Large N, small N, andadiabatic continuity
Aleksey Cherman
UMN

Summarizes work by/with many people: D. Dorigoni, G. Basar, E. Poppitz, M. Shifman,
M.Unsal, L. Yaﬀe, …

Big picture
Goal: understand some physically interesting quantity 𝒪
This is a resurgence workshop. We think in terms ofin QM, QFT, string theory…
But what is λ in QFT context?

• Usually λ is a running coupling• Often it isn’t small at the energy scales of interest
How do we accomplish anything, then?

The challenge
Idea: find control parameter C, use it to make λ ≪ 1, compute.

petwave.com

But then what do we learn about the original physics?
• If there are no phase transitions as a function of C, then welearn a lot.
• With phase transitions we get a disaster (in practice)!
• Have to understand full resurgence behavior as aprerequisite for making even qualitative predictions.

Two approaches• Supersymmetry• Studied since ~1980s (in the relevant context)• Adiabatic Compactification• Studied since ~2010s, due to Mithat Unsal andcollaborators: Shifman, Yaffe, Poppitz, Dunne, Schafer,
Sulejmanpasic, Tanizaki, Misumi, AC,…

SUSYSupersymmetry often naturally gives a control parameter C• C = < VEV of fundamental scalar field >• Asymptotically-freedom ⇒ weak coupling for large C• SUSY holomorphy ⇒ results for all C• Very nice in its own right!
• Loss of control if SUSY is broken• Have to hope there are no phase transitions• Whatever you learn might be tied to specifics of SUSY
setting: specifics of matter content & interactions…
• Resurgence structure often very different from generic
expectations.
• Cancellations hidden, have to be decoded.

Adiabatic compactification

Unsal, Yaffe, Shifman,… 2008-onward

Idea: break 4D Lorentz, but as little as possible!
S1
R3

If circle size L is small, can get weak coupling by asymptotic
freedom3
NB:
non-compactℝ⇒ symmetries can break spontaneously.
•• Large L: some symmetries preserved, others spontaneously
broken!
• In practice the small L limit is useful only if we get same
symmetry breaking pattern at large L and small L• Assume symmetry breaking pattern doesn’t change atintermediate L - checkable by lattice simulations.

Plan of the talk
Focus will be on adiabatic compactification
1. Fixed N - already done by Mithat, so I’ll be brief.
• Reminder about mass gap and remark on renormalons …
2. ’t Hooft large N limit: volume independence, and Hagedorn
3. Small-circle large N limit: emergent extra dimension, and thefate of the mass gap.

Part 1
Small L, fixed N.

Self-Higgsing
S1
R3

When gauge theories are compactified on S1, tr(Polyakov loop)
is an observable

Eigenvalues are determined dynamically. Their distribution isvery important!

Confinement and center symmetry
Heuristically, Polyakov loop associated to confinement

Confinement ~ infinite cost to have k ≠ N excess fundamental
quarks.
• N quarks make a baryon, and baryon has finite energy.
• So expect confinement to be associated with

Indeed, YM (without fundamental quarks) has ZN center symmetry

unbroken center ⇒ ⟨tr Ωk⟩ = 0, k ≠ 0 mod N

Center symmetry and self-Higgsing

If ⟨tr Ωk⟩ = 0 ⇒ "⟨A3⟩ ≠ 0". A3 ~ compact adjoint Higgs field!
Non-coincident eigenvalues for Ω“broken” gauge group SU(N) → U(1)N-1 in long-distance 3D EFT

tr

[

]

=0

W-boson mass scale: mW = 2π/NL

Coupling flows with center symmetry on R3 x S1
g2

1
Flow for NL  1

(NL)

-1



Q

The NLΛ ≫ 1 regime is strongly-coupled at long distances for all L!

Coupling flows with center symmetry on R3 x S1
g2

Flow for NL  1
g2 (1/NL)
Semiclassicallycalculable
regime



(NL)

-1

Q

The NLΛ ≪ 1 regime gives a weakly-coupled theory at all scales!

