Non-perturbative effects in ABJM theory
Seyed Morteza Hosseini
University of Milano-Bicocca
October 9, 2015
Seyed Morteza Hosseini
Non-perturbative effects in ABJM theory
University of Milano-Bicocca
1 / 43
Outline
1. Non-perturbative effects
1.1 General aspects
1.2 Non-perturbative aspects of string theory
1.3 Non-perturbative effects in M-theory
2. Non-perturbative effects in ABJM theory
2.1 A short review of ABJM theory
2.2 The ‘t Hooft expansion
2.3 The M-theory expansion
Based on
Marcos Mariño, Non-perturbative effects in string theory and AdS/CFT
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Non-perturbative effects in ABJM theory
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Non-perturbative effects
General aspects
Why non-perturbative effects are important?
Most perturbative series in quantum mechanics are not convergent.
Simplest example: quartic oscillator in QM
p2
q2
+
|2 {z 2}
H=
harmonic oscillator
E0 (g) ∼
=
X
+
g 4
q
4
|{z}
.
stationary perturbation theory
an g n
n≥0
1 3 g 21 g 2 333 g 3
+
−
+
+ O(g 4 ) = ϕ(g) .
2 4 4
8 4
16 4
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Non-perturbative effects in ABJM theory
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General aspects
an ∼
n
3
n! ,
4
grow factorially for n 1 .
Non-perturbative definition of the ground state energy
H |ψn i = En |ψn i
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Non-perturbative effects in ABJM theory
n = 0, 1, 2, . . . .
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General aspects
General definitions
Given an asymptotic series
ϕ(z) =
X
an z n ,
n≥0
we say that a well-defined function f (z) provides a non-perturbative
definition of ϕ(z) if f (z) has ϕ(z) as its asymptotic series,
f (z) ∼ ϕ(z) .
f (z) +
−A/z
|e {z }
.
non-perturbative ambiguity
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Non-perturbative effects in ABJM theory
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General aspects
Extract information: optimal truncation
Find the partial sum
ϕN (z) =
N
X
an z n .
n=0
which gives the best possible estimate of f (z).
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Non-perturbative effects in ABJM theory
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General aspects
Extract information: optimal truncation
Find the partial sum
ϕN (z) =
N
X
an z n .
n=0
which gives the best possible estimate of f (z).
Typically, optimal truncation gives an exponentially small error,
proportional to
e−A/z .
Seyed Morteza Hosseini
Non-perturbative effects in ABJM theory
University of Milano-Bicocca
6 / 43
General aspects
Extract information: optimal truncation
Find the partial sum
ϕN (z) =
N
X
an z n .
n=0
which gives the best possible estimate of f (z).
Typically, optimal truncation gives an exponentially small error,
proportional to
e−A/z .
⇒ First indication of a NP effect!
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Non-perturbative effects in ABJM theory
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General aspects
Improving optimal truncation
=⇒
Borel resummation
Borel transform
ϕ(z) 7→ ϕ(ζ)
b
=
∞
X
an n
ζ .
n!
n=0
The series ϕ(ζ)
b
has a finite radius of convergence ρ = |A| and it
defines an analytic function in the circle ζ < |A|.
Example:
ϕ(z) =
∞
X
(−1)n An n! z n
n=0
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Non-perturbative effects in ABJM theory
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General aspects
Improving optimal truncation
=⇒
Borel resummation
Borel transform
ϕ(z) 7→ ϕ(ζ)
b
=
∞
X
an n
ζ .
n!
n=0
The series ϕ(ζ)
b
has a finite radius of convergence ρ = |A| and it
defines an analytic function in the circle ζ < |A|.
Example:
ϕ(z) =
∞
X
Borel transform
b
=
(−1)n An n! z n −−−−−−−−−−→ ϕ(ζ)
n=0
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Non-perturbative effects in ABJM theory
∞
X
(−1)n An ζ n =
n=0
1
1+
ζ
A
.
