Grand Partition Functions of Little Matrix Models with ABCD

0.37 MB·English

by  Hironobu Kihara

KIAS-P08027
Grand Partition Functions of Little Matrix Models with ABCD
Hironobu Kihara
Korea Institute for Advanced Study
207-43 Cheongnyangni 2-dong, Dongdaemun-gu,
Seoul 130-722, Republic of Korea
(Dated: March 28, 2008)
Abstract
Itoyama-Tokura type USp matrix model is discussed. Non-Abelian Berry’s phases in a T-dualized model of IT model were
reconsidered. Thesephasesdescribethehigherdimensionalmonopoles; Yangmonopoleandnine-dimensionalmonopole. They
are described by theconnections of theBPST instanton on S4 and the Tchrakian-GKS instanton on S8, respectively.
As a preparation to understand their effect in original zero-dimensional model, we consider partition function of simplified
8 matrix models. We compute partition functions of SU, SO and USp reduced matrix models. Groups SO and USp appear in
0 low energy effective theories of string against orientifold background. In this evaluation we chose different poles from that of
0 Moore-Nekrasov-Shatashvili and ourprevious result.
2 Theposition of polesexplainbranes’andtheorientifold’s configurations. Thereisabranewhichissittingontheorientifold
in the SO(2N) model, while in USp(2N) and SO(2N +1) model there are no branes on the orientifold. The grand partition
y
functionsofthesemodelsareconsidered. Theyfollowtolinearsecondorderordinarydifferentialequationsandtheirsingularities
a
areq=0,∞. Theirsolutionscanbeanalyticallycontinuedtowholeq plane. WeshowtheexpectationvaluesofthenumberN
M
of AandCcases as examples. There isan ambiguitycoming from theproblem onsign. Grand partition functionswith minus
sign give effective actions which havecusp singularities.
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I. INTRODUCTION
In 1997,Itoyama and Tokura considereda matrix model which we call USp matrix model. The model was studied
in order to understand the dynamics of string against the orientifold background [1]. There is another USp matrix
model by S.J. Rey and N.G. Kim [2] and their work must be important for us. Because the eleven-dimensional
minimalcouplingsupergravitytheorycontainsrank-threeantisymmetrictensorfields,thetheoryhasbeenconsidered
asthe low-energyeffective theoryofthe extendedobject“membrane”whosetrajectoriesofmovementare,in general,
2+1-dimensional space. The mass spectrum of the supersymmetric membrane is described by the Hamiltonian of
the dimensionally reduced model of the ten-dimensional supersymmetric Yang-Mills theory to one dimension.
Type IIA superstring theoryhas infinitely many BPSstates whichcouple with the Ramond-Ramondone-formand
these BPS states are identified with configurations of D-particles. The tower consisting of these configurations of
D-particles can be interpreted into the Kaluza-Klein modes accompanied by the circular compactification along the
eleventhdirection. This viewpointleads people to the discussiononthe duality betweenthe theory ofmembraneand
IIA superstring theory [3, 4]. The matrix theory has the time direction and it is not clear whether the model has
covarianceornot. Ishibashi,Kawai,Kitazawa,andTsuchiyahaveproposeda matrixmodelwhosemasslessspectrum
isthesameasthatoftypeIIBstringtheory[5]. Itwasconjecturedthatthelarge-N reducedmodeloften-dimensional
super Yang-Mills theory can be regarded as a constructive definition of string theory.
∞
1 1
Z = [dX(N)dΨ(N)]exp( S ) , S =α Tr[X(N),X(N)]2 TrΨ¯(N)Γµˆ[X(N),Ψ(N)] +βTr1 ,
− N N −4 µˆ νˆ − 2 µˆ
N=0Z (cid:18) (cid:19)
X
whereX(N) andΨ(N) arebosonicandfermionicN N Hermitianmatrices,respectivelyandµˆ =0,1, ,9. α=1/g2
µˆ × ···
is the inverse of string tension or gauge coupling and β is the chemical potential which is needed for insertion of one
instanton.