Preservation of center symmetry
Mithat already explained that preserving center symmetry at small
L is hard.
• To the extent L = 1/T, center symmetry “wants” to break!
• This can be avoided using several ingredients:• Double-trace deformations• Light adjoint fermions with periodic BCs

With you favorite method, you can ensure center symmetry ispreserved at small L. Then what?

Small L effective field theory
Suppose N is fixed and LΛ ≪ 1, with center preserved.
• Thanks to adjoint Higgsing, lots of stuff is heavy:mW /Λ ∼ 1/(NLΛ) → ∞• Integrate out the manifestly heavy stuff! What remains?
• N − 1 Cartan gluons, from 3D components of gluon fieldstrength• Working out their fate is crucial!

Small L limit in perturbation theory
N - 1 Cartan gluons are classically gapless.

iσ• shift symmetry ⟺ conservation of magnetic charge.

• But there are no magnetic monopoles in perturbation theory.
i• So σ are massless to all orders in perturbation theory.

Finite-action field configurations
Since SU(N) → U(1)N−1, 4D BPST instanton breaks up into N
2‘monopole-instantons’ with action SI /N = 8π /λN − 1 have Q = 1/N, magnetic charges ± 1 under nearest- Lee, Yi;
Kraan, vanthneighbor U(1)’s. The N one is ‘Kaluza-Klein’ monopole.
Baal; 1998
’t Hooft
amplitudeλ ≪ 1 when NLΛ ≪ 1, so dilute gas approximation is justified.

• Contrast with usual IR disasters with instantons in YM!

Weak coupling confinement

Unsal,
Yaffe, Shifman, Poppitz,
Sulejmanpasic,…

Dual photons get a mass gap:
, p = 1, … , N - 1.

Concrete realization of old Mandelstam, ’t Hooft, Polyakov dreams:
mass gap driven by proliferation of magnetic monopoles.
String tension also calculable, and is finite. Behaves just as
expected from YM.
Poppitz, Erfan S. T, Anber, … 2017 onward

Can also profitably study θ dependence.
Unsal, Yaffe, Tanizaki, Misumi, Fukushima, AC, Poppitz, Schafer, …

Resurgent ambiguities in adiabatic compactification
Beyond leading order in semi-classical expansion,
neutral bion amplitudes are (usually) ambiguous:⇥<latexit sha1_base64="AuTCIx5ts6DGraNMkwY4cihd/8k=">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</latexit>⇤

Mi M i ⇠ ±i e

16⇡ 2 /

(exactly massless adjoint fermions change the story)

This ambiguity does not vanish exponentially with N.
Arises from quasi-zero mode integration. A kind of generalized
instanton effect, so it should be possible to relate to someproliferation of Feynman diagrams.
3
1
4
Feynmandiagramson
Feynmandiagrams
on
≠
ℝ
×
S
ℝ
•
1
• Color-sums related to S momentum sums
Eguchi, Kawai; Gross, Kitazawa, …

Comments on renormalons
What is a renormalon?
My preferred definition: it is an ambiguity in the Borel resummationof perturbation theory, with a size which doesn’t vanish at large N.
• Other definitions are used in some of the literature. I think thisone is better, for reasons I’ll explain next.
What is the interplay of renormalons with adiabatic
compactification?
4D:
2D:

Argyres, Unsal; Anber, Sulejmanpasic;
Ashie, Ishikawa, Takaura, Morikawa, Suzuki, Takeuchi, …
Dunne, Unsal; Fujimori, Kamata, Misumi, Nitta, Sakai;…

Comments on renormalons
On ℝ4, renormalons come from diagrams like this:Renormalons arise from an IR divergence in these diagrams, giverise to ambiguities in Borel summed perturbation theory, so in YM
ambiguity ∼ ± ie −#/λ# is such that it can be cancelled by an ambiguity in some24
−8π 2/(λ⋅11/3)‘condensate’, e.g. ⟨trFμν⟩ ∼ Λ . Remember: Λ ∼ μe

• Has to be like this for consistency! Λ is the only scale.

Comments on renormalons
What should we expect with adiabatic compactification?
Adiabatic compactification eliminates IR divergences by design!
• Are renormalons gone? The Feyman diagrams that gave them4

on ℝ aren’t divergent any more.
• If we define renormalon = certain Feynman diagrams withcertain divergences, then, yes they’re gone.
• My view: not a good definition. Number and value ofindividual Feynman diagrams is not a physical invariant!

