Non-perturbative effects in ABJM theory
|
|
- Jemimah Banks
- 6 years ago
- Views:
Transcription
1 October 9, 2015 Non-perturbative effects in ABJM theory 1 / 43
2 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
3 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 q2 2 }{{} harmonic oscillator + g 4 q4 }{{} stationary perturbation theory. E 0 (g) n 0 = a n g n ( g ) 21 ( g ) ( g ) O(g 4 ) = ϕ(g) Non-perturbative effects in ABJM theory 3 / 43
4 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
5 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
6 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
7 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
8 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. First indication of a NP effect! Non-perturbative effects in ABJM theory 6 / 43
9 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
10 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 Borel transform ϕ(ζ) = ( 1) n A n ζ n = ζ A n=0. Non-perturbative effects in ABJM theory 7 / 43
11 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
12 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
13 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
14 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
15 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
16 General aspects Example: double-well potential in QM H = p2 2 + W (q), W (q) = g ( q 2 1 ) 2, g > g Ground state energy ϕ 0 (g) = 1 2 g 9 2 g 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
17 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
18 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
19 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
20 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. Non-perturbative effects in ABJM theory 15 / 43
21 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
22 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
23 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 Non-perturbative effects in ABJM theory 17 / 43
24 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
25 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
26 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
27 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
28 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
29 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
30 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
31 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
32 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
33 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 The gauge theory is providing a non-perturbative definition of this asymptotic expansion. Non-perturbative effects in ABJM theory 23 / 43
34 N and k are integers. λ and g st are continuous! Non-perturbative effects in ABJM theory 24 / 43
35 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
36 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
37 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
38 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
39 M-theory regime: N, k fixed. Non-perturbative effects in ABJM theory 27 / 43
40 M-theory regime: N, k fixed. t Hooft regime: N, N k = λ fixed. Non-perturbative effects in ABJM theory 27 / 43
41 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) ( ) 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 Non-perturbative effects in ABJM theory 27 / 43
42 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) ( ) 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 N k = λ ( L l s ) 4 e ( L ls )2 worldsheet instantons. Non-perturbative effects in ABJM theory 27 / 43
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
44 F g (λ) can be computed recursively. F g (λ) = 1 λ g 3 2 polynomial in λ cg + l 1 e 2πl 2ˆλf g,l ( ) 1. π 2ˆλ Non-perturbative effects in ABJM theory 29 / 43
45 F g (λ) can be computed recursively. F g (λ) = 1 λ g 3 2 polynomial in λ 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
46 F g (λ) can be computed recursively. F g (λ) = 1 λ g 3 2 polynomial in λ 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. Leading NP effect: exp ( A st /g st ) ia st (λ) = 2π 2 ( ) 2λ + π 2 i + O e 2π 2λ, λ 1. Non-perturbative effects in ABJM theory 29 / 43
47 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
48 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
49 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
50 The M-theory expansion: N, k fixed Non-perturbative effects in ABJM theory 31 / 43
51 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
52 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
53 Grand canonical ensemble Non-perturbative effects in ABJM theory 33 / 43
54 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
55 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
