Matrix Models of 2D String Theory in Non-trivial Backgrounds

University of Utrecht

MSc Thesis (Theoretical Physics)

Matrix Models of 2DString 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—INTRODUCTION 2/31

Partition function of string theory

Euclidean Polyakov action + Einstein–Hilbert action

S(E)P =

14πα′

∫Σ

dτdσ√h(habGµν ∂aX

µ ∂bXν + α′νR

)world–sheet Σ swept out by the string, with curvature R in Euclidean signature.

Formal partition function of string theory:

Z =∑

Topologies

∫D%(hab)

∫DXµ e−S

(E)P [Xµ,hab,R]

Can view as partition function of 2D quantum gravity coupled to matter fields Xµ.

1—INTRODUCTION 3/31

Topological Expansion

For Σ a 2D Riemann surface

14π

∫Σ

dτdσ√h νR = νχ = ν(2− 2g) (Euler–Poincare characteristic)

⇒ Z is a sum over genus g closed orientable surfaces!

String coupling: κ ∼ eν

Z =∞∑

g=0

κ−χZg = κ2 + + κ−2 + κ−4 + · · ·

Zg =∫D%(hab)

∫DXµ exp

(− 1

4πα′

∫Σ

dτdσ√hhabGµν ∂aX

µ ∂bXν

)∫D%(hab) dealt with in terms of discretised surfaces −→ Matrix Models

2—CRITICAL STRINGS IN BACKGROUNDS 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 stringworld–sheet action.

For the closed bosonic string, massless modes are:

Gµν(Xρ) traceless, symmetric graviton,Bµν(Xρ) antisymmetric axion,Φ(Xρ) scalar dilaton.


Critical Strings in Background Fields II

Adding those fields to the action, we get a sigma model in D dimensions

Sσ =1

4πα′

∫d2σ√h[ (habGµν(X) + iεabBµν(X)

)∂aX

µ ∂bXν

+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 groupβ–function

=⇒ Theory should possess conformal invariance =⇒ β–functions vanish


β–functions

To zeroth order in α′

βGµν = Rµν + 2DµDνΦ−

14HµσρHν

σρ = 0,

βBµν = DρHµν

ρ − 2HµνρDρΦ = 0,

βΦ =1α′D − 2648π2

+1

16π2

[4(DµΦ)2 − 4DµD

µΦ−R+112HµνρH

µνρ

]= 0,

Hµνρ = ∂µBνρ + ∂ρBµν + ∂ν Bρµ

Critical string theory:

Gµν = ηµν, Bµν = 0, Φ = ν ⇒ D = 26

Solution of interest: Linear dilaton background

Gµν = ηµν, Bµν = 0, Φ = lµXµ.


Problem: Divergent String Coupling

Linear dilaton background in present form leads to a divergent string coupling.

String couplingκ ∝ eν = eΦ(Xρ) = elµXµ

=⇒ For some Xµ, coupling diverges!

Solution: add a potential to the sigma model

Scosmoσ =

14πα′

∫d2σ√hT (Xµ)


Cosmological Constant Term I

New action (with Bµν = 0):

Sσ =1

4πα′

∫d2σ√h[habGµν(X) ∂aX

µ ∂bXν + α′RΦ(X) + T (Xµ)

]New β–functions:

βGµν = Rµν + 2DµDνΦ−DµTDνT = 0,

βT = −2DµDµT + 4DµΦDµT − 4T = 0,

βΦ =D − 26

3α′−R+ 4(DµΦ)2 − 4DµD

µΦ + (DµT )2 − 2T 2 = 0,

These β–functions are the Euler–Lagrange equations of an effective target spaceaction.


Cosmological Constant Term II — Effective Action

“tachyon” part of effective action is

Stacheff = −1

2

∫dDX

√−Ge−2Φ

((DµT )2 − 4

α′T 2

)

1. Substitute L.D. background: Gµν = ηµν, Bµν = 0, Φ = lµXµ

2. Equation of motion on shell: ∂2 T − 2lµ ∂µ T + 4α′T = 0

3. Solution: T = µ exp(kµXµ), (kµ − lµ)2 = 2−D

6α′

T is a tachyon field in the target space. (in 2D, the tachyon is massless)

So we add “cosmological constant term”

Scosmoσ =

14πα′

∫d2σ√hT (Xµ) =

14πα′

∫d2σ√h µ exp(kµX

µ)

to the sigma model.


Liouville Wall

How does Scosmoσ improve the situation?

