Introduction
Results from Large N
The torus
The sphere
Conclusions

Large quantum number expansion in O(2N)
vector model and Resurgence
I. P. Kalogerakis1
1 Institute for Theoretical Physics

University of Bern

Talk for ”IPMU focus week”, Bern, September 2021
Based on the arXiv:2102.12488v2

Kalogerakis

Conference Talk

Introduction
Results from Large N
The torus
The sphere
Conclusions

Outline1

Introduction

2

Results from Large N

3

The torus

4

The sphere

5Conclusions

Kalogerakis

Conference Talk

Introduction
Results from Large N
The torus
The sphere
Conclusions

Outline1Introduction

2

Results from Large N

3The torus

4The sphere

5Conclusions

Kalogerakis

Conference Talk

Introduction
Results from Large N
The torus
The sphere
Conclusions

What is the Large quantum number expansion?

1

It is limited to theories with global symmetries.

2

It allows the analytic treatment of otherwise inaccessible
strogly coupled systems.
The idea
Study subsectors of the theory with fixed quantum number Q.
In each sector, a large Q is the controlling parameter in a
perturbative expansion.

Kalogerakis

Conference Talk

Introduction
Results from Large NThe torus
The sphere
Conclusions

In this talkConsider the O(2N) vector model in 2 + 1 dimensions.
Fine tuned, in the IR it flows to a conformal fixed point.
Use large charge to compute the scaling dimension of thelowest primary operator.
c3/2√∆Q = √ Q 3/2 + 2 πc1/2 Q 1/2 − 0, 094 + O(Q −1/2 ).
2 πKalogerakis

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Introduction
Results from Large N
The torus
The sphereConclusions

What you should remember
12

The 0.094 is a prediction of the theory.
The large charge expansion seems to work for small charges!

Why?
Kalogerakis

Conference TalkIntroduction
Results from Large N
The torus
The sphere
Conclusions

No reason for the large charge expansion to work at small Q
from the EFT point of view.
Can add another controlling parameter, e.g. large N and go
beyond the EFT.
Use the double scaling limit:
Q → ∞, N → ∞, Q/(2N) = q̂ = constantto solve the problem exactly.
The expansion is asymptotic.
Asymptotic series = non-perturbative phenomena = resurgent
asymptotics.

Kalogerakis

Conference Talk

Introduction
Results from Large NThe torus
The sphere
Conclusions

Outline1

Introduction

2

Results from Large N

3The torus

4The sphere

5Conclusions

Kalogerakis

Conference Talk

Introduction
Results from Large NThe torus
The sphere
Conclusions

Start with the Landau-Ginzburg model for N complex fields
on R × M
S[φi ] =N ZX

dtdM gµνu(∂µ φi ) (∂ν φi ) + r φ∗i φi + (φ∗i φi )2
∗



2i=1

The system flows to a Wilson-Fisher fixed point in the IR, i.e.
u → ∞, when r is fined tuned to the conformal coupling, i.e.
r = R/8.
Work in sector of fixed charge Q.

Kalogerakis

Conference Talk

IntroductionResults from Large NThe torus
The sphere
Conclusions

In the limit:
N → ∞,Q → ∞,q̂ = Q/(2N) = fixed

express the free energy as the Legendre transform of a zetafunction.
f (q̂) = F /(2N) = sup(mq̂ − ω(µ))

Free energy per dofµd(ω(µ))dµ11ω(µ) = − ζ − M, µ22q̂ =Charge
Grand potential

where µ is the chemical potential.
Kalogerakis

Conference Talk

Introduction
Results from Large N
The torus
The sphere
Conclusions• ζ s M, µ is the Hurwitz zeta function of the operator −∆ + µ2 .
• In the Mellin representationζ s M, µ =

1
Γ(s)

Z∞


2dt t s−1 e −µ t Tr e ∆t .

