The Large Quantum-Number Expansion:
Review Plus Some New Stuff About It
Simeon Hellerman
Kavli Institute for the Physics and Mathematics of the Universe
The University of Tokyo
S.H., Orlando, Reffert, & Watanabe, 1505.01537
S.H., Maeda, & Watanabe, arXiv:1706.05743
S.H. & Maeda, arXiv:1710.07336
S.H., Maeda, Orlando, Reffert, & Watanabe, arXiv:1804.01535
Strings 2018, Okinawa Institute for Science and Technology
Onna, Okinawa, Japan, June 29, 2018

Note to self

REMEMBER TO THANK THE ORGANIZERS!!!

The Large Quantum Number Expansion

This talk is about the simplification of otherwise-strongly-coupled
quantum systems in the limit of large quantum number, which I’ll
refer to generically as ”J”.

The Large Quantum Number Expansion

This talk is about the simplification of otherwise-strongly-coupled
quantum systems in the limit of large quantum number, which I’ll
refer to generically as ”J”.

The Large Quantum Number Expansion

By "otherwise strongly coupled" I’ll mean outside of any
simplifying limit where the theory becomes semiclassical for otherreasons or possibly in a simplifying limit but with the quantum
number taken so large that the system behaves differently than you
might have expected despite being weakly coupled.

The Large Quantum Number Expansion

By "otherwise strongly coupled" I’ll mean outside of any
simplifying limit where the theory becomes semiclassical for otherreasons or possibly in a simplifying limit but with the quantum
number taken so large that the system behaves differently than you
might have expected despite being weakly coupled.

The Large Quantum Number Expansion

The primary question in such a talk is, is this even a subject?

The Large Quantum Number Expansion

The primary question in such a talk is, is this even a subject?

The Large Quantum Number Expansion

The answer is, yes, and in some sense it’s an old one; many
examples have appeared in the literature going far back into thepast. Recently there have been a number of groups focusing on
systematizing this point of view and applying it more broadly.

The Large Quantum Number Expansion
Pre-history:
I Atomic hypothesis [Democritus]
I Correspondence principle [Bohr]I Large spin in hadron spectrum [Regge]
I Macroscopic limit [Deutsch] [Srednicki]
History:
I N = 4 SYM at large R-charge [Bernstein, Maldacena, Nastase]
I and large spin [Belistsky, Basso, Korchemsky, Mueller], [Alday,
Maldacena]

I Large-spin expansion in general CFT from light-conebootstrap [Komargodski-Zhiboedov], [Fitzpatrick, Kaplan, Poland,Simmons-Duffin], [Alday 2016]

I Large-spin expansion in hadrons [SH, Swanson], [SH, Maeda,
Maltz, Swanson], [Caron-Huot, Komargodski, Sever, Zhiboedov],[Sever, Zhiboedov]

The Large Quantum Number Expansion
Modern:I Large-charge expansion in generic systems with abelian global
symmetries: [SH, Orlando, Reffert, Watanabe 2015], [Monin
2016], [Monin, Pirtskhalava, Rattazzi, Seibold 2016], [Loukas 2016]
I Nonabelian symmetries: [Alvarez-Gaume, Loukas, Orlando,
Reffert 2016], [Loukas, Orlando, Reffert 2016], [SH, Kobayashi,
Maeda, Watanabe 2017], [Loukas 2017], [SH, Kobayashi, Maeda,
Watanabe 2018]
I Charge AND spin: [Cuomo, de la Fuente, Monin, Pirtskhalava,
Rattazzi 2017]
I Topological charge: [Pufu, Sachdev 2013] [Dyer, Mezei, Pufu,
Sachdev 2015 ], [de la Fuente 2018]
I EFT connection with bootstrap: [Jafferis, Mukhametzhanov,
Zhiboedov 2017]
I Large charge limit in gravity: [Nakayama, Nomura 2016],[Loukas, Orlando, Reffert, Sarkar 2018]

The Large Quantum Number Expansion

Vacuum manifolds ⇔ chiral rings at large-R-charge:
I D = 4, N ≥ 2 theories : [SH, Maeda, Watanabe 2016]
I D = 4, N ≥ 2 theories : [SH, Maeda 2017], [SH, Maeda,
Orlando, Reffert, Watanabe 2017]

I Double-scaling limit in lagrangian N ≥ 2 theories: [Bourget,
Rodriguez-Gomez, Russo 2018]

Large quantum number: What do we hope to learn?

