Non-perturbative effects in ABJM theory

October 9, 2015 Non-perturbative effects in ABJM theory 1 / 43

Outline 1. Non-perturbative effects 1.1 General aspects 1.2 Non-perturbative aspects of string theory 1.3 Non-perturbative effects in M-theory 2. Non-perturbative effects in ABJM theory 2.1 A short review of ABJM theory 2.2 The t Hooft expansion 2.3 The M-theory expansion Based on Marcos Mariño, Non-perturbative effects in string theory and AdS/CFT Non-perturbative effects in ABJM theory 2 / 43

Non-perturbative effects General aspects Why non-perturbative effects are important? Most perturbative series in quantum mechanics are not convergent. Simplest example: quartic oscillator in QM H = p 2 2 + q2 2 }{{} harmonic oscillator + g 4 q4 }{{} stationary perturbation theory. E 0 (g) n 0 = 1 2 + 3 4 a n g n ( g ) 21 ( g ) 2 333 ( g ) 3 + + O(g 4 ) = ϕ(g). 4 8 4 16 4 Non-perturbative effects in ABJM theory 3 / 43

General aspects a n ( ) n 3 n!, grow factorially for n 1. 4 Non-perturbative definition of the ground state energy H ψ n = E n ψ n n = 0, 1, 2,.... Non-perturbative effects in ABJM theory 4 / 43

General aspects General definitions Given an asymptotic series ϕ(z) = n 0 a n z n, we say that a well-defined function f(z) provides a non-perturbative definition of ϕ(z) if f(z) has ϕ(z) as its asymptotic series, f(z) + f(z) ϕ(z). } e A/z {{}. non-perturbative ambiguity Non-perturbative effects in ABJM theory 5 / 43

General aspects Extract information: optimal truncation Find the partial sum N ϕ N (z) = a n z n. n=0 which gives the best possible estimate of f(z). Non-perturbative effects in ABJM theory 6 / 43

General aspects Extract information: optimal truncation Find the partial sum N ϕ N (z) = a n z n. n=0 which gives the best possible estimate of f(z). Typically, optimal truncation gives an exponentially small error, proportional to e A/z. Non-perturbative effects in ABJM theory 6 / 43

General aspects Improving optimal truncation = Borel resummation Borel transform ϕ(z) ϕ(ζ) = n=0 a n n! ζn. The series ϕ(ζ) has a finite radius of convergence ρ = A and it defines an analytic function in the circle ζ < A. Example: ϕ(z) = ( 1) n A n n! z n n=0 Non-perturbative effects in ABJM theory 7 / 43

General aspects Borel sum Let us suppose that ϕ(ζ) has an analytic continuation to a neighborhood of the positive real axis, in such a way that the Laplace transform s(ϕ)(z) = 0 e ζ ϕ(zζ) dζ = z 1 e ζ/z ϕ(ζ) dζ, exists in some region of the complex z-plane. In this case, we say that the series ϕ(z) is Borel summable and s(ϕ)(z) is called the Borel sum of ϕ(z). s(ϕ)(z) = z 1 n 0 a n n! 0 0 e ζ/z ζ n dζ = n 0 a n z n. Non-perturbative effects in ABJM theory 8 / 43

General aspects Example: ϕ(z) = A n n! z n n=0 Borel transform ϕ(ζ) = n=0 A n ζ n = 1 1 ζ. A Borel resummation does NOT exist due to the singularity in the positive real axis. One can then deform the contour of integration! Non-perturbative effects in ABJM theory 9 / 43

General aspects Lateral Borel resummations s ± (ϕ)(z) = z 1 C ± e ζ/z ϕ(ζ) dζ, In this case one gets a complex number, whose imaginary piece is also O(exp( A/z)). Non-perturbative effects in ABJM theory 10 / 43

General aspects In many cases the (lateral) Borel resummation of the perturbative series does not reproduce the right answer! Non-perturbative effects in ABJM theory 11 / 43

General aspects In many cases the (lateral) Borel resummation of the perturbative series does not reproduce the right answer! Something else should be added to the perturbative series! ϕ l (z) = z b l e la/z n 0 a n,l z n, l = 1, 2,.... ϕ l (z) encodes the non-perturbative effects due to l-instantons. Non-perturbative effects in ABJM theory 11 / 43