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General aspects
Borel sum
Let us suppose that ϕ(ζ)
b
has an analytic continuation to a
neighborhood of the positive real axis, in such a way that the Laplace
transform
Z ∞
Z ∞
−ζ
−1
e−ζ/z ϕ(ζ)
b dζ ,
e ϕ(zζ)
b
dζ = z
s(ϕ)(z) =
0
0
exists in some region of the complex z-plane. In this case, we say that
the series ϕ(z) is Borel summable and s(ϕ)(z) is called the Borel sum
of ϕ(z).
s(ϕ)(z) = z −1
X an Z ∞
X
e−ζ/z ζ n dζ =
an z n .
n! 0
n≥0
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Non-perturbative effects in ABJM theory
n≥0
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General aspects
Example:
ϕ(z) =
∞
X
n=0
Borel transform
An n! z n −−−−−−−−−−→ ϕ(ζ)
b
=
∞
X
n=0
An ζ n =
1
1−
ζ
A
.
Borel resummation does NOT exist due to the singularity in the
positive real axis.
One can then deform the contour of integration!
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Non-perturbative effects in ABJM theory
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General aspects
Lateral Borel resummations
s± (ϕ)(z) = z
−1
Z
C±
e−ζ/z ϕ(ζ)
b dζ ,
In this case one gets a complex number, whose imaginary piece is also
O(exp(−A/z)).
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Non-perturbative effects in ABJM theory
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General aspects
In many cases the (lateral) Borel resummation of the perturbative
series does not reproduce the right answer!
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General aspects
In many cases the (lateral) Borel resummation of the perturbative
series does not reproduce the right answer!
Something else should be added to the perturbative series!
X
ϕ` (z) = z b` e−`A/z
an,` z n ,
` = 1, 2, . . . .
n≥0
ϕ` (z) encodes the non-perturbative effects due to `-instantons.
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Non-perturbative effects in ABJM theory
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General aspects
Example: double-well potential in QM
H=
p2
+ W (q) ,
2
W (q) =
g
2
q2 −
1
4g
2
,
g > 0.
Ground state energy
9
89
1
− g − g2 − g3 − . . . .
2
2
3
This series is obtained by doing a path integral around the constant
1
trajectory q± = ± 2√
g.
ϕ0 (g) =
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Non-perturbative effects in ABJM theory
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General aspects
Saddle-point of the Euclidean path integral
1
t0
q±
= ± √ tanh
2 g
t − t0
2
.
This gives a non-perturbative contribution to the ground state energy,
1/2 −1/6g
2
e
√
ϕ1 (g) = −
(1 + O(g)) .
g
2π
One should then consider a trans-series of the form
Φ(z) = ϕ0 (z) +
∞
X
C ` ϕ` (z) .
`=1
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Non-perturbative effects in ABJM theory
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Non-perturbative aspects of string theory
Two coupling constants
I
I
`s →
gst →
Worldsheet instantons:
Spacetime instantons:
Seyed Morteza Hosseini
Non-perturbative effects in ABJM theory
exp −Aws /`2s ,
exp (−Ast /gst ) .
University of Milano-Bicocca
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Non-perturbative aspects of string theory
Two coupling constants
I
I
`s →
gst →
Worldsheet instantons:
Spacetime instantons:
exp −Aws /`2s ,
exp (−Ast /gst ) .
Focus on: the total partition function of superstring/M-theory in an
AdS background.
Total free energy
F (λ, gst ) =
X
2g−2
Fg (λ) gst
,
λ = function of
g≥0
L
.
`s
The functions Fg (λ) have a finite radius of convergence.
Seyed Morteza Hosseini
Non-perturbative effects in ABJM theory
University of Milano-Bicocca
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Non-perturbative aspects of string theory
It turns out that Fg (λ) are essentially analytic functions at λ = 0 (in
many examples).
Once we fix |λ| < |λ∗ |:
Fg (λ) ∼ (2g)! (Ast (λ))
−2g
,
where Ast is a spacetime instanton action.
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Non-perturbative effects in ABJM theory
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Non-perturbative aspects of string theory
It turns out that Fg (λ) are essentially analytic functions at λ = 0 (in
many examples).
Once we fix |λ| < |λ∗ |:
Fg (λ) ∼ (2g)! (Ast (λ))
−2g
,
where Ast is a spacetime instanton action.