56 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
57 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
58 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). What can we learn from the ABJM partition function in the Fermi gas formalism? Non-perturbative effects in ABJM theory 35 / 43
59 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
60 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
61 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) Non-perturbative effects in ABJM theory 37 / 43
62 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
63 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
64 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
65 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
66 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
67 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
68 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
69 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
70 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. Beyond the leading order of the WKB expansion J 1 (µ) = µ O ( µ 2 e 2µ). Non-perturbative effects in ABJM theory 40 / 43
71 Non-renormalization theorem J n (µ) = A n + O ( µ 2 e 2µ), n 2. Non-perturbative effects in ABJM theory 41 / 43
72 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 = Non-perturbative effects in ABJM theory 41 / 43
73 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
74 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/ 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
75 Thank you for your attention! Non-perturbative effects in ABJM theory 43 / 43
Non-perturbative effects in string theory and AdS/CFT
Preprint typeset in JHEP style - PAPER VERSION Non-perturbative effects in string theory and AdS/CFT Marcos Mariño Département de Physique Théorique et Section de Mathématiques, Université de Genève, Genève,
More informationLarge N Non-Perturbative Effects in ABJM Theory
Strings 2015@Bengaluru Large N Non-Perturbative Effects in ABJM Theory Yasuyuki Hatsuda (DESY) Collaborators: A. Grassi, M. Honda, M. Marino, S. Moriyama & K. Okuyama Basic Flow in Localization Path Integrals
More informationarxiv: v3 [hep-th] 7 May 2015
Preprint typeset in JHEP style - PAPER VERSION Resumming the string perturbation series arxiv:1405.4214v3 [hep-th] 7 May 2015 Alba Grassi, Marcos Mariño and Szabolcs Zakany Département de Physique Théorique
More informationQuantum gravity at one-loop and AdS/CFT
Quantum gravity at one-loop and AdS/CFT Marcos Mariño University of Geneva (mostly) based on S. Bhattacharyya, A. Grassi, M.M. and A. Sen, 1210.6057 The AdS/CFT correspondence is supposed to provide a
More informationarxiv: v1 [hep-th] 7 Jun 2013
Preprint typeset in JHEP style - PAPER VERSION DESY 3-096, TIT/HEP-67 Non-perturbative effects and the refined topological string Yasuyuki Hatsuda a, Marcos Mariño b, Sanefumi Moriyama c and Kazumi Okuyama
More informationM-theoretic Matrix Models
M-theoretic Matrix Models Alba Grassi Université de Genève Mostly based on: A.G., M. Mariño, 1403.4276 Outline The ABJM theory: it has been possible to compute exactly the full partition function which
More informationNon-perturbative Effects in ABJM Theory from Fermi Gas Approach
Non-perturbative Effects in ABJM Theory from Fermi Gas Approach Sanefumi Moriyama (Nagoya/KMI) [arxiv:1106.4631] with H.Fuji and S.Hirano [arxiv:1207.4283, 1211.1251, 1301.5184] with Y.Hatsuda and K.Okuyama
More informationMatrix models from operators and topological strings
Preprint typeset in JHEP style - PAPER VERSION Matrix models from operators and topological strings arxiv:150.0958v [hep-th] 3 Jun 015 Marcos Mariño and Szabolcs Zakany Département de Physique Théorique
More informationABJM 行列模型から. [Hatsuda+M+Okuyama 1207, 1211, 1301]
ABJM 行列模型から 位相的弦理論へ Sanefumi Moriyama (NagoyaU/KMI) [Hatsuda+M+Okuyama 1207, 1211, 1301] [HMO+Marino 1306] [HMO+Honda 1306] Lessons from '90s String Theory As Unified Theory String Theory, NOT JUST "A
More informationarxiv: v2 [hep-th] 12 Jan 2012
Preprint typeset in JHEP style - PAPER VERSION Imperial-TP-011-ND-01 Nonperturbative aspects of ABJM theory arxiv:1103.4844v [hep-th] 1 Jan 01 Nadav Drukker a, Marcos Mariño b,c and Pavel Putrov c a The
More informationResurgent Transseries: Beyond (Large N) Perturbative Expansions
Resurgent Transseries: Beyond (Large N) Perturbative Expansions Ricardo Schiappa ( Instituto Superior Técnico ) CERN, October 25, 2013 Ricardo Schiappa (Lisbon, IST) Resurgent Transseries and Large N CERN,
More information1 What s the big deal?
This note is written for a talk given at the graduate student seminar, titled how to solve quantum mechanics with x 4 potential. What s the big deal? The subject of interest is quantum mechanics in an