The IKKTmatrixmodelwasconstructedasthe unitedtheoryofstring. If infactthe modelunifies stringtheories,
the modelshouldnotdependonthebackground. Thebackgroundindependence isthe mostsignificantimplicationin
theinclusionofquantumgravityinstringtheory. Thegeometryofspace-timeshouldnotbesetupapriori,ratheritis
generatedbyahighlynonperturbativeeffect,thecondensationofstrings. Thereforeunderstandingofthebackground
independence is promisingly the key ingredient to seek the underlying principle of nonperturbative string theory.
We can include another type of background. Orientifolds are generalized orbifolds. In the orbifold construction,
discrete internalsymmetries of the world-sheettheoryare gauged. In the orientifold,products of internalsymmetries
with world-sheet parity reversal are also gauged. Roughly speaking, these symmetries yield restrictions of gauge
groups to unitary symplectic (USp) or orthogonal(SO) groups.
In this article, we will consider partition functions of reduced matrix models. Severalpeople evaluate the quantity.
The quantity relates to the Witten index [6] of matrix quantum mechanics and to the dynamics of D-particles in the
context of M-theory [7, 8, 9, 10]. Monte-Carlosimulationhas been considered[11, 12, 13] and the convergenceof the
integralwasstudiedin[14]. Moore-Nekrasov-Shatashviliobtainedtheresult1/N2byusingthedeformationmethod[9]
and the Monte-Carloresultby Krauth-Staudacher[11] agreewith the MNS result. We start fromthe MNS’s contour
integral and obtain the different result from the result 1/N2 for SU(N) case. There are four types of the partition
functionsZ ,Z ,Z ,Z . Thetruepartitionfunctionofthereducedmatrixmodelwithsuitablenormalization
RMM MC MNS X
factor is denoted Z , the result of the Monte-Carlosimulation which might be used different normalizationfactor
RMM
is Z , the MNS result is Z and ours is Z . The partition function Z is a finite list, however it agrees with
MC MNS X MC
Z for SU(2) and SU(3) cases. Let us start from doubt of the equality between Z and Z . It implies that
MNS RMM MNS
we do not believe the equivalence between Z and Z . For SU(2) case, our result agrees with their result 1/4,
RMM MC
while SU(3) case there exist the difference of factor 2. Two quantities Z , Z are different with each other. The
MNS X
difference is occurred from the difference of the choice of the contour. In this paper, we will not discuss the justice
of these results, but will discuss the meaning of the multiplicity. Our resultant effective potential for SU series has
slightly similar shape with a graph in [15], however our independent variable is the expectation value of the matrix
size and their variable is temperature.
There are two kinds of bosonic sector of Type IIB superstring theory; i) NS-NS sector: dilaton φ, graviton g ,
µˆνˆ
two-form field BF ; ii) R-R sector: axion χ, two-form field BD , self-dual four-form field A+ . The combination of
µˆνˆ µˆνˆ µˆνˆρˆτˆ
thesescalarfieldsτ =χ+iexp( φ)isregardedasamodulusoftorusfiberT2. Thenthedoublet(BF ,BD )becomes
− µˆνˆ µˆνˆ
a SL(2,Z) multiplet. Therefore IIB superstring on a compactified space B is considered as a compactification of F
theory over M, where fiber bundle π : M B (π−1(z) T2) is defined by the moduli function τ(z). One famous
example of this configuration is M =K3 a→nd B =CP1 ≃S2. In this case, F-theory over K3 is dual to IIB over S2.
≃
HereK3issmoothCalabi-Yau2-fold. EspeciallyFtheoryoverK3neartheorbifoldlimitofK3describesthedynamics
of orientifoldof Type IIB over T2. On the other hand, orientifold of Type IIB on T2 is T-dual to Type I on T2. One
2
RealizationofK3(notgeneric)isaquarticsurfaceinCP3;V(f)= Z CP3 f(Z)=0,degf =4,f : homogeneous .
The cohomologyclassis obtainedby the K¨ahler2-formonCP3; (1{+J∈)4/(1|+4J) 1+6J2. Fromthis, we canrea}d
∼
off that the Euler number is equal to 24.