M-theory perspective
In M-theory, worldsheet instantons and D-brane instantons can be
unified in terms of membrane instantons.
Seyed Morteza Hosseini
Non-perturbative effects in ABJM theory
University of Milano-Bicocca
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Non-perturbative effects in M-theory
One coupling constant
M-theory compactification −→ type IIA superstring theory
1/3
`p = gst `s
,
R11 = gst `s .
M2-branes
A membrane wrapped around a three-cycle S leads to an
exponentially small effect of the form
vol(S)
.
exp − 3
`p
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Non-perturbative effects in ABJM theory
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Non-perturbative effects in M-theory
I
Wrap the compact, eleventh dimension −→ fundamental strings.
vol(S) = R11 vol(Σ) ,
where Σ is a cycle in ten dimensions. Then,
vol(S)
vol(Σ)
exp − 3
= exp − 2
.
`p
`s
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Non-perturbative effects in ABJM theory
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Non-perturbative effects in M-theory
I
Wrap the compact, eleventh dimension −→ fundamental strings.
vol(S) = R11 vol(Σ) ,
where Σ is a cycle in ten dimensions. Then,
vol(S)
vol(Σ)
exp − 3
= exp − 2
.
`p
`s
I
Do not wrap the compact dimension −→ D2 branes.
vol(S) = vol(M) ,
where M is a three-dimensional cycle in ten dimensions. Then,
"
#
vol(M)/`3s
vol(S)
= exp −
exp − 3
.
`p
gst
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Non-perturbative effects in ABJM theory
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Non-perturbative effects in ABJM theory
A short review of ABJM theory
It describes N M2 branes on C4 /Zk .
W =
U (N )
4π
Tr Φ1 Φ†2 Φ3 Φ†4 − Φ1 Φ†4 Φ3 Φ†2 .
k
U (N )
Φi=1,...,4
Freund-Rubin background
One of the most important aspects of ABJM theory is that, at large
N, it describes a nontrivial background of M theory
X11 = AdS4 × S7 /Zk .
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Non-perturbative effects in ABJM theory
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Non-perturbative effects in ABJM theory
If we represent S7 inside C4 as
4
X
i=1
|zi |2 = 1
the action of Zk is given by
zi → e
2πi
k
zi .
The metric on AdS4 × S7
ds2 = L2
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Non-perturbative effects in ABJM theory
1 2
ds
+ ds2S7 /Zk
4 AdS4
.
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Non-perturbative effects in ABJM theory
The AdS/CFT correspondence
ABJM theory (3d N =6 SCFT)
M-theory on AdS4 × S7 /Zk
k, Chern-Simons level
Zk , purely geometric interpretation
N, rank of the gauge group
N, number of M2 branes
L
`p
6
= 32π 2 kN .
Thermodynamic limit
M-theory description:
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Non-perturbative effects in ABJM theory
N → ∞,
k fixed .
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Non-perturbative effects in ABJM theory
Type IIA dual
S1
S7
Hopf fibration:
CP3
reduction on S1
M-theory
X11 = AdS4 × S7 /Zk
type IIA theory
X10 = AdS4 × CP3
We need the circle to be small, and this is achieved when k is large.
1
gst = 2
k
L
`s
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Non-perturbative effects in ABJM theory
2
,
N
1
=λ=
k
32π 2
L
`s
4
.
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Non-perturbative effects in ABJM theory
Perturbative regime of the type IIA superstring corresponds to the
‘t Hooft limit,
N
fixed .
N → ∞,
λ=
k
gst
SUGRA
planar limit
λ
≈
1
λ≫1
λ
point-particle limit
strongly coupled non-linear σ − model
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Non-perturbative effects in ABJM theory
Simplest prediction of AdS/CFT
ZABJM (S3 ) = ZM-theory/string theory (X11 /X10 ) .
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Non-perturbative effects in ABJM theory
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Non-perturbative effects in ABJM theory
Simplest prediction of AdS/CFT
ZABJM (S3 ) = ZM-theory/string theory (X11 /X10 ) .
At large distances:
ZM-theory (X11 ) ≈ ZSUGRA (X11 ) .