I The goals of the LQNE are largely to answer the samequestions as the conformal bootstrap:
I Learn to systematically and efficiently analyze QFT (inpractice usually CFT) that have no exact solution in terms of
explicit functions.
I One target is high numerical precision for known CFT.
Yesterday David Simmons-Duffin nicely reviewed the amazing
accomplishments of the conformal bootstrap in its modern
renaissance.
I I’d like to address Daniel Harlow’s question in the Q & A afterthat talk:Large-Scale Structure of Theory SpaceI We’d all like to know "what does theory space look like":
Generic theories, generic amplitudes.
I This is a very consequential question for field theory,
mathematics, quantum gravity, and cosmology.
I Most theories are not integrable, and we need to learn how to
attack them in general circumstances.
I "Direct" numerical bootstrap methods are remarkably
efficient, power-law in number of operators.

Large-Scale Structure of Theory SpaceI Since number of operators grows exponentially with dimension/ central charge / other quantum number, direct numericalattack is still intractable in extreme limits.
I Fortunately, known "extreme limits" appear to have
simplifying limits in many (all?) known circumstances. Thisis broadly a generalization of the notion of "duality".
I In the case of large spin in a single plane, the limit has been
analyzed within the bootstrap itself.
I The relative ease of this is related to the fact that theconformal blocks themselves carry the quantum number.

Critique of Pure Bootstrap

I For other quantum numbers, this is not the case. For
instance, there is no known analytic bootstrap method toattack the case of large spin in multiple planes in D ≥ 4.
I The same is true∗ for internal global symmetries of various
kinds.
I (∗) (Though see [Jafferis, Mukhamezhanov, Zhiboedov 2017].)

Bootstrap-EFT duality?

I As David mentioned, in many cases such limits are accessible
to some new kinds of EFT in regions where bootstrap
methods slow down.
I As we’ll see, there’s also a excellent agreement for one
prediction where the two methods overlap .
I Where does this leave us? What do we hope to accomplish ?

Squad Goals
I (*) Most modestly: Translate EFT behavior into bootstrap
terms, say what it means for CFT data. Operator dimensions
and OPE coefficients.
I (***) Most grandiosely: Derive EFT behavior from bootstrap
equations, and use it to solve everything in every limit where
direct numerical methods break down.
I (**) Intermediate: Use some small subset of EFT inputs,
and obtain some subset of CFT data not directly numericallyaccessible.
I Grandiose goal (***) appears out of reach for now. (I tried!)
I For progress on the intermediate goal (**) see[Jafferis-Mukhametzhanov-Zhiboedov 2017].

I This talk is about progress on modest goal (*).

Large charge J in the O(2) model

I Simplest example: The conformal Wilson-Fisher O(2) modelat large O(2) charge J.
I Canonical question: What is the dimension ∆J of the lowest
operator OJ at large J?
I Translated via radial quantization: Energy of lowest state ofcharge J on unit S 2 ?
I Renormalization-group analysis reveals the low-lying
large-charge sector is described by an EFT of a single
compact scalar χ, which can be thought of as the phasevariable of the complex scalar φ in the canonical UV
completion of the O(2) model.