Adding Scosomoσ to the action creates a potential “wall” in the target space.

1

2

CoupledWeakly

Strongly

StringsCoupled

X

X

µ1log

Strings


Critical string theory with L.D. background— Summary —

Action for critical string theory in D dimensions, in a linear dilaton background:

SLDσ =

14πα′

∫d2σ√h[hab ∂aX

µ ∂bXµ + α′RlµXµ + µekµXµ]

(kµ − lµ)2 =2−D6α′

This is an exact, well defined conformal field theory.

3—NON–CRITICAL STRING THEORY 12/31

Noncritical String Theory INo Weyl invariance.

Fix a conformal gauge: hab = eγφ(σ,τ)hab

Fixed metric: hab

Dynamical Liouville field φ(σ, τ)Gauge fixes world–sheet diffeomorphism symmetry.

Action [Polyakov ’81]

SCFT =1

4πα′

∫dσ2

√h

(hab ∂aX

µ ∂bXµ + hab ∂a φ∂b φ− α′QRφ

+µeγφ + ghost

terms

), X1, . . . , Xd

Q =

√25− d

6α′, γ = − 1√

6α′

(√25− d−

√1− d

)Describes Liouville gravity coupled to c = d matter.


Non–critical ↔ Critical String Theory

Now, compare:

SCFT =1

4πα′

∫dσ2

√h(hab ∂aX

µ ∂bXµ + hab ∂a φ∂b φ

− α′QRφ + µeγφ), X1, . . . , Xd

SLDσ =

14πα′

∫d2σ√h(hab ∂aX

µ ∂bXµ

+ α′RlµXµ + µekµXµ), X1, . . . , XD

Make the following associations:

D = d+ 1; XD = φ, lµ =

0; µ 6= D−Q, µ = D

; kµ =

0, µ 6= Dγ, µ = D

Then, SCFT = SLDσ


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 coupledto c = 1 matter, or non–critical string theory in 1 dimension (α′ = 1):

SCFT =14π

∫d2σ√h(hab ∂aX ∂bX + hab ∂a φ∂b φ− 2Rφ+ µe−2φ

)


Time dependant Backgrounds

We wish to study more general time dependant backgrounds

Perturb action by tachyon vertex operators

S = SCFT +∑n 6=0

tnVn, Vn =∫d2σ e−in(t+φ)e−2φ

With only t±1 6= 0 and the rest all zero, get “sine–Liouville CFT”

The T–Dual of this theory is a CFT perturbed by vortex operators generatingwinding excitations.

Conjectured to be equivalent to a WZW model which describes a target spacewith a Euclidean black hole (“cigar”) background.


Cigar Manifold

Asymptotically, the metric looks like

ds2 = −(1− e−2Qr

)dt2 +

11− e−2Qr

dr2, Φ = %0 −Qr.

Curvature is RG = 4Q2e−2Qr. Performing analytic continuation to Euclidean timet 7→ iθ, get the Euclidean cigar:

3—MATRIX MODELS 17/31

Matrix Models — Definition

A matrix model consists of

Symmetry group GEnsemble of N ×N matrices M invariant under GProbability law (partition function)

Z =∫dM exp [−N Tr V (M)] , V (M) =

∑k>0

gk

kMk

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 Feynmandiagrams from the partition function (N ∼ −1).


Propagators and Vertices

From quadratic term in action, we write down a fat propagator

=1

Ng2δilδjk

Higher order terms give fat vertices

=1

Ngkδj1i2δj2i3 · · · δjki1


Discretised SurfacesDuality: Feynman graph network←→ Discretised surface

Discretised surface

Dual Propogator


Free energy

We can express the free energy F = logZ as a sum over connected diagrams.

This is dual to a sum over genus g surfaces

F =∞∑

g=0

NχFg = N2 + + N−2 + N−4 + · · ·

Fg =∑

Discretisationsof genus g

1sg−P2

∏k>2

(−gk)nk.

P edges, nk k–polygons, s =symmetry factor.

We are interested in continuous surfaces, where the number of polygons tends toinfinity.


Double Scaling Limit

Continuum Limit: Number of polygons diverges when couplings approach criticalvalues gk → gc.

Spherical Limit: Genus zero surfaces dominate in the limit N →∞.

If surface is string world–sheet, then only get contributions from genus zerowhen 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 infinitethe matrix label, which is then continuous, is interpreted as time.