Comments on renormalonsDistinction between Borel singularities “from IR divergences” or“from number of individual diagrams” is not physical.
• What matters: size of effect, how to understand it, how it fitswith other dynamics, and so on.
Hence my preferred definition: renormalons are an ambiguity in the
Borel resummation of perturbation theory, with a size which
doesn’t vanish at large N.
• This is the definition assumed in the Argyres-Unsal and DunneUnsal papers that kicked off modern QFT resurgence.
So what should we expect about renormalons at small L?

Renormalons in adiabatic compactification
With adiabatic compactification, there is another scale in addition
to Λ, namely 1/L. The physics depends on both!
• Renormalons should depend on 1/L as well, and indeed they do:
−#/λ
p
k
±
∼ Λ (NLΛ)
ambiguity ∼ ie
see e.g. Dunne, Shifman, Unsal 2015

More precisely: we know non-perturbative quantities haveambiguities like this, due to “neutral bions”.
• It’s harder to explicitly match it to perturbation theory…
but see talk by O. Morikawa on Thursday (Wed in US)!
In any case, this perspective implies that location of Borel
singularities must flow as a function of NLΛ .

Part 2’t Hooft large N limit

Large N vs adiabatic compactification
Suppose N → ∞ with λ and all other parameters fixed in a theory
with adiabatic circle-size dependence.
• What does it imply?

• Dependence on L is very adiabatic: no dependence at all• “Eguchi-Kawai reduction”/“large N volume independence”• in the adjoint QCD example: spectacular Bose-Fermi
cancellations• More subtle (and not well understood) with double-trace
deformations

Hagedorn instability
Put any confining large N theory on R3 x S1β

Once β < βH, energy integral diverges! No big change if ℝ3
replaced by some compact manifold M.

⇒ phase transition at or below TH to a phase where ρ(E)scales diﬀerently.

This is the deconfinement transition to the quark-gluon plasmaphase!

Confinement versus Hagedorn
Can deconfinement as a function of β be avoided?

‘all’ we need are precise-enough cancellations between
ρB(E) and ρF(E).
With SUSY ρB(E) = ρF(E), E > 0, so no problem.
What about without SUSY?

Required cancellations
Expect Hagedorn scaling for both ρB and ρF. More precisely:

All terms with positive exponentials must be identical to avoid
an instability!

Avoiding Hagedorn
This degree of conspiracy between bosons andfermions seem fanciful without supersymmetry.
And indeed, it doesn't work in large N QCD.
Fermionic states — baryons —only for odd N,
which are heavy at large N.
F βHZ̃QCD(β) = tr (−1) e= ZQCD(β) = tr eβHHave to look further afield for a working example.

A special non-SUSY QFT
Consider SU(N) YM coupled to 1 ≤ NF ≤ 5 flavors of masslessadjoint Majorana fermions: adjoint QCD
4ℝ
Confiningon
if NF is not too close to 5.
•

NF = 1 ⇒ 𝒩 = 1 super YM
Otherwise, no SUSY.

Adjoint QCD
Adjoint QCD has lots of light fermionic states for any NF
2
2
tr (Fμνλa) | 0⟩, tr (Fμνλaλbλc) | 0⟩, …
2

But no SUSY, because there are 2(N − 1) microscopic
2
bosons and 2NF(N − 1) microscopic fermionsPunchline 1
F βH

No phase transitions with Z̃(β) = tr (−1) eeven when

1 < NF ≤ 5.

• Originally proposed by Kovtun, Unsal, Yaﬀe, 2007.
• Later analytic and many numerical lattice analyses basically
proved it.

At large N, KUY observation has a striking implication:
all Hagedorn instabilities cancel, without SUSY.
Basar, AC, Dorigoni, Unsal,
2013

Punchline 2

AC, Shifman,
Unsal 2018

What could be left after the Hagedorn cancellations?
Naive guess: a few particles worth of 4d degrees of freedom.

