University of Utrecht
MSc Thesis (Theoretical Physics)
Matrix Models of 2D
String Theory in
Non–trivial Backgrounds
Arnaud Koetsier
Supervisors: Prof. G. ’t Hooft and Dr. S. Alexandrov

1/31
Overview of Talk
1. Introduction — the partition function
2. Critical strings in background fields
3. Noncritical strings and Liouville gravity
4. Matrix models and discretised surfaces
5. Matrix quantum mechanics (MQM)
6. MQM in the chiral representation
7. Conclusion and outlook

1—I NTRODUCTION 2/31
Partition function of string theory
Euclidean Polyakov action + Einstein–Hilbert action
(E) 1 √
SP = dτ dσ h habGµν ∂ a X µ ∂ b X ν + α νR
4πα Σ
world–sheet Σ swept out by the string, with curvature R in Euclidean signature.
Formal partition function of string theory:
(E)
µ −SP [X µ ,hab ,R]
Z= D (hab) DX e
Topologies
Can view as partition function of 2D quantum gravity coupled to matter fields X µ.

1—I NTRODUCTION 3/31
Topological Expansion
For Σ a 2D Riemann surface
1 √
dτ dσ h νR = νχ = ν(2 − 2g) (Euler–Poincare characteristic)
´
4π Σ
⇒ Z is a sum over genus g closed orientable surfaces!
String coupling: κ ∼ eν
∞
Z= κ−χZg = κ2 + + κ−2 + κ−4 + ···
g=0
µ 1 √
Zg = D (hab) DX exp − dτ dσ h habGµν ∂ a X µ ∂ b X ν
4πα Σ
D (hab) dealt with in terms of discretised surfaces −→ Matrix Models

2—C RITICAL S TRINGS IN B ACKGROUNDS 4/31
Critical Strings in Background Fields I
Background fields emerge out of string dynamics
In turn, string dynamics is determined by background.
Non–trivial background: include the excitation modes of the string in the string
world–sheet action.
For the closed bosonic string, massless modes are:
Gµν (X ρ) traceless, symmetric graviton,
Bµν (X ρ) antisymmetric axion,
Φ(X ρ) scalar dilaton.

2—C RITICAL S TRINGS IN B ACKGROUNDS 5/31
Critical Strings in Background Fields II
Adding those fields to the action, we get a sigma model in D dimensions
1 2
√
Sσ = d σ h habGµν (X) + i ab
Bµν (X) ∂ a X µ ∂ b X ν
4πα
+α RΦ(X)
NB: Coupling constants on the world–sheet become fields in the target space. e.g.
ν → Φ(X).
Gµν , Bµν and Φ so far unconstrained.
Each “running coupling” (target–space field) has a renormalization group
β–function
=⇒ Theory should possess conformal invariance =⇒ β–functions vanish

2—C RITICAL S TRINGS IN B ACKGROUNDS 6/31
β–functions
To zeroth order in α
1
G
βµν = Rµν + 2DµDν Φ − HµσρHν σρ = 0,
4
B
βµν = DρHµν ρ − 2Hµν ρDρΦ = 0,
1 D − 26 1 1
βΦ = + 4(DµΦ)2 − 4DµDµΦ − R + HµνρH µνρ = 0,
α 48π 2 16π 2 12
Hµνρ = ∂ µ Bνρ + ∂ ρ Bµν + ∂ ν Bρµ
Critical string theory:
Gµν = ηµν , Bµν = 0, Φ=ν ⇒ D = 26
Solution of interest: Linear dilaton background
Gµν = ηµν , Bµν = 0, Φ = lµ X µ .

