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Resurgence in η-deformed Principal Chiral Models
Received: May
Resurgence in -deformed Principal Chiral Models
Saskia Demulder 0 1 3
Daniele Dorigoni 0 1 2
Daniel C. Thompson 0 1 3
0 Durham DH1 3LE , U.K
1 Pleinlaan 2 , 1050, Brussels , Belgium
2 Centre for Particle Theory & Department of Mathematical Sciences, Durham University
3 Theoretische Natuurkunde, Vrije Universiteit Brussel and The International Solvay Institutes
We study the SU(2) Principal Chiral Model (PCM) in the presence of an integrable -deformation. We put the theory on R S1 with twisted boundary conditions and then reduce the circle to obtain an e ective quantum mechanics associated with the Whittaker-Hill equation. Using resurgent analysis we study the large order behaviour of perturbation theory and recover the fracton events responsible for IR renormalons. The fractons are modi ed from the standard PCM due to the presence of this -deformation but they are still the constituents of uniton-like solutions in the deformed quantum
Nonperturbative E ects; Sigma Models; Integrable Field Theories; Solitons
-
eld
theory. We also
nd novel SL(2; C) saddles, thus strengthening the conjecture that the
semi-classical expansion of the path integral gives rise to a resurgent transseries once written
as a sum over Lefschetz thimbles living in a complexi cation of the eld space. We conclude
by connecting our quantum mechanics to a massive deformation of the N = 2 4-d gauge
theory with gauge group SU(2) and Nf = 2.
1 Introduction
1.1
1.2
1.3
1.4
2.2
2.3
3.1
3.2
1.5 Summary of results
1.6 Outline
2 The -deformed PCM
2.1 PCM background
The -deformed PCM generalities
Deformed-uniton solution
2.4 Complex uniton solution
3 Fractionalization and reduction to QM
Twisted spatial compacti cation
Unitons on R
S1
3.3 Non-perturbative saddles in the reduced quantum mechanics
4 Perturbation theory in the reduced quantum mechanics
4.1
WKB Perturbation theory and Borel singularities
4.2 Uniform WKB
5 Resurgence analysis
5.1
Large order behaviour in quantum mechanics
5.2 The role of complex saddles
6 Stokes phenomena and Seiberg-Witten theory
7 Conclusions
A Algebraic details
A.1 Conventions
A.2 Drinfeld double and the R-matrix
{ 1 {
1.1
Introduction
some real and positive coupling g2,
Consider, for example, a ground state energy obtained as a perturbative expansion in
HJEP07(216)8
Epert =
1
n=0
X cn(g2)n :
Due to the proliferation of Feynman diagrams it is quite generic to nd factorial growth
cn
Ann! and the perturbative expansion, whilst initially providing increasingly accurate
results, will begin to diverge. Borel summation can provide a way to attach a meaning
to such asymptotic series. In this method the factorial growth is essentially traded for an
integral via n! = R01 dte ttn. One rst constructs the Borel transform,1
B[Epert](t) =
n=0
X1 cn n
t ;
n!
S[Epert](g2) =
1 Z 1
g2
0
B[Epert](t)e g2 dt :
t
which will have a nite radius of convergence and then one can perform the resummation,
There is some danger here! The integral (1.3) may not be well de ned; the Borel
transformed series can have poles along the integration path. It is oft said that this occurs
when the coe cients cn are non-alternating but even an alternating series can be
nonBorel summable and indeed we will encounter this. In this case it is natural to deform the
integration contour by extending t to the complex plane and integrating along a ray i.e.,
S [Epert](g2) =
decays su ciently fast at in nity), then S [Epert](g2) de nes an analytic function in the
(and
If B[Epert](t) has singularities along the direction arg(t) =
, this is called a Stokes
direction. For example if the original direction of integration (1.3), arg(t) = 0, is a Stokes
1In practice one might not have access to the full in nite series (1.1) but only a truncation in which case
direction we can dodge the singularities by picking
=
> 0 small and positive, in this
way we avoid poles in the original integration cycle and we can calculate the well de ned
S [Epert](g). However we could equally choose
=
If B[Epert](t) has singularities in the complex wedge
< 0 and calculate S
[Epert](g).
arg( t)
, these two lateral
[Epert](g), are generically di erent but still yield the same asymptotic
perturbative expansion (1.1) once expanded at weak coupling. This jumping as one deforms
an integration cycle is known as the Stokes phenomenon.
Generically, the original real positive coupling arg(g2) = 0 lies on a Stokes line | the
location at which such a jump takes place. Thus we have traded our initial asymptotic series
expression for the ground state energy for a
nite but ambiguous result. Evidently this is
still unsatisfactory; physical observables surely shouldn't be blighted with such ambiguity.
We have of course missed the vital part of the physics; the non-perturbative sector.
In these cases one should include in the path integral contributions from non-perturbative
saddle points and the
uctuations around them. These non-perturbative contributions
quite cer (...truncated)