<latexit sha1_base64="JAqLFHCET2oKxs1NLW6iS21ly+M=">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</latexit>

Claim: actually get at most a 2d density of states<latexit sha1_base64="FXU+eOMJJvauZxW8TRu1V3X1G8w=">AAAHXHiclVXbbhMxEN1yC4RbAYkXXixSpCIhkkVIICEQ0FLxAG2RWkDUUeR4Z7MG76W2Ay2Wv46v4IVXxCv8ALO73ja0JAVL2Z2cOXM8M77ssJBCm17v69yx4ydOnmqdPtM+e+78hYvzly6/1vlYcdjkuczV2yHTIEUGm0YYCW8LBSwdSngz/LBU+t98BKVFnm2Y3QL6KRtlIhacGYQG831qhIzAUpXkbvHZTUK1SAmFnWIRzW1lLAUpyTN3kxD6gL6A2CgxSgxTKv9EHxAq8xHxGu9cEy1llw7BsMF8p3e7Vw1y2Ai90Qn8WB9cOvmdRjkfp5AZLpnWW2GvMH3LlBFcgmvTsYaC8Q9sBFtoZiwF3bdVHxy5gUhE4lzhLzOkQicjLEu13k2HyEyZSfRBXwn+zbc1NvH9vhVZMTaQ8XqieCyJyUnZVBIJBdzIXTQYVwJzJTxhinGDrW//MQ3oj0h1bRw0gphuKEvTYb5jNxRxzoMGQQM7xhrl6C3XgCv76ABXLCUrk94nB71Povf7/lVuV2uYuwaKGyhuoHTbpgO73fxdw9ywGZxJu9ZgL3BzRezV0rL1Zi2BQJX+MuC6KXiJYWsFKGZyrEUDT5ytnlMYXJeM6jmFwRRHAWf9ezqL56ZiLeF7BkvXWkt6tpZOvJiemZnIal5lTOcZ5nmVMbPSpCl1Fq9uW13GTF5umjrKLe63mcYTlpQbtRPiMc/0WEG53JbMHjfa+MA1Nbhtew9pUu7eBetQxC24hkPQTxhBX3UgOyGpoiKBRxuDPkW9I2YpNVgWkRHqCKOJFp9hYuLQT9x1XjQsRcNDEmV4GUrymHSreBEj3yfyyMdOTwETF5oMxWgEql2DVElW2Gr68gbwUgjEAu+98uncXhdywvEuA0W6RGRlO7xI3a8je7DXhkIJvNI6YVUCSA3/ElqHd/+tgrCuoFnJ/UImq6gT+J8xmX69ALFAWfw0hAc/BIeN13duh2i/utt5/NR/JE4H14LrwWIQBveCx8HzYD3YDHjwJfgR/Ax+nfrWOtE62zpfU4/N+ZgrwR+jdfU3rEtNRw==</latexit>

Just like in SUSY QFT, despite lack of SUSY!

Punchline 2
More precisely, in SU(N) x U(1) ~ U(N) adjoint QCD in
the large N limit we get1c3 ∼ 2N
If we take N large with geometry fixed,<latexit sha1_base64="DHp0qsZZzVueuW/3p/LCLHZusHs=">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</latexit>

No sharp cancellations at finite N. But at large N, cancellations
are as strong as in SUSY theories.

Origin of cancellationsFirst, how do we know whether or not center is broken
when L is small?
• One-loop calculations giving<latexit sha1_base64="xJvId+Ki6JPQXcgK2Pp6uLrQULQ=">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</latexit>(NF
V (⌦) ⇠
L4

1
X
1)
1

n
n=1

n 2
|tr
⌦
|
4

What is the condition for results like this to be valid?

Center symmetry in YM theory
Polyakov loop size ∼ β. When βΛ ≪ 1, quantum corrections are
small, can compare center-broken and center symmetric extrema.
• If this was false, we wouldn’t even know that hot YM isdeconfined!
More careful argument: IR divergences, magnetic and electric screening
scales, turning on non-trivial holonomy reduces IR divergences.

