A new look at instantons
in THE large-N limit
Masanori Hanada(YITP, Kyoto)(mainly at SITP, Stanford U., Oct2013〜)
Fujita (UW, Seattle→IPMU)-Hoyos(Tel Aviv)-M.H., PRD(2012)
Azeyanagi(Harvard→ENS, Paris)-Fujita-M.H., PRL (2013)
Azeyanagi-M.H.-Honda(KEK→HRI, India)-Matsuo(KEK)-Shiba(KEK), 1307.0809[hep-th]

28 Sept 2013 @ KEKIntroduction & summary•••••

The standard folklore on the large-N limit is wrong.
Nonperturbative effects can be treated straightforwardly inthe large-N limit.
Now you can easily derive full instanton partition functionsof a class of non-SUSY theories.
A conceptual difficulty for the gauge/gravity duality with theM-theory gravity dual is resolved.
Now we have an appropriate set up to play withinstantons/monopoles with 11d gravity dual -- qualitatively
new applications of the gauge/gravity duality to QCD!

what is “large-N” ?
usual
2answer λ=g N fixed, N→∞ (’t Hooft limit)why?

• 1/N expansion = genus expansion. (stringtheory!)• perturbative series may have a finite radius ofconvergence at large-N → analytic
continuation to strong coupling ?

• Various nice properties (factorization,vertex 〜 N
index loop 〜 N
propagator 〜 1/N
planar diagram

N2× N-3×N3 =2N

nonplanar diagram(genus one)

2-3

1

N × N ×N =vertex 〜 N〜triangle/rectangle
index loop 〜 N〜vertex
propagator 〜 1/N〜sidetorus

triangulation of torus

Euler number(#triangles)−(#sides)+(#vertices)=2−3+1=0
more generally,(#triangles)−(#sides)+(#vertices)=2−2gwhereg = (#genus)

two-sphere (g=0)

4 triangles
6 sides
4 vertices
4−6+4 = 2 = 2−2g

6 squares
12 sides 6−12+8 = 2 = 2−2g
8 veticesdiagram which can be drawn
genus-g diagram =on genus-g surface
g closed string loops
(1/N)

2g-2
2

2g= gs1/N = gsg

classical gravity = planar limit

In AdS/CFT,
1/N correction = gs correction
1/λ correction = α’ correction
But what about M-theory?
(gYM) 〜12

Let’s consider Another large-N limit:
g 〜 N ;α=1 is the ‘t Hooftlimit
α<1: ‘very strongly
coupled’why?
It is possible.
2-α•2• application to M-theory.(gYM) 〜1• Instanton effect remains
finite.
exp(-8π2/gYM2) = O(1)λ is N-dependent.
1/N expansion and genus expansion are different.

Our conjecture• The very strongly coupled large-N limit iswell-defined and essentially the same as the ‘t
Hooft limit.

• More precisely: large-N limit and strong
coupling limit commute.

• When there is no ‘phase transition’ (or as

long as one considers the same point in themoduli space), the analytic continuation fromthe planar limit gives the right answer.

N=∞‘IIA SUGRA’‘11d SUGRA’
α=1/2
α=1

α=0

The large-N limit(simply PLANAR)

2λ=gYM N(Azeyanagi-Fujita-M.H., Phys.Rev.Lett. 110 (2013))

A typical (wrong) objection•Planar and nonplanar diagrams are mixed in such alimit!

Nonplanar diagrams contribute as well,so the limits do not commute!
Why are you using a perturbative
expression at strong coupling???????

Example (1)4d N=4 SYM

AdS5/CFT4 conjectre(Maldacena, 1997)

Bunch of D3-branes4d N=4 SYM
equivalent
gYM 〜gstring,
2IIB string on
AdS5×S52α’/RAdS 〜-1/2λperturbative string picture is valid when2gYM <<1, λ>>1

The right picture[Azeyanagi-Fujita-M.H., Phys.Rev.Lett. 110 (2013)]

When there is a gravity dual:(gs〜gYM 〜λ/N, α’〜λ2-1/2

for 4d N=4 SYM)
2

f0,0 dominates as long as gYM <<1 and λ>>1
The same expression at 1<< λ<<N, simply supergravity!
(By using the S-dual, we can show it even at N < λ.)•••••

perturbative string picture is valid when
2gYM <<1, λ>>1

Usually one takes the ‘t Hooft limit first and then consider
strong ‘t Hooft coupling. (tree-level string)
Or one consider large-but-finite-N with λ=O(1), so that
1/N expansion and string loop expansion coincide.
But according to Maldacena’s conjecture, such limit is not
required for the validity of the weakly coupled gravity
description.
So, the very strongly coupled limit exists, and at λ << N it
is simply the same as the planar limit: supergravity!
Analytic continuation to λ >> N can be confirmed by using
S-duality.

Example (2)ABJM theory(Analytic continuation to M-theory)ABJM theory(Aharony-Bergman-Jafferis-Maldacena, 2008)

AB

JM

3d U(N)k×U(N)-k Superconformal Chern-Simons-Matter theory(developed out of earlier works by Schwarz, Bagger-Lambert, etc etc...)M-theory

1/N

perturbative
gauge
theoryquantum
stringtree-level string
IIASUGRA(α’ correction)
λ=N/kPrediction from gravity side

• Free energy in IIA string region5(k<<N<<k )• Free energy in M-theory region (N >> k )5the same expression→ analytic continuationfrom λ=O(1) to λ = O(N)N-dependent
AdS/CFT tells us that, at strong coupling,α’-expansion (1/λ-expansion) is good,at least in IIA string region.
Only the leading term in each Fg is important.
(g=0),(g>0)

g=0 (planar) dominates even outside the planar limit.