2—C RITICAL S TRINGS IN B ACKGROUNDS 7/31
Problem: Divergent String Coupling
Linear dilaton background in present form leads to a divergent string coupling.
String coupling
ν Φ(X ρ ) lµ X µ
κ∝e =e =e
=⇒ For some X µ, coupling diverges!
Solution: add a potential to the sigma model
cosmo 1 2
√
Sσ = d σ hT (X µ)
4πα

2—C RITICAL S TRINGS IN B ACKGROUNDS 8/31
Cosmological Constant Term I
New action (with Bµν = 0):
1 2
√
Sσ = d σ h habGµν (X) ∂ a X µ ∂ b X ν + α RΦ(X) + T (X µ)
4πα
New β–functions:
G
βµν = Rµν + 2DµDν Φ − DµT Dν T = 0,
βT = −2DµDµT + 4DµΦDµT − 4T = 0,
D − 26
βΦ = − R + 4(DµΦ)2 − 4DµDµΦ + (DµT )2 − 2T 2 = 0,
3α
These β–functions are the Euler–Lagrange equations of an effective target space
action.

2—C RITICAL S TRINGS IN B ACKGROUNDS 9/31
Cosmological Constant Term II — Effective Action
“tachyon” part of effective action is
tach 1 D
√ −2Φ 4 22
Seff =− d X −Ge (DµT ) − T
2 α
1. Substitute L.D. background: Gµν = ηµν , Bµν = 0, Φ = lµX µ
2. Equation of motion on shell: ∂ 2 T − 2lµ ∂ µ T + α T = 0
4
2−D
3. Solution: T = µ exp(kµX µ), (kµ − lµ)2 = 6α
T is a tachyon field in the target space. (in 2D, the tachyon is massless)
So we add “cosmological constant term”
cosmo 1 2
√ µ 1 2
√
Sσ = d σ hT (X ) = d σ h µ exp(kµX µ)
4πα 4πα
to the sigma model.

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Adding Sσ
cosomo
to the action creates a potential “wall” in the target space.
How does Sσ
cosmo
improve the situation?
Liouville Wall
10/31 B ACKGROUNDS IN 2—C RITICAL S TRINGS

2—C RITICAL S TRINGS IN B ACKGROUNDS 11/31
Critical string theory with L.D. background
— Summary —
Action for critical string theory in D dimensions, in a linear dilaton background:
LD 1 2
√ ab µ µ kµ X µ
Sσ = d σ h h ∂ a X ∂ b Xµ + α RlµX + µe
4πα
2 2−D
(kµ − lµ) =
6α
This is an exact, well defined conformal field theory.

3—N ON – CRITICAL S TRING T HEORY 12/31
Noncritical String Theory I
No Weyl invariance.
ˆ
Fix a conformal gauge: hab = eγφ(σ,τ )hab
ˆ
Fixed metric: hab
Dynamical Liouville field φ(σ, τ )
Gauge fixes world–sheet diffeomorphism symmetry.
Action [Polyakov ’81]
1 ˆ ˆ ˆ
SCFT = dσ 2 h hab ∂ a X µ ∂ b Xµ + hab ∂ a φ ∂ b φ − α QRφ
4πα
ghost
+µeγφ + terms , X 1, . . . , X d
25 − d 1 √ √
Q= , γ = −√ 25 − d − 1 − d
6α 6α
Describes Liouville gravity coupled to c = d matter.

3—N ON – CRITICAL S TRING T HEORY 13/31
Non–critical ↔ Critical String Theory
Now, compare:
1 ˆ ˆ ˆ
SCFT = dσ 2 h hab ∂ a X µ ∂ b Xµ + hab ∂ a φ ∂ b φ
4πα
− α QRφ + µeγφ , X 1, . . . , X d
LD 1 2
√
Sσ = d σ h hab ∂ a X µ ∂ b Xµ
4πα
µ kµ X µ
+ α RlµX + µe , X 1, . . . , X D
Make the following associations:
0; µ = D 0, µ=D
D = d + 1; X D = φ, lµ = ; kµ =
−Q, µ = D γ, µ=D
LD
Then, SCFT = Sσ

3—N ON – CRITICAL S TRING T HEORY 14/31
Non–critical ↔ Critical String Theory
— Summary —
d Dimensional noncritical string theory
=
d + 1 Dimensional critical string theory in L.D. background.
2D bosonic string theory in a linear dilaton background = Liouville gravity coupled
to c = 1 matter, or non–critical string theory in 1 dimension (α = 1):
1 ˆ ˆ ˆ
SCFT = d2 σ h hab ∂ a X ∂ b X + hab ∂ a φ ∂ b φ − 2Rφ + µe−2φ
4π