General aspects Example: double-well potential in QM H = p2 2 + W (q), W (q) = g ( q 2 1 ) 2, g > 0. 2 4g Ground state energy ϕ 0 (g) = 1 2 g 9 2 g2 89 3 g3.... This series is obtained by doing a path integral around the constant trajectory q ± = ± 1 2 g. Non-perturbative effects in ABJM theory 12 / 43

General aspects Saddle-point of the Euclidean path integral q± t0 = ± 1 ( ) t 2 g tanh t0. 2 This gives a non-perturbative contribution to the ground state energy, ϕ 1 (g) = ( ) 1/2 2 e 1/6g (1 + O(g)). g 2π One should then consider a trans-series of the form Φ(z) = ϕ 0 (z) + C l ϕ l (z). l=1 Non-perturbative effects in ABJM theory 13 / 43

Non-perturbative aspects of string theory Two coupling constants l s Worldsheet instantons: exp ( A ws /l 2 s ), g st Spacetime instantons: exp ( A st /g st ). Non-perturbative effects in ABJM theory 14 / 43

Non-perturbative aspects of string theory Two coupling constants l s Worldsheet instantons: exp ( A ws /l 2 s ), g st Spacetime instantons: exp ( A st /g st ). Focus on: the total partition function of superstring/m-theory in an AdS background. Total free energy F (λ, g st ) = g 0 F g (λ) g 2g 2 st, λ = function of L l s. The functions F g (λ) have a finite radius of convergence. Non-perturbative effects in ABJM theory 14 / 43

Non-perturbative aspects of string theory It turns out that F g (λ) are essentially analytic functions at λ = 0 (in many examples). Once we fix λ < λ : F g (λ) (2g)! (A st (λ)) 2g, where A st is a spacetime instanton action. M-theory perspective In M-theory, worldsheet instantons and D-brane instantons can be unified in terms of membrane instantons. Non-perturbative effects in ABJM theory 15 / 43

Non-perturbative effects in M-theory One coupling constant M-theory compactification type IIA superstring theory l p = g 1/3 st l s, R 11 = g st l s. M2-branes A membrane wrapped around a three-cycle S leads to an exponentially small effect of the form ( exp vol(s) ) l 3. p Non-perturbative effects in ABJM theory 16 / 43

Non-perturbative effects in M-theory Wrap the compact, eleventh dimension fundamental strings. vol(s) = R 11 vol(σ), where Σ is a cycle in ten dimensions. Then, ( exp vol(s) ) ( l 3 = exp vol(σ) ) p l 2. s Do not wrap the compact dimension D2 branes. vol(s) = vol(m), where M is a three-dimensional cycle in ten dimensions. Then, ( exp vol(s) ) [ ( )] vol(m)/l 3 l 3 = exp s. p g st Non-perturbative effects in ABJM theory 17 / 43

A short review of ABJM theory It describes N M2 branes on C 4 /Z k. W = 4π ) (Φ k Tr 1 Φ 2 Φ 3Φ 4 Φ 1Φ 4 Φ 3Φ 2. U(N) U(N) Φ i=1,...,4 Freund-Rubin background One of the most important aspects of ABJM theory is that, at large N, it describes a nontrivial background of M theory X 11 = AdS 4 S 7 /Z k. Non-perturbative effects in ABJM theory 18 / 43

If we represent S 7 inside C 4 as the action of Z k is given by 4 z i 2 = 1 i=1 z i e 2πi k zi. The metric on AdS 4 S 7 ( ) 1 ds 2 = L 2 4 ds2 AdS 4 + ds 2 S 7 /Z k. Non-perturbative effects in ABJM theory 19 / 43

The AdS/CFT correspondence ABJM theory (3d N =6 SCFT) M-theory on AdS 4 S 7 /Z k k, Chern-Simons level Z k, purely geometric interpretation N, rank of the gauge group N, number of M2 branes ( ) 6 L = 32π 2 kn. l p Thermodynamic limit M-theory description: N, k fixed. Non-perturbative effects in ABJM theory 20 / 43

Type IIA dual Hopf fibration: M-theory reduction on S 1 S 1 S 7 CP 3 type IIA theory X 11 = AdS 4 S 7 /Z k X 10 = AdS 4 CP 3 We need the circle to be small, and this is achieved when k is large. g st = 1 k 2 ( L l s ) 2, N k = λ = 1 32π 2 ( ) 4 L. l s Non-perturbative effects in ABJM theory 21 / 43