0

• Large q̂ is large µ. Can be written in terms of Seeley-DeWitt
coefficients:



VR
∆t
Tr e
∼
1 + t + ... .
4πt
12Kalogerakis

Conference Talk

Introduction
Results from Large NThe torus
The sphere
Conclusions

Outline1

Introduction

2

Results from Large N

3The torus

4The sphere

5Conclusions

Kalogerakis

Conference Talk

Introduction
Results from Large NThe torus
The sphere
Conclusions• First let’s take M = T 2 .
• All but the first Seeley-DeWitt coefficients vanish, hence:
L2Tr e ∆t ∼+ O(e −1/t ),4πt L2 µ2(1−s)ζ s T 2, µ =+ O(e µ )4π(s − 1)• Use these to derive:
L2 µ3
ω(µ) =,12π

L2 µ2q̂ =,4π√
4 π 3/2f (q̂) =q̂ .
3L• We can do better, closed form expression of the corrections.

Kalogerakis

Conference Talk

Introduction
Results from Large NThe torus
The sphereConclusions• Through the spectrum:4π 2  22spec ∆T 2 = − 2 k1 + k2 k1 , k2 ∈ Z .
L• The heat kernel trace is2
2X − 4π2 k 2 +k 2 t − 4π2 t∆t1
22Tr e= θ3 (0, e L )=e L
k1 ,k2 ∈Z

the square of a theta function.
• For t small we can use Poisson resummation
XXZh(n) =dρh(ρ)e 2πikρk∈Z Rn∈Z

and expand around t → 0+ .
Kalogerakis

Conference Talk

Introduction
Results from Large NThe torus
The sphereConclusions• Then  L 
X k 2 L2 2Xkkk2 L2L2− 4t∆t− 4t1+eTr e1+e= √=4πt4πtk∈Z
k∈Z2• Now we can find the subleading contribution to the grandpotential and free energyf (q̂)

=

ω(µ)

=

√

√
X e −kkk 4πq̂
4 π 3/2q̂
1−+.
.
.
,
3L
8kkk2 π q̂µkX e −kkkµL 1  1 2 L 2 µ31 − ζ − T ,µ =
1+1+.
2212π
kkk2 µ2 L2kkkµL

sup(mq̂ − ω(µ)) =kKalogerakis

Conference Talk

Introduction
Results from Large N
The torus
The sphere
Conclusions

Outline1Introduction

2

Results from Large N

3The torus

4The sphere

5Conclusions

Kalogerakis

Conference Talk

Introduction
Results from Large NThe torus
The sphere
Conclusions• Now let’s study the sphere of radius r , M = S 2 .
• The spectrum is`(` + 1)`∈N ,spec(∆S 2 ) = −r2
with multiplicity 2` + 1.
• Again use Poisson resummation to write the heat kernel asfollows
i X∞h∆ − 1 t− t (`+ 1 )2Tr e S 2 4r 2=(2` + 1)e r 2 2`=02= 2
X2|k|πr 3 πr |k|rr√+(−1)k−F(),ttt 3/2tk∈Z

whereF (z) = e −z2Z z√
2dte −t =0Kalogerakis

π −z 2e erfi(z).
2Conference Talk

Introduction
Results from Large N
The torus
The sphere
Conclusions• For small t, we can use the asymptotic expansion of F (z)F (z) ∼∞
X(2n − 1)!!  1 2n+1
n=02n+1z.

andi∞∞
h1r 2 X (−1)n (1 − 21−2n )r 2 X  t nn−1Tr e ∆S 2 − 4r 2 t = −Bt≡
αn 2 .
2nt n=1n!r 2n−2t n=0r
• The series is asymptotic since the Seeley-DeWitt coefficients diverge
like n!:B2n = (−1)n+1(−1)n+1 (1 − 21−2n )2(2n)!
2 n−1/2
√
ζ(2n)
→
α=B
∼n!.
n2n(2π)2nn!
π π 2n• We assume that this series can be completed into a resurgent trans-series.
Kalogerakis

Conference Talk

Introduction
Results from Large NThe torus
The sphereConclusions• In general a trans-series solution with a small parameter z hasthe form for z → 0:X1/βΦ(σk , z) = Φ(0) (z) +σk e −Ak /z k z −bk /βk Φ(k) (z).
k6=0

• The coefficients of the non-perturbative part are encoded in thelarge-order behavior of the perturbative series:
αn ∼βkX Skk∞X

2πi Anβk +bkk(k)α` A`k Γ(βk n + bk − `),`=0where Sk are Stokes constants.