So for pure YM theory with LΛ ≪ 1, one gets<latexit sha1_base64="WJmcMIf+HuikDnvPvl6VXzA8Yso=">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</latexit>

V (⌦) ⇠

1
X
1
1

L4 n=1 n

n 2

|tr
⌦
|
⇠
4
2 −3

This implies log Z ∼ N β

1
X
1
1

2
N
2
|tr
1
|
⇠
N
L4 n=1 n4
L4⋅ (spatial volume)

Center symmetry in adjoint QCD
Things change in adjoint QCD with periodic boundary conditions
One-loop calculation:<latexit sha1_base64="DxVPmhRDBr9kIxK0vE+5Lje6x2s=">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</latexit>

Potential flips! Minima turn into maxima, and vice versa.

ℤN center symmetry preserved on new minimum<latexit sha1_base64="RNwKdEBZS0xjY2+NgtPlHcjS8dE=">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</latexit>

Center symmetry in adjoint QCD
Evaluate eﬀective potential on its minimum to get log Z̃

Physically, results from +/- grading for physical states.
What happened to the individual quarks and gluons?
Quarks and gluons contribute with ℤN phases
due to Ω. Huge cancellations!

Only with a trivial — center-breaking — Polyakov loop do all
fields contribute with one sign!

Center symmetry in adjoint QCD
To see how it works pick a gauge where A4 = const.
<latexit sha1_base64="LOFFEy+aGgFD7/9OXl2qRpaNygo=">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</latexit>

Holonomy ~ imaginary chemical potential for colored fields<latexit sha1_base64="gXtEZ5m/6oDi12kdE+a43SEfFBM=">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</latexit>

Implies that contributions weighed by phases in general.
2

When Ω ∼ 1N, no phases ⇒ N scaling
But with center symmetry, massive cancelations that convert
2

N → 1/N2

Center symmetry in adjoint QCD
Do cancelations work beyond one loop? Yes.
AC, Shifman,
Unsal 2018<latexit sha1_base64="1YxP0VJwIdns/UAS0ZfeS0pVDAs=">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</latexit><latexit sha1_base64="GXc+LpNBQ0RAi/L87+KYdBm10fw=">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</latexit>

Write fg as sums of terms with diﬀerent number of color traces
summed over windings.
• Finiteness of Veﬀ requires certain properties of these sums
• Some special terms can be estimated explicitly.
• Combination of these features implies all—order result:<latexit sha1_base64="DHp0qsZZzVueuW/3p/LCLHZusHs=">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</latexit>

Beyond perturbation theory
Discussion so far has been perturbative.
• Non-perturbative eﬀects are irrelevant!
• Non-perturbative eﬀects weighed by powers of e-1/λ• Dimensional transmutation turns this into powers of thestrong scale Λ• Dimensional analysis ⇒ non-perturbative physics can’t aﬀect
coeﬃcient of β−3

in log Z̃

Perturbation theory is all that matters!

Summary of part 2
If we take a standard large N limit in theories with adiabatic
continuity, fun and magical things happen:• Volume dependence disappears entirely• In best-understood example, adjoint QCD, we find amazing
SUSY-like spectral cancellations.
• Would be nice to understand in detail how twisted EguchiKawai and double-trace deformed YM manage to avoid
deconfinement from this perspective…
Of course, there is also something less fun:• We can’t calculate almost anything at large N!
• Field-theory semiclassical methods seem useless.
• Is there any easy way around that?

Part 3
Small-circle large N limit

Paths to large N
2’t Hooft limit: fix λ = g N at the UV cutoff as well as all physicalparameters, and take N → ∞We can’t solve most interesting 4d gauge theories in this limit.
• But we do have adiabatic compactification that lets us solve
some of them on small circles at finite N• If we take an ’t Hooft large N limit, we lose control.
• Large N volume independence forces this!
• Is there some other large N limit where we can keep
computational control?

Small circle large N limit
To keep control, we have to ensure mW ≫ Λ.

NLΛ ≪ 1
g2

Flow for NL  1
g2 (1/NL)



(NL)

-1

Q

Small circle large N limit
To keep control, we have to ensure mW ≫ Λ.