Perturbative regime of the type IIA superstring corresponds to the t Hooft limit, N, λ = N k fixed. g st SUGRA planar limit λ 1 point-particle limit λ 1 λ strongly coupled non-linear σ model Non-perturbative effects in ABJM theory 22 / 43

Simplest prediction of AdS/CFT Z ABJM (S 3 ) = Z M-theory/string theory (X 11 /X 10 ). Non-perturbative effects in ABJM theory 23 / 43

Simplest prediction of AdS/CFT Z ABJM (S 3 ) = Z M-theory/string theory (X 11 /X 10 ). At large distances: Z M-theory (X 11 ) Z SUGRA (X 11 ). Non-perturbative effects in ABJM theory 23 / 43

Simplest prediction of AdS/CFT At large distances: Z ABJM (S 3 ) = Z M-theory/string theory (X 11 /X 10 ). Z M-theory (X 11 ) Z SUGRA (X 11 ). In the case of string theory we know a little bit more, { } Z string theory (X 10 ) exp F g (λ)g 2g 2 st. g=0 Non-perturbative effects in ABJM theory 23 / 43

N and k are integers. λ and g st are continuous! Non-perturbative effects in ABJM theory 24 / 43

N and k are integers. λ and g st are continuous! If we define F (N, k) = log Z(N, k), one finds, from the supergravity approximation to M-theory F (N, k) π 2 3 k1/2 N 3/2, N 1. Non-perturbative effects in ABJM theory 24 / 43

N and k are integers. λ and g st are continuous! If we define F (N, k) = log Z(N, k), one finds, from the supergravity approximation to M-theory F (N, k) π 2 3 k1/2 N 3/2, N 1. Planar free energy of ABJM theory at strong t Hooft coupling lim N 1 N 2 F (N, λ) π 2 3 λ, λ 1. F 0 (λ) λ 2 log λ, perturbation theory. Non-perturbative effects in ABJM theory 24 / 43

One can obtain explicit formulae for the genus g free energies appearing in the 1/N expansion: F (λ, g s ) = g=0 F g (λ) g 2g 2 s, g s = 2πi k. Non-perturbative effects in ABJM theory 25 / 43

The matrix integral describing Z ABJM (S 3 ) Z ABJM (N, k) = 1 [ d N µ d N ν i<j N! 2 (2π) N (2π) N [ ] ik N exp (µ 2 i νi 2 ). 4π i=1 2 sinh i,j ( µi µ j 2 [ 2 cosh )] 2 [ 2 sinh )] 2 ( µi ν j 2 ( νi ν j 2 )] 2 Non-perturbative effects in ABJM theory 26 / 43

M-theory regime: N, k fixed. Non-perturbative effects in ABJM theory 27 / 43

M-theory regime: N, k fixed. t Hooft regime: N, N k = λ fixed. Non-perturbative effects in ABJM theory 27 / 43

M-theory regime: N, k fixed. t Hooft regime: N, N k = λ fixed. The t Hooft expansion F 0 (ˆλ) = 4π3 2 ˆλ 3/2 + ζ(3) 3 2 + ( ) e 2πl 2ˆλf 1 l. l 1 π 2ˆλ f l (x) is a polynomial in x of degree 2l 3 (for l 2) and ˆλ = λ 1 24. Non-perturbative effects in ABJM theory 27 / 43

AdS 4 CP 3 CP 3 l CP 1 Area = 2π 2λ S 2 Spherical strings (genus zero sector) Non-perturbative effects in ABJM theory 28 / 43

F g (λ) can be computed recursively. F g (λ) = 1 λ g 3 2 polynomial in λ 1 2 + cg + l 1 e 2πl 2ˆλf g,l ( ) 1. π 2ˆλ Non-perturbative effects in ABJM theory 29 / 43

F g (λ) can be computed recursively. F g (λ) = 1 λ g 3 2 polynomial in λ 1 2 + cg + l 1 e 2πl 2ˆλf g,l ( ) 1. π 2ˆλ General arguments suggest that this series diverges factorially, F g (λ) c g (2g)! A st (λ) 2g, g 1. Non-perturbative effects in ABJM theory 29 / 43

Membrane instanton/d2-brane instanton ( exp 2πk 1/2 N 1/2), in M-theory variables: N ( ) 3 L exp. l p ( L l p ) 6 Non-perturbative effects in ABJM theory 30 / 43

