Instanton Eects in Matrix Models
Joost Pluijmen
July 14, 2016
Student number: 6044557
Report Bachelor Project Physics and Astronomy, 15 EC University of Amsterdam (UvA)
Faculty of Science (FNWI)
The Institute for Theoretical Physics Amsterdam (ITFA) Conducted between 01-04-2016 and 14-07-2016
Supervisor: Dr. M.L. Vonk Second Assessor: Dr. A. Castro Anich
Summaries
Abstract
Instantons are the tunneling processes in which eigenvalues travel from one mininum to another. In this thesis I want to nd the instanton action in the Gaussian matrix model with the Gaussian W (M ) = M22 potential, a cubic matrix model with the W (M) = M −M33 potential and a quartic matrix model with the W (M) = M2
2 −
M4
4 potential. For each of these potentials I calculated the
one instanton action in the one-cut solution. This was done with the help of two techniques, the saddle-point analysis and the method of orthogonal polynomials. The main goal of achievement is to deduce the same instanton action with both techniques for each of the matrix models. There are two types of instanton action, one in which the eigenvalues tunnel from one critical value to another and the so-called multi-sheeted instanton action in which the eigenvalues travel between dierent sheets in a unfolded world. What eventually became clear is that the instanton action and the multi-sheeted instanton action have values in the complex plane with contributions of πin except for the Gaussian matrix model where the instanton action only has a contribution for the action between dierent sheets.
Populair Wetenschappelijke Samenvatting
In gevorderde natuurkundige theoriëen zoals snaartheorie komen na berekeningen allerlei eecten naar voren die van te voren niet per se verwacht werden. In het geval van mijn scriptie gaat het om instantoneecten. Een instanton kan gezien worden als de oplossing van de bewegingsvergelijkin-gen in bijvoorbeeld de quantummechanica. Deze oplossing kan in de theorie vervolbewegingsvergelijkin-gens omschreven worden als een bewegend deeltje. Zo ook komen instantonen voort uit de oplossingen van matrix-modellen. Een matrixmodel is een natuurkundig model waarin de natuurkundige variabelen voor bijvoorbeeld plaatsbepaling vervangen zijn door nuldimensionele matrices.
In mijn scriptie ga ik in op de vraag hoe instantonen zich gedragen in het geval van de volgende potentialen: W (M) = M2 2 te zien in guur 3, W (M) = M − M3 3 te zien in guur 4 en W (M) = M2 2 − M4
4 te zien in guur 5. In de guren die hier weergegeven zijn, zie je steeds plekken waar
de potentiaallijn verbogen wordt, zogenaamde zadelpunten zadelpunten. In het geval van een matrixmodel bewegen de eigenwaarden zich onder invloed van een bepaald potentiaal door de ruimte. Wanneer de eigenwaarden van een matrixmodel `tunnelen' van het ene minimum naar het andere minimum is er sprake van een instanton. Dit tunnelen is een term die staat voor het ontsnappen van een deeltje uit een potentiaalput. Het doel van mijn onderzoek is om de kans te berekenen die een eigenwaarde heeft om te tunnelen in de potentialen die ik genoemd heb.
In het geval van instantonen in matrix modellen gaat het echter niet om een van de reeds bekende deeltjes uit de natuurkunde zoals bijvoorbeeld elektronen of protonen, maar gaat het om deeltjes die alleen in de theorie ontdekt zijn. Omdat deze theorie van matrix modellen een soort van versimpelde basis vormt van de veel verder gevorderde zogenaamde `topologische snaartheorie' is het een manier om mee te rekenen bij benaderingen van onderdelen in deze theorie.
Contents
Introduction 4
1 Matrix Models 5
1.1 Basics . . . 5
1.2 Instanton Eects in Matrix Models . . . 8
1.2.1 Saddle-Point Analysis . . . 8
1.2.2 Orthogonal Polynomials . . . 11
2 Instanton Eects in Simple Matrix Models 14 2.1 Instanton Eects in the Gaussian Matrix Model . . . 14
2.1.1 Saddle-Point Analysis . . . 14
2.1.2 Orthogonal Polynomials . . . 16
2.2 Instanton Eects in the Cubic Matrix Model . . . 17
2.2.1 Saddle-Point Analysis . . . 17
2.2.2 Orthogonal Polynomials . . . 19
2.3 Instanton Eects in the Quartic Matrix Model . . . 21
2.3.1 Saddle-Point Analysis . . . 22
2.3.2 Orthogonal Polynomials . . . 23
Discussion 26
Conclusion 27
Introduction
As the title suggests there are two mathematical terms that form a signi cant basis in this thesis, `Instanton Eects' and `Matrix Models'. Before outlining these terms and the use of them in this thesis, it is useful to underline that both terms appear in the theoretical research of devel-oped quantum mechanical theories such as for example string theory. As a basis of these highly developed theories the subject of this thesis is a good starting point in understanding some of the hardest physical subjects. In this introduction I will rst give a general de nition of matrix models and instanton eects with an overview of which aspects of them I use in my own research. Further I will outline the components that my own research consists of and the total structure of the thesis.
As a kind of 'play'-model for nonpertubative eects it is possible to use matrix models. In matrix models the physical quantities are represented by matrices. Matrix models can be explained as quantum gauge theories in zero dimensions which for example underly the theory of topological strings. In these models there exist nonperturbative e−N corrections in the large N limit which
are called instanton eects. These nonperturbative corrections need to be added to divergent perturbative series in order to obtain convergent series. Instantons are solutions to the equations of motion in quantum mechanics and quantum eld theory. The action A ∼ Ngswhich represents
the chance of tunneling between critical values is called instanton action and is expressed in showing the N dependent exponential corrections as e−A
gs terms. Instanton action can be thought of as the
height of the barrier of a certain potential under which an eigenvalue is tunneling from one critical value of the potential to another critical value of the potential. A property of this instanton action is that we can regard it as an action on a plane where multiple sheets were collapsed onto each other. Now we can lift this degeneracy by rewriting the instanton action to a world where the collapsed sheets are fully unfolded in a so-called multi-sheet domain. The action that we can now call the multi-sheeted instanton action represents an action in which the eigenvalue travels from one sheet to another sheet.