Let us talk about Itoyama-TokuraModel which is a kind of reduced matrix model. The model is written in terms
of dimensional reduction of the d = 4, = 2 USp(2N) supersymmetric gauge theory to zero dimension. The model
N
has i) one adjoint vectormultiplet; ii) one anti-symmetric hypermultiplet; iii) N fundamental hypermultiplets. The
f
total action of this model consists of two parts; S =S +S . Here the closed sector S is obtained by orientifold
Tot C O C
projection from IKKT action and the open sector S is from space-time filling D-brane. Three representations of
O
USp group appear. The adjoint and the antisymmetric representations are, in a sense, parts of Hermitian matrices;
u(2N)= iX M(2N,C)X† =X , usp(2N)= iX u(2N)XTJ +JX =0
{ ∈ | } { ∈ | }
0 I
asym(2N)= iX u(2N)XTJ JX =0 , J = N
{ ∈ | − } IN 0
(cid:18)− (cid:19)
Thereforeu(2N)is adirectsumofadjointandantisymmetric;u(2N) usp(2N) asym(2N). LetiX be anelement
≃ ⊕
of usp(2N) and iY be an element of asym(2N). The block notation show us their construction
A B C D
X = , Y = ,
B† AT D† CT
(cid:18) − (cid:19) (cid:18) (cid:19)
where A† =A, C† =C, BT =B, DT = D.
−
The orientifold projection is described as follows. The =4 SU(2N) supersymmetric Yang-Mills theory consists
N
of four multiplets V,Φ ,Φ ,Φ which are all adjoint representation of SU(2N). Let us project them into = 2
1 2 3
N
USp(2N) supersymmetric gaugetheorywith adjointrepresentationsV,Φ and antisymmetricrepresentationsΦ ,Φ .
1 2 3
We also add N fundamental representations Q ,Q˜ by hand. In other words, a ten-dimensional vector splits into
f i i
six-dimensional vector and four scalars; Xˆ su(2N) X usp(2N) (µ = 0,1, ,4,7) X asym(2N)
µˆ µ a
∈ ⇒ ∈ ··· ⊕ ∈
(a = 5,6,8,9). Splitting of a Majorana-Weyl fermion Ψˆ which belongs to the SU(2N) adjoint representation can
be considered; Ψ = ψ+λ, where ψ is a USp(2N) adjoint eight component spinor and λ is an anti-symmetric eight
component spinor. The action of the closed sector, S , is
C
1 1
S = Tr[X ,X ]2 TrΨ¯Γµˆ[X ,Ψ].
C −4g2 µˆ νˆ − 2g2 µˆ
Let us omit the explanation of the open sector S . In this model, the classical configuration is given by diagonal
O
matrices X and all fermions are set to zero; Ψ=0, Q,Q˜ =0.
M
X =diag(x1, ,xN, x1, , xN) , X =diag(x1, ,xN,x1, ,xN)
µ µ ··· µ − µ ··· − µ a a ··· a a ··· a
They form N pairs of ten-dimensional vectors; (xi,xi),( xi,xi) . In each pair, one vector is a mirror
{ µ a − µ a }i=1,···,N
image of the other with respect to the orientifold. In order to reveal the physics of the fermion of the fundamental
matter, we computed the non-Abelian Berry’s phases against the classical background [16, 17]. It was computed as
thefermionicintegrationofthematrixmodel. Inthepaper,wedidnotconsidertheeffectfromthefermionicdiagonal
terms. In order to make the discussion precise, we should take account into the bilinear terms with respect to the
fermionic diagonalelements, which areyielded from the quadraticcompletion of fermionic off-diagonalterms. In this
paper we will not treat this problem. In fact such a term has influence on the integration of the bosonic off-diagonal
degrees. Suppose that X is diagonal. We split the fermion into diagonal and off-diagonalterms; Ψ=Ψ +Ψ . As
µˆ D O
we mentioned above, we neglect Ψ . Fermionic part of the whole action is
D
1 nf
S = TrΨ¯ Γµˆ(adX )Ψ + χ¯(I)J(γµX + )χ(I)+χ¯(II)J(γµX + )χ(II) ,
F 2g2 O µˆ O i µ Mi i i µ Mi i
Xi=1n o
where γµ are six-dimensional Dirac matrices and the eight-component spinors χ(I),χ(II) are components of the
i i
fundamental hypermultiplets Q ,Q˜ , respectively. The n n matrix is the mass matrix of the fundamental
i i f f
× M
matter and please permit to restrict our discussion in the case that the mass matrix is diagonal. The off-diagonal
M
part can be written in the components using roots and weights;
Ψ = ψ T + λ T ,
O α α ω ω
α∈∆ ω∈A′
X X
3
where ∆ is the root system of usp(2N) and A′ is a set of non-zero weights with respect to the anti-symmetric
representation asym(2N). T and T are generators of usp(2N) and weight vector of asym(2N), respectively. They
α ω
arerepresentedasunitarymatrices. LetH =diag(h , ,h , h , , h ) be anelementofa Cartansubalgebra.