NLΛ ≪ 1New large N limit: fix η = NLΛ, λUV = g 2 N and take N → ∞.
• If η ≫ 1 we have no control - but it should look volume
independent.
• If η ≪ 1, we have complete control. Can calculate everything:mass gap, symmetry breaking, renormalons, theta dependence,…
• Solvable large N limits are extremely rare, so we should takethem very seriously.

Large N super-YM at small η = NLΛI’ll explain how things go for 4d 𝒩 = 1 pure SU(N) super-YM
theory• No phase transitions for any L.
• Confines and spontaneously breaks discrete chiralsymmetry at any L.
• Weakly coupled when η ≪ 1, even at large N.
What’s the low energy spectrum at large N?

Light fields at small L
For long distances ℓ ≫ mW = Λη−1, dynamics is Abelian:Without SUSY, only classically massless fields are the N - 1“Cartan gluons”.
NB: they’re physical!

(added fictitious j = 0 mode for notational simplicity; decouples exactly.)‘color’ label j = Fourier transform of winding label p.

Light fields at small L
For long distances ℓ ≫ mW = Λη−1, dynamics is Abelian:With SUSY, classically massless fields in 3D EFT
are:• N-1 Cartan gluons, from 3D components of gluon
field strength• + N-1 φa scalar fields (A3 fluctuations)• + N-1 ψa fermion fields (Cartan gluinos)• All these fields sit in the same supermultiplet
I’ll focus on the Cartan gluons - the other fields come along forthe ride.
What is the effective action?

Small L limit in perturbation theory
The Cartan gluons are classically gapless.

σi shift symmetry ⟺ conservation of magnetic charge.

No magnetic monopoles in perturbation theory
σi are massless to all orders in loop expansion

Non-perturbative mass gap
Since SU(N) → U(1)N−1, 4D BPST instanton breaks up into N
2‘monopole-instantons’ with action SI /N = 8π /λMonopole-instantons have Q = 1/N,
and magnetic charges +1,-1 under nearest-neighbor U(1)’s’t Hooft amplitude

λ ≪ 1 when NLΛ ≪ 1, dilute gas approximation justified at small L.
Contrast with usual IR disasters with instantons in YM!

Monopole-instantons and related excitations induce mass gap.

Long-distance EFT for 𝒩 = 1 SYM
N−1Since SU(N) → U(1), 4D BPST instanton breaks up into N‘monopole-instantons’ with action SI /N = 8π 2 /λsuper-YM has massless adjoint fermionson monopoles;

2 fermion zero modes

Monopole-instantons give interactions for light fermionsContribute to fermionic potential, not the bosonic one.

Long-distance EFT for 𝒩 = 1 SYM
Any field configuration with fermion zero modes can’t contributeto bosonic potential.
• Since every BPS configuration has fermion zero modes,
can’t get bosonic potential from any BPS objects.
• Mass gap is induced by topologically-trivial solutions withfinite action and magnetic charge: “magnetic bions”

Unsal 2007

Making sense of this is fun challenge for SUSY connoisseurs.

Emergent dimension
More careful look at long-distance EFT for 𝒩 = 1 sYM:

N minima due to discrete chiral symmetry breaking ℤ2N → ℤ2

• Canonically normalize, expand around any given vacuum:Complete EFT for 𝒩 = 1 SYM
Package scalars as Φj = ϕj + iσj, diagonalize quadratic action:

Mass gap vanishes at large N!
What’s going on?

Emergent dimension‘color’ label i = position, difference operators = derivatives
Large circular extra dimension L̃ = Na

Disappearing mass gap ⟺ decompactification of gapless fields

Emergent dimension
On O(N0) distance scales L̃ ≫ ℓ ≫ aspatial Lifshitz scale invariance with z = 2!

Summary
Took 4D 𝒩 = 1 super-YM, turned on “relevant deformation” —the circle.
• At least if S1 is small, resulting flow is to a non-trivial IR fixed
point!
• Is this some weird SUSY thing?

Emergent dimension far from SUSY point

Fourier-space “mass” ~ M | sin(πp/N) |Long distance theory again scale invariant, but now with z = 1• Tuning to SUSY point tunes IR theory to z = 2.
• SUSY isn’t necessary: just need massless adjoint fermions.