To come to concrete calculations of instanton action there is a range of techniques to solve matrix models. Two of these techniques which I will apply in this thesis are the saddle-point analysis and the method of orthogonal polynomials. The saddle-point analysis, as the word itself already suggests, involves an analysis of the saddle-points of the potential used in a matrix model. We can see these saddle-points in the potential as so-called potential wells in which the eigenvalues can reside. By using the saddle-point analysis it is for instance possible to nd an expression for the instanton action near a saddle-point. The method of orthogonal polynomials is a recursive method which gives a recursion relation that can be expanded in the large N limit to nd a trans-series that can for example lead to the instanton action of a certain potential. In the calculations to be made in this thesis I will only use the saddle-point analysis and the method of orthogonal polynomials to nd the instanton action for one instanton in the so-called one cut solution. This one instanton action in the one cut solution represents a solution where you calculate the instanton action for only one instanton around only one cut in the given potential. A cut in the potential can be regarded as one of the saddle-points of the potential.
In this thesis I want to calculate the instanton action in three dierent matrix models in order to nally compare the results I gathered. First of all I will give an introduction in the theory of matrix models with the help of the most important equations needed for my own research. After this short introduction I will show how to generally solve matrix models with the two dierent techniques I mentioned before. With the help of the basics gathered in previous articles I will evaluate the calculations I did on nding the instanton action of three simple matrix models. These models are the Gaussian matrix model with the Gaussian W (M) = M2
2 potential, a cubic matrix model
with the W (M) = M − M3
3 potential and a quartic matrix model with the W (M) = M2
2 −
M4 4
potential. The main goal for my calculations will be to nd the same instanton action by the method of saddle-point analysis as by the method of orthogonal polynomials. The calculation of the density of eigenvalues can also be made to switch from the method of orthogonal polynomials to the method of saddle-point analysis. The results I gather in my calculations are comparable to
each other and so form an interesting point of discussion. Conclusions that I can make regarding my results will mostly be based on the action that I calculated in the three simple matrix models.
1 Matrix Models
1.1 Basics
The theory of matrix models is based on a N × N matrix M that works as a basic eld in the theory. For an introduction see (Marino 2004). Now we can consider a potential V (M)
V (M ) =1 2M 2+X p≥3 gp pM p (1.1.1)
of which the acton is de ned by 1 gs W (M ) = 1 gs Tr(V (M)) = 1 2gs Tr(M2) + 1 gs X p≥3 gp pTr(M p) (1.1.2)
where gsand gp are couplig constants. The traces of this above de ned action can be interpreted
as
Tr(Mn) = X
i1,i2,...,in
Mi1i2Mi2i3· · · Mini1. (1.1.3)
This action has a gauge symmetry given by
M → U M U† (1.1.4)
where U is a matrix dependent on N. Normally in a gauge symmetry U should have been dependent on for instance x but since we are working in zero-dimensional theory U is dimensionless. Considering the action, the partition function is given by
Z = 1
vol(U(N)) Z
dM e−gs1W (M ) (1.1.5)
with dM the Haar measure
dM = 2N (N −1)2 N Y i=1 dMii Y 1≤i<j≤N dReMijdImMij (1.1.6)
and vol(U(N)) the volume factor of the gauge group which should be divided out after xing the gauge. The perturbative expansion of the free energy can eventually be computed by
F = log (Z). (1.1.7)
To represent the matrix elds we would expect Feynman diagrams to do the work. But standard Feynman diagrams are not good to keep track of powers of N, which is necessary since there is a N dependence coming from the perturbative expansion of the free energy. To keep track of powers of N we can seperate dierent pieces of a normal Feynman diagram by writing them as so-called fatgraphs from which an example is given in gure 1.
Figure 2: The cubic vertex in the fatgraph representation (Marino 2004).
An example of a cubic vertex is given in gure 2. This vertex is given by
g3 gsTr(M 3) = g3 gs X i,j,k MijMjkMki. (1.1.8)
A fatgraph will give a factor
gsE−VNh
Y gVp
p (1.1.9)
to the free energy with E the number of edges, Vp the number of vertices with p legs and h the
number of closed loops. The total number of vertices is V = PpVp . Now a power of gsis given
by each propagator and a power of gp
gs by each vertex with p legs. The genus g of a fatgraph's
surface is given by the relation
2g − 2 = E − V − h (1.1.10)
which changes the before named factor to gs2g−2+hNhYgVp p = g 2g−2 s t hY gVp p (1.1.11)
with the 't Hooft parameter
t = N gs. (1.1.12)
These conditions ultimately give the perturbative expansion of the free energy F = ∞ X g=0 ∞ X h=1 gs2g−2thFg,h (1.1.13)
and the genus expansion of the free energy of the matrix model F =
∞
X
g=0
where Fg(t)is de ned by Fg(t) = ∞ X h=1 thFg,h. (1.1.15)
For later calculations with matrix models it is useful to write the partition function in a dierent way. To get to this other expression it is necessary to reduce the amount of parameters N2 in the
original matrix model to only N parameters which can be achieved by diagonalizing the matrix M
M → U M U† = D (1.1.16)
where D = diag(λ1, · · · , λN)and λi are the eigenvalues. With this diagonalization it is possible
to get the δ-function constraint
δ(U M U†) =Y i<j δ(2)(U MijU†). (1.1.17) If we now introduce ∆−2(M ) = Z dU δ(U M U†) (1.1.18)
it follows that the integral of a gauge-invariant function f(M) can be written as Z dM f (M ) = Z dM f (M )∆2(M ) Z dU δ(U M U†) = ΩN Z N Y i=1 dλi∆2(λ)f (λ) (1.1.19) where ΩN = Z dU . (1.1.20)
and the Vandermonde determinant given by ∆(λ) =Y
i<j
(λi− λj). (1.1.21)
By now using the Gaussian matrix integral which can be exactly calculated we get Z dM e−2gs1 TrM 2 = (2πgs) N 2 2 (1.1.22)
which by 1.1.19 should equal ΩN Z N Y i=1 dλi∆2(λ)e− 1 2gs PN i=1λ 2 i. (1.1.23)
Now we can exactly solve the last integral Z N Y i=1 dλi∆2(λ)e− 1 2gs PN i=1λ 2 i = g N 2 2 s (2π) N 2G2(N + 2) (1.1.24)
where the Barnes function G2(z)is de ned by
From combining these results eventually it follows that ΩN =
(2π)12N (N −1)
G2(N + 2) (1.1.26)
If we now use that
vol(U(N)) = (2π) 1 2N (N +1) G2(N + 1) (1.1.27) we see that 1 vol(U(N)) Z dM f (M ) = 1 N ! 1 (2π)N Z N Y i=1 dλi∆2(λ)f (λ) (1.1.28)
leading to the general partition function Z = 1 N ! Z N Y i=1 dλi 2π ∆2(λ)e−gs1 PN i=1W (λi). (1.1.29)
1.2 Instanton Eects in Matrix Models
To computate an approximation to a variety of aspects in matrix models, there are two technologies that can be used. These technologies are the saddle-point analysis and the method of orthogonal polynomials. In the following two sections I will review both of these technologies in a general way where I will focus on the one-cut solution. In these sections I will mainly use the equations retrieved by (Marino 2004), (Marino 2008), (Pasquetti & Schiappa 2009) and (Marino 2012). In the research on instanton eects in matrix models I present later on I used both methods for my calculations.