1 N 1 N
Then the dual basis is defined as e (H)=h (i=1,2,··· ,N).− ··· −
i i
···
∆= (e e ), (e +e ),i<j,2e ,i=1, ,N
i j i j i
{± − ± ··· }
A′ = (e e ), (e +e ),i<j
i j i j
{± − ± }
Components ψ , λ from roots and weights which are explained like (e e ), (e + e ) form sixteen-
α ω i j i j
component spinors and components from 2e form eight component spin±ors. −In othe±r words, the composition
i
±
(ψ±(ei−ej),λ±(ei−ej)) and (ψ±(ei+ej),λ±(ei+ej)) form sixteen-component spinors, while (ψ±2ei,0) are still eight-
component spinors. Therefore we obtain two types of action; i) S [ξ,z] = ξ¯γµz ξ where ξ is a 8 component spinor,
I µ
and z is a 6-dimensional vector; ii) S [Ξ,y]= Ξ¯Γµˆy Ξ where Ξ is a 16 component spinor and y is a 10-dimensional
II µˆ
vector. Howeverthis model is a model on zero-dimensionalspace (a set of discrete points). Let us consider T-duality
in order to “make” time direction and let us concentrate on the “one-particle state”. Then Hamiltonian becomes
H [z]=γµz , H [y]=Γµˆy .
I µ II µˆ
We set y = 0 and z = 0 and evaluate their Berry’s phases in consideration of degeneracy. These Hamiltonians
0 0
are higher dimensional generalizations of the system where spin couple to background magnetic field. In the case of
H [z], the background magnetic field are given by the Hodge dual of four form field strength and the corresponding
I
gaugefieldsarethreeformfield. BecauseofthedegeneracyoftheseHamiltonians,weobtainnon-Abelianconnections
which are related to the generalized Monopoles. They are generalization of the Dirac monopole in five-dimensional
space and nine-dimensional space. Berry’s connections are
z =(z ,z ,z ,z ,z )t , y =(y , ,z )t , R2 =zt z , R2 =yt y ,
1 2 3 4 5 1 ··· 9 z · y ·
1 τ2
Z = z +i(z σ +z σ +z σ ) , τ = R2 z2 , A = dZZ−1 , λ=R +z . (1)
τ { 4 1 1 2 2 3 3 } z − 5 Yang τ2+λ2 z 5
q
Hypersurfaces defined by the condition λ = const are hyperboloid and the hypersurfaces shrink to the Dirac string
after taking limit λ 0. This five-dimensional monopole is called Yang monopole [18]. If we transform this space
→
to a space where the hypersurfaces λ=const. become hyperplanes. There appears a mirage four plane where λ=0.
The metric on the hyperplanes, which is induced from Euclidean metric on the original five-dimensional space, has
non-trivial curvature. In fact, such a hyperboloid is a curved space. We omit the exhibition of the Tchrakian-GKS
type solution. Thus the Hamiltonian H gives BPST instanton connection on S4 and the connection satisfies the
I
self-dual equation; F = F. While remaining Hamiltonian H gives the Tchrakianor GKS connection on S8 and
II
±∗
theconnectionsatisfiesthegeneralizedself-dualequationF F = F F [19,20]. TheyaregeneralizationsofDirac
∧ ±∗ ∧
monopoleandthe latteris generalizationofthe selfduality. Innine dimensionalspace,the shape ofsingularitieslook
like a point. These branes whichare denoted by solidlines come fromfundamental multiplets and their distances are
determined by the mass matrix.
