Large N matrix models, gravity and strings

Marcos MariñoUniversity of Geneva

Three key problems in modern theoretical physics (in order of importance):

1. Quantizing gravity

2. Understanding the strong-coupling behavior of gauge theories

The AdS/CFT correspondence gives a framework in which these problems can be addressed in a

quantitative way, at least for some toy models (with supersymmetry). Essentially, the first two problems

are each other’s solution!

3. Defining strings/M-theory non-perturbatively

AdS/CFT postulates an exact equivalence:

These are theories with two quantum parameters:

(quantum) U(N) gauge theory =(quantum) string theory/ M-theory/gravity

1 / N

! = g2N

gst

‘t Hooft coupling

L/!s

Take the simplest quantity in a gauge theory:the free energy (i.e. the log of the partition function) on a

compact manifold M

F (M) = log!

DA · · · e!S(A,··· ) would give the thermal free energy

M = S2 ! S1!

‘t Hooft expansion: N !", ! fixed

F (M) = N2F0(!) +O(N0)

planar free energy/free energy of strings at genus zero (sphere)

expansion in powers of 1/N

To reach this strongly coupled regime we must resum all planar diagrams in the gauge theory. Therefore, in AdS/CFT, gravity is an emergent phenomenon: it emerges from strong

gauge interactions

In the limit where strings are spherical and point-like we recover classical (super)gravity. But in gauge theory this is

planar but strongly coupled limit

+ · · ·F0(!) =

! !2! = 0

Perturbation theory is an

expansion around +

N !", !!"

In this talk I will review some recent developments in which this general picture has been made very

precise at a quantitative level

This involves an AdS/CFT “dual pair” in which one can explicitly resum the gauge theory diagrams to

see how classical gravity emerges in the strong coupling regime of the gauge theory.

Moreover, one can use the gauge theory to study quantum and stringy corrections to classical gravity

Basic building block: Chern-Simons theory and its supersymmetric extensions

SCS = ! k

4!

!

Mtr

"A " dA +

2i3

A "A "A#

CS level (must be an integer)

Susy extensions use the standard N=2, 3d vector multiplet, built on the gauge connection, and the N=2, 3d

matter hypermultiplet

auxiliary

A B J M theory

U(N1)k ! U(N2)!k

ABJM theory: 2 CS theories + 4 hypers in the bifundamental

ABJM theory is a 3d superconformal field theory

two ‘t Hooft couplings

!i =Ni

k

In this talk we will mostly restrict ourselves to the “ABJM slice”

!1 = !2 = ! =N

k

U(N1) U(N2)

!i=1,··· ,4

Large N string theory dual [Maldacena 1998]

IIA superstring theory on the AdS compactification

The dictionary relating string and gauge theory parameters is

!L

!s

"4

= 32"2#

gst =1k

!L

!s

"! "5/4

N

! =N

k

AdS4 ! CP3

ds2 = L2

!14ds2

AdS4+ ds2

CP3

"

(common) radius

Metric:

The parameter space of string/gauge dualities

g st

classicalSUGRAgauge

loops

! e!Ast(!)/gst

spacetime loops,spacetime instantons

! = 0 !! 1

worldsheet loops ! 1/"

!

worldsheet instantons! e!

"!

1N

quan

tum

SUG

RA

But this “background” also sheds light on the mysterious...

M-theory dualThis is classically 11d SUGRA, and after compactification on a

circle leads to the type IIA superstring. In our case, the SUGRA background is

AdS4 !X7 X7 = S7/Zk

In M-theory, k is a geometric parameter and it realizes the strong coupling limit of type IIA string. N is related to the radius (in

Planck length units)

1k

!L

!p

"6

= 32"2N

ds2 = L2

!14ds2

AdS4+ ds2

X7

"

L/!p

L/!p ! 1 L/!p ! 1

Planckian sizes, strong quantum gravity effects

weak curvature, classical SUGRA is a good

approximation

N large, k fixedN small, k fixed

The natural expansion in M-theory is in powers of !p/L

This leads to the M-theory expansion of ABJM theory: a 1/N expansion at fixed k. This is not the ‘t Hooft expansion, which is an expansion in powers of 1/N at fixed ‘t Hooft parameter

“thermodynamic limit”

! =N

k

An AdS/CFT prediction at strong coupling

At large N and fixed k, one can use classical (super)gravity and just evaluate the gravitational AdS action on-shell (as is standard

in Euclidean quantum gravity). One finds, after using the AdS/CFT dictionary and regularization:

