FROM WEAK TO STRONG COUPLING IN ABJM THEORY Marcos Mariño University of Geneva Based on [M.M.-Putrov, 0912.1458] [Drukker-M.M.-Putrov, 1007.1453] [in progress]
Two well-known virtues of large N string/gauge theory dualities: • The large radius limit of string theory is dual to the strong coupling regime in the gauge theory R ≫ 1 ↔ λ ≫ 1 ℓ s • The genus expansion of the string theory can be in principle mapped to the 1/N expansion of the gauge theory
These virtues have their counterparts: • It is hard to test the duality, since one has to do calculations at strong ‘t Hooft coupling in the gauge theory. More ambitiously, one would like to have results interpolating between weak and strong coupling • It is hard to obtain information beyond the planar limit, even in the gauge theory side.
In this talk I will report on some recent progress on these problems in ABJM theory and its string dual. In particular, I will present exact results (interpolating functions) for the planar 1/2 BPS Wilson loop vev and for the planar free energy on the thee-sphere. The strong coupling limit is in perfect agreement with the AdS dual, and in particular provides the first quantitative test of the behaviour of the M2 brane N 3 / 2 theory
Moreover, I will show that it is possible to calculate explicitly the free energy for all genera (very much like in non-critical string theory). This makes possible to address some nonperturbative aspects of the genus expansion in a quantitative way (large order behavior, Borel summability, spacetime instantons...)
We will rely on the following “chain of dualities”, which relates a sector of ABJM theory to a topological gauge/string theory via a matrix model: Topological Strings large N CS on S 3 / Z 2 on local P 1 × P 1 localization CS matrix model localization Type IIA superstring ABJM theory on AdS 4 × CP 3 large N
A B J M theory 2 hypers U ( N 1 ) k × U ( N 2 ) − k N 1 N 2 CS theories + 4 hypers C in the bifundamental; related to 2 twisted hypers supergroup theory U ( N 1 | N 2 ) via [Gaiotto-Witten] two ‘t Hooft λ i = N i couplings k This is a 3d SCFT which (conjecturally) describes M2 min( N 1 , N 2 ) branes probing a singularity, with fractional C 4 / Z k | N 1 − N 2 | branes Note: “ABJM slice” refers to λ 1 = λ 2 = λ
Gravity dual Hopf type IIA theory / AdS 4 × P 3 reduction M-theory on d s 2 = R 2 AdS 4 × S 7 / Z k d s 2 AdS 4 + 4d s 2 � � CP 3 4 ℓ 2 s � R � 2 � 1 / 2 Gauge/gravity � 32 π 2 ˆ λ = dictionary: ℓ s � 1 / 4 g st = 1 � 32 π 2 ˆ λ k B = λ 1 − λ 2 + 1 2 Warning! � � λ = λ 1 − 1 B 2 − 1 − 1 ˆ shifted charges 2 4 24 [Bergman-Hirano, Aharony et al.]
Wilson loops Circular 1/6 BPS Wilson loops: they involve only one of the gauge connections, but they know about the other node through the bifundamentals � �� � � W 1 / 6 A 1 · ˙ x + | ˙ x | CC = Tr R P exp i R 1/2 BPS Wilson loops constructed by [Drukker-Trancanelli]. They exploit the hidden supergroup structure, and they are symmetric in the two nodes � � � A 1 · ˙ �� x + · · · W 1 / 2 = sTr R P exp i R − A 2 · ˙ x + · · · rep U ( N 1 | N 2 ) circle U ( N 1 ) connection U ( N 2 ) connection
Two string/gravity predictions 1) 1/2 BPS Wilson √ 2ˆ loop from � W 1 / 2 � planar ∼ e π λ fundamental string 2) The planar free energy of the Euclidean theory on S 3 should be given by the (regularized) Euclidean Einstein-Hilbert action on AdS4 d s 2 = d ρ 2 + sinh 2 ( ρ ) d Ω 2 S 3 , √ 2 π = π k 2 ˆ ˆ λ 3 / 2 , − F ( N, k ) ≈ S AdS 4 = λ ≫ 1 , g st ≪ 1 2 G N 3 [Emparan-Johnson-Myers] Nonzero and probing the 3/2 growth! using universal counterterms
Exact interpolation from a matrix model Similar story: 1/2 BPS Wilson loop in N=4 SYM. The string prediction at strong coupling can be derived in the gauge theory from an exact interpolating function [Ericksson-Semenoff-Zarembo, Drukker-Gross] √ λ λ ≫ 1 ∼ e √ 1 2 √ N � W � planar = I 1 ( λ ) λ 1 + λ λ ≪ 1 8 + · · · λ = g 2 YM N Rationale: the path integral calculating of the vev of the Wilson loop reduces to a Gaussian matrix model � W R � = 1 � λ Tr M 2 Tr R e M d M e − 2 N Z
This is the simplest matrix model , and the planar density of eigenvalues is the famous Wigner semicircle distribution � √ λ 1 N � W � planar = ρ ( z )e z d z √ λ − ρ ( z ) = 2 � λ − z 2 πλ One can also compute 1/N corrections systematically This conjecture was finally proved by using localization techniques [Pestun].