1.2.1 Saddle-Point Analysis
In this section I will show how with the technique of saddle-point analysis it is eventually possible to calculate a good approximation for the instanton action. To begin with we rewrite the partition function (1.2.27) after reduction to eigenvalues as
Z = 1 N ! Z N Y i=1 dλi 2π eN2Sef f(λ) (1.2.1)
where it is now written as an integral over N eigenvalues. In this `new' partition function the eective action is Se(λ) = − 1 tN N X i=1 W (λi) + 2 N2 X i<j log |λi− λj|. (1.2.2)
Varying Se(λ) with respect to λi gives
1 2tW 0(λ i) = 1 N X j6=i 1 λi− λj (1.2.3) which leads to an eective potential
We(λi) = W (λi) − 2t N X j6=i log |λi− λj|. (1.2.4)
This eective potential can be understood as the potential an eigenvalue λi is subject to when the
the eigenvalues λj where j 6= i are chosen concretely. By using the standard rule
1 N N X i=1 f (λi) → Z C f (λ)ρ(λ)dλ (1.2.5)
where C is the interval outside of which the eigenvalue distribution for nite N ρ(λ) = 1 N N X i=1 δ(λ − λi) (1.2.6) vanishes and Z C ρ(λ)dλ = 1 (1.2.7)
we retrieve the saddle-point equation 1 2tW
0(λ) =PZ ρ(λ0)dλ0
λ − λ0 (1.2.8)
where P expresses the principal value of the integral. The density of eigenvalues can be calculated by solving this saddle-point equation. The way to solve the saddle-point equation is to introduce a function called the resolvent
ω(p) = 1 N Tr 1 p − M (1.2.9) which is the same as
ω(p) = 1 N ∞ X k=0 TrMk p−k−1. (1.2.10)
Given that the resolvent can be seen as an expansion like ω(p) =
∞
X
g=0
g2gs ωg(p) (1.2.11)
and that averages in the planar limit satisfy 1 N TrMl = Z C dλλlρ(λ) (1.2.12)
we obtain the genus zero piece
ω0(p) =
Z
dλ ρ(λ)
p − λ. (1.2.13)
This is analytic on the whole complex plane except on C since if λ ∈ C there is a singularity at λ = pand due to (1.2.7) it behaves asymptotically as
ω0(p) ∼
1
p, p → ∞. (1.2.14)
Now we nd following from discontinuity of ω0(p)the key equation for the density of eigenvalues
ρ(λ) = − 1
By now looking at the right hand side of (1.2.8) we nd that ω0(p + i) + ω0(p − i) = −
1 tW
0(p). (1.2.16)
When C is given by b ≤ λ ≤ a we obtain for the resolvent ω0(p) = 1 2t I C dz 2πi W0(z) p − z (p − a)(p − b) (z − a)(z − b) 12 . (1.2.17)
Requiring (1.2.14) the above expression leads to I C dz 2πi W0(z) p(z − a)(z − b) = 0 I C dz 2πi zW0(z) p(z − a)(z − b) = 2t. (1.2.18)
When W (z) is a polynomial and we deform the rst contour integral by picking up a pole at z = p and one at z = ∞ we get
ω0(p) = 1 2tW 0(p) − 1 2t p (p − a)(p − b)M (p) (1.2.19)
where the moment function is de ned by M (p) = I 0 dz 2πi W0(1z) 1 − pz 1 p(1 − az)(1 − bz). (1.2.20)
An important tool of this moment function is that it can determine a value for z0 that we need
to calculate the instanton action by setting M(p) = 0 which gives one or more value for p that we call z0. To simplify the calculations we can de ne the spectral curve by
y(p) = W0(p) − 2tω0(p) ≡ M (p)
p
(p − a)(p − b) (1.2.21)
leading to the equation
We0 (p) = 1
2(y(p + i) + y(p − i)). (1.2.22)
If we now call the real part of the eective potential the so-called holomorphic eective potential
We(p) = <Wh;e(p; t) (1.2.23)
it turns out that
Wh;e(p) = Z
dpy(p). (1.2.24)
These results eventually lead to the instanton action A = Wh,e(z0) − Wh,e(b) =
Z z0
b
y(p)dp (1.2.25)
where z0 is de ned by the points where the moment function (1.2.20) equals zero.
Another instanton action that we can calculate is the so-called multisheeted instanton action as descriped in the introduction. The method of calculation this instanton action is by changing the variables in the original holomorphic eective potential in such a way that you retrieve an answer that is written in terms of 3 in.