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Non-perturbative effects in ABJM theory
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Non-perturbative effects in ABJM theory
Simplest prediction of AdS/CFT
ZABJM (S3 ) = ZM-theory/string theory (X11 /X10 ) .
At large distances:
ZM-theory (X11 ) ≈ ZSUGRA (X11 ) .
In the case of string theory we know a little bit more,
(∞
)
X
2g−2
Zstring theory (X10 ) ∼ exp
Fg (λ)gst
.
g=0
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Non-perturbative effects in ABJM theory
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Non-perturbative effects in ABJM theory
Simplest prediction of AdS/CFT
ZABJM (S3 ) = ZM-theory/string theory (X11 /X10 ) .
At large distances:
ZM-theory (X11 ) ≈ ZSUGRA (X11 ) .
In the case of string theory we know a little bit more,
(∞
)
X
2g−2
Zstring theory (X10 ) ∼ exp
Fg (λ)gst
.
g=0
The gauge theory is providing a non-perturbative definition of this
asymptotic expansion.
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Non-perturbative effects in ABJM theory
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Non-perturbative effects in ABJM theory
N and k are integers. λ and gst are continuous!
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Non-perturbative effects in ABJM theory
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Non-perturbative effects in ABJM theory
N and k are integers. λ and gst are continuous!
If we define
F (N, k) = log Z(N, k) ,
one finds, from the supergravity approximation to M-theory
√
π 2 1/2 3/2
k N
F (N, k) ≈ −
,
N 1.
3
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Non-perturbative effects in ABJM theory
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Non-perturbative effects in ABJM theory
N and k are integers. λ and gst are continuous!
If we define
F (N, k) = log Z(N, k) ,
one finds, from the supergravity approximation to M-theory
√
π 2 1/2 3/2
k N
F (N, k) ≈ −
,
N 1.
3
Planar free energy of ABJM theory at strong ‘t Hooft coupling
√
1
π 2
√
F
(N,
λ)
≈
−
,
N →∞ N 2
3 λ
lim
F0 (λ) ≈ −λ2 log λ ,
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Non-perturbative effects in ABJM theory
λ 1.
perturbation theory.
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Non-perturbative effects in ABJM theory
One can obtain explicit formulae for the genus g free energies
appearing in the 1/N expansion:
F (λ, gs ) =
∞
X
Fg (λ) gs2g−2 ,
g=0
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Non-perturbative effects in ABJM theory
gs =
2πi
.
k
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Non-perturbative effects in ABJM theory
The matrix integral describing ZABJM (S3 )
ZABJM (N, k)
=
1
N !2
Z
N
N
d µ d ν
(2π)N (2π)N
"
Q
i<j
h
i2 h
i2
µ −µ
ν −ν
2 sinh i 2 j
2 sinh i 2 j
i2
Q h
µi −νj
i,j 2 cosh
2
#
N
ik X 2
exp
(µ − νi2 ) .
4π i=1 i
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Non-perturbative effects in ABJM theory
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Non-perturbative effects in ABJM theory
M-theory regime: N → ∞, k fixed.
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Non-perturbative effects in ABJM theory
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Non-perturbative effects in ABJM theory
M-theory regime: N → ∞, k fixed.
‘t Hooft regime: N → ∞,
Seyed Morteza Hosseini
Non-perturbative effects in ABJM theory
N
k
= λ fixed.
University of Milano-Bicocca
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Non-perturbative effects in ABJM theory
M-theory regime: N → ∞, k fixed.
‘t Hooft regime: N → ∞,
N
k
= λ fixed.
The ‘t Hooft expansion
√
√
4π 3 2 3/2 ζ(3) X −2π` 2λ̂
F0 (λ̂) =
λ̂ +
+
e
f`
3
2
`≥1
π
1
p
!
.
2λ̂
f` (x) is a polynomial in x of degree 2` − 3 (for ` ≥ 2) and λ̂ = λ −
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Non-perturbative effects in ABJM theory
1
24 .
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Non-perturbative effects in ABJM theory
M-theory regime: N → ∞, k fixed.
‘t Hooft regime: N → ∞,
N
k
= λ fixed.
