Numerical results in ABJM theory

Transcription

1 Numerical results in ABJM theory Riccardo Conti Friday 3ʳᵈ March 2017 D. Bombardelli, A. Cavaglià, R. Conti and R. Tateo (in progress)

2 Overview Purpose: Solving the Quantum Spectral Curve (QSC) equations to find the conformal dimensions of ABJM operators with high precision. N. Gromov, V. Kazakov, S. Leurent and D. Volin (2013) A. Cavaglià, D. Fioravanti, N. Gromov and R.Tateo (2014) C. Marboe and D. Volin (2014) N. Gromov, F. Levkovich Maslyuk and G. Sizov (2015) Applications: o Finding the strong coupling expansion coefficients of ; o Studying the analytic continuation in the complex plane of the coupling around the branch point at (still not fully understood also in 4SYM)

3 ABJM model Aharony, Bergman, Jafferis, Maldacena (2008) super Chern Simons matter theory, also called ABJM model, o is a 3dimensional CFT; o is a gauge theory with gauge group ; o possesses superconformal symmetry o is dual to type IIA superstring theory on in the planar limit;

4 Nonlinear Riemann Hilbert problem: QSC : Introduction D. Bombardelli, A. Cavaglià, D. Fioravanti, N. Gromov and R. Tateo (2017) System of functional equations in the complex domain of the spectral parameter Metric tensor,,,,,,,,, Building blocks,, Non elementary objects,,,, Transition object, Notation:, functions

5 QSC : Analytic properties functions o are multivalued functions of which live on an infinite sheet Riemann surface; o have branch points (square root type) in the plane whose positions depend on the coupling constant, namely 2. o have a single cut in the first sheet and an infinite ladder in the second sheet. QSC equations relate the discontinuities of the functions across the cuts.

6 Zhukovsky map The Zhukovsky map is defined implicitly as plane linear cut 2; 2 (first sheet) (second sheet) plane unit circle 0 In the plane all the cuts in the second sheet are inside the unit circle. Analytic continuation around a branch point on the first sheet corresponds to.

7 Gluing Conditions D. Bombardelli, A. Cavaglià, R. Conti and R. Tateo (in progress) It is convenient to consider the QSC equations written in this form: Shift notation:, Gluing Conditions: exact relations between on the first sheet and on the second sheet

8 Algorithm: Introduction o Expand in power of around,, 1,, 6 C. Marboe and D. Volin (2014) Important fact:, depends on.find by inverting,. The series converges everywhere outside the barred circle. Thisis important because we will impose gluing conditions on the unit circle.

9 Algorithm: Main picture N. Gromov, F. Levkovich Maslyuk and G. Sizov (2015) Ansatz for the s,,, Find the s from the s solving the QSC equations,, Fix, and by imposing gluing conditions (Newton s method),

10 Match with TBA predictions F. Levkovich Maslyuk (2011) With TBA it is difficult to go beyond 1.Maybedueto singularities crossing the integration contours. P. Dorey and R. Tateo (1998) G. Arutyunov, S. Frolov and R. Suzuki (2009) There is no such problem in the QSC formalism.

11 Strong coupling behaviour Find the coefficients of the strong coupling expansion of M. Beccaria, G. Macorini, C. A. Ratti and S. Valatka (2013) N. Gromov, G. Sizov (2014) / / /, 2πlog2,, No analytic predictions for the 1,1case Need more strong coupling data to fix the remaining coefficients!

12 What is the difficulty in going higher in? As increases the branch points approach the unit circle (in particular the points 1), thus reducing the convergence radius of the power expansion of the.

13 Analytical continuation in Considering pure imaginary coupling, weird things happen when approaches The cuts are well separated The cuts merge together The cuts overlap Numerical evidences suggest that has a square root type branch point at

14 What happens when we move around? Impossibility to overcome the imaginary axis for ; is a branch cut o The branch points leave the unit circle. o There is more than a single branch cut in the first sheet of the s. The working hypothesis isn t true anymore!

15 Idea: cuts deformation A sin shaped deformation (as an example) prevents the overlapping of the cuts when In this way the branch points in the second sheet can t reach the first sheet.

16 Doubled Zhukovsky transformation: 1 2 maps circles in the plane into ellipses in the plane. Now expand in power of around,, 1,, 6 o Blue dotted curve: deformed cut; o Green ellipse: convergence region of the s; The deformed cut is completely inside the convergence region of the new expansion of the.

17 Preliminary results Idea: explore numerically the second sheet of

18 Summary & Conclusions Summary: o QSC is a powerful tool to compute the spectrum with high precision; o The algorithm is effective also at strong coupling; o Trick to perform analytic continuation around the branch point at ; o Plan: explore the second sheet numerically; Open problems: o Can we find a similar trick to simplify the computation at strong coupling? o What about the other branch points at, 1,, in the complex plane of? o What is the physical interpretation of the second sheet in?

19 Thank you for the attention!