1.2.2 Orthogonal Polynomials
In this section I will show how to retrieve the same instanton action as in the previous section by the method of orthogonal polynomials. To start with we have to look to the partition function (1.2.27) and de ne
dµ = e−gs1W (λ)dλ
2π (1.2.26)
so that the partition function changes to Z = 1 N ! Z N Y i=1 dµi∆2(λ). (1.2.27)
Now it is possible to introduce orthogonal polynomials pn(λ) de ned by
Z
dµpn(λ)pm(λ) = hnδmn, n ≥ 0 (1.2.28)
where the polynomials pn(λ) are normalized by requiring that pn(λ) = λn+ · · ·. Now the
Van-dermonde determinant (1.1.21) can be expressed as
∆(λ) = det pj−1(λi). (1.2.29)
By expanding the determinant
X
σ∈SN
(−1)(σ)Y
k
pσ(k)−1(λk) (1.2.30)
where (s) is the permutation, the partition function becomes Z = N −1 Y i=0 hi= hN0 N Y i=1 rN −ii (1.2.31)
with the coe cients
ri=
hi
hi−1
, i ≥ 1. (1.2.32)
The orthogonal polynomials must satisfy the following recursion relation
(λ + sn)pn(λ) = pn+1(λ) + rnpn−1(λ). (1.2.33)
which can be shown by substituting the above expression in the equality hn+1=
Z
dµpn+1(λ)λpn(λ). (1.2.34)
This substitution leads to hn+1= Z dµpn+1(λ)(pn+1(λ) − snpn(λ) + rnpn−1(λ)) = Z dµpn+1(λ)pn+1(λ). (1.2.35) We can now note that
from which it follows that
nhn−1=
Z
dµp0n(λ)pn−1(λ). (1.2.37)
Then by partially integrating the right hand side of the above formula we retrieve nhn−1=
1 gs
Z
dµ(λ)W0(λ)pn(λ)pn−1(λ). (1.2.38)
To nd the instanton action from the method of orthogonal polynomials in the N → ∞ limit, rk
becomes a function of ξ = k
N that is a continuous variable 0 ≤ ξ ≤ 1 and t
gsk → ξt, 0 ≤ ξt ≤ t (1.2.39)
so that
rk → R(ξt, gs). (1.2.40)
Now we want to obtain the trans-series solutions which we do with the ansatz that R(ξt, gs) = ∞ X l=0 ClR(l)(ξt, gs) (1.2.41) where for l = 0 R(0)(ξt, gs) = ∞ X s=0 g2ss R0,2s(ξt) (1.2.42) and for l ≥ 1 R(l)(ξt, gs) = e −lA(ξt) gs Rl,1 1 + ∞ X n=1 gsnRl,n+1(ξt) ! . (1.2.43)
In above equations the l stands for the amount of instantons. So in the case of a one instanton solution that I try to recover in the next sections it su ces to nd a relation consisting of only l ≤ 1coe cients. From this relation it will eventually be possible to calculate A(ξt).
As in the saddle-point analysis it is also possible to recover the density of eigenvalues from orthog-onal polynomials. To do so we have to introduce the orthonormal polynomials
Pn(λ) = 1 √ hn pn(λ) (1.2.44) which satisfy (λ + sn)Pn(λ) = √ rn+1Pn+1(λ) + √ rnPn−1(λ). (1.2.45) If we now consider TrMl = 1 N !Z Z N Y i=1 e−gs1W (λi) dλi 2π ∆2(λ) N X i=1 λli ! (1.2.46) and use the new de nition for the Vandermonde determinant we can see that this is equal to
N −1 X j=0 Z dµλlP2 j(λ). (1.2.47)
By introducing the Jacobi matrix J = 0 r 1 2 1 0 0 · · · r 1 2 1 0 r 1 2 2 0 · · · 0 r 1 2 2 0 r 1 2 3 · · · · · · · (1.2.48)
and the diagonal matrix
S = s0 0 0 0 · · · 0 s1 0 0 · · · 0 0 s2 0 · · · · · · · (1.2.49) it follows that TrMl =Tr(J − S)l. (1.2.50)
To evaluate now the planar approximation at large N Jl nn∼ l! l 2! 2r l 2 n (1.2.51)
where we can change the representation l! l 2!2 = Z 1 −1 dy π (2y)l p 1 − y2 (1.2.52) we recover 1 N TrMl = Z 1 0 dξ Z 1 −1 dy π 1 p 1 − y2 2yR 1 2 0(ξ) − s(ξ) l (1.2.53) where R0(ξ)and s(ξ) are the limits of rn and sn respectively when N → ∞. By comparing with
1 N TrMl = Z C dλλlρ(λ) (1.2.54)
we nd the density of eigenvalues ρ(λ) = Z 1 0 dξ Z 1 −1 dy π 1 p 1 − y2δ λ −2yR 1 2 0(ξ) − s(ξ) = Z 1 0 dξ π θ 4R0(ξ) − (λ + s(ξ))2 p4R0(ξ) − (λ + s(ξ))2 (1.2.55)
where θ is the step function. We can now see that ρ(λ) is only supported on the interval [b(t), a(t)], where
b(t) = −2pR0(1) − s(1)
a(t) = 2pR0(1) − s(1).
2 Instanton Eects in Simple Matrix Models
In the following three sections I show my partly gathered partly calculated results for respectively the Gaussian, cubic and quartic matrix models. These results are mainly based on the calculations made in (Francesco et al. 1993), (Francesco 1999), (Marino 2004), (Marino 2008), (Pasquetti & Schiappa 2009) and (Marino 2012).
2.1 Instanton Eects in the Gaussian Matrix Model
In this section I will focus on nding the instanton action in the Gaussian matrix model with the potential W (M) = M2
2 visualized in gure 3. By using the methods of saddle-point analysis and
orthogonal polynomials for a one-instanton model in the one-cut solution I will try to nd the same instanton action both ways. With mathematical methods that are shown in the previous sections I will show in a few steps how to come to a good result.
Figure 3: Visualization of the potential W (M) = M2 2 .
