Numerical results in ABJM theory
Riccardo Conti
Friday 3ʳᵈ March 2017
D. Bombardelli, A. Cavaglià, R. Conti and R. Tateo (in progress)
Overview
Purpose:
Solving the Quantum Spectral Curve (QSC) equations to find the conformal dimensions ∆
ABJM operators with high precision.
of 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
(still not fully understood also in 4 SYM)
point at
around the branch
ABJM model
Aharony, Bergman, Jafferis, Maldacena (2008) super Chern‐Simons‐matter theory , also called ABJM model,
o is a 3
dimensional 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;
QSC : Introduction
D. Bombardelli, A. Cavaglià, D. Fioravanti, N. Gromov and R. Tateo (2017)
Nonlinear Riemann‐Hilbert problem:
System of functional equations in the complex domain of the spectral parameter
Metric tensor
, , , , ,
,
,
Building blocks
Non‐elementary objects
∈
,
∈
Transition object ⇆
,
∈
|
Notation: , ⟶
∈
functions
,
, , , ,
∘
,
∈
,
QSC : Analytic properties
functions
o are multivalued functions of
surface;
which live on an infinite‐sheet Riemann
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.
Zhukovsky map
The Zhukovsky map is defined implicitly as
∶
plane
linear cut
2 ; 2
∞ (first sheet)
∞ (second sheet)
In the
→
∶
2
2
→
plane
unit circle
∞
0
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
⟶ .
2
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
Algorithm: Introduction
o Expand in power of around
∞
,
Important fact:
, ∆ .
,
depends on ∆
. Find ∆
,
by inverting
The series converges everywhere outside the barred circle. This is
important because we will impose gluing conditions on the unit
circle.
1, … , 6
C. Marboe and D. Volin (2014)
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)
|
,
|
|
•
,
∆
|
Match with TBA predictions
F. Levkovich‐Maslyuk (2011)
With TBA it is difficult to go beyond
1. Maybe due to
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.
Strong coupling behaviour
M. Beccaria, G. Macorini, C. A. Ratti and S. Valatka (2013)
N. Gromov, G. Sizov (2014)
Find the coefficients of the strong coupling expansion of ∆
∆
/
∆
∆
/
∆
/
∆
⋯ ,
,
log 2
,
∆
∆
2π
0
2
1.999956
3.4 · 10
1
‐0.5
‐0.4999952
4.8 · 10
2
1.5625
1.562509
9.1 · 10
3
‐2.439979…
‐2.439924
5.5 · 10
∆
∆
∆
0
2
∆
1.8 · 10
1.999964
No analytic predictions for the 1,
Need more strong coupling data to fix the remaining coefficients!
1 case
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 .
Analytical continuation in Considering pure imaginary coupling, weird things happen when
1
4
The cuts are well separated
approaches
1
4
1
4
The cuts merge together
Numerical evidences suggest that ∆
The cuts overlap
has a square root type branch point at
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! ?
Idea: cuts deformation
A sin shaped deformation (as an example) prevents the overlapping of the cuts when
1
4
1
4
In this way the branch points in the second sheet can’t reach the first sheet.
1
4
’’Doubled‐Zhukovsky’’ transformation: 1
2
maps circles in the
Now expand
plane into ellipses in the
in power of
around
,
,
plane.
∞
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 .
Preliminary results
Idea: explore numerically the second sheet of ∆
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 ∆
?
Thank you for the attention!