More informationResurgence Structure to All Orders of Multi-bions in Deformed SUSY Quantum Mechanics
Resurgence Structure to All Orders of Multi-bions in Deformed SUSY Quantum Mechanics Toshiaki Fujimori (Keio University) based on arxiv:1607.04205, Phys.Rev. D94 (2016) arxiv:1702.00589, Phys.Rev. D95
More informationHow to resum perturbative series in supersymmetric gauge theories. Masazumi Honda ( 本多正純 )
How to resum perturbative series in supersymmetric gauge theories Masazumi Honda ( 本多正純 ) References: M.H., Borel Summability of Perturbative Series in 4D N=2 and 5D N=1 Supersymmetric Theories, PRL116,
More informationN = 2 CHERN-SIMONS MATTER THEORIES: RG FLOWS AND IR BEHAVIOR. Silvia Penati. Perugia, 25/6/2010
N = 2 CHERN-SIMONS MATTER THEORIES: RG FLOWS AND IR BEHAVIOR Silvia Penati Perugia, 25/6/2010 Motivations AdS 4 /CFT 3 correspondence states that the strong coupling dynamics of a N = 6 Chern-Simons theory
More informationThree-sphere free energy for classical gauge groups
MIT-CTP-4520 arxiv:32.0920v3 [hep-th] 3 Dec 205 Three-sphere free energy for classical gauge groups Márk Mezei and Silviu S. Pufu Center for Theoretical Physics, Massachusetts Institute of Technology,
More informationAdS 6 /CFT 5 in Type IIB
AdS 6 /CFT 5 in Type IIB Part II: Dualities, tests and applications Christoph Uhlemann UCLA Strings, Branes and Gauge Theories APCTP, July 2018 arxiv: 1606.01254, 1611.09411, 1703.08186, 1705.01561, 1706.00433,
More informationTHE IMAGINARY CUBIC PERTURBATION: A NUMERICAL AND ANALYTIC STUDY JEAN ZINN-JUSTIN
THE IMAGINARY CUBIC PERTURBATION: A NUMERICAL AND ANALYTIC STUDY JEAN ZINN-JUSTIN CEA, IRFU (irfu.cea.fr), IPhT Centre de Saclay 91191 Gif-sur-Yvette Cedex, France and Shanghai University E-mail: jean.zinn-justin@cea.fr
More informationTOPIC V BLACK HOLES IN STRING THEORY
TOPIC V BLACK HOLES IN STRING THEORY Lecture notes Making black holes How should we make a black hole in string theory? A black hole forms when a large amount of mass is collected together. In classical
More informationAsymptotic series in quantum mechanics: anharmonic oscillator
Asymptotic series in quantum mechanics: anharmonic oscillator Facultat de Física, Universitat de Barcelona, Diagonal 645, 0808 Barcelona, Spain Thesis supervisors: Dr Bartomeu Fiol Núñez, Dr Alejandro
More informationConsidering our result for the sum and product of analytic functions, this means that for (a 0, a 1,..., a N ) C N+1, the polynomial.
Lecture 3 Usual complex functions MATH-GA 245.00 Complex Variables Polynomials. Construction f : z z is analytic on all of C since its real and imaginary parts satisfy the Cauchy-Riemann relations and
More informationQuark-gluon plasma from AdS/CFT Correspondence
Quark-gluon plasma from AdS/CFT Correspondence Yi-Ming Zhong Graduate Seminar Department of physics and Astronomy SUNY Stony Brook November 1st, 2010 Yi-Ming Zhong (SUNY Stony Brook) QGP from AdS/CFT Correspondence
More informationHolographic QCD at finite (imaginary) chemical potential
Holographic QCD at finite (imaginary) chemical potential Università di Firenze CRM Montreal, October 19, 2015 Based on work with Francesco Bigazzi (INFN, Pisa), JHEP 1501 (2015) 104 Contents: The Roberge-Weiss
More informationds/cft Contents Lecturer: Prof. Juan Maldacena Transcriber: Alexander Chen August 7, Lecture Lecture 2 5
ds/cft Lecturer: Prof. Juan Maldacena Transcriber: Alexander Chen August 7, 2011 Contents 1 Lecture 1 2 2 Lecture 2 5 1 ds/cft Lecture 1 1 Lecture 1 We will first review calculation of quantum field theory
More informationPlanar Limit Analysis of Matrix Models
Planar Limit Analysis of Matrix Models Submitted in partial fulfilment of the requirements of the degree of Bachelor of Technology by Sarthak Bagaria 10060006 Supervisor : Prof. P. Ramadevi Department
More informationBrane decay in curved space-time
Brane decay in curved space-time Dan Israël, IAP, Univ. Pierre et Marie Curie From D. I. & E. Rabinovici, JHEP 0701 (2007) D. I., JHEP 0704 (2007) D. Israël, Brane decay in curved space-time 1 Outline
More informationBPS Black holes in AdS and a magnetically induced quantum critical point. A. Gnecchi
BPS Black holes in AdS and a magnetically induced quantum critical point A. Gnecchi June 20, 2017 ERICE ISSP Outline Motivations Supersymmetric Black Holes Thermodynamics and Phase Transition Conclusions
More informationM-Theory and Matrix Models
Department of Mathematical Sciences, University of Durham October 31, 2011 1 Why M-Theory? Whats new in M-Theory The M5-Brane 2 Superstrings Outline Why M-Theory? Whats new in M-Theory The M5-Brane There
More informationMembranes and the Emergence of Geometry in MSYM and ABJM