2.1.1 Saddle-Point Analysis Taking W (z) = z2
2 and starting with the equations in (1.2.18), where W
0(z) = z and deforming
the contour to in nity and changing z → 1
z where the contour is now around z = 0 these become
I 0 dz 2πi 1 z2 1 p(1 − az)(1 − bz) = 0 I 0 dz 2πi 1 z3 1 p(1 − az)(1 − bz) = 2t. (2.1.1)
By expressing the square root fractions in their Taylor expansion I 0 dz 2πi (1 +az 2 − 3(az)2 8 + · · · )(1 + bz 2 − 3(bz)2 8 + · · · ) z2 = 0 I 0 dz 2πi (1 +az2 −3(az)8 2 + · · · )(1 +bz2 −3(bz)8 2 + · · · ) z3 = 2t. (2.1.2)
Cauchy's residue theorem leads to a + b 2 = 0 1 8 3a 2+ 2ab + 3b2 = 2t. (2.1.3) which by lling in a = −b in the second equation leads to
a2
2 = 2t (2.1.4)
and eventually to
a = 2√t = −b. (2.1.5)
Now we want to nd the resolvent with (1.2.19) where the moment function (1.2.20) M (p) = I 0 dz 2πi 1 z 1 − pz 1 p(1 − az)(1 − bz) = 1 (2.1.6) leaving no signi cant value for z0. Now the resolvent
ω0(p) =
1 2t
p −pp2− 4t (2.1.7)
from which we can derive the density of eigenvalues with (1.2.15) ρ(λ) = 1
2πt p
4t − λ2. (2.1.8)
With the help of (1.2.21) we can nd the spectral curve
y(p) =pp2− 4t. (2.1.9)
Then the holomorphic eective potential is de ned by Wh;e(p) = Z dpy(p) =1 2p p p2− 4t − 2t log p + p p2− 4t −2√t ! (2.1.10)
where the result is normalized such that Wh;e(p = b) = 0. Now I can nd the instanton action
A(t) = Z z0
b
dpy(p) = Wh;e(z0) − Wh;e(b)
= 0.
(2.1.11) Because there is no known value for z0 this instanton action is not possible to calculate, but we
will see that is possible to nd a multisheeted eective potential which we will see next.
Now I want to nd the multisheeted instanton action. To start with we change the variables from p to
so that we obtain the following eective potential Wh;e(u) = 1
8e
2u− 2t2e−2u− 2tu + 2t log(−2√t). (2.1.13)
Now the critical points of the potential are located at
e2u= 4t → u(n)± = log (±2√t) + iπn, n ∈ Z (2.1.14) giving the multisheeted instanton action
An(t) = Wh;e(u (n) + ) − Wh;e(u ? +) = −2πint. (2.1.15) 2.1.2 Orthogonal Polynomials With the potential W (λ) = λ2
2 the equation in 1 gs Z dµ(λ)W0pn(λ)pn−1(λ) = nhn−1 (2.1.16) becomes 1 gs Z dµ(λ)λpn(λ)pn−1(λ) = nhn−1. (2.1.17)
By using the recursion relation 1.2.33 of the orthogonal polynomials and knowing that all sn= 0
for even potentials, we get 1 gs Z dµ(λ)rnpn−1(λ)pn−1(λ) = nhn−1 (2.1.18) where Z dµ(λ)rnpn−1(λ)pn−1(λ) = rnhn−1 (2.1.19)
from which follows that
rn = ngs. (2.1.20)
In the large N limit this means according to 1.2.41 that
R(ξt, gs) = ξt (2.1.21)
and the planar part
R0= ξt. (2.1.22)
Now we want to nd the trans-series which leads to
cosh (A0(ξt)) = 1 (2.1.23)
giving the instanton action
A(ξt) = − Z
dR0cosh−1(1)
= 0
(2.1.24) as in the saddle-point approximation.
Now we nd the density of eigenvalues with 1.2.55 where R0(ξ) = tξand s(ξ) = 0
ρ(λ) = Z 1 0 dξ π θ 4ξt − λ2 p 4ξt − λ2 = 1 2πt p 4t − λ2 (2.1.25) which reproduces the same result as in the saddle-point approximation.
2.2 Instanton Eects in the Cubic Matrix Model
In this section I will focus on nding the instanton action of a cubic matrix model with the potential W (M) = M −M3
3 visualized in gure 4. By using the methods of saddle-point analysis
and orthogonal polynomials for a one-instanton model in the one-cut solution I will try to nd the same instanton action both ways. With mathematical methods that are shown in the rst sections I will show in a few steps how to come to a good result.
Figure 4: Visualization of the potential W (M) = M −M3 3 .
2.2.1 Saddle-Point Analysis Taking W (z) = z −z3
3 where W
0(z) = 1 − z2 and deforming the contour to in nity and changing
z → 1z where the contour is now around z = 0 the equations in (1.2.18) become I 0 dz 2πi 1 z 1 p(1 − az)(1 − bz)− I 0 dz 2πi 1 z3 1 p(1 − az)(1 − bz) = 0 (2.2.1) I 0 dz 2πi 1 z2 1 p(1 − az)(1 − bz)− I 0 dz 2πi 1 z4 1 p(1 − az)(1 − bz) = 2t (2.2.2) leaving 1 − 1 8(3a 2+ 2ab + 3b2) = 0 (2.2.3) a + b 2 − 1 16(5a 3+ 3a2b + 3ab2+ 5b3) = 2t. (2.2.4)
Here the rst equation gives two solutions for a a = 1
3(−b ± 2 √
2p3 − b2). (2.2.5)
By now using a new variable z0 and using the variance δ we change the variables so that
a = −z0+ δ
b = −z0− δ.