Large charge J in the O(2) model
I The leading-order Lagrangian of the EFT is remarkably
simple:Lleading−order = b|∂χ|3
I The coefficient b is not something we know how to computeanalytically; nonetheless the simple structure of this EFT has
sharp and unexpected consequences.
I The immediate consequence of the structure of the EFT isthat the lowest operator is a scalar, of dimension
3∆J ' c 3 J 2 ,2where c 3 has a simple expression in terms of b.
2

Large charge J in the O(2) model

I The leading-order EFT predicts more than just the leading
power law, because quantum loop effects in the EFT aresuppressed at large J, so the EFT can be quantized as aweakly-coupled effective action with effective loop-counting
3
parameter J − 2 .
I For instance we can compute the entire spectrum of low-lying
excited primaries.
I The dimensions, spins, and degeneracies of the excited
primaries, are those of a Fock space of oscillators of spin `,with ` ≥ 2.

Large charge J in the O(2) model

I The propagation speed of the χ-field is equal to √1 times the2
speed of light.
I So the frequencies of the oscillators are
1 pω` = √`(` + 1) ,2`≥1.

I The ` = 1 oscillator is also present, but exciting it only gives
descendants; the leading-order condition for a state to be a
primary is that there be no ` = 1 oscillators excited.
I So for instance, the first excited primary of charge J always√(1)
has spin ` = 2 and dimension ∆J = ∆J + 3.

Large charge J in the O(2) model

I Subleading terms can be computed as well.
I These depend on higher-derivative terms in the effective
action with powers of |∂χ| in the denominator .
I These counterterms have a natural hierarchical organizationin J :

Large charge J in the O(2) model

I At any given order in derivatives, there are only a finite
number of such terms.
I As a result, at a given order in the large-J expansion, only afinite number of these terms contribute.
I Since there are far more observables than effective terms,
there are an infinite number of theory-independent relations
among terms in the asymptotic expansions of various
observables.

Large charge J in the O(2) model

I Our gradient-cubed term is the only term allowed by the3symmetries at order J 2 , and there is only one other term
contributing with a nonnegative power of J, namely(∂ |∂χ|)2
L + 12 = b 1 |∂χ| Ric3 + 2
J2|∂χ|I In particular, there are no terms in the EFT of order J 0 , withthe result that the J 0 term in the expansion of ∆J is
calculable, independent of the unknown coefficients in theeffective lagrangian.

Large charge J in the O(2) model

I Specifically, the formula for ∆J takes the form
3

∆J = c 3 J + 2
21+ c 1 J + 2 −0.0937256 · · ·2

up to terms vanishing at large J.

Large charge J in the O(2) model

I Specifically, the formula for ∆J takes the form
3

∆J = c 3 J + 2
21+ c 1 J + 2 −0.0937256 · · ·2

up to terms vanishing at large J.

Large charge J in the O(2) model

I Specifically, the formula for ∆J takes the form
3

∆J = c 3 J + 2
21+ c 1 J + 2 −0.0937256 · · ·2

up to terms vanishing at large J.

Large charge J in the O(2) model

I Specifically, the formula for ∆J takes the form
3

∆J = c 3 J + 2
21+ c 1 J + 2 −0.0937256 · · ·2

up to terms vanishing at large J.

Large charge J in the O(2) model

I Specifically, the formula for ∆J takes the form
3

∆J = c 3 J + 2
21+ c 1 J + 2 −0.0937256 · · ·2

up to terms vanishing at large J.

Large charge J in the O(2) model

I Specifically, the formula for ∆J takes the form
3

∆J = c 3 J + 2
21+ c 1 J + 2 −0.0937256 · · ·2

up to terms vanishing at large J.

Large charge J in the O(2) model

I Specifically, the formula for ∆J takes the form
3

∆J = c 3 J + 2
21+ c 1 J + 2 −0.0937256 · · ·2

up to terms vanishing at large J.

Large charge J in the O(2) model

I Specifically, the formula for ∆J takes the form
3

∆J = c 3 J + 2
21+ c 1 J + 2 −0.0937256 · · ·2

up to terms vanishing at large J.