The ‘t Hooft expansion
√
√
4π 3 2 3/2 ζ(3) X −2π` 2λ̂
F0 (λ̂) =
λ̂ +
+
e
f`
3
2
`≥1
π
1
p
!
.
2λ̂
f` (x) is a polynomial in x of degree 2` − 3 (for ` ≥ 2) and λ̂ = λ −
4
L 2
N
L
=λ∝
⇒ e−( `s ) worldsheet instantons.
k
`s
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Non-perturbative effects in ABJM theory
1
24 .
University of Milano-Bicocca
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Non-perturbative effects in ABJM theory
AdS4 × CP3
CP3
CP1
√
Area = 2π 2λ
ℓ
S2
Spherical strings (genus zero sector)
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Non-perturbative effects in ABJM theory
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Non-perturbative effects in ABJM theory
Fg (λ) can be computed recursively.
Fg (λ) =
1
3
λg− 2
1
2
polynomial in λ + cg +
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Non-perturbative effects in ABJM theory
X
`≥1
e
√
−2π`
2λ̂
fg,`
1
p
π 2λ̂
!
.
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Non-perturbative effects in ABJM theory
Fg (λ) can be computed recursively.
Fg (λ) =
1
3
λg− 2
1
2
polynomial in λ + cg +
X
e
√
−2π`
2λ̂
fg,`
`≥1
1
p
π 2λ̂
!
.
General arguments suggest that this series diverges factorially,
Fg (λ) − cg ∼ (2g)!|Ast (λ)|−2g ,
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Non-perturbative effects in ABJM theory
g 1.
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Non-perturbative effects in ABJM theory
Fg (λ) can be computed recursively.
Fg (λ) =
1
3
λg− 2
1
2
polynomial in λ + cg +
X
e
√
−2π`
2λ̂
fg,`
`≥1
1
p
π 2λ̂
!
.
General arguments suggest that this series diverges factorially,
Fg (λ) − cg ∼ (2g)!|Ast (λ)|−2g ,
g 1.
Leading NP effect: exp (−Ast /gst )
√ √
−iAst (λ) = 2π 2 2λ + π 2 i + O e−2π 2λ ,
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Non-perturbative effects in ABJM theory
λ 1.
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Non-perturbative effects in ABJM theory
Membrane instanton/D2-brane instanton
∼ exp −2πk 1/2 N 1/2 ,
∼ exp
L
`p
in M-theory variables: N ∼
L
`p
6
3
.
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Non-perturbative effects in ABJM theory
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Non-perturbative effects in ABJM theory
Membrane instanton/D2-brane instanton
∼ exp −2πk 1/2 N 1/2 ,
∼ exp
L
`p
in M-theory variables: N ∼
L
`p
6
3
.
The genus expansion is Borel summable.
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Non-perturbative effects in ABJM theory
University of Milano-Bicocca
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Non-perturbative effects in ABJM theory
Membrane instanton/D2-brane instanton
∼ exp −2πk 1/2 N 1/2 ,
∼ exp
L
`p
in M-theory variables: N ∼
L
`p
6
3
.
The genus expansion is Borel summable.
F (N, k) 6= Borel summation of Fg (λ)
Complex saddle point! The membrane instantons are lack in our
analysis!
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Non-perturbative effects in ABJM theory
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Non-perturbative effects in ABJM theory
The M-theory expansion: N → ∞ , k fixed
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Non-perturbative effects in ABJM theory
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Non-perturbative effects in ABJM theory
The M-theory expansion: N → ∞ , k fixed
The Fermi gas approach
Cauchy identity:
i2 h
i2
νi −νj
2
sinh
i<j
2
1
= detij
i2
Q h
µ −ν
µi −νj
2 cosh i 2 j
2
cosh
i,j
2
X
Y
1
(σ)
.
=
(−1)
µi −νσ(i)
i 2 cosh
σ∈SN
2
Q
h
2 sinh
µi −µj
2
SN is the permutation group of N elements, and (σ) is the signature
of the permutation σ.