D-particle
Nf D4-brane
ff
Orientifold 4-plane
↑
m
f
↓
FIG. 1: D-particles are from theHamiltonian HII and orientifold and D4-branesare from HI.
Our notation in [17] owed to [21] which was introduced by Itoyama. The non-Abelian generalization of Berry and
Simon [22, 23] is argued by Wilczek and Zee [24]. The derivation of BPST instanton as Berry’s phase is discussed in
[25].
4
II. TCHRAKIAN’S MONOPOLE
Because the Yang monopole is a singular object like Dirac monopole, we cannot determine its mass or energy.
In this section we will consider the Tchrakian’s monopole [19, 26] as a possibility of the regularization of the Yang
monopole. He constructed five-dimensional finite energy monopole solution which is an analogue of the ’t Hooft-
Polyakov monopole. The computation of the Berry’s connection was done with adiabatic approximation. If the
parameter z is near the origin,the adiabatic approximationis not valid because the gapof the spectrum of H is not
I
large enough. Such a gap is needed to avoid the transition between different energy eigen states. Therefore in order
to clarify the behavior around the singular point, we need such a regularization.
Let us work on the five-dimensional Euclidean Clifford algebra: γ ,γ = 2δ , where a,b = 1,2,3,4,5 and
a b ab
{ }
σ = [γ ,γ ]/2 are SO(5) generators. These gauge fields are brought together in one differential form which takes
ab a b
value in the Lie algebra; A = (1/2)Abcσ dxa. The scalar fields are also represented as a matrix φ = φaγ . Field
a bc a
strength two form is F =dA+gA2 where g is a gauge coupling constant. The energy is defined as
1 1
E = Tr (F F) (F F)+ Dφ Dφ+λV(φ)d5x .
8 4! ∧ ∧∗ ∧ 8 ∧∗
Z (cid:26) · (cid:27)
The symbol is the Hodge dual operator with respect to the Euclidean metric on R5. From this energy, the Bo-
∗
gomol’nyi equation with the Prasad-Sommerfeld limit becomes Dφ = (F F). Let us consider the Hedge-Hog
±∗ ∧
solution;
1 K(r)
e=xaγ , r2 =xaxa, A= − ede, φ=H U(r)e,
a 0
2g
whereH isthe vacuumexpectationvalue. Unknownfunctions areK(r)andU(r) andtheir boundaryconditionsare
0
U(0)=0,K(0)=1,U( )= 1,K( )=0. After change of the variable a3 =2g2H /3, s=ln(ar), X(s)=K2, the
0
∞ ± ∞
equation becomes an autonomous differential equation:
1 dX dY
Y(X)= , X(1 X)Y =2XY2+3Y 2 .
X(1 X) ds − dX −
−
This equation is the Abel differential equation of the second kind. Let us put Z = dY/dX. Points (X,Y,Z)
are sitting on a surface defined by a quartic polynomial. The surface is singular and has a line singularity. The
differential equation does not have the same property as that of Kovalevskaja. Fortunately numerical evaluation
shows the existence of a flow which connect two boundary points. The effective theory against this background are
written in terms of three form gauge field.
III. PARTITION FUNCTION OF LITTLE SO AND USP MATRIX MODEL
In this part we will compute the partition function of matrix models whose actions are given by the dimensional
reductionofd=4 =1supersymmetricYang-Millstheories. Heregaugegroupsareclassicalgroupsandwelistsome
N
properties of Lie groups in the appendix. Low dimensional counterparts of IKKT matrix model have been discussed
in[27]andmore. Weusetheterminology“litte”matrixmodelborrowedfromthem. Thepartitionfunctionofmatrix
models are considered in the context of the discussion about the existence of D-particle bound states; [7, 8, 9, 10].
They evaluated the Witten index of matrix quantum mechanics [6]. We will follow to the equivariant deformation
method which was used in [9]. Once we performed the computation [28]. In this paper we will obtain different result
from them.
Thepartitionfunctionofareducedmatrixmodelisgivenasamatrixintegral. Moore-Nekrasov-Shatashviliobtained
the following result for d=10 =1 supersymmetric Yang-Mills theory.