Large charge J in the O(2) model

I This universal term and the other universal large-J relationsin the O(2) model don’t have any fudge factors or adjustableparameters;
I Given the identification of the universality class, these values
and relations are universal and absolute;
I Similar predictions have been made for OPE coefficients[Monin, Pirtskhalava, Rattazzi, Seibold 2016]

FAQ

I You might think that there is something ”weird” or”inconsistent” or ”uncontrolled” about a Lagrangian like
L = |∂χ|3 .
I So, let me anticipate some frequently asked questions:FAQ
I Q: Isn’t this Lagrangian singular?? It is a nonanalytic
functional of the fields, so when you expand it around χ = 0,
you will get ill-defined amplitudes.
I A: Yes, but you aren’t supposed to use the Lagrangian there.
It is only meant to be expanded around the large charge
vacuum , which at large J is the classical solution
χ = µt,with1√µ = O( ρ) = O(J 2 ) .

I The expansion into vev and fluctuations carries a suppressionof µ−1 or more for each fluctuation.

FAQ

I (parenthetical comment:) There are already many well-known
effective actions of this kind, including the Nambu-Gotoaction, which I would hope my fellow attendees as a strings
conference might be familiar with.

FAQ

I Q: Isn’t this effective theory ultraviolet-divergent ? Thatmeans that loop corrections are incalculable and observables
are meainingless beyond leading order.
I A: No. The EFT is quantized in a limit where loop corrections
are small . Our UV cutoff Λ for the EFT is taken to satisfy
EIR = RS−1
2

<<

Λ

<< EUV =√31

ρ ∝ J + 2 RS−1
2

I Loop divergences go as powers of Λ3 /ρ 2 << 1, and areproportional to nonconformal local terms which are to be
subtracted off to maintain conformal invariance of the EFT.

FAQ

I Q: OK but then don’t the counterterms ruin everything?
Don’t they render the theory incalculable?
I A: No. As usual in EFT the counterterm ambiguities ofsubtraction correspond one-to-one with terms in the originalaction allowed by symmetries;
I As we’ve mentioned there are only a finite and small numberof those contributing at any given order in the expansion, andat some orders there are no ambiguities at all.

FAQ
I Q: You’re saying that every CFT with a conserved global
charge has this exact same asymptotic expansion . But here’s
a counterexample! describes theory SH didn’t say
anything about Doesn’t that mean you’re a crackpot?
I A: No. I didn’t make any claim that broad. Our RG analysis
applies to many but not all CFT with a conserved global
charge. More generally, CFT can be organized intolarge-charge universality classes.
I For instance, free complex fermions as well as free complex
scalars in D = 3 are in different large-J universality classes.
I The large-J universality class of the O(2) model contains
many other interesting theories, such as
I The CIP(n) models at large topological charge ;I The D = 3, N = 2 superconformal fixed point for a chiralsuperfield with W = Φ3 superpotential, at large R-charge;
I Probably others ◦ ◦ ◦Other large-J universality classes

I Many other interesting universality classes in D = 3:
I Large Noether charge in the higher Wilson-Fisher O(N)
[Alvarez-Gaumé , Loukas, Reffert, Orlando 2016] and U(N) models;
I Also the CIP(n) [de la Fuente] and higher Grassmanian models
real and complex ; [Loukas, Reffert, Orlando 2017]
I Large baryon charge in the SU(N) Chern-Simons-mattertheories;
I Large monopole charge in the U(N) Chern-Simons-mattertheories;
I Of course these last two are dual to one another and would
be interesting to investigate.

Vacuum moduli spaces and the large-R-charge limit

I Among the most tractable universality classes are large
R-charge in extended superconformal theories with moduli
spaces of supersymmetric vacua.
I Simplest case is the N = 2, D = 3 superconformal fixed pointof three chiral superfields with superpotential W = XYZ .
I Its vaccum manifold has three one-complex-dimensional
branches: X , Y , Z 6= 0.
I WLOG consider the X −branch.

Vacuum moduli spaces and the large-R-charge limit

I The X -branch has coordinate ring spanned by X J , J ≥ 0.
I These BPS scalar chiral primary operators are the (X -branchpart of the) chiral ring of the theory.
I The dimension of X J is exactly equal to its R-charge J andprotected from all quantum corrections: In this case theformula for the dimension ∆J is boring :∆J = 1 · J⇐ BORING!