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Non-perturbative effects in ABJM theory
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Non-perturbative effects in ABJM theory
Canonical partition function of a free Fermi gas with N particles
Z(N, k) =
Z
dN x Y
1 X
(σ)
(−1)
ρ xi , xσ(i) ,
N
N!
(2π)
i
σ∈SN
with
ρ (x1 , x2 ) =
1
1
1
1
.
2πk 2 cosh x1 1/2 2 cosh x2 1/2 2 cosh x1 −x2 1/2
2
2
2
The canonical density matrix is related to the Hamiltonian operator
Ĥ in the usual way,
ρ(x1 , x2 ) = hx1 | ρ̂ |x2 i ,
ρ̂ = e−Ĥ .
where the inverse temperature β = 1 is fixed.
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Non-perturbative effects in ABJM theory
University of Milano-Bicocca
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Non-perturbative effects in ABJM theory
Grand canonical ensemble
Seyed Morteza Hosseini
Non-perturbative effects in ABJM theory
University of Milano-Bicocca
33 / 43
Non-perturbative effects in ABJM theory
Grand canonical ensemble
Grand canonical partition function
Ξ(µ, k) = 1 +
X
Z(N, k)eN µ .
N ≥1
Here, µ is the chemical potential.
Seyed Morteza Hosseini
Non-perturbative effects in ABJM theory
University of Milano-Bicocca
33 / 43
Non-perturbative effects in ABJM theory
Grand canonical ensemble
Grand canonical partition function
Ξ(µ, k) = 1 +
X
Z(N, k)eN µ .
N ≥1
Here, µ is the chemical potential.
Grand canonical potential
J (µ, k) = log Ξ(µ, k) = −
X (−κ)`
`≥1
`
Z` ,
where
κ = eµ (fugacity) ,
Seyed Morteza Hosseini
Non-perturbative effects in ABJM theory
Z` = Tr ρ̂` .
University of Milano-Bicocca
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Non-perturbative effects in ABJM theory
The canonical partition function is recovered from the grand
canonical one by integration,
I
dκ Ξ(µ, k)
Z(N, k) =
.
2πi κN +1
The spectrum of the Hamiltonian Ĥ is defined by,
ρ̂ |ϕn i = e−En |ϕn i .
n = 0, 1, . . . ,
or equivalently by the integral equation associated to the kernel,
Z
ρ(x, x0 )ϕn (x0 )dx0 = e−En ϕn (x) .
n = 0, 1, . . . .
The spectrum is discrete and the energies are real.
Seyed Morteza Hosseini
Non-perturbative effects in ABJM theory
University of Milano-Bicocca
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Non-perturbative effects in ABJM theory
The grand canonical partition function is given by the Fredholm
determinant,
Y
1 + κe−En .
Ξ(µ, k) = det(1 + κρ̂) =
n≥0
In terms of the density of eigenvalues
X
ρ(E) =
δ(E − En ) ,
n≥1
we also have the standard formula
Z ∞
J (µ, k) =
dEρ(E) log 1 + κe−E .
0
Seyed Morteza Hosseini
Non-perturbative effects in ABJM theory
University of Milano-Bicocca
35 / 43
Non-perturbative effects in ABJM theory
The grand canonical partition function is given by the Fredholm
determinant,
Y
1 + κe−En .
Ξ(µ, k) = det(1 + κρ̂) =
n≥0
In terms of the density of eigenvalues
X
ρ(E) =
δ(E − En ) ,
n≥1
we also have the standard formula
Z ∞
J (µ, k) =
dEρ(E) log 1 + κe−E .
0
What can we learn from the ABJM partition function in the Fermi
gas formalism?
Seyed Morteza Hosseini
Non-perturbative effects in ABJM theory
University of Milano-Bicocca
35 / 43
Non-perturbative effects in ABJM theory
Density matrix
1
1
ρ̂ = e− 2 U (x̂) e−T (p̂) e− 2 U (x̂) ,
with
h
x i
U (x) = log 2 cosh
,
2
h
p i
T (p) = log 2 cosh
.
2
x̂, p̂ are canonically conjugate operators,
[x̂, p̂] = i~ ,
Seyed Morteza Hosseini
Non-perturbative effects in ABJM theory
~ = 2πk .