in MSYM and ABJM Department of Mathematics University of Surrey Guildford, GU2 7XH, UK Mathematical Physics Seminar 30 October 2012 Outline 1 Motivation 2 3 4 Outline 1 Motivation 2 3 4 Gauge theory /
More informationInstanton Effects in Orbifold ABJM Theory
Instanton Effects in Orbifold ABJM Theory Sanefumi Moriyama (NagoyaU/KMI) [Honda+M 1404] [M+Nosaka 1407] Contents 1. IntroducOon 2. ABJM 3. GeneralizaOons Message The Exact Instanton Expansion of The ABJM
More informationDuality Chern-Simons Calabi-Yau and Integrals on Moduli Spaces. Kefeng Liu
Duality Chern-Simons Calabi-Yau and Integrals on Moduli Spaces Kefeng Liu Beijing, October 24, 2003 A. String Theory should be the final theory of the world, and should be unique. But now there are Five
More informationHolographic Geometries from Tensor Network States
Holographic Geometries from Tensor Network States J. Molina-Vilaplana 1 1 Universidad Politécnica de Cartagena Perspectives on Quantum Many-Body Entanglement, Mainz, Sep 2013 1 Introduction & Motivation
More informationScaling dimensions at small spin
Princeton Center for Theoretical Science Princeton University March 2, 2012 Great Lakes String Conference, Purdue University arxiv:1109.3154 Spectral problem and AdS/CFT correspondence Spectral problem
More informationOne Loop Tests of Higher Spin AdS/CFT
One Loop Tests of Higher Spin AdS/CFT Simone Giombi UNC-Chapel Hill, Jan. 30 2014 Based on 1308.2337 with I. Klebanov and 1401.0825 with I. Klebanov and B. Safdi Massless higher spins Consistent interactions
More informationt Hooft Loops and S-Duality
t Hooft Loops and S-Duality Jaume Gomis KITP, Dualities in Physics and Mathematics with T. Okuda and D. Trancanelli Motivation 1) Quantum Field Theory Provide the path integral definition of all operators
More informationκ = f (r 0 ) k µ µ k ν = κk ν (5)
1. Horizon regularity and surface gravity Consider a static, spherically symmetric metric of the form where f(r) vanishes at r = r 0 linearly, and g(r 0 ) 0. Show that near r = r 0 the metric is approximately
More informationIntegrability and Finite Size Effects of (AdS 5 /CFT 4 ) β
Integrability and Finite Size Effects of (AdS 5 /CFT 4 ) β Based on arxiv:1201.2635v1 and on-going work With C. Ahn, D. Bombardelli and B-H. Lee February 21, 2012 Table of contents 1 One parameter generalization
More informationElements of Topological M-Theory
Elements of Topological M-Theory (with R. Dijkgraaf, S. Gukov, C. Vafa) Andrew Neitzke March 2005 Preface The topological string on a Calabi-Yau threefold X is (loosely speaking) an integrable spine of
More informationSTOCHASTIC QUANTIZATION AND HOLOGRAPHY
STOCHASTIC QUANTIZATION AND HOLOGRAPHY WORK WITH D.MANSI & A. MAURI: TO APPEAR TASSOS PETKOU UNIVERSITY OF CRETE OUTLINE CONFORMAL HOLOGRAPHY STOCHASTIC QUANTIZATION STOCHASTIC QUANTIZATION VS HOLOGRAPHY
More information10 Interlude: Preview of the AdS/CFT correspondence
10 Interlude: Preview of the AdS/CFT correspondence The rest of this course is, roughly speaking, on the AdS/CFT correspondence, also known as holography or gauge/gravity duality or various permutations
More informationField Theory: The Past 25 Years
Field Theory: The Past 25 Years Nathan Seiberg (IAS) The Future of Physics A celebration of 25 Years of October, 2004 The Nobel Prize in Physics 2004 David J. Gross, H. David Politzer and Frank Wilczek
More informationTwistors, amplitudes and gravity
Twistors, amplitudes and gravity From twistor strings to quantum gravity? L.J.Mason The Mathematical Institute, Oxford lmason@maths.ox.ac.uk LQG, Zakopane 4/3/2010 Based on JHEP10(2005)009 (hep-th/0507269),
More informationThermodynamics of black branes as interacting branes
Thermodynamics of black branes as interacting branes Shotaro Shiba (KEK, Japan) East Asia Joint Workshop on Fields and Strings on May 29, 2016 Branes in supergravity 4d Einstein gravity Blackhole solutions
More informationIntroduction to Path Integrals
Introduction to Path Integrals Consider ordinary quantum mechanics of a single particle in one space dimension. Let s work in the coordinate space and study the evolution kernel Ut B, x B ; T A, x A )
More informationThéorie des cordes: quelques applications. Cours IV: 11 février 2011
Particules Élémentaires, Gravitation et Cosmologie Année 2010-11 Théorie des cordes: quelques applications Cours IV: 11 février 2011 Résumé des cours 2009-10: quatrième partie 11 février 2011 G. Veneziano,
More informationarxiv: v4 [hep-th] 17 Mar 2009
Preprint typeset in JHEP style - PAPER VERSION CERN-PH-TH/2008-86 Multi Instantons and Multi Cuts arxiv:0809.269v4 [hep-th] 7 Mar 2009 Marcos Mariño, Ricardo Schiappa 2 and Marlene Weiss 3,4 Section de
More information1. The COMPLEX PLANE AND ELEMENTARY FUNCTIONS: Complex numbers; stereographic projection; simple and multiple connectivity, elementary functions.