3—N ON – CRITICAL S TRING T HEORY 15/31
Time dependant Backgrounds
We wish to study more general time dependant backgrounds
Perturb action by tachyon vertex operators
S = SCFT + tn V n , Vn = d2σ e−in(t+φ)e−2φ
n=0
With only t±1 = 0 and the rest all zero, get “sine–Liouville CFT”
The T–Dual of this theory is a CFT perturbed by vortex operators generating
winding excitations.
Conjectured to be equivalent to a WZW model which describes a target space
with a Euclidean black hole (“cigar”) background.

3—N ON – CRITICAL S TRING T HEORY 16/31
Cigar Manifold
Asymptotically, the metric looks like
1
ds2 = − 1 − e−2Qr dt2 + dr2, Φ= 0 − Qr.
1 − e−2Qr
Curvature is RG = 4Q2e−2Qr . Performing analytic continuation to Euclidean time
t → iθ, get the Euclidean cigar:

3—M ATRIX M ODELS 17/31
Matrix Models — Definition
A matrix model consists of
Symmetry group G
Ensemble of N × N matrices M invariant under G
Probability law (partition function)
gk k
Z= dM exp [−N Tr V (M )] , V (M ) = M
k
k>0
We are interested in unitary matrix ensembles: G = U(N ), M hermitian.
As in QFT, we can write down a perturbation expansion in terms of Feynman
diagrams from the partition function (N ∼ −1).

3—M ATRIX M ODELS 18/31
Propagators and Vertices
From quadratic term in action, we write down a fat propagator
1
= δilδjk
N g2
Higher order terms give fat vertices
1
= δj1i2 δj2i3 · · · δjk i1
N gk

3—M ATRIX M ODELS 19/31
Discretised Surfaces
Duality: Feynman graph network ←→ Discretised surface
Discretised surface
Dual Propogator

3—M ATRIX M ODELS 20/31
Free energy
We can express the free energy F = log Z as a sum over connected diagrams.
This is dual to a sum over genus g surfaces
∞
F = N χ Fg = N 2 + + N −2 + N −4 + ···
g=0
1 −P
Fg = g2 (−gk )nk .
s
Discretisations k>2
of genus g
P edges, nk k–polygons, s =symmetry factor.
We are interested in continuous surfaces, where the number of polygons tends to
infinity.

3—M ATRIX M ODELS 21/31
Double Scaling Limit
Continuum Limit: Number of polygons diverges when couplings approach critical
values gk → gc.
Spherical Limit: Genus zero surfaces dominate in the limit N → ∞.
If surface is string world–sheet, then only get contributions from genus zero
when N → ∞. Contributions from all genera are included by taking the
Double Scaling Limit: Take both limits simultaneously with string coupling fixed
N → ∞, gk → gc, κ−1 = N (gc − gk )(2−γstr)/2

4—MQM 22/31
Matrix Quantum Mechanics
MQM refers to a matrix model when
the number of distinct N × N matrices is infinite
the matrix label, which is then continuous, is interpreted as time.
Mi, −→ M (t)
Partition function of MQM
1
ZN = DM (t) exp −N Tr dt (∂ t M (t))2 + V [M (t)]
2
It models 2D gravity coupled to c = 1 matter (scalar field t(σ, τ )).

4—MQM 23/31
Hamiltonian Analysis
Diagonalize the matrices
M (t) = Ω†(t)x(t)Ω(t), x(t) = diag{x1(t), . . . , xN (t)}, , Ω†(t)Ω(t) = 1
Quantum Hamiltonian of the system:
N
− 2 ∂2 1 Π2 + Π2
ij ij
HMQM = ∆(x) + V (xi) + .
i=1
2∆(x) ∂ x2
i 2 i<j
(xi − xj )2
Write partition function as
b
− 1 T HMQM
ZN = Tr e , T →∞
In this limit, only ground state contributes to the free energy
E0
F =−

4—MQM 24/31
Non–interacting Fermions
So we just need ground state.
Symmetry considerations show it belongs to singlet representation of SU(N )
and is a Slater determinant.
N
1
ΨGS(x) = √ det ψi(xj ), E0 = εi ,
N! i,j
i=1
Hamiltonian splits into a sum of single–particle Hamiltonians
singlet N
sector (sing)
2
∂2
HMQM −→ HMQM = Hi , Hi = − 2 + V (xi )
i=1
2 ∂ xi
MQM reduced to a system of N non–interacting fermions moving in a potential
V (x).