University of Milano-Bicocca
36 / 43
Non-perturbative effects in ABJM theory
Density matrix
1
1
ρ̂ = e− 2 U (x̂) e−T (p̂) e− 2 U (x̂) ,
with
h
x i
U (x) = log 2 cosh
,
2
h
p i
T (p) = log 2 cosh
.
2
x̂, p̂ are canonically conjugate operators,
[x̂, p̂] = i~ ,
~ = 2πk .
Note that ~ is the inverse coupling constant of the gauge
theory/string theory.
Seyed Morteza Hosseini
Non-perturbative effects in ABJM theory
University of Milano-Bicocca
36 / 43
Non-perturbative effects in ABJM theory
The potential U (x) is a confining one, and at large x it behaves
linearly,
|x|
U (x) ∼
,
|x| → ∞ .
2
When N → ∞, the typical energies are large, and we are in the
semiclassical regime,
Hcl (x, p) = U (x) + T (p) ∼
Seyed Morteza Hosseini
Non-perturbative effects in ABJM theory
|x| |p|
+
.
2
2
(1)
University of Milano-Bicocca
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Non-perturbative effects in ABJM theory
The potential U (x) is a confining one, and at large x it behaves
linearly,
|x|
U (x) ∼
,
|x| → ∞ .
2
When N → ∞, the typical energies are large, and we are in the
semiclassical regime,
Hcl (x, p) = U (x) + T (p) ∼
|x| |p|
+
.
2
2
(1)
Fermi surface: Hcl (x, p) = E
For large values of the energies, the Fermi surface is very well
approximated by the polygon (1).
Vol(E)
8µ2
∂J (µ, k)
= hN (µ, k)i ≈
≈
.
∂µ
Vol. of an elementary cell
2π~
Seyed Morteza Hosseini
Non-perturbative effects in ABJM theory
University of Milano-Bicocca
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Non-perturbative effects in ABJM theory
J (µ, k) ≈
Grand canonical potential:
2µ3
3π 2 k
.
At large N, the contour integral
I
Z(N, k) =
dκ Ξ(µ, k)
.
2πi κN +1
can be computed by a saddle-point approximation.
Seyed Morteza Hosseini
Non-perturbative effects in ABJM theory
University of Milano-Bicocca
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Non-perturbative effects in ABJM theory
J (µ, k) ≈
Grand canonical potential:
2µ3
3π 2 k
.
At large N, the contour integral
I
Z(N, k) =
dκ Ξ(µ, k)
.
2πi κN +1
can be computed by a saddle-point approximation.
Free energy: F (N, k) ≈ J (µ∗ , k) − N µ∗ ,
where µ∗ is the function of N and k defined by,
√
2 1/2 1/2
πk N
.
µ∗ ≈
2
Seyed Morteza Hosseini
Non-perturbative effects in ABJM theory
University of Milano-Bicocca
38 / 43
Non-perturbative effects in ABJM theory
In the strict large N limit
√
π 2 3/2 1/2
−F (N, k) ≈
N k .
3
Seyed Morteza Hosseini
Non-perturbative effects in ABJM theory
University of Milano-Bicocca
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Non-perturbative effects in ABJM theory
In the strict large N limit
√
π 2 3/2 1/2
−F (N, k) ≈
N k .
3
The WKB expansion of the grand potential reads,
X
J WKB (µ, k) =
Jn (µ)k 2n−1 .
n≥0
Seyed Morteza Hosseini
Non-perturbative effects in ABJM theory
University of Milano-Bicocca
39 / 43
Non-perturbative effects in ABJM theory
In the strict large N limit
√
π 2 3/2 1/2
−F (N, k) ≈
N k .
3
The WKB expansion of the grand potential reads,
X
J WKB (µ, k) =
Jn (µ)k 2n−1 .
n≥0
Leading term n = 0
Z` = Tr e
−`Ĥ
Seyed Morteza Hosseini
Non-perturbative effects in ABJM theory
≈
Z
dxdp −`Hcl (x,p)
e
,
2π~
k → 0.