Complex Analysis Qualifying Examination 1 The COMPLEX PLANE AND ELEMENTARY FUNCTIONS: Complex numbers; stereographic projection; simple and multiple connectivity, elementary functions 2 ANALYTIC FUNCTIONS:
More informationChern-Simons Theory & Topological Strings
1 Chern-Simons Theory & Topological Strings Bonn August 6, 2009 Albrecht Klemm 2 Topological String Theory on Calabi-Yau manifolds A-model & Integrality Conjectures B-model Mirror Duality & Large N-Duality
More informationQuantization of gravity, giants and sound waves p.1/12
Quantization of gravity, giants and sound waves Gautam Mandal ISM06 December 14, 2006 Quantization of gravity, giants and sound waves p.1/12 Based on... GM 0502104 A.Dhar, GM, N.Suryanarayana 0509164 A.Dhar,
More informationA supermatrix model for ABJM theory
A supermatrix model for ABJM theory Nadav Drukker Humboldt Universität zu Berlin Based on arxiv:0912.3006: and arxiv:0909.4559: arxiv:0912.3974: N.D and D. Trancanelli A. Kapustin, B. Willett, I. Yaakov
More informationFluctuations for su(2) from first principles
Fluctuations for su(2) from first principles Benoît Vicedo DAMTP, Cambridge University, UK AdS/CFT and Integrability Friday March 14-th, 2008 Outline Semiclassical quantisation The zero-mode problem Finite
More informationEmergent geometry: seeing further from the shoulders of giants.
Emergent geometry: seeing further from the shoulders of giants. David Berenstein, UCSB. Chapel Hill, May 8, 2014 Based mostly on arxiv:1301.3519 + arxiv:1305.2394 w. E. Dzienkowski + work in progress.
More informationBoost-invariant dynamics near and far from equilibrium physics and AdS/CFT.
Boost-invariant dynamics near and far from equilibrium physics and AdS/CFT. Micha l P. Heller michal.heller@uj.edu.pl Department of Theory of Complex Systems Institute of Physics, Jagiellonian University
More informationHeterotic Torsional Backgrounds, from Supergravity to CFT
Heterotic Torsional Backgrounds, from Supergravity to CFT IAP, Université Pierre et Marie Curie Eurostrings@Madrid, June 2010 L.Carlevaro, D.I. and M. Petropoulos, arxiv:0812.3391 L.Carlevaro and D.I.,
More informationHolographic Entanglement Beyond Classical Gravity
Holographic Entanglement Beyond Classical Gravity Xi Dong Stanford University August 2, 2013 Based on arxiv:1306.4682 with Taylor Barrella, Sean Hartnoll, and Victoria Martin See also [Faulkner (1303.7221)]
More informationLecture 9: RR-sector and D-branes
Lecture 9: RR-sector and D-branes José D. Edelstein University of Santiago de Compostela STRING THEORY Santiago de Compostela, March 6, 2013 José D. Edelstein (USC) Lecture 9: RR-sector and D-branes 6-mar-2013
More informationSUPERSTRING REALIZATIONS OF SUPERGRAVITY IN TEN AND LOWER DIMENSIONS. John H. Schwarz. Dedicated to the memory of Joël Scherk
SUPERSTRING REALIZATIONS OF SUPERGRAVITY IN TEN AND LOWER DIMENSIONS John H. Schwarz Dedicated to the memory of Joël Scherk SOME FAMOUS SCHERK PAPERS Dual Models For Nonhadrons J. Scherk, J. H. Schwarz
More informationEric Perlmutter, DAMTP, Cambridge
Eric Perlmutter, DAMTP, Cambridge Based on work with: P. Kraus; T. Prochazka, J. Raeymaekers ; E. Hijano, P. Kraus; M. Gaberdiel, K. Jin TAMU Workshop, Holography and its applications, April 10, 2013 1.