Kalogerakis

Conference Talk

Introduction
Results from Large NThe torus
The sphereConclusions• In this case we have complete knowledge of the αn and we write
them in the suggestive form:
Γ(n + 12 )1 X
αn = − √(−1)k.
(πk)2nπk6=0• Comparing the two expressions
β = 1,bk =1,2

Ak = (πk)2 ,√Sk (k)α = (−1)k+1 |k| π,
2πi 0(k)α>0 = 0.

• The series around each exponential are truncated to only one term andthe heat trace has to contain the terms
i πr 2  32
h12Tr e ∆S 2 − 4r 2 t ⊃ 2i(−1)k+1 |k|e −(kπr ) /t .
t
• The result is defined up to a k-dependent complex constant hence
resurgence leaves us with an ambiguity in the non-perturbativecontribution.
Kalogerakis

Conference Talk

Introduction
Results from Large NThe torus
The sphereConclusions• The ambiguity can be resolved in two ways:
A resurgent analysis of the Dawson’s function using the Borel
resummation.
Using a geometric intepretation in terms of worldline
instantons.
• For the later, the key is to write the heat trace as a path integral
over closed loops
h i Z
Tr e ∆t ≡Dx µ e −S[x] ,x(t)=x(0)whereS[x] =14

Z tdτ gµν (x)ẋ µ (τ )ẋ ν (τ ).

0Kalogerakis

Conference Talk

Introduction
Results from Large N
The torus
The sphereConclusions• In the case of the sphere x µ = (θ, φ) the EOMs are
Zir2 1 h 2
S =dτ θ̇ + sin2 θφ̇2 ,4t 0
φ̈ + 2 cot(θ)θ̇φ̇ = 0,θ̈ − φ̇2 sin(2θ) = 0.
• We can solve them to find the classical solutions and then add
fluctuations around them.
• Solving these we can see that there is a zero mode and multiple
negative modes.

Kalogerakis

Conference Talk

Introduction
Results from Large NThe torus
The sphereConclusions• In both cases the final, real and unambiguous result is 2 3 Z 2 3 X
 i2 2h22k 2 π2 r 22rπr
ζe −ζ r /t
∆ − 1 tTr e S 2 4r 2
= √±idζ(−1)k+1 |k|e − t
π tsin ζtC±k6=02= √π 2 3Z
2 22r
ζe −ζ r /tP.V.
dζtsin ζC±• Now we can write the exact expression of the grand potential andnumerically compare it with the convergent small-charge expansion. Theyagree to at least eight digits!

Kalogerakis

Conference Talk

Introduction
Results from Large NThe torus
The sphere
Conclusions

Outline1

Introduction

2

Results from Large N

3The torus

4The sphere

5Conclusions

Kalogerakis

Conference Talk

Introduction
Results from Large N
The torus
The sphere
Conclusions

The large-charge expansion of the Wilson–Fisher point isasymptotic.
In the double-scaling limit Q → ∞, N → ∞ we control theperturbative expansion.
We have a geometric intepretation of the non-perturbative
corrections.
We can propose an exact form of the grand potential valid forany value of q̂.
The fact that non-perturbative corrections are finite-volume
effects motivates us to extend our results to large charge but
finite N.
We conjecture that the large-charge expansion is always√
asumptotic with an optimal truncation of N ∗ = O( Q),consistent with the lattice results.
Kalogerakis

Conference Talk

Introduction
Results from Large N
The torus
The sphere
Conclusions

Thank you!

Kalogerakis

Conference Talk