We will look at a “supersymmetric” quantity: the partition function/free energy on the three-sphere. If AdS/CFT holds the

partition functions should be equal [Witten]:

ZABJM(S3;N, k) = ZM(AdS4 !X7) " e!ISUGRA

FABJM(S3) ! "!#

23

k1/2N3/2

“anomalous” scaling [Klebanov-Tseytlin]

Equivalently, we can use the string picture, and consider classical gravity as the point-particle limit of spherical strings. Therefore, the above gravity calculation can be regarded as a

prediction for the planar free energy at strong ‘t Hooft coupling

F0(!) ! ""#

23#

!!!"

Is it possible to resum all planar diagrams in this theory?

In supersymmetric theories one can often reduce the path integral to an integral over supersymmetric configurations/

vacua. On spherical spacetimes the vacua are just zero modes and the gauge theory path integral reduces to a matrix model

(i.e. an integration over constant matrices)

Example: in N=4 SYM the path integral calculating the vev of the 1/2 BPS Wilson loop reduces to a Gaussian, Hermitian

matrix model [Ericksson-Semenoff-Zarembo, Drukker-Gross, Pestun]

Localization and matrix models

Reduction to a matrix model in ABJM

Localization techniques were applied to the ABJM theory in a beautiful paper by [Kapustin-Willett-Yaakov]. The partition function on

is given by the following matrix integral:

contribution 4 hypers

S3

contribution 2 vectors

ZABJM(S3) =1

N !2

! N"

i=1

dµi

2!

d"i

2!

"

1!i<j!N

#2 sinh

#µi ! µj

2

$$2

""

1!i<j!N

#2 sinh

#"i ! "j

2

$$2 "

i,j

#2 cosh

#µi ! "j

2

$$"2

e"1

2gs(P

i(µ2i"!2

i ))

gs =2!ik

Calculating at large N

We have then reduced our problem drastically, from a field theory path integral to a matrix integral. However, in order to extract the gravitational/stringy physics, we have to study this

integral in the limit of large N

‘t Hooft expansion N !", ! fixed

makes contact with string theory

M-theory expansion N !", k fixedmakes contact with M-theory

One has to develop tools to study this integral in these two different regimes [M.M.+Putrov, Drukker, Klemm, ...]

The ‘t Hooft expansion

The planar free energy can be obtained, as an exact function of the ‘t Hooft parameter, by adapting standard large N technology for matrix models [Brezin-Itzykson-Parisi-Zuber 1978...].

0.1 0.2 0.3 0.4 0.5

10

20

30

40

50

60

!

exact

gravity

!!

!"2F0(")

"

string instantons

CP3

CP1

This result, being exact, contains an infinite series of “stringy” corrections to the point-particle limit. They are conjecturally due

to embeddings of spherical string worldsheets inside the projective space :

One can also calculate subleading 1/N corrections in the ‘t Hooft expansion, which are exact in the ‘t Hooft parameter

F0(!) = !""

23"

!+O

!e!2!

"2"

"

! O!e!L2/!2s

"

The M-theory expansion

One way to obtain results in the M-theory expansion is to re-express the matrix integral as the partition function of a one-

dimensional ideal Fermi gas [M.M.-Putrov]. One obtains

ZABJM ! Ai

!"!2k

2

#1/3 "N " 1

3k" k

24

#$+O

%e!"

Nk&

Airy function originally [Fuji, Hirano, Moriyama]

This expression resums the full perturbative quantum gravity partition function of M-theory/11d SUGRA and includes the full

perturbative series in !p/L

The exponentially small effects are of the form

and correspond to membranes wrapping three-dimensional submanifolds of the compactification space

! O!e!L3/!3p

"

These results in the M-theory expansion can be extended to 3d quantum gauge theories generalizing ABJM theory. These

theories are dual to M-theory on other backgrounds, and their partition function on the three-sphere can be reduced to a matrix model. In the large N limit one “reconstructs” the

background geometry

Conclusions and prospects

• Interesting supersymmetric gauge theories reduce sometimes to matrix models. These encode, at large N, not only classical AdS gravity, but also quantum/stringy corrections

• We have in particular matrix model predictions for quantum gravity effects, as well as an exact partition function in M-theory (up to exponentially small effects)

• Can we reproduce these results directly in the gravity side? What can we learn about quantum gravity by using these exact results?