More informationarxiv: v1 [hep-th] 5 Jan 2015
Prepared for submission to JHEP YITP-SB-14-55 Finite N from Resurgent Large N arxiv:151.17v1 [hep-th] 5 Jan 215 Ricardo Couso-Santamaría, a Ricardo Schiappa, a Ricardo Vaz b a CAMGSD, Departamento de Matemática,
More informationAnomalies, Gauss laws, and Page charges in M-theory. Gregory Moore. Strings 2004, Paris. Related works: Witten , , ,
Anomalies, Gauss laws, and Page charges in M-theory Gregory Moore Strings 2004, Paris Related works: Witten 9609122,9610234,9812012,9912086 Diaconescu, Moore, and Witten 00 Diaconescu, Freed, and Moore
More informationQuantum Dynamics of Supergravity
Quantum Dynamics of Supergravity David Tong Work with Carl Turner Based on arxiv:1408.3418 Crete, September 2014 An Old Idea: Euclidean Quantum Gravity Z = Dg exp d 4 x g R topology A Preview of the Main
More informationBispectrum from open inflation
Bispectrum from open inflation φ φ Kazuyuki Sugimura (YITP, Kyoto University) Y TP YUKAWA INSTITUTE FOR THEORETICAL PHYSICS K. S., E. Komatsu, accepted by JCAP, arxiv: 1309.1579 Bispectrum from a inflation
More informationM-theory S-Matrix from 3d SCFT
M-theory S-Matrix from 3d SCFT Silviu S. Pufu, Princeton University Based on: arxiv:1711.07343 with N. Agmon and S. Chester arxiv:1804.00949 with S. Chester and X. Yin Also: arxiv:1406.4814, arxiv:1412.0334
More informationGauge-string duality in lattice gauge theories. Sourav Chatterjee
Yang Mills theories Maxwell s equations are a set of four equations that describe the behavior of an electromagnetic field. Hermann Weyl showed that these four equations are actually the Euler Lagrange
More informationAsymptotic Expansion of N = 4 Dyon Degeneracy
Asymptotic Expansion of N = 4 Dyon Degeneracy Nabamita Banerjee Harish-Chandra Research Institute, Allahabad, India Collaborators: D. Jatkar, A.Sen References: (1) arxiv:0807.1314 [hep-th] (2) arxiv:0810.3472
More informationIntroduction to Instantons. T. Daniel Brennan. Quantum Mechanics. Quantum Field Theory. Effects of Instanton- Matter Interactions.
February 18, 2015 1 2 3 Instantons in Path Integral Formulation of mechanics is based around the propagator: x f e iht / x i In path integral formulation of quantum mechanics we relate the propagator to
More information8.821 F2008 Lecture 8: Large N Counting
8.821 F2008 Lecture 8: Large N Counting Lecturer: McGreevy Scribe: Swingle October 4, 2008 1 Introduction Today we ll continue our discussion of large N scaling in the t Hooft limit of quantum matrix models.
More informationAbstract. PACS number(s): Sq, Ge, Yz
Classical solution of the wave equation M. N. Sergeenko The National Academy of Sciences of Belarus, Institute of Physics Minsk 007, Belarus, Homel State University, Homel 6699, Belarus and Department
More informationIntroduction to String Theory ETH Zurich, HS11. 9 String Backgrounds
Introduction to String Theory ETH Zurich, HS11 Chapter 9 Prof. N. Beisert 9 String Backgrounds Have seen that string spectrum contains graviton. Graviton interacts according to laws of General Relativity.
More informationIntegration in the Complex Plane (Zill & Wright Chapter 18)
Integration in the omplex Plane Zill & Wright hapter 18) 116-4-: omplex Variables Fall 11 ontents 1 ontour Integrals 1.1 Definition and Properties............................. 1. Evaluation.....................................