N
1 1
= [dXdΨ]exp( S) , = .
Z vol(G) − ZN k2
Z Xk|N
Here the summation with respect to k is taken over the set which consists of all positive factors of N. In order to
obtain these result, Moore-Nekrasov-Shatashviliused equivariant deformation method.
Recently we have applied the deformation method to = 1, d = 4 in the case of SO and USp gauge theories.
N
In this paper, we retry the calculation and will study their grand partition functions. This work is a preparation to
evaluate the integral ob the USp matrix model.
5
Q φ φ¯ η χ B vI ψI
1 2 −2 −1 −1 0 0 1
TABLE I:The assignment of theghost charge.
Let us consider a model whose action is
1 1
[A ]=Tr [v ,v ]2 Λ¯γm[v ,Λ] .
S N−1 −4g2 m n − 2g2 m
(cid:26) (cid:27)
Here the gauge group is SU(N)/Z and suppose that the metric is Lorentzian m,n = 0,1,2,3. This action is a low
N
dimensional counterpart of the IKKT action. The signature of metric should cause a problem on its convergence.
Suppose that the integration path of φ¯ is rotated. We will follow to the MNS’s method and the matrix integral
reduce to the residue calculus. We will skip the detail of the derivation. The degrees of freedom are v ,Λ which are
m
N N Hermitian bosonic and fermionic matrices, respectively. Let T be generators of SU(N). The
a a=1,···,N2−1
ma×trices are expanded by these generators: v = vaT , Λ= Λa{T .}The components Λa are Majorana spinors:
m m a a
Λa = (λa,λ¯a)t. These are not fields, but constant matrices. Let us rearrange these matrices; φ = (v +v ) ,φ¯ =
0 3
P P −
v v ,λ1 =(ψ +iψ )/√2 ,λ2 =η/2√2+iχ. Let us use an auxiliary field B and modify the action [B].
0 3 1 2
− S →S
2
1 1 1
[B]= Tr ([φ,v ][φ¯,v ] ψ [φ¯,ψ ]+ψ [v ,η]) [φ,φ¯]2 (2)
S −kg2 (2 I I − I I I I − 8
I=1
X
1
B2+i B+ η[φ,η]+χ[φ,χ]+√2χ([v ,ψ ]+[ψ ,v ]) . (3)
1 2 1 2
− E 8
(cid:27)
Here = √2i[v ,v ]. A BRST-like charge is defined as φ = 0, φ¯ = ξ , ξ = [φ,φ¯] , v = ψ , ψ =
1 2 I I I
[φ,v ]E, χ=B , B =[φ,χ] , where is nilpQotent up to tracQeless partQ: 2φ=0Q,Q2φ¯=[φ,φ¯],QQ2v =[φ,vQ] ,etc
I I I
Q Q Q Q
(I =1,2). The action [B] is written in the closed form,
S Q
2
1 1 1
[B]=Q , = ψ [v ,φ¯]+ η[φ,φ¯]+χB iχ . (4)
S P P kg2 (2 I I 8 − E)
I=1
X
Thus, [B] is -exact. The partition function is not a path integral, but an ordinary integration,
S Q
1
= [dv][dλ][dB]exp(iQ ) . (5)
Z vol(G) P
Z
All fermionic contents form a finite-dimensional Grassmann algebra which are generated by Λa . The Grassmann
{ α}
algebra are graded and we call the number of multiplied Λa rank. Let us understand that the fermionic integration
α
pick up the highest rank elements. This integralmight be divergentbecause there are flat directions, though we take
account into the Euclideanization of φ¯ and there is the vanishing determinant term which comes from the fermionic
part. Let us change our space to Euclidean space [B] :=i [B].