Reduction to a matrix model in ABJM Localization techniques were extended to the ABJM theory in a beautiful paper by [Kapustin-Willett-Yaakov]. The partition function on is given by the following matrix integral: S 3 contribution CS gauge fields Z ABJM ( N 1 , N 2 , g top ) �� 2 � �� 2 � � � µ i − µ j ν i − ν j � 2 sinh 2 sinh N 1 N 2 1 d µ i d ν j � 2 g top ( j ) i<j 2 2 i µ 2 j ν 2 1 P i − P � � − = e �� 2 N 1 ! N 2 ! 2 π 2 π � � µ i − ν j � 2 cosh i =1 j =1 i,j 2 g top = 2 π i contribution 4 hypers k We “just” need the planar solution, but exact in the ‘t Hooft parameters, in order to go to strong coupling
Relation to Chern-Simons matrix models Shortcut: relate this to CS matrix models [M.M. building on Lawrence-Rozansky] [AKMV, Halmagyi-Yasnov] U(N) (pure!) CS theory on : S 3 N �� 2 � � µ i − µ j Z S 3 ( N, g top ) = 1 � d µ i i µ 2 1 P � � 2 sinh e − i 2 g top N ! 2 π 2 i =1 i<j can be rederived with SUSY localization [Kapustin et al.] U(N) (pure!) � Z L (2 , 1) ( N, g top ) = Z L (2 , 1) ( N 1 , N 2 , g top ) CS theory on N 1 + N 2 = N L(2,1)= S 3 / Z 2 sum over flat connections
N 1 N 2 �� 2 � �� 2 � � µ i − µ j � ν i − ν j 1 � d µ i d ν j � � � Z L (2 , 1) ( N 1 , N 2 , g top ) = 2 sinh 2 sinh N 1 ! N 2 ! 2 π 2 π 2 2 i =1 j =1 i<j �� 2 � � µ i − ν j 2 g top ( j ) i µ 2 j ν 2 1 P i + P � − 2 cosh e × 2 i,j This is a two-cut model with two ‘t Hooft parameters t i = g top N i z Z = e z N 2 ρ 2 ( ν ) z = π i − b − 1 /b 1 /a a ρ 1 ( µ ) N 1 z = 0 Superficially similar to the matrix model describing ABJM...
1/N expansion � g 2 g − 2 F ( N 1 , N 2 , g top ) = top F g ( t 1 , t 2 ) g ≥ 0 Fact : The ABJM MM is the supermatrix version of the L(2,1) MM. They are related by t 2 → − t 2 [M.M.-Putrov] i.e. t 1 = 2 π i λ 1 , t 2 = − 2 π i λ 2 ABJM theory CS matrix model The 1/N expansion of the lens space matrix model gives the 1/N expansion of the ABJM free energy on the three-sphere
Planar solution: matrix model approach The planar solution of the CS lens space matrix model has been known for some time [AKMV, Halmagyi-Yasnov] . The solution is elegantly encoded in a resolvent or spectral curve � e − t/ 2 �� �� � ω 0 ( z ) = 2 log ( Z + b )( Z + 1 /b ) − ( Z − a )( Z − 1 /a ) 2 C 2 C 1 t = t 1 + t 2 discontinuity across 1 ρ k ( z ) = − t 2 π i ( ω 0 ( z + i ǫ ) − ω 0 ( z − i ǫ )) the cuts=densities t k
All the planar information is given by period integrals of the resolvent 1 � t i = ω 0 ( z ) dz, i = 1 , 2 4 π i C i � 1 /a ∂ F 0 − ∂ F 0 − π i t = − ω 0 ( z ) dz ∂ t 1 ∂ t 2 − 1 /b We have to understand what are the weak and the strong coupling limit in terms of the geometry of the curve (i.e. the location of the cuts). In ABJM theory we also want the ‘t Hooft parameters to be imaginary
We first consider the ABJM slice. It turns out that all quantities are naturally expressed in terms of a real variable , closely related to the positions of the cuts κ weak coupling strong coupling κ = 0 κ = ∞ a, b ∼ 1 a ∼ i κ , b ∼ − i κ λ ∼ log 2 ( κ ) λ ∼ κ ∂ λ F 0 ∼ log κ ∂ λ F 0 ∼ κ log κ F 0 ( λ ) ∼ λ 3 / 2 , λ ≫ 1
We can in fact write very explicit interpolating functions: 2; − κ 2 � 1 � 2 , 1 2 , 1 2; 1 , 3 λ ( κ ) = κ 8 π 3 F 2 16 � − κ 2 1 1 1 � � � ( λ ) = κ 2 , 2 , 4 G 2 , 3 ∂ λ F orb � + 4 π 3 i λ 2 − 1 � 0 3 , 3 0 , 0 , 16 2 exact 60 ∂ λ F orb 0 50 40 30 20 SUGRA 10 0.1 0.2 0.3 0.4 0.5 λ
Planar limit from topological strings Pure CS theory on L(p,1) has a large N topological string dual [AKMV] . The genus g free energies (for a fixed, generic flat connection) are equal to the genus g free energies of topological string theory on a toric CY manifold F CS ( t i = g s N i ) = F TS ( t i = moduli) g g O ( − 1) ⊕ O ( − 1) → S 2 For p=1 (i.e. M= ) this is the S 3 original Gopakumar-Vafa large N duality. The toric CY is the resolved conifold (single) ‘t Hooft parameter= (complexified) area of two-sphere
For p=2 (i.e. the lens space L(2,1)) the CY target is local . P 1 × P 1 It has two complexified Kahler moduli measuring the sizes of the two-spheres A 1 S 2 S 2
Topological Strings on local P 1 × P 1 CS matrix model ABJM theory F ABJM ( λ 1 , λ 2 ) = F CS (2 π i λ 1 , − 2 π i λ 2 ) = F TS (2 π i λ 1 , − 2 π i λ 2 ) g g g topological large relation between N duality matrix models In particular, the planar free energy in ABJM is just the prepotential of the topological string! (a standard calculation in mirror symmetry)
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