Bootstrapping the N = 2 landscape

Transcription

1 Bootstrapping the N = 2 landscape Pedro Liendo DESY Hamburg November Simons Foundation Meeting 2017 The N = 2 bootstrappers: C. Beem, M. Cornagliotto, M. Lemos, W. Peelaers, I. Ramirez, L. Rastelli, J. Seo, B. van Rees. Pedro Liendo (DESY) N = 2 Bootstrap November / 18

2 Outline 1 Find general constraints 2 Solve individual models Pedro Liendo (DESY) N = 2 Bootstrap November / 18

3 A vast zoo of N = 2 SCFTs N = 2 superconformal dynamics is rich. Ample catalog of theories: Lagrangian theories built with vectors and hypers such that β(g) = 0. Theories obtained through Seiberg-Witten theory: some strongly interacting with no known Lagrangian. Low-energy limit of string theory on D3-branes. Gaiotto theories obtained by compactifying 6d (2, 0) theories on Riemann surfaces. Pedro Liendo (DESY) N = 2 Bootstrap November / 18

4 Is there an underlying structure? Having such a vast zoo, instead of solving theories one by one, maybe we should try to find the underlying principles, and attempt to classify N = 2 SCFTs. Work in this direction includes Classification of Lagrangian theories. (Bhardwaj,Tachikawa) Classification of scale-invariant Coulomb branch geometries. (Argyres, Lotito, Lu, Martone, Xie, Yau) Classification of Gaiotto theories. (Chacaltana, Distler, Tachikawa, Trimm) The Superconformal Bootstrap. (The N = 2 bootstrappers) Pedro Liendo (DESY) N = 2 Bootstrap November / 18

5 Crossing Symmetry Invariance under the interchange of x 1 x 3 gives crossing symmetry: v φ (1 + O λ 2 Og O (u, v)) = u φ (1 + O λ 2 Og O (v, u)) It can be represented pictorially, O O = O O Overconstrained system for the CFT data, but what to do with it? Pedro Liendo (DESY) N = 2 Bootstrap November / 18

6 A solvable truncation Let s go back to crossing 1 + O ( ) λ 2 z z φ(1 OG O (z, z) = + λ 2 (1 z)(1 z) OG O (1 z, 1 z)) where we have defined z z = u, (1 z)(1 z) = v. In N = 2 SCFTs there is a solvable truncation. λ 2 Oh O (z) λ 2 Oh O (1 z) O O The correlators in the subsector are meromorphic: O(0)O(1)O(z, z)o( ) f (z) They can be thought of as correlators describing a 2d chiral algebra. O Pedro Liendo (DESY) N = 2 Bootstrap November / 18

7 Why? The N = 1 chiral ring In N = 1 theories there are chiral operators that satisfy [Q α, φ(x)] = 0 The translation generators are Q-exact: P α α = {Q α, Q α } Derivatives of φ are also Q-exact: α α φ(x) = [P α α, φ(x)] = {Q α, X α } One can then prove φ(x 1 )φ(x 2 ) φ(x n ) = 0 The correlator doesn t care about Q-exact terms. Pedro Liendo (DESY) N = 2 Bootstrap November / 18

8 4d N = 2 SCFT 2d chiral algebra Any 4d N = 2 SCFT contains a protected subsector Global SL(2, R) Virasoro Global flavor Affine Symmetry 4d SCFT 2d Chiral Algebra J R (z, z) T (z), c 2d = 12 c 4d. M ij (z, z) J(z), k 2d = 1 2 k 4d. The 4d stress tensor and flavor current are T µν (Q QJ R ) µν and J F µ (Q QM) µ. Pedro Liendo (DESY) N = 2 Bootstrap November / 18

9 The four-point function of T (z) is in textbooks! T (0)T (z)t (1)T ( ) 1 + z 4 (1 z) c 2d z A truncation of the full correlator J R J R J R J R is solvable. We can expand it in 4d conformal blocks λ 2 OG O (z, z) and solve for an infinite number of OPE coefficients. Assuming absence of higher-spin currents (Maldacena, Zhiboedov) ( λ 2 O 0 = 2 11 ) 15c 4d O Pedro Liendo (DESY) N = 2 Bootstrap November / 18