4—MQM 25/31
Fermi Sea
The phase space of MQM: Fermi seas (right) corresponding to potentials (left).
V p
15 4
10 2
5
x
4 2 2 4
x
4 2 2 4 2
5
4
V p
4
2
2
1
x
4 2 2 4
x
4 2 2 4 2
1
4

4—MQM 26/31
Inverse Oscillator Potential
Continuum limit corresponds to energy levels reaching top of potential.
V
εC
εF
x
xC x1 x2
In the double scaling limit
the exact form of the potential is unimportant
MQM reduces to a problem of fermions in an inverse oscillator potential
x2
V (x) = − 2

5—MQM IN THE C HIRAL R EPRESENTATION 27/31
Chiral Coordinates
MQM can be reformulated in terms of left and right “light cone” matrix variables.
[Alexandrov,Kazakov,Kostov ’02] . In the singlet sector:
x±p ∂
x± = √ , p ≡ −i
2 ∂x
Many advantages follow from the fact that the chiral Hamiltonian is linear
± 1 ∂ 1
H0 = (p2 − x2) = i(x± + )
2 ∂ x± 2
Introduce tachyon perturbations by acting on ground state wave function, not
perturbing the Hamiltonian.

5—MQM IN THE C HIRAL R EPRESENTATION 28/31
Tachyon Perturbations
Perturbed state is
E ϕ± (x± ;E) E E
ψ± (x± ) = e ψ± (x± ) = W±ψ± (x± )
Perturbing phase ϕ± (x± ; E) contains MQM realizations of tachyon matrix operators
∞
V± (x± ) = t±k x±/R,
k
k=1
In quasiclassical limit µ → ∞ we get constraint equations
∂
x± x = −E± + x± ϕ
∂ x± ±
which we solve for the profile of the Fermi sea.

5—MQM IN THE C HIRAL R EPRESENTATION 29/31
Results
Constraint equations for two non–zero couplings t±1, t±2 = 0
χ 1 2
− 2R
x± = e ω ±1 1 + a±1ω R + a±2ω R
R−1/2 2R−3/2
1 χ R2 χ 2−R
t 1 Re +t 2 t±1 e R2
R3
a±1 =
χ 2R−2 2−R 2
1 − t±2t 2e R 2
R3
t 2 χ R−1
a±2 = e R2
R
R−1 R−1
2−R χ R2 2−R χ R2
χ t−1 + t−2t1 R2 e t1 + t2t−1 R2 e 1 − R χ 2R−1
1 = µe + R
2R−2 2
e R2
2−R 2 χ R2 R3
1 − t−2t2 R2
e
2 − R χ 2R−2
+ t2t−2 3 e R2
R

5—MQM IN THE C HIRAL R EPRESENTATION 30/31
Profiles
2 4
Plot of Fermi Sea for t 1 2, t 2 0, R Plot of Fermi Sea for t 1 2, t 2 2, R
3 3
Χ 0 solutions for Μ 1,359 Χ 0 solutions for Μ 3,363
60
x 75
40
50
x
20
25
0 x 0 x
25
20
x
50
40
x 75
10 20 30 40 50 60 20 40 60 80
First couplings only First and second couplings

5—MQM IN THE C HIRAL R EPRESENTATION 31/31
Conclusions and Outlook
The CFT describing string theory in a linear dilaton background was perturbed
with first and second couplings
2
S = SCFT + tn V n ,
n=−2
and described as an MQM model.
Problems and outlook:
Target space interpretation is complicated
Free energy of MQM not found explicitly
Critical points in the moduli space of the perturbed theory correspond to a class
of minimal CFTs. [Kazakov ’89]
M(atrix) theory?