Fundamental matter
If we add nF ≪ N fundamental quarks, have to pick their BCs.

What’s effect on emergent dimension story?
• Use index theorems to see how ψa couple to σi
Poppitz, Unsal, 2009; AC, Schafer, Unsal, 2016; AC, Poppitz 2016

Each fundamental quark field brings in two fermion zeromodes;
they sit on one of the N monopole-instantons.
• Which monopole gets the fundamental zero modes depends
on BCs.
• Result: fundamentals don’t propagate into extra dimension!
Fundamental fermions live on 3d branes in a 4d bulk spacetime

R3

…

…

AC, Poppitz 2016

ψ1

σ

ψ2

Emergent direction yWait, what?

AC, Poppitz 2016

Took close look at rare case of solvable large N limit.
Got two startling things.
(1) Long-distance theory is non-trivially scale-invariant.
(2) Put 4D QFT on circle. Pushed circle to be small, butthen somehow NP dynamics generated a large circle!
Adjoint matter lives macroscopic 4d bulk,
fundamental matter lives on branes.
How reliable are these conclusions?
Why is this happening?

Gaplessness
Is

AC, Poppitz 2016

allowed in long-distance EFT?

No.
Compactness of SU(N) ⟺ magnetic charge quantization
Potential only has differences of σi, σj; generates derivatives!

Impossible to get mass term for emergent 4D σ scalar!

Gaplessness
So far neglected “Kahler potential” - in general it isn’t trivial!

Can’t gap out large N theory!
From extra dimension point of view, this is wave function
renormalization anomalous dimensions!

• Resulting anomalous dimensions vanish as NLΛ → 0• So at large N, get non-trivial Lifshitz scale-invariant IR
fixed points, which are weakly-coupled when NLΛ ≪ 1.

Interpreting extra dimension
Re-examine how extra dimension appears in e.g. deformed
YM caseThe mass eigenstates are labeled by the Fourier dual of “color”
index j:But the dual of the index j is the winding number! So the lattice
momentum quantum number is the holonomy winding number.

Interpreting extra dimension
Tempting but incomplete interpretation follows.
We took large N confining theory, put it on tiny circle.
Confining string winding modes become light: T-duality
S1
R3

Interpreting extra dimension
Since “T-dual” dimension comes from confining string, this extra
dimension must be a discretized one!
• Wind N times = no winding, because baryons aren’t confined.
• Upper cutoff on winding = upper cutoff on emergent
momentumThis formula only valid for p ≪ N.
This is of course just what we see!

Interpreting extra dimension
Truth in advertising:Marketoonist.com

Problems with T-duality interpretation
Truth in advertising:• No stringy interpretation of the Lifshitz scaling.
• Relating size of extra dimension to string tension confusing.
3 string tension ≠ S1 string tension.
At
small
L,
R
•

Extra factor of N unexpected from T-duality
1• But S strings are very short, so not best definition of α′S1 isn’t
obvious• T-duality interpretation very tempting, but not proven.

Gaplessness
Big open question: what is the IR behavior when NLΛ ≫ 1?

• Naively expect smooth behavior in η = NLΛ, with volume

independence setting in smoothly for large η• Three possibilities occur to me:• Behavior really is smooth - then the theory has a gaplesssector even for large η!
• Large N phase transition at some critical η to a gappedphase• T-duality picture is actually correct, and somehow allows
physics to be smooth in η, without implying large ηgaplessness

Curvature?
• The emergent dimension is flat when center symmetry is
unbroken• We could make center symmetry break spontaneously or
explicitly.
• Does the extra dimension pick a warp factor, so that the
emergent spacetime becomes curved?
• Work in progress!

AC, Andy Sheng, Baiyang Zhang, 2020?

Conclusions
Yang-Mills and QCD still surprise us a lot.
Weak-coupling insight into confinement dynamics in 4d
New insights into renormalons
Working examples of large N volume independence
Remarkable Bose-Fermi correlations without supersymmetry
Emergent dimension within field theory
Gapless confining theory
Fundamental fields ↔ branes in an emergent bulk,
all from QFT

Lots more to understand…

Thank you for your attention!