Now we can nd from the second equation that −z0− 1 16(5(−z0+ δ) 3+ 3(−z 0+ δ)2(−z0− δ) + 3(−z0+ δ)(z0− δ)2+ 5(−z0− δ)3) = 2t (2.2.7) which leads to −z0(1 − 3 2δ 2 − z20) = 2t. (2.2.8)
If we now look at the equality (a + z0)2= (b + z0)2= δ2 it is possible to express δ in terms of z0
so that the solutions of the quadratic equation give a and b as the expressions in new variables given before. This can be done by stating that
δ2= 2(1 − z02). (2.2.9)
So that the previous expression reduces to
z0(1 − z02) = t. (2.2.10)
Now we want to nd the resolvent ω0(p) with (1.2.19) where in this case the moment function
M (p)becomes M (p) = I 0 dz 2πi 1 − 1 z2 1 − pz 1 p(1 − az)(1 − bz) = −p −a + b 2 (2.2.11) so that ω0(p) = 1 2t 1 − p2−p(p − a)(p − b) −p −a + b 2 (2.2.12) Filling in the values for a and b this gives
ω0(p) = 1 2t 1 − p2−p (p + z0)2− δ2(−p + z0) (2.2.13) from which we can derive the density of eigenvalues with (1.2.15) to
ρ(λ) = 1
2πt(−λ + z0) p
δ2− (λ + z
0)2. (2.2.14)
Now the spectral curve by (1.2.21) becomes
y(p) =p(p − a)(p − b) −p −a + b 2 = (−p + z0) p (p + z0)2− δ2 (2.2.15) Then the holomorphic eective potential is de ned as
Wh;e(p) = Z dpy(p) = Z dp (−p + z0) p (p + z0)2− δ2 = 1 3(3z0(p + z0) − (p + z0) 2+ δ2)p (p + z0)2− δ2 −z0δ2log p + z0+p(p + z0)2− δ2 −δ ! (2.2.16)
where the result is normalized such that Wh;e(p = b) = 0. Now it is possible to nd the instanton
action
A(t) = Z z0
b
dpy(p) = Wh;e(z0) − Wh;e(b)
=1 3(δ 2(t) + 2z2 0(t)) q 4z2 0(t) − δ2(t) − 2t log 2z0(t) +p4z20(t) − δ2(t) −δ(t) ! . (2.2.17)
Now I want to nd the multisheeted instanton action. Changing the variables from p to eu= p + z0+ p (p + z0)2− δ2 (2.2.18) we obtain Wh;e(u) = 1 24 δ
6e−3u− 3δ4e−u+ 3δ2eu− e3u− 6δ4e−2uz
0+ 6e2uz0+ 24δ2z0(log (δ) − u) .
(2.2.19) Now the critical points of the potential are located at
eu= ±δ ∨ 2z0±
q 4z2
0− δ2
→ u(n)± = log (±δ) + 2iπn ∨ log (2z0±
q 4z2
0− δ2) + 2iπn, n ∈ Z
(2.2.20) which gives the multisheeted instanton action by changing the variables back to t
An(t) = Wh;e(u (n) + ) − Wh;e(u ? +) = 4πint. (2.2.21) 2.2.2 Orthogonal Polynomials With the potential W (λ) = λ −λ3
3 the equation (1.2.28) becomes
1 gs
Z
dµ(λ)(1 − λ2)pn(λ)pn−1(λ) = nhn−1. (2.2.22)
by using the recursion relation of the orthogonal polynomials (1.2.33) twice we get λ2pn = λ(pn+1(λ) − snpn(λ) + rnpn−1(λ))
= pn+2(λ) − (sn+1+ sn)pn+1(λ)+
(rn+1+ rn+ s2n)pn(λ) − rn(sn+ sn−1)pn−1(λ) + rnrn−1pn−2(λ).
(2.2.23) Inserting this result gives only a contribution with the pn−1terms such that
1 gs Z dµ(λ)rn(sn+ sn−1)pn−1(λ)pn−1(λ) = nhn−1 (2.2.24) where Z dµ(λ)rn(sn+ sn−1)pn−1(λ)pn−1(λ) = rn(sn+ sn−1)hn−1 (2.2.25)
from which follows that
Now by seeing that
Z
dµ(λ)p0n(λ)pn(λ) = 0 (2.2.27)
which gives by partially integrating Z
dµ(λ)W0(λ)pn(λ)pn(λ) = 0 (2.2.28)
the same as
Z
dµ(λ)(1 − λ2)pn(λ)pn(λ) = 0. (2.2.29)
By lling in the pn terms we see that
− Z
dµ(λ)(rn+1+ rn+ s2n− 1)pn(λ)pn(λ) = 0 (2.2.30)
which gives
1 − rn+1− rn− s2n = 0. (2.2.31)
So to solve the matrix model we have to solve the recursion relations rn(sn+ sn−1) = ngs 1 − rn+1− rn− s2n= 0 (2.2.32) giving rn p1 − rn+1− rn+p1 − rn− rn−1 = ngs. (2.2.33)
If we now take the limit where N → ∞ and we take the variables ξ = n
N and t = Ngswith (1.2.41) this becomes R(ξt)p1 − R(ξt + gs) − R(ξt) + p 1 − R(ξt) − R(ξt − gs) = ξt (2.2.34)
where the planar part is given by
2R0
p
1 − 2R0= ξt. (2.2.35)
Here we can see a similarity with the expression given in the saddle-point analysis where z0(1 −
z0)2= t. If we now change this expression to an expression dependent on ξt we keep
2R0(ξt)s(ξt) = z0(ξt)
δ2(ξt)
2 . (2.2.36)
It is now obvious for further calculations that this leads to the equalities 4R0(ξt) = δ2(ξt)
s(ξt) = z0(ξt).
(2.2.37) Now we want to nd the instanton action by using the the recursion relations. We nd a system of recursive dierence equations for R(l)(ξt, g
s) by lling in the trans-series ansatz (1.2.41) and
using (1.2.42) and (1.2.43) we get at leading order in gsto
eA0(ξt)+ e−A0(ξt)+10R0− 4 R0
leaving
cosh (A0(ξt)) = 2 − 5R0 R0
. (2.2.39)
Now the instanton action is
A(ξt) = − Z dR0cosh−1 2 − 5R0 R0 2 − 6R 0 √ 1 − 2R0 = 4R0 3√1 − 2R0 s 1 R0 − 3 1 R0 − 2 − 2p1 − 2R0R0cosh−1 2 − 5R0 R0 =1 3(δ 2(ξt) + 2z2 0(ξt)) q 4z2 0(ξt) − δ2(ξt) − 2ξt log 2z0(ξt) +p4z02(ξt) − δ2(ξt) −δ(ξt) ! (2.2.40)
where we changed the variable R0 back to z0as in the saddle-point approximation.