Membrane instanton/d2-brane instanton ( exp 2πk 1/2 N 1/2), in M-theory variables: N ( ) 3 L exp. l p ( L l p ) 6 The genus expansion is Borel summable. Non-perturbative effects in ABJM theory 30 / 43

Membrane instanton/d2-brane instanton ( exp 2πk 1/2 N 1/2), in M-theory variables: N ( ) 3 L exp. l p ( L l p ) 6 The genus expansion is Borel summable. F (N, k) Borel summation of F g (λ) Complex saddle point! The membrane instantons are lack in our analysis! Non-perturbative effects in ABJM theory 30 / 43

The M-theory expansion: N, k fixed Non-perturbative effects in ABJM theory 31 / 43

The M-theory expansion: N, k fixed The Fermi gas approach Cauchy identity: [ i<j 2 sinh = i,j ( µi µ j 2 σ S N ( 1) ɛ(σ) [ 2 cosh i )] 2 [ ( )] 2 νi ν 2 sinh j 2 1 )] 2 = det ij ( µi ν j 2 2 cosh 1 ( µi ν σ(i) 2 2 cosh ( µi ν j 2 S N is the permutation group of N elements, and ɛ(σ) is the signature of the permutation σ. ). ) Non-perturbative effects in ABJM theory 31 / 43

Canonical partition function of a free Fermi gas with N particles with Z(N, k) = 1 N! ρ (x 1, x 2 ) = 1 2πk σ S N ( 1) ɛ(σ) 1 [ ( 2 cosh x1 )] 1/2 2 d N x (2π) N ρ ( ) x i, x σ(i), 1 [ ( 2 cosh x2 )] 1/2 2 i 1 [ 2 cosh ( x1 x 2 2 )] 1/2. The canonical density matrix is related to the Hamiltonian operator Ĥ in the usual way, ρ(x 1, x 2 ) = x 1 ˆρ x 2, ˆρ = e Ĥ. where the inverse temperature β = 1 is fixed. Non-perturbative effects in ABJM theory 32 / 43

Grand canonical ensemble Non-perturbative effects in ABJM theory 33 / 43

Grand canonical ensemble Grand canonical partition function Ξ(µ, k) = 1 + Z(N, k)e Nµ. N 1 Here, µ is the chemical potential. Non-perturbative effects in ABJM theory 33 / 43

Grand canonical ensemble Grand canonical partition function Ξ(µ, k) = 1 + Z(N, k)e Nµ. N 1 Here, µ is the chemical potential. Grand canonical potential J (µ, k) = log Ξ(µ, k) = ( κ) l Z l, l l 1 where κ = e µ (fugacity), Z l = Tr ˆρ l. Non-perturbative effects in ABJM theory 33 / 43

The canonical partition function is recovered from the grand canonical one by integration, dκ Ξ(µ, k) Z(N, k) = 2πi κ N+1. The spectrum of the Hamiltonian Ĥ is defined by, ˆρ ϕ n = e En ϕ n. n = 0, 1,..., or equivalently by the integral equation associated to the kernel, ρ(x, x )ϕ n (x )dx = e En ϕ n (x). n = 0, 1,.... The spectrum is discrete and the energies are real. Non-perturbative effects in ABJM theory 34 / 43

The grand canonical partition function is given by the Fredholm determinant, Ξ(µ, k) = det(1 + κˆρ) = n 0 ( 1 + κe E n ). In terms of the density of eigenvalues ρ(e) = n 1 δ(e E n ), we also have the standard formula J (µ, k) = 0 deρ(e) log ( 1 + κe E). Non-perturbative effects in ABJM theory 35 / 43

Density matrix ˆρ = e 1 2 U(ˆx) e T (ˆp) e 1 2 U(ˆx), with [ ( x )] [ ( p U(x) = log 2 cosh, T (p) = log 2 cosh. 2 2)] ˆx, ˆp are canonically conjugate operators, [ˆx, ˆp] = i, = 2πk. Non-perturbative effects in ABJM theory 36 / 43

Density matrix ˆρ = e 1 2 U(ˆx) e T (ˆp) e 1 2 U(ˆx), with [ ( x )] [ ( p U(x) = log 2 cosh, T (p) = log 2 cosh. 2 2)] ˆx, ˆp are canonically conjugate operators, [ˆx, ˆp] = i, = 2πk. Note that is the inverse coupling constant of the gauge theory/string theory. Non-perturbative effects in ABJM theory 36 / 43