More informationOne-loop Partition Function in AdS 3 /CFT 2
One-loop Partition Function in AdS 3 /CFT 2 Bin Chen R ITP-PKU 1st East Asia Joint Workshop on Fields and Strings, May 28-30, 2016, USTC, Hefei Based on the work with Jie-qiang Wu, arxiv:1509.02062 Outline
More informationarxiv: v1 [hep-th] 6 Nov 2012
Dual description of a 4d cosmology Michael Smolkin and Neil Turok Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada. Abstract M-theory compactified on S 7 /Z k allows for a
More information1 Infinite-Dimensional Vector Spaces
Theoretical Physics Notes 4: Linear Operators In this installment of the notes, we move from linear operators in a finitedimensional vector space (which can be represented as matrices) to linear operators
More informationMIFPA PiTP Lectures. Katrin Becker 1. Department of Physics, Texas A&M University, College Station, TX 77843, USA. 1
MIFPA-10-34 PiTP Lectures Katrin Becker 1 Department of Physics, Texas A&M University, College Station, TX 77843, USA 1 kbecker@physics.tamu.edu Contents 1 Introduction 2 2 String duality 3 2.1 T-duality
More informationIntroduction to AdS/CFT
Introduction to AdS/CFT D-branes Type IIA string theory: Dp-branes p even (0,2,4,6,8) Type IIB string theory: Dp-branes p odd (1,3,5,7,9) 10D Type IIB two parallel D3-branes low-energy effective description:
More informationM5-branes and Wilson Surfaces! in AdS7/CFT6 Correspondence
M5-branes and Wilson Surfaces! in AdS7/CFT6 Correspondence Hironori Mori (Osaka Univ.) based on arxiv:1404.0930 with Satoshi Yamaguchi (Osaka Univ.) 014/05/8, Holographic vistas on Gravity and Strings
More informationDualities and Topological Strings
Dualities and Topological Strings Strings 2006, Beijing - RD, C. Vafa, E.Verlinde, hep-th/0602087 - work in progress w/ C. Vafa & C. Beasley, L. Hollands Robbert Dijkgraaf University of Amsterdam Topological
More informationWHY BLACK HOLES PHYSICS?
WHY BLACK HOLES PHYSICS? Nicolò Petri 13/10/2015 Nicolò Petri 13/10/2015 1 / 13 General motivations I Find a microscopic description of gravity, compatibile with the Standard Model (SM) and whose low-energy
More informationCold Holographic matter in top-down models
Cold Holographic matter in top-down models A. V. Ramallo Univ. Santiago NumHol016, Santiago de Compostela June 30, 016 Based on 1503.0437, 160.06106, 1604.03665 with Y. Bea, G. Itsios and N. Jokela Motivation
More informationGRAPH QUANTUM MECHANICS
GRAPH QUANTUM MECHANICS PAVEL MNEV Abstract. We discuss the problem of counting paths going along the edges of a graph as a toy model for Feynman s path integral in quantum mechanics. Let Γ be a graph.
More informationString-Theory: Open-closed String Moduli Spaces
String-Theory: Open-closed String Moduli Spaces Heidelberg, 13.10.2014 History of the Universe particular: Epoch of cosmic inflation in the early Universe Inflation and Inflaton φ, potential V (φ) Possible
More informationResurgence, Trans-series and Non-perturbative Physics
Resurgence, Trans-series and Non-perturbative Physics Gerald Dunne University of Connecticut UK Theory Meeting, December 16, 2014 GD & M. Ünsal, 1210.2423, 1210.3646, 1306.4405, 1401.5202 GD, lectures
More informationWhat is F-theory? David R. Morrison. University of California, Santa Barbara
University of California, Santa Barbara Physics and Geometry of F-theory 2015 Max Plack Institute for Physics, Munich 25 February 2015 arxiv:1503.nnnnn Inspired in part by Grassi Halverson Shaneson arxiv:1306.1832
More informationAdS/CFT duality. Agnese Bissi. March 26, Fundamental Problems in Quantum Physics Erice. Mathematical Institute University of Oxford
AdS/CFT duality Agnese Bissi Mathematical Institute University of Oxford March 26, 2015 Fundamental Problems in Quantum Physics Erice What is it about? AdS=Anti de Sitter Maximally symmetric solution of
More informationNUMERICAL CALCULATION OF RANDOM MATRIX DISTRIBUTIONS AND ORTHOGONAL POLYNOMIALS. Sheehan Olver NA Group, Oxford
NUMERICAL CALCULATION OF RANDOM MATRIX DISTRIBUTIONS AND ORTHOGONAL POLYNOMIALS Sheehan Olver NA Group, Oxford We are interested in numerically computing eigenvalue statistics of the GUE ensembles, i.e.,