More on the planar dominancein the M-theory limit

Large-N orbifold equivalence (Kachru-Silverstein 1998)
4d N =4

AdS/CFTU(kN) SYM
orbifoldequivalence

IIB stringon5AdS5×S
orbifold

equivalence

AdS/CFTIIB string[U(N)] SYM
on5AdS×S/Z5• In the gravity side, Z -invariant modes do notkkkdistinguish these two theories.

•In the gauge theories, correlation functions of Zkinvariant operators coincide with the counterparts inthe orbifolded theory.

Large-N orbifold equivalence (Kachru-Silverstein 1998)
4d N =4AdS/CFTU(kN) SYM
orbifold
equivalence

IIB stringon5
AdS5×S
orbifold

equivalence

AdS/CFTIIBstring[U(N)] SYM
on5/Zklanguage.
In the planar limit, it can be proven in theAdS
field5×S
theoryk(Bershadsky-Johansen ’98, Kovtun-Unsal-Yaffe ’06,...)

From the gauge theory point of view,the equivalence is gone as soon as thenonplanar diagrams are taken into account.
However, though the planar limit is always assumed, it is notreally necessary; classical gravity description is the key.

orientifold

Orbifold equivalence inABJM(1)(Fujita-M.H.-Hoyos 2012)

U(2N)2k×U(2N)-2k

AdS/CFT

7

AdS4×S /Z2k
orientifoldprojection

O(2N)2k×USp(2N)-kAdS/CFT

7

AdS4×(S /Z2k)/Z2It provides us with a natural generalization of
the planar equivalence to the M-theory region.

orientifold

Orbifold equivalence inABJM(2)(Fujita-M.H.-Hoyos 2012)

U(2N)2k×U(2N)-2k

AdS/CFTIIA onAdS×CP
4
orientifold3equivalenceO(2N)2k×USp(2N)-kAdS/CFT

IIA onAdS4×(CP3/Z2)
5

k<<N<<korientifold

Orbifold equivalence inABJM(3)(Fujita-M.H.-Hoyos 2012)

U(2N)2k×U(2N)-2k

AdS/CFT

M on
7AdS
orientifold
4×S /Z2kequivalenceO(2N)2k×USp(2N)-kAdS/CFT

5N>>k

M on
7AdS4×(S /Z2k)/Z2

orientifold

Orbifold equivalence inABJM(4)(Fujita-M.H.-Hoyos 2012)• Planar equivalence, which does not hold if thenonplanar contribution is taken into account, isnaturally generalized to the M-theory region.
5• No discontinuity at N〜k (←loclization)
Why can it hold?
The planar dominance outside the planar limit!

More evidence• Any field theory with a gravity dual
• 2d pure Yang-Mills• strong coupling expansion of the lattice(solvable thanks to Migdal)

gauge theory• large-N reduction(Eguchi-Kawai 1982)

in the M-

theory limit (Honda-Yoshida 2012, Ishiki-Ohta-Shimasaki-Tsuchiya,private communication)

Instantons(Azeyanagi-M.H.-Honda-Matsuo-Shiba, 1307.0809 [hep-th])instantons• The same argument is valid at each instanton(Azeyanagi-M.H.-Honda-Matsuo-Shiba, 1307.0809 [hep-th])sector.

• Therefore ‘planar dominance’ holds there.
• Can be confirmed in various theories with N=2
SUSY by using the Nekrasov formula for thepartition functions.
Nice properties in the planar limit holds
where the instanton weight is finite!
22exp(-8π /gYM ) = O(1)orbifold equivalence between
instanton patition functions(※ It is just one of various examples, which isalmost trivial from our new viewpoint.)‘parent’
4d N=2 SYM

Zk orbifolding

Nekrasov formula‘daughter’
YM with less SUSY
you can take it
to be non-SUSY!

Zp-inst, parent = (Zp-inst,daughter)in THE large-N limit

kWhen the daughter keeps N=2 SUSY,
you can easily confirm it by using the Nekrasov formula.
(Azeyanagi-M.H.-Honda-Matsuo-Shiba, 1307.0809 [hep-th])U(2N)

Z2 projection

2NAμ 00 Bμ[U(N)]

NAμk1-inst

2NBμk2-inst(k1+k2)-inst
Z(k1+k2)-inst, parent = (Z(k1+k2)-inst,2)
daughter
The equivalence holds at each instanton sector.

Speculations

M-theoretic holography(in progress)• 11-d SUGRA should know the instantons,
2monopoles, etc at large-N with g fixed.
This is nothing but ‘planar’ in the gauge
theory side.

• It should be possible to study thedynamics of solitons by using 11d SUGRA!

•Asafirstnontrivial
test:On-shell action of 11d SUGRAFree energy of 4d N=2 gauge(Gaiotto-Maldacena geometry)=theory (Gaiotto theory)(conjecture; now checking it.)Instantons in QCD(not even in progress)• The coupling constant runs with the scale.
• The ’t Hooft coupling diverges when instanton
size is of order 1/ΛQCD. →The very strong
coupling limit can be realized.

• Small instantons (g 〜N →∞) are naturally-2#suppressed.

• So instantons with the radius 1/ΛQCD can bedominant. Looks consistent with lattice data!

Conclusion

N=∞‘IIA SUGRA’‘11d SUGRA’

α=1

α=02

λ=gYM N