectrum at k = 5 Order E 0 E 1 e 4E/ e 16E/ e 32E/ e 48E/ Numerical value The topological string solves the spectral problem! 18

23 Summary Refined Topological String on Local P 1 P 1 1 = 2 = 4 k 1 =2 k, 2! 0 Worldsheet Instantons Membrane Instantons Large µ Expansion of J Large N Expansion of Z Quantization Condition 19

24 Other Topics around Fermi-Gas Weak coupling (genus) expansion [Drukker, Marino & Putrov 10] (Borel resum) (Exact result) [Grassi, Marino & Zakany 14] More general circular quiver CS [Moriyama & Nosaka 14; YH, Honda & Okuyama 15] Spectral theory and topological strings Exact spectral determinant and QC [Grassi, YH & Marino 14] Beautiful structure in QC [Wang, Zhang & Huang 14] 3d mirror symmetry [Assel, Drukker & Felix 15] 20

25 Thank you!

26 Appendix

27 Saddle Point Approx. Saddle point equation r kn J 0 (µ ) N =0 µ 2 Free energy at large N F (N) J(µ )+µ N p 2k 3 N 3/2 Exponentially suppressed corrections O(e 2µ ) O(e p 2kN ) O(e 2 2p 2 /g s ) O(e 4µ k ) O(e 2 p 2N/k ) O(e 2 p2 ) 23

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