To nd the density of eigenvalues with (1.2.55) I use the change in variables to obtain ρ(λ) = Z 1 0 dξ π θ 4R0(ξ) − (λ + s(ξ))2 p4R0(ξ) − (λ + s(ξ))2 = Z 1 0 dz0(1 − 3z20) πt θ δ2− (λ + z 0)2 pδ2− (λ + z 0)2 = 1 2πt(−λ + z0) p δ2− (λ + z 0)2 (2.2.41)
which gives the same result as in the saddle-point approximation.
2.3 Instanton Eects in the Quartic Matrix Model
In this section I will search for instanton eects in a quartic matrix model de ned by the potential W (M ) = M22−M4
4 visualized in gure 5. As in the preceding sections I will rst investigate these
eects by following the saddle-point approximation after which I will use the method of orthogonal polynomials to investigate these eects.
Figure 5: Visualization of the potential W (M) =M2 2 −
M4 4 .
2.3.1 Saddle-Point Analysis The potential W (z) = z2
2 −
z4
4 has three critical values. The equations given in (1.2.18) will give
after deforming the contour to in nity and changing z → 1
z where the contour is now around z = 0
I 0 dz 2πi 1 z2 1 p(1 − az)(1 − bz)− I 0 dz 2πi 1 z4 1 p(1 − az)(1 − bz) = 0 (2.3.1) I 0 dz 2πi 1 z3 1 p(1 − az)(1 − bz)− I 0 dz 2πi 1 z5 1 p(1 − az)(1 − bz) = 2t (2.3.2) leading to the equalities
a + b 2 − 1 16(5a 3+ 3a2b + 3ab2+ 5b3) = 0 (2.3.3) 1 8(3a 2+ 2ab + 3b2) − 1 128(35a
4+ 20a3b + 18a2b2+ 20ab3+ 35b4) = 2t. (2.3.4)
The rst equation leads to the following results for a
a = −b a = 1
5
b ±p40 − 28b2 (2.3.5)
that all express the values of a around three dierent cuts. To keep the calculations more or less simple we are only investigating the cut around z = 0 where a = −b. Filling this result in into the second equation this gives
a2= b2=2 3(1 ±
√
1 − 12t). (2.3.6)
Leaving this result for later, I now want to nd the resolvent ω0(p)with (1.2.19). Where in this
case the moment function
M (p) = I 0 dz 2πi 1 z− 1 z3 1 − pz 1 √ 1 − a2z2 = 1 − 1 2a 2− p2 (2.3.7) leaving the resolvent
ω0(p) = 1 2t p − p3−pp2− a2 1 −1 2a 2− p2 (2.3.8) from which we can derive the density of eigenvalues with (1.2.15)
ρ(λ) = 1 2πt 1 −1 2a 2 − λ2 p a2− λ2. (2.3.9)
Now the spectral curve by (1.2.21) becomes y(p) =pp2− a2 1 − 1 2a 2− p2 (2.3.10) leading to the holomorphic eective potential
Wh;e(p) = Z dpy(p) = Z dppp2− a2 1 − 1 2a 2− p2 = 1 8 −p p p2− a2(a2+ 2p2− 4) − (4a2− 3a4) log p + p p2− a2 b !! (2.3.11)
that I normalized so that Wh;e(p = b) = 0. Now I come to the calculation of the instanton action
A(t) = Z z0
b
dpy(p) = Wh;e(z0) − Wh;e(b)
= 1 4 s 1 − a 2(t) 2 1 −3a 2(t) 2 − 2t log q 1 − a22(t)+ q 1 −3a22(t) −a(t) . (2.3.12)
The way to nd the instanton action for a multisheeted potential can be done by changing the variables
eu= p +pp2− a2. (2.3.13)
This change in variables leads to the following holomorphic eective potential
Wh;e(u) = 1
64 a
8e−4u− 8a4e−2u+ 4a6e−2u+ 8e2u− 4a2e2u− e4u− 32a2(u − log (−a)) + 24a4(u − log (−a)) (2.3.14) where the critical points of the potential are now located at
e2u= a2 ∨ 2 − 2a2±p
4 − 8a2+ 3a4
→ u(n)± = log (±a) + iπn ∨ log ± q 2 − 2a2±p4 − 8a2+ 3a4 + iπn with n ∈ Z (2.3.15)
Which gives the same multisheeted instanton action with using the earlier found value for a2
An(t) = Wh;e(u (n) + ) − Wh;e(u ? +) = 16iπn(1 ∓ √ 1 − 12t). (2.3.16) 2.3.2 Orthogonal Polynomials With the potential W (λ) = λ2
2 −
λ4
4 the equation in (1.2.28) becomes
1 gs
Z
dµ(λ)(λ − λ3)pn(λ)pn−1(λ) = nhn−1. (2.3.17)
By using the recursion relation of the orthogonal polynomials (1.2.33) three times and knowing that all sn= 0 for even potentials, we get
λ3pn= λ2(pn+1(λ) + rnpn−1(λ)) = λ(pn+2+ rn+1pn) + rnλ(pn+ rn−1pn−2) = λ(pn+2+ (rn+1+ rn)pn+ rnrn−1pn−2) = pn+3+ rn+2pn+1+ (rn+1+ rn)(pn+1+ rnpn−1) + rnrn−1(pn−1+ rn−2pn−3) = pn+3+ (rn+2+ rn+1+ rn)pn+1+ (rn+1+ rn+ rn−1)rnpn−1+ rnrn−1rn−2pn−3. (2.3.18)
Inserting this result gives only a contribution with the pn−1terms such that
1 gs Z dµ(λ)(1 − rn+1− rn− rn−1)rnpn−1(λ)pn−1(λ) = nhn−1 (2.3.19) where Z dµ(λ)(1 − rn+1− rn− rn−1)rnpn−1(λ)pn−1(λ) = (1 − rn+1− rn− rn−1)rnhn−1 (2.3.20)
from which follows that
(1 − rn+1− rn− rn−1)rn= ngs. (2.3.21)
If I now take the limit where N → ∞ and I take the variables ξ = n
N and t = Ngs and together
with (1.2.41) this becomes
R(ξt) (1 − R(ξt + gs) − R(ξt) − R(ξt − gs)) = ξt (2.3.22)
Where the planar part is given by
R0(1 − 3R0) = ξt (2.3.23)
from which it follows that
R0=
1 6
1 ±p1 − 12ξt. (2.3.24)
Here we can see an analogy with the value found for a2(t)in the saddle-point analysis. When we
change the variable in a from t to ξt we retrieve
a2(ξt) = 4R0(ξt). (2.3.25)
Now we want to nd the instanton action by using the the recursion relations. We nd a system of recursive dierence equations for R(l)(ξt, g
s)by lling in the trans-series ansatz (1.2.41) in (2.3.22)
and using (1.2.42) and (1.2.43) we get R(l)(ξt, gs) = l X k=0 Rl−k(ξt, gs) R(k)(ξt + gs, gs) + R(k)(ξt, gs) + R(k)(ξt − gs, gs) . (2.3.26) This leads for l = 1 in the one instanton solution to
R(1)(ξt + gs, gs) + R(1)(ξt − gs, gs) +R (1)(ξt, g s) R(0)(ξt, g s) R(0)(ξt + gs, gs) + 2R(0)(ξt, gs) + R(0)(ξt − gs, gs) − 1 = 0 (2.3.27)
which leads at leading order in gsto
eA0(ξt)+ e−A0(ξt)+ 4 − 1 R0 = 0 (2.3.28) leaving cosh (A0(ξt)) = 1 − 4R0 2R0 . (2.3.29)
Now the result is
A(ξt) = − Z dR0cosh−1 1 − 4R0 2R0 (1 − 6R0) = 1 4 p (1 − 6R0)(1 − 2R0) + (3R0− 1) cosh−1 1 − 4R0 2R0 =1 4 s 1 −a 2(ξt) 2 1 − 3a 2(ξt) 2 − 2ξt log q 1 − a2(ξt)2 + q 1 −3a22(ξt) −a(ξt) (2.3.30)