Mi,−→M(t)

Partition function of MQM

ZN =∫DM(t) exp

[−N Tr

∫dt

(12(∂tM(t))2 + V [M(t)]

)]

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) = diagx1(t), . . . , xN(t), ,Ω†(t)Ω(t) = 1

Quantum Hamiltonian of the system:

HMQM =N∑

i=1

(−2

2∆(x)∂2

∂ x2i

∆(x) + V (xi)

)+

12

∑i<j

Π2ij + Π2

ij

(xi − xj)2.

Write partition function as

ZN = Tr e−1T bHMQM, T → ∞

In this limit, only ground state contributes to the free energy

F = −E0

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.

ΨGS(x) =1√N !

deti,j

ψi(xj), E0 =N∑

i=1

εi,

Hamiltonian splits into a sum of single–particle Hamiltonians

HMQM

singletsector−→ H

(sing)MQM =

N∑i=1

Hi, Hi = −2

2∂2

∂ x2i

+ V (xi)

MQM reduced to a system of N non–interacting fermions moving in a potentialV (x).

4—MQM 25/31

Fermi Sea

The phase space of MQM: Fermi seas (right) corresponding to potentials (left).

-4 -2 2 4x

-1

1

2

V

-4 -2 2 4x

-4

-2

2

4p

-4 -2 2 4x

-4

-2

2

4p

-4 -2 2 4x

-5

5

10

15

V

-4 -2 2 4x

-4

-2

2

4

p

-4 -2 2 4x

-4

-2

2

4

p

4—MQM 26/31

Inverse Oscillator Potential

Continuum limit corresponds to energy levels reaching top of potential.

V

Cε

x xx

εF

C 1 2

x

In the double scaling limit

the exact form of the potential is unimportantMQM reduces to a problem of fermions in an inverse oscillator potentialV (x) = −x2

2

5—MQM IN THE CHIRAL REPRESENTATION 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± =x± p√

2, p ≡ −i ∂

∂ x

Many advantages follow from the fact that the chiral Hamiltonian is linear

H±0 =

12(p2 − x2) = ∓i(x±

∂

∂ x±+

12)

Introduce tachyon perturbations by acting on ground state wave function, notperturbing the Hamiltonian.


Tachyon Perturbations

Perturbed state is

ψE± (x±) = e∓ϕ±(x±;E)ψE

± (x±) =W±ψE± (x±)

Perturbing phase ϕ±(x±;E) contains MQM realizations of tachyon matrix operators

V± (x±) =∞∑

k=1

t±kxk/R

± ,

In quasiclassical limit µ→∞ we get constraint equations

x±x∓ = −E± + x±∂

∂ x±ϕ±

which we solve for the profile of the Fermi sea.


Results

Constraint equations for two non–zero couplings t±1, t±2 6= 0

x± = e−χ2Rω±1

(1 + a±1ω

∓ 1R + a±2ω

∓ 2R

)

a±1 =t∓1

1Re

χR−1/2

R2 + t∓2t±1eχ

2R−3/2

R2(2−RR3

)1− t±2t∓2e

χ2R−2R2(2−RR3

)2a±2 =

t∓2

Re

χR−1R2

1 = µeχR +

(t−1 + t−2t1

2−RR2 e

χR−1R2

)(t1 + t2t−1

2−RR2 e

χR−1R2

)[1− t−2t2

(2−RR2

)2e

χ2R−2R2

]2 1−RR3

eχ2R−1

R2

+ t2t−22−RR3

eχ2R−2

R2


Profiles

10 20 30 40 50 60

-40

-20

0

20

40

60

x

Plot of Fermi Sea for t±1 = 2, t±2 = 0, R=23

HΧ < 0 solutions for ΜÎ@-1,359DL

x+

x-

20 40 60 80

-75

-50

-25

0

25

50

75

x

Plot of Fermi Sea for t±1 = 2, t±2 = 2, R=43

HΧ < 0 solutions for ΜÎ@3,363DL

x+

x-

First couplings only First and second couplings


Conclusions and Outlook

The CFT describing string theory in a linear dilaton background was perturbedwith first and second couplings

S = SCFT +2∑

n=−2

tnVn,

and described as an MQM model.

Problems and outlook:

Target space interpretation is complicatedFree energy of MQM not found explicitlyCritical points in the moduli space of the perturbed theory correspond to a classof minimal CFTs. [Kazakov ’89]M(atrix) theory?

Matrix Models of 2D String Theory in Non-trivial Backgrounds

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