E
−S S
1 1
= [dφ][dφ¯][dB][dv ][dη][dχ][dψ ]exp( Q )= [dφ]exp( ) . (6)
E I I eff
Z vol(G) − P vol(G) −S
Z Z
Let us assign ghost charge +1 to Q. The Lagrangian should have ghost charge 0. If the ghost charge of χ is
ν, the auxiliary field B has ghost charge ν +1. Then the term QTrχB should have charge 0 and it means that
1+ν+(ν+1) = 0. So ν = 1. The matrix φ have the same charge as that of Q2. The charge of φ is 2 and φ¯ has
−
2. From the term QTrχ , we can read off the charge of v as 0. Let us deform the BRST charge with respect to
I
− E
the little group SO(2): ,
ε
Q→Q
v =ψ , φ¯=ξ , χ=B , φ=0 ,
ε I I ε ε ε
Q Q Q Q
ψ =[φ,v ]+ε Ev , ξ =[φ,φ¯] , B =[φ,χ]+Eχ .
ε I I IJ J ε ε
Q Q Q
Here E is the deformation parameter and let us consider the deformed action: = . Let us assign the ghost
ε ε
S Q P
charge of E as 0. Then the additional terms coming from this deformation have ghost charge 1 and it does not
−
matter. Therefore we can consider this deformation does not change the value of integration.
6
Localizationtechnique shows that the integralreduces to counting of the fixed points. The integralis Gaussianfor
almostall fermions and B and φ¯. We add mass terms for the convergenceof zero-modes’integrations by hand. After
thatintheclassicallimit(thisprocedurepicksupthefixedpoint),wecanintegratethemandobtainthesimpleresult
1 1
= [dφ]
Z vol(G) Det (adφ+E)
Z adj.
where we neglect various numerical factors coming from the Gaussian integrations. We expect that such a numerical
factorcancelsforpairsofbosonandfermionexceptforφ. Inaddition,thoughthefirstintegralconverges,theresultant
integraldiverges. Itmeansthatthereexistsaspacetodiscussthevalidityofthedeformationmethod. Letusconsider
this integral with some regularization, which may not be equal to the original integral. The integrand is invariant
under a transformation φ φ′ = UφU−1 with a unitary matrix U. Let us use the Weyl integration formula. These
integrals reduce to integra→ls over the Cartan subalgebra. The volume of group is given as vol(SU(N)/Z ). The
N
integral can be reduced to a simple integral over diagonal traceless matrices.
N (φ φ )2
i j
[A ]= [dφ] − ,
Z N−1 EN−1N! (φ φ )2 E2
I i<j i− j −
Y
where N in the numerator is order of the center of SU(N) and N! is the order of its Weyl group. Generalizations
to other groups are obtained immediately. Let Φ be one of root systems A ,B ,C ,D . The integral for Φ is
N−1 N N N
deduced as
#Z (α,φ)
[Φ]= [dφ] .
Z Er#W (α,φ) E
I α∈Φ −
Y
Here in order to obtain finite result, we close the contour with additional paths. We will talk the detail later. The
symbolαdenotesaroot,thevariableφisanelementofCSA,Z isthecenteroftheuniversalcoveringgroupofG,and
W istheWeylgroup. Thisintegraldivergesbecausepolessitting onthe contour. Inordertoavoidsuchadivergence,
let us shift the parameter E to the imaginary direction Im(E) > 0. Then we count poles where (α,φ) > 0 for all
positive roots α because we chose the upper contour on each complex φ -plane. This region is a Weyl chamber. Let
i
us define quantities p = (s ,φ) where s (i = 1, ,r) are simple roots. The linear transformation φ (p ) yields a
i i i i
··· →
Jacobian 1/C where C is the determinant of the Cartan matrix and C = #Z. Our previous work shows that each
nontrivialpole which exists inside of a Weyl chamber is given by p =E for all i for all classicalgauge groups, where
i
p are defined by the fundamental root system obtained by the Weyl chamber. Let us put ourselves on a skeptical
i
ground. In this article we do not treat the result by MNS as exact. Our selection of contour excludes the additional
multiplicity. This is not a truncation. This differs fromthat of MNS. This contour is, in a sense,a fundamental cycle
and MNS count (N 1)! cycles. Remaining poles are sitting on boundaries of Weyl chambers. The considerationon
−
polesonboundariesaremoresensitive. Inthispaperweignoresuchapoleonboundaries. Suchapolesmightexistin
SO and USp cases, while in SU model, there are no poles on boundaries. Concrete calculation show that the residue
at poles on boundaries in some cases in SO and USp models vanish. This is the reason why we do not consider the
contribution of poles on boundaries [29]. In addition, diagrams which are not connected give points on boundaries.