With the help of (1.2.55) the same density of eigenvalues as in the saddle-point analysis can be reproduced ρ(λ) = Z 1 0 dξ π θ 23 1 ±√1 − 12ξt − λ2 q 2 3 1 ± √ 1 − 12ξt − λ2 → ρ(λ) = 1 2πt 1 3 2 ∓ √ 1 − 12t − λ2 r 2 3 1 ± √ 1 − 12t − λ2 → ρ(λ) = 1 2πt 1 − 1 2a 2− λ2 p a2− λ2. (2.3.31)
Discussion
By entering the fascinating world of matrix models I got to learn an to me unknown aspect of theoretical physics in which I was going to do research on instanton action. In this thesis the focus was put on retrieving the one instanton action in the one-cut solution of three dierent matrix models. Using the techniques obtained in the literature mentioned in the text it was possible to obtain comparable results to the results worked out in the articles. In most of the steps made it was needed to calculate the desired values totally autonomously by using the well known mathematical computation program `Mathematica'. Mathematica was a program which I had to use for the rst time in a research project, making it an extra task for me to understand it properly but also a useful tool for later research. By comparing the results following from the computed mathematics in Mathematica with previous work there were clear analogies to observe which made the results obtained very reliable.
It must be clear that my research was based on a very small range of matrix model theory. There are a lot of dierent subjects to be studied next and a lot of questions to be asked of which I will adress a few in the following lines.
A thing that was not explained and calculated regarding matrix models is the way to obtain the free energy F from the partition function Z by the relation F = log (Z). While this was done and shown in the literature as an important aspect within the calculation of instanton eects it was not done in this thesis because of the possibility to immediatly retrieve the instanton action from the holomorphic eective potential Wh;e. Since the instanton action was the key subject of
explaining in this thesis we chose to not spend further attention to calculation of the free energy. For further research there are some cases to nd out or explain more closely. For example there is the physical interpretation of instantons in the Gaussian matrix model, which is hard to understand because of the single potential well in the model. By looking at my calculations it seems plausible that instantons do not exist in the folded domain but that they can exist in the case where we look at the multi-sheeted domain.
Next to the example of the interpretation of the instantons in the Gaussian matrix model there is another important mathematical method that was not discussed in this thesis called `resurgence'. The method of resurgence can be used to calculate the instanton eects from the large order behaviour of the rn coe cients.
Conclusion
The goal to obtain values for instanton action in three dierent matrix models for one instanton in the one cut solution has been achieved with the help my supervisor dr. M.L. Vonk and with the knowledge gathered in the literature that I mentioned in the text.
By mutually comparing the results calculated in this thesis I can see a clear similarity which is expressed in the form of the instanton action which in the cases of the cubic and quartic matrix models consists a −2t log (x) term. In this term the variable x was expressed dierently both times but it returns each time so that it becomes clear that you can expect such a result for each matrix model with a potential of the form W (M) = Pi=1
Mi
i . It was notable to see that the instanton
action in the Gaussian matrix model was equal to 0, what must come forth from the fact that there is no possiblity for an instanton to tunnel to another saddle-point in the potential. While this `normal' instanton action doesn't exist, there is an obvious multisheeted instanton action of 2πint which shows the chance of tunneling between the dierent sheets in the Gaussian one-cut potential.
For further work on these matrix models it is possible to calculate more aspects of the instanton eects in matrix models. First of all it is possible to go from a one cut solution for one instanton to a multi-cut solution for multi-instantons. By doing so the computing techniques become heavier but have already been discussed in some articles. Then there is the application of matrix models in topological string theory which is an interesting subject to be studied.
Finally I want to thank my supervisor for his advice and help on the studied matter and for the feedback given during my writing period.
References
Francesco, P. D. 1999, Matrix Model Combinatorics: Applications to Folding and Coloring Francesco, P. D., Ginsparg, P., & Zinn-Justin, J. 1993, 2D Gravity and Random Matrices Marino, M. 2004, Les Houches lectures on matrix models and topological strings
Marino, M. 2008, Nonperturbative eects and nonperturbative de nitions in matrix models and topological strings
Marino, M. 2012, Lectures on non-perturbative eects in large N gauge theories, matrix models and strings
Pasquetti, S. & Schiappa, R. 2009, Borel and Stokes Nonperturbative Phenomena in Topological String Theory and c=1 Matrix Models