Vacuum moduli spaces and the large-R-charge limit

I The formula for the dimension of the second-lowest primaryof JR = JX = J is it lies an a protected scalar semishort
representation with only 12 Poincaré superpartners:
(+1)

∆J

=1·J +1⇐ also boring!

I Nonetheless we would like to see this explicitly in a large-J
expansion, and also be able to compute non-protected large-J
quantities such as third-lowest operator dimensions and also
OPE coefficients.

Vacuum moduli spaces and the large-R-charge limit

I The effective theory describing the lowest state of
JX = JR = J, is simply the moduli space effective action ,appearing in the same role as the gradient-cubed theory for
the O(2) model.
I Unlike the O(2) model EFT, here the leading effective action
is simply free :Z3L=d 2 θ d 2 θ̄ Φ† Φ ,Φ = (const.) × X 4 + · · · ,where the · · · are higher-derivative D-terms .

Vacuum moduli spaces and the large-R-charge limit
I To compute operator dimensions, quantize the theory aroundthe lowest classical solution with given large J on an S 2
spatial slice:
I Here, the classical solution is
φ = v exp (iµt) ,
1
,
µ=
2R

r
v=

J
.
2πR

I Note here the frequency of the solution (chemical potential )is determined by supersymmetry (the BPS bound on operator
dimensions) rather than the unknown coefficients in the
Lagrangian.

Vacuum moduli spaces and the large-R-charge limit
I The results of the direct diagrammatic quantization are as
follows, for the lowest and second-lowest states:∆J = J+0 × J 0 + 0 × J −1 + 0 × J −2 + 0 × J −3
+O(J −4 )⇐ three loops!

∆J (+1) = J + 1 × J 0+0 × J −1 + 0 × J −2 + 0 × J −3
+O(J −4 )⇐ two loops! ,confirming the predictions of supersymmetry to the order we
can calculate .

Vacuum moduli spaces and the large-R-charge limit
I The third-lowest primary is a non-BPS scalar, with dimension
(+2)

∆J

= J + 2 · J0+0 × J −1 + 0 × J −2
−κ × 192 π 2 × J −3
+O(J −4 )⇐ one loop! ,where κ the coefficient of the leading interaction term in the
EFT .

Vacuum moduli spaces and the large-R-charge limit
I The form of the leading interaction term is a D-term,
consisting of a four-derivative bosonic component
L−1 ≡ +4 κFTP

|∂φ|4
,
|φ|6

plus conformally and superconformally completing terms
worked out by many authors [Fradkin, Tseytlin] [Paneitz][Riegert] [Kuzenko].

I We don’t know the value of κ for the XYZ model, but we doknow its sign :κ>0(superluminality constraint)[Adams, Arkani-Hamed, Dubovsky, Nicolis]

I So the first nonprotected operator dimension gets a
contribution of order J −3 with a negative coefficient of
unknown magnitude .

Vacuum moduli spaces and the large-R-charge limit

I It is more fun to compute quantities which are both nontrivial
in the large-J expansion and checkable in principle by exact
supersymmetric methods.
I One nice example is the two-point functions of chiral primary
operators in 8-supercharge theories.
I The technically simplest class of examples are the chiralprimaries spanning the Coulomb branch chiral ring in
D = 4, N = 2 theories, in the special case the gauge grouphas rank one .

Vacuum moduli spaces and the large-R-charge limit

I Examples include
I N = 4 SYM with G = SU(2),I N = 2 SQCD with Nc = 2, Nf = 4,
I Many rank-one nonlagrangian Argyres-Douglas theories with
one-dimensional Coulomb branch,
I including the recently discovered N = 3 examples.

I Some of these are Lagrangian theories with marginal
coupling, and some of them are non-Lagrangian theories withmore abstract descriptions, but we can treat them all on an
equal footing.