More informationF-theory effective physics via M-theory. Thomas W. Grimm!! Max Planck Institute for Physics (Werner-Heisenberg-Institut)! Munich
F-theory effective physics via M-theory Thomas W. Grimm Max Planck Institute for Physics (Werner-Heisenberg-Institut) Munich Ahrenshoop conference, July 2014 1 Introduction In recent years there has been
More informationSupersymmetric Gauge Theories in 3d
Supersymmetric Gauge Theories in 3d Nathan Seiberg IAS Intriligator and NS, arxiv:1305.1633 Aharony, Razamat, NS, and Willett, arxiv:1305.3924 3d SUSY Gauge Theories New lessons about dynamics of quantum
More informationFrom Strings to AdS4-Black holes
by Marco Rabbiosi 13 October 2015 Why quantum theory of Gravity? The four foundamental forces are described by Electromagnetic, weak and strong QFT (Standard Model) Gravity Dierential Geometry (General
More informationInverse square potential, scale anomaly, and complex extension
Inverse square potential, scale anomaly, and complex extension Sergej Moroz Seattle, February 2010 Work in collaboration with Richard Schmidt ITP, Heidelberg Outline Introduction and motivation Functional
More informationTESTING ADS/CFT. John H. Schwarz STRINGS 2003
TESTING ADS/CFT John H. Schwarz STRINGS 2003 July 6, 2003 1 INTRODUCTION During the past few years 1 Blau et al. constructed a maximally supersymmetric plane-wave background of type IIB string theory as
More informationContinuum limit of fishnet graphs and AdS sigma model
Continuum limit of fishnet graphs and AdS sigma model Benjamin Basso LPTENS 15th Workshop on Non-Perturbative QCD, IAP, Paris, June 2018 based on work done in collaboration with De-liang Zhong Motivation
More information7 Asymptotics for Meromorphic Functions
Lecture G jacques@ucsd.edu 7 Asymptotics for Meromorphic Functions Hadamard s Theorem gives a broad description of the exponential growth of coefficients in power series, but the notion of exponential
More informationQuantization of the open string on exact plane waves and non-commutative wave fronts
Quantization of the open string on exact plane waves and non-commutative wave fronts F. Ruiz Ruiz (UCM Madrid) Miami 2007, December 13-18 arxiv:0711.2991 [hep-th], with G. Horcajada Motivation On-going
More informationPutting String Theory to the Test with AdS/CFT
Putting String Theory to the Test with AdS/CFT Leopoldo A. Pando Zayas University of Iowa Department Colloquium L = 1 4g 2 Ga µνg a µν + j G a µν = µ A a ν ν A a µ + if a bc Ab µa c ν, D µ = µ + it a
More informationFirst Problem Set for Physics 847 (Statistical Physics II)
First Problem Set for Physics 847 (Statistical Physics II) Important dates: Feb 0 0:30am-:8pm midterm exam, Mar 6 9:30am-:8am final exam Due date: Tuesday, Jan 3. Review 0 points Let us start by reviewing
More information4.3 Classical model at low temperature: an example Continuum limit and path integral The two-point function: perturbative
Contents Gaussian integrals.......................... Generating function.......................2 Gaussian expectation values. Wick s theorem........... 2.3 Perturbed gaussian measure. Connected contributions.......
More informationThe Sommerfeld Polynomial Method: Harmonic Oscillator Example
Chemistry 460 Fall 2017 Dr. Jean M. Standard October 2, 2017 The Sommerfeld Polynomial Method: Harmonic Oscillator Example Scaling the Harmonic Oscillator Equation Recall the basic definitions of the harmonic
More informationT-reflection and the vacuum energy in confining large N theories
T-reflection and the vacuum energy in confining large N theories Aleksey Cherman! FTPI, University of Minnesota! with Gokce Basar (Stony Brook -> U. Maryland),! David McGady (Princeton U.),! and Masahito
More informationQGP, Hydrodynamics and the AdS/CFT correspondence
QGP, Hydrodynamics and the AdS/CFT correspondence Adrián Soto Stony Brook University October 25th 2010 Adrián Soto (Stony Brook University) QGP, Hydrodynamics and AdS/CFT October 25th 2010 1 / 18 Outline
More informationABJM Baryon Stability at Finite t Hooft Coupling
ABJM Baryon Stability at Finite t Hooft Coupling Yolanda Lozano (U. Oviedo) Santiago de Compostela, October 2011 - Motivation: Study the stability of non-singlet baryon vertex-like configurations in ABJM
More information