The potential U(x) is a confining one, and at large x it behaves linearly, U(x) x 2, x. When N, the typical energies are large, and we are in the semiclassical regime, H cl (x, p) = U(x) + T (p) x 2 + p 2. (1) Fermi surface: H cl (x, p) = E For large values of the energies, the Fermi surface is very well approximated by the polygon (1). J (µ, k) µ = N(µ, k) Vol(E) Vol. of an elementary cell 8µ2 2π. Non-perturbative effects in ABJM theory 37 / 43

Grand canonical potential: J (µ, k) 2µ3 3π 2 k. At large N, the contour integral Z(N, k) = dκ Ξ(µ, k) 2πi κ N+1. can be computed by a saddle-point approximation. Non-perturbative effects in ABJM theory 38 / 43

Grand canonical potential: J (µ, k) 2µ3 3π 2 k. At large N, the contour integral Z(N, k) = dκ Ξ(µ, k) 2πi κ N+1. can be computed by a saddle-point approximation. Free energy: F (N, k) J (µ, k) Nµ, where µ is the function of N and k defined by, 2 µ 2 πk1/2 N 1/2. Non-perturbative effects in ABJM theory 38 / 43

In the strict large N limit F (N, k) π 2 3 N 3/2 k 1/2. Non-perturbative effects in ABJM theory 39 / 43

In the strict large N limit F (N, k) π 2 3 N 3/2 k 1/2. The WKB expansion of the grand potential reads, J WKB (µ, k) = n 0 J n (µ)k 2n 1. Non-perturbative effects in ABJM theory 39 / 43

In the strict large N limit F (N, k) π 2 3 N 3/2 k 1/2. The WKB expansion of the grand potential reads, J WKB (µ, k) = n 0 J n (µ)k 2n 1. Leading term n = 0 Z l = Tr e lĥ dxdp 2π e lh cl(x,p), k 0. Non-perturbative effects in ABJM theory 39 / 43

where J 0 (µ) = 4µ3 3π 2 + µ 3 + 2ζ(3) π 2 + J0 M2 (µ), J M2 0 (µ) = ( a0,l µ 2 ) + b 0,l µ + c 0,l e 2lµ. l=1 Non-perturbative effects in ABJM theory 40 / 43

where J 0 (µ) = 4µ3 3π 2 + µ 3 + 2ζ(3) π 2 + J0 M2 (µ), J M2 0 (µ) = ( a0,l µ 2 ) + b 0,l µ + c 0,l e 2lµ. l=1 Fermi gas approach makes it possible to go beyond the strict large N limit. e 2µ e 2πk 1/2 N 1/2 e ( ) 3 L lp. Non-perturbative effects in ABJM theory 40 / 43

Non-renormalization theorem J n (µ) = A n + O ( µ 2 e 2µ), n 2. Non-perturbative effects in ABJM theory 41 / 43

Non-renormalization theorem J n (µ) = A n + O ( µ 2 e 2µ), n 2. J WKB (µ, k) = J (p) (µ) + J M2 (µ, k). The perturbative piece J (p) (µ) is given by J p (µ) = C(k) 3 µ3 + B(k)µ + A(k), C(k) = 2 π 2 k, B(k) = 1 3k + k 24, A(k) = n 0 A n k 2n 1, where A 0 = 2ζ(3) π 2, A 1 = 1 12. Non-perturbative effects in ABJM theory 41 / 43

We find that, up to exponentially small corrections, Z(N, k) 1 ( ) exp J (p) (µ) µn dµ 2πi = 1 [ ] C(k) exp 2πi 3 µ3 + (B(k) N) µ + A(k) dµ C { } = e A(k) C(k) 1/3 Ai C(k) 1/3 [N B(k)]. Non-perturbative effects in ABJM theory 42 / 43

We find that, up to exponentially small corrections, Z(N, k) 1 ( ) exp J (p) (µ) µn dµ 2πi = 1 [ ] C(k) exp 2πi 3 µ3 + (B(k) N) µ + A(k) dµ C { } = e A(k) C(k) 1/3 Ai C(k) 1/3 [N B(k)]. M-theory regime F (N, k) 1 384π 2 k ζ3/2 + 1 6 log where ζ = 32π 2 k [N B(k)]. ( π 3 k 3 ζ 3/2 ) +A(k) + d n+1 π 2n k n ζ 3n/2, n=1 Non-perturbative effects in ABJM theory 42 / 43

Thank you for your attention! Non-perturbative effects in ABJM theory 43 / 43