Vacuum moduli spaces and the large-R-charge limit

I The Coulomb branch chiral ring in a rank-one theory is
spanned byn
OJ ≡ O∆,J = n∆ ,where the(∗) generator O∆ of the chiral ring has U(1)R -charge
JR = ∆.
I This assumes the chiral ring is freely generated; there are no known
counterexamples, but see recent work [Argyres, Martone 2018] forcounterexamples in higher rank.

I At large charge in radial quantization these correspond toclassical solutions on the sphere where√ the Coulomb branchscalar â gets a vev proportional to J/R.

Vacuum moduli spaces and the large-R-charge limit

I For Lagrangian theories the generator O is tr(â2 ) and ∆ = 2.
I For non-Lagrangian theories the dimension ∆ of thegenerator can take certain other values.
I These are constrained to some extent and recently it wasproven that ∆ is always rational [Argyres, Martone 2018]
I We can write the large-J effective action in terms of an1
effective field φ ≡ (O∆ ) ∆ . The singularity in the change ofvariables is invisible in large-J perturbation theory because
the quantum state field is supported far away from φ = 0.

Vacuum moduli spaces and the large-R-charge limit
I The leading-order action is again the free action for φ, andthe leading interaction term is the anomaly termcompensating the difference in Weyl a- anomaly andU(1)R -anomalies between the underlying interacting SCFT
and the free vector multiplet.
I The leading interaction term isZ
Lanom ≡ αd 4 θ d 4 θ̄ log(φ) log(φ̄)+(curvature and U(1)R connection terms) ,I where the coefficient α is proportional to the Weyl-anomaly
mismatch:
α = +2 (aCFT − aEFT )[AEFGJ units]

Vacuum moduli spaces and the large-R-charge limit
I Some comments on this interaction term:
I It was first written down by [Dine, Seiberg 1997] as the uniquefour-derivative term in the Coulomb branch EFT of an N = 2gauge theory;

I It is formally an N = 2 D- term, i.e. a full-superspace
integrand · · ·I · · · but only formally, since it is non-single-valued; its
single-valued version can be obtained as an F -term, i.e. an
integral over only the θ’s and not the θ̄’s.
I Its bosonic content comprises the famous Wess-Zumino term
for the Weyl a-anomaly that was used [Komargodski,
Schwimmer] to prove the a-theorem in four dimensions.

I This is why its coefficient α is proportional to the a-anomaly
mismatch.

Vacuum moduli spaces and the large-R-charge limit
I One other remarkable fact about rank-one theories, is thatthe anomaly term is that it is unique as a (quasi-)F -term on
conformally flat space.
I That is, there are an infinite number of higher-derivative
D-terms, but there are no higher-derivative F -terms one canconstruct out of a single vector multiplet in a superconformal
N = 2 theory.
I The simple explanation: An N = 2 superconformal theory issuper-Weyl invariant, with the super-Weyl transformation
parametrized by a chiral superfield Ω:
φ → exp (Ω) · φ .
I In the regime of the validity of the effective theory, φ has anonzero vev, and in flat space we can super-Weyl transformthe vector multiplet to 1.

Vacuum moduli spaces and the large-R-charge limit
I The EFT is therefore(∗)
L = Lfree + Lanomaly + Lhigher D−term
I For quantities insensitive to D-terms, this simple, two-term
effective action, can be quantized meaningfully, and gives
unambiguous answers to all orders in J1 perturbation theory.
I Note that the dimension ∆ of the generator of the chiral ring
does not enter into the EFT at all, nor does the marginal
coupling τ or any other parameter .
I In other words, any purely F-term-dependent observable hasa large-J expansion that is uniquely determined by theanomaly coefficient α and nothing else, for a one-dimensional
Coulomb branch of an N = 2 gauge theory.