University of Milano-Bicocca
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Non-perturbative effects in ABJM theory
J0 (µ) =
where
J0M2 (µ) = −
Seyed Morteza Hosseini
Non-perturbative effects in ABJM theory
4µ3
µ 2ζ(3)
+ +
+ J0M2 (µ) ,
3π 2
3
π2
∞
X
a0,` µ2 + b0,` µ + c0,` e−2`µ .
`=1
University of Milano-Bicocca
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Non-perturbative effects in ABJM theory
J0 (µ) =
where
J0M2 (µ) = −
4µ3
µ 2ζ(3)
+ +
+ J0M2 (µ) ,
3π 2
3
π2
∞
X
a0,` µ2 + b0,` µ + c0,` e−2`µ .
`=1
Fermi gas approach makes it possible to go beyond the strict large N limit.
e−2µ ∼ e−
Seyed Morteza Hosseini
Non-perturbative effects in ABJM theory
√
2πk1/2 N 1/2
∼e
3
− `Lp
.
University of Milano-Bicocca
40 / 43
Non-perturbative effects in ABJM theory
J0 (µ) =
where
J0M2 (µ) = −
4µ3
µ 2ζ(3)
+ +
+ J0M2 (µ) ,
3π 2
3
π2
∞
X
a0,` µ2 + b0,` µ + c0,` e−2`µ .
`=1
Fermi gas approach makes it possible to go beyond the strict large N limit.
e−2µ ∼ e−
√
2πk1/2 N 1/2
∼e
3
− `Lp
.
Beyond the leading order of the WKB expansion
J1 (µ) =
Seyed Morteza Hosseini
Non-perturbative effects in ABJM theory
µ
1
−
+ O µ2 e−2µ .
24 12
University of Milano-Bicocca
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Non-perturbative effects in ABJM theory
Non-renormalization theorem
Jn (µ) = An + O µ2 e−2µ ,
Seyed Morteza Hosseini
Non-perturbative effects in ABJM theory
n ≥ 2.
University of Milano-Bicocca
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Non-perturbative effects in ABJM theory
Non-renormalization theorem
Jn (µ) = An + O µ2 e−2µ ,
n ≥ 2.
⇒ J WKB (µ, k) = J (p) (µ) + J M2 (µ, k) .
The “perturbative” piece J (p) (µ) is given by
C(k) 3
µ + B(k)µ + A(k) ,
3
2
1
k
C(k) = 2 ,
B(k) =
+
,
π k
3k 24
J p (µ) =
where
A0 =
Seyed Morteza Hosseini
Non-perturbative effects in ABJM theory
2ζ(3)
,
π2
A1 = −
A(k) =
X
An k 2n−1 ,
n≥0
1
.
12
University of Milano-Bicocca
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Non-perturbative effects in ABJM theory
We find that, up to exponentially small corrections,
Z
1
Z(N, k) ≈
exp J (p) (µ) − µN dµ
2πi
Z
1
C(k) 3
=
exp
µ + (B(k) − N ) µ + A(k) dµ
2πi
3
C
n
o
= eA(k) C(k)−1/3 Ai C(k)−1/3 [N − B(k)] .
Seyed Morteza Hosseini
Non-perturbative effects in ABJM theory
University of Milano-Bicocca
42 / 43
Non-perturbative effects in ABJM theory
We find that, up to exponentially small corrections,
Z
1
Z(N, k) ≈
exp J (p) (µ) − µN dµ
2πi
Z
1
C(k) 3
=
exp
µ + (B(k) − N ) µ + A(k) dµ
2πi
3
C
n
o
= eA(k) C(k)−1/3 Ai C(k)−1/3 [N − B(k)] .
M-theory regime
F (N, k) ≈
1
1
ζ 3/2 + log
384π 2 k
6
π3 k3
ζ 3/2
+A(k)+
∞
X
dn+1 π 2n k n ζ −3n/2 ,
n=1
where ζ = 32π 2 k [N − B(k)] .
Seyed Morteza Hosseini
Non-perturbative effects in ABJM theory
University of Milano-Bicocca
42 / 43
Thank you for your attention!
Seyed Morteza Hosseini
Non-perturbative effects in ABJM theory
University of Milano-Bicocca
43 / 43