10 The four-point function of T (z) is in textbooks! T (0)T (z)t (1)T ( ) 1 + z 4 (1 z) c 2d z A truncation of the full correlator J R J R J R J R is solvable. We can expand it in 4d conformal blocks λ 2 OG O (z, z) and solve for an infinite number of OPE coefficients. Assuming absence of higher-spin currents (Maldacena, Zhiboedov) ( λ 2 O 0 = 2 11 ) 15c 4d O Unitarity implies c 4d Pedro Liendo (DESY) N = 2 Bootstrap November / 18

11 The SU(2) landscape We can also look at T (0)T (z)t (1)T ( ) T (0)T (z)j(1)j( ) J(0)J(z)J(1)J( ) Free Hyper H1 (2) H0 (2)L =4 SYM c4 d /k4 d Figure: The landscape of N = 2 SCFTs with flavor group SU(2). Pedro Liendo (DESY) N = 2 Bootstrap November / 18

12 The simplest N = 2 SCFT There is an Argyres-Douglas theory that saturates the bound. Its chiral algebra is the Yang-Lee model. c 2d = 12c 4d c 2d = 22 5 It has no Higgs branch. Its Coulomb branch is parameterized by the vev a single scalar φ φ is chiral with φ = r 0 = 6 5 The central charges are known (Aharony,Tachikawa) c = a = Pedro Liendo (DESY) N = 2 Bootstrap November / 18

13 Bootstrapping the AD theory Let us look at φ(x 1 ) φ(x 2 )φ(x 3 ) φ(x 4 ) The blocks are known (Fitzpatrick et al.). The antichiral OPE reads φ φ 1 + T +... Here T is the stress tensor. The chiral OPE reads φ φ φ 2 + C l +... The operators C l is a family of protected multiplets. The OPE coefficients are not protected and there is always a gap. Pedro Liendo (DESY) N = 2 Bootstrap November / 18

14 Central charge bound Let us be agnostic about the value of c cmin /Λ Figure: Minimum central charge for r 0 = 6 5. Pedro Liendo (DESY) N = 2 Bootstrap November / 18

15 Bounds on φ 2 Let us put rigorous upper and lower bounds on the OPE coefficient of φ 2 2 λ E12/ λ E12/ c /Λ Figure: Numerical upper and lower bounds on the OPE coefficient squared of φ λ 2 φ Pedro Liendo (DESY) N = 2 Bootstrap November / 18

16 The C l semi-short multiplets Let us now look at the C l multiplets, recall Let us also recall φ φ φ 2 + C l +... φ φ 1 + T +... In addition to numerics, there is also an inversion formula [ (z z) r 0 ( ) ] C(z, β) ddisc (1 z) r 1 + λ 0 (1 z) r 0 φ φt 2 G T (1 z, 1 z) Pedro Liendo (DESY) N = 2 Bootstrap November / 18

17 Numerics vs the inversion formula λ 2 2 Rl -λ gfft l λ 2 2 Rl -λ gfft l Figure: The dashed line shows the result of CH, where we considered only the contribution of the identity and stress-tensor operators in the non-chiral channel, and thus is an approximate result for sufficiently large spin. Pedro Liendo (DESY) N = 2 Bootstrap November / 18

18 Summary Every N = 2 SCFT contains a subsector of operators with meromorphic correlators described by a 2d chiral algebra. Studying the interplay between 4d and 2d drescriptions we obtained general analytic bounds valid for any N = 2 SCFT with the given flavor group. We gave the first step towards solving individual models. To do: More numerical bootstrap, develop block technology! Pedro Liendo (DESY) N = 2 Bootstrap November / 18

19 Thank you. Pedro Liendo (DESY) N = 2 Bootstrap November / 18