Vacuum moduli spaces and the large-R-charge limit
I One set of such observables are the Coulomb branchcorrelation functions


n
2J
n
exp (qn ) ≡ Zn ≡ ZS 4 × |x − y |(O(x)∆ ) (O(y )∆ )S4

I The insertions φJ (x) and φ̄J (y ) can be taken into the
exponent asSsources ≡ −J log φ(x) − J log φ̄(y )I This quantity Zn = exp (qn ) is partition function of the EFTwith sources:Z
Zn =DΦ DΦ† exp (−SEFT − Ssources )Vacuum moduli spaces and the large-R-charge limit

I This quantity is scheme-dependent, and dependent on thenormalization of O, but these dependences cancel out in the
double difference observables
Zn+1 Zn−1= exp (qn+1 − 2qn + qn−1 ) .
Zn2
I These can now in principle be evaluated straightforwardly as
functions of J and α using Ferynman diagrams, with nofurther input from the underlying CFT, as long as we are inlarge-J perturbation theory.

Vacuum moduli spaces and the large-R-charge limit
I The form of the expansion is
qn = A n + B + J log(J ) +

1
α+
2


log(J )+

X K̂m (α)
m≥1Jm

I The first two terms are the scheme and normalization
ambiguities, the third term is the classical value of thesource term, one loop free term , and classical anomaly term
contributions.
I The last is the series of power-law corrections coming fromloop diagrams with interaction vertices coming from thesource term and the anomaly term, with the anomaly term
vertices carrying powers of α.
I The structure of the EFT makes the polynomials K̂m (α) apolynomial in α of order m + 1:
K̂m (α) =

m+1
X
`=0

K̂m,` α` .

.

Vacuum moduli spaces and the large-R-charge limit

Vacuum moduli spaces and the large-R-charge limit

I Of course, actually directly evaluating multiloop diagrams in
an EFT is hard ;I To evaluate the power-law corrections, my collaborators and I
used a combination of
I Direct evaluation of some low-order diagrams;
I Use of known data for some theories such as the free vector
multiplet and N = 4 SYM ;I Supersymmetric recursion relations [Papadodimas 2009];
I Embedding of the Coulomb-branch EFT into nonunitary UVcompletions invoving ghost hypermultiplets to apply therecursion relations to arbitrary versions of α.

Vacuum moduli spaces and the large-R-charge limit

I With this combination of tricks, we were able to solve all thepower-law corrections for any value of α, with the result:qn = A n + B + log J + α + 1+smaller than any power of J .
I I’ll comment on those exponentially small corrections in a
moment.

Confirmation of the large-J expansion
I But first, let me talk about some evidence for this picture oflarge-J self-perturbatization of strongly coupled theories.
I Starting with our predictions for the O(2) model, where we
predicted a formula3∆J = ∆J = c 3 J + 2
21+ c 1 J + 2 −0.0937256 · · ·2

I It would be good to compare with bootstrap calculations in
the O(2) model; at the moment bootstrap methods can only
reach J ≤ 2 with any precision. [Kos, Poland, Simmons-Duffin2013].

I It would be good if bootstrap methods could be developed tothe point of being able to confirm our results, or add
something substantial to them.
I But at the moment that hasn’t happened, so let’s move on toother avenues of confirmation.

Confirmation of the large-J expansion

I The first really nontrivial confirmation came from a Monte
Carlo analysis up to J = 15 in the O(2) model, independently
computing charged operator dimensions and estimating theleading Lagrangian coefficient b from the energies of charged
ground states on the torus .
I These results are from a PRL by [Banerjee, Orlando,
Chandrasakhran 2017].

Monte Carlo numerics [Banerjee, Chandrasekharan, Orlando 2017]

Figure: Operator dimensions with the c3/2 , c1/2 coefficients in the EFT√√
prediction fit to data, giving c3/2 = 1.195/ 4π and c1/2 = 0.075 4π.

Monte Carlo numerics [Banerjee, Chandrasekharan, Orlando 2017]

Figure: Note the coefficients are fit with high-J data for operator
dimensions and torus energies, and yet the leading-order prediction
extrapolates extremely well down to J = 2.

Confirmation of the large-J expansion

I Though precise bootstrap results only exist up to J = 2, notethat the values of the EFT parameters calculated from Monte
Carlo calculation give
∆J=2 = 1.236(1)[MonteCarlo + large − J]

which one can compare to the bootstrap result∆J=2 = 1.236(3)[bootstrap] .

I There are other high-precision agreements between large-J
theory and MC simulation in [Banerjee, Chandrasekharan,
Orlando 2017].

Confirmation of the large-J expansion
I Moving beyond the O(2) case to other models in the samelarge- J universality class, one can look at dimensions ofoperators carrying topological charge J in the CIP(n) models.
I This analysis was done by [de la Fuente 2018], using a
combination of large-N methods and numerical methods, with theresult1
3
1
CIP(n)∆J= c 32 (n) J 2 + c 12 (n) J 2 + c0 + O(J − 2 ) ,where the first two coefficients depend on the n of the model, but
the J 0 term does not; in particular he finds
c0 = −0.0935 ± 0.0003 ,as compared to the EFT prediction
c0 = −0.0937 · · · .

I So the error bars are less than one percent , and our EFT
prediction sits right in the middle of them.

Confirmation of the large-J expansion
I Now let’s move on to our predictions for D = 4, N = 2
superconformal theories with one-dimensional Coulomb
branch.
I For the case of free Abelian gauge theory and N = 4 SYM
with G = SU(2) our all-orders-in-J formula agrees with theexact expression:
(EFT)

Zn

(EFT)

Zn

(CFT)

= Zn

(CFT)

= Zn

= n! ,= (2n + 1)! ,free vector multiplet ,N = 4 SYM .

In these cases, there are no exponentially small corrections to
the formula.

Confirmation of the large-J expansion
I For other cases, the correlation functions are D-term
independent and can be evaluated by exact supersymmetricmethods involving localization [Pestun 2007] andsupersymmetric recursion relations [Papadodimas 2009],[Gerchkovitz, Gomis, Komargodski 2014] · · ·I · · · though at present these methods are limited to theories
with a marginal coupling.
I Even using these methods, the recursion relations grow morechallenging in application to compute corelators of higher J
owing to the complication of the sphere partition function as
a function of the coupling .
I Nonetheless we have been able to carry the recursion relations
to J ∼ 76 in the case of N = 2 SQCD with Nc = 2, Nf = 4.

Numerics (Localization)

Confirmation of the large-J expansion
I It is interesting to try to understand the disagreement between
the all-orders- J1 formula and the exact localization results.
I Our framework for large-J analysis dictates that any
disagreement must be smaller than any power of J andassociated with a breakdown of the Coulomb-branch EFT .
I The natural candidate for such an effect would be propagationof a massive particle over the infrared scale R = |x − y |.
I Therefore we would expect the leading difference between thelocalization result and the EFT prediction, to be of the form
(loc)

qn

(EFT)

− qn

∼ const. × exp (−MBPS particle × R)s
= const. × exp −(const.)

J
Im(τ )!
.

Confirmation of the large-J expansion

I For fun, we compared the difference between EFT and exact
results in the scaling limit of [Bourget, Rodriguez-Gomez, Russo
2018], where J is taken large with this exponent held fixed and fit
it to this virtual-BPS-dyon ansatz for the exponentially small
correction .

I We found the difference qn(loc) − qn(EFT) fits very well to
(loc)

qn

(EFT)

− qn

1

' 1.6 e − 2

λ ≡ 2πJ /Im(τ ) .

√

πλ

,

Numerics (Localization)

Conclusions
I The large-J expansion gives an analytically controlled way tocompute CFT data outside of any other sort of simplifying
limit, particularly illuminating simple behavior in regimes
where numerical bootstrap methods cannot currently access,despite formal similarity of the expansions.
I The large- J predictions in cases such as the O(2) model andvarious D = 4, N = 2 superconformal theories with
one-dimensional Coulomb branch, agree extremely well evenat low J with Monte Carlo, bootstrap, and exact
supersymmetric methods.
I These results have greatly improved our quantitative control
and conceptual understanding of even the simplest
strongly-coupled CFT.
I Analysis of more examples is sure to yield further interesting
surprises about the large-scale structure of theory space .
I Thank you.