Everyrootαiswrittenasalinearsumofsimplerootswithintegralcoefficientsandthisfactinheritstothevariable
(α,φ).
r r
α= u(α) s , (α,φ)= u(α) p . (7)
i i i i
i=1 i=1
X X
Therefore the integral is written as
r
1 u(α) u(α)
i i
[Φ]= .
Z #W  u(α)i 1 u(α)i+1!
α∈Φ+Y,α∈/{αj}Yi=1 − αY∈Φ+
 
In order to obtain the result 1/N2, we need multiplicity (N 1)! for SU(N) cases. We do not insert the multiplicity
−
in this paper. We show several value for low rank groups in Table II. These poles are written as
A :φ =(N +1)/2 i, B :φ =N i+1, C :φ =N i+1/2, D :φ =N i, (8)
N−1 i N i N i N i
− − − −
7
A1 C1 A1⊕A1 D2 B2 C2 A3 D3
1/4 1/4 1/4×1/4 1/16 3/64 3/64 1/96 1/96
TABLE II: This table show that theresulting residues correspond with theisomorphic algebra.
where 1 i N. We can interpret these poles as a configuration of branes. Residues are computed for all classical
≤ ≤
groups.
1 1 (2N 3)!! 1 (2N 1)!!
[A ]= , [D ]= − , [B ]= [C ]= − , (9)
Z N−1 N N! Z N 2NN N!(2N 4)!! Z N Z N 2N+1N N!(2N 2)!!
· · − · −
Equivalence relations, [B ]= [C ] and (N +1)2 [D ]=N [C ], are derived.
N N N+1 N
Z Z Z Z
A. Grand Partition Functions; A,C
Now we can sum up them. Let us define corresponding grand partition functions;
∞ ∞ 1 qN
Θ[A;q]:=1+ [A ]qN =1+ (10)
N−1
Z N N!
N=2 N=2
X X
∞ ∞ 1 (2N 3)!! 1 q2 N
Θ[D;q]:=1+ [D ]q2N =1+ − , (11)
N
Z N (2N 4)!!N! 2
N=2 N=2 − (cid:18) (cid:19)
X X
∞ ∞ (2N 1)!! 1 q2 N
Θ[B;q]:=q+ [B ]q2N+1 =q+q − (12)
N
Z (2N)!! N! 2
N=1 N=1 (cid:18) (cid:19)
X X
∞ ∞ (2N 1)!! 1 q2 N
Θ[C;q]:=1+ [B ]q2N =1+ − (13)
N
Z (2N)!! N! 2
N=1 N=1 (cid:18) (cid:19)
X X
where q = e−β and β is the chemical potential which couples to the matrix size N. The convergent radii of these
functions are infinite. These functions satisfy the relation Θ[B;q]=qΘ[C;q] and written in terms of the exponential
integral function Ei(z) and the confluent hypergeometric function Ψ(s,t;z) defined in the appendix. Here we study
about grand partition functions of A and C series;
q ex 1
Θ[A;q] =1 q+ − dx=Ei(q) C ln(q) q+1 , Θ[C;q] =Φ(1/2,1;q2/2) , (14)
+ +
− x − − −
Z0
These functions, Θ[A;q] and Θ[C;q] , obey the following differential equations. The meaning of the subscript + is
+ +
explained later.
d2 d d2 d
(1 q−1) Θ[A;q] =1 , (q q−1) 1 Θ[C;q] =0 (15)
dq2 − − dq + dq2 − − dq − +
(cid:20) (cid:21) (cid:20) (cid:21)
Theseequationsaresecondorderlineardifferentialequationwithpolynomialcoefficients. The equationofΘ[A;q] is
+
inhomogeneous while the equation of Θ[C;q] is homogeneous. Our series solutions are particular solutions of these
+
equations. The corresponding homogeneous equations are
d2 d d2 d
(1 q−1) Θ[A;q] =0 , (q q−1) 1 Θ[C;q] =0 (16)
dq2 − − dq + dq2 − − dq − +
(cid:20) (cid:21) (cid:20) (cid:21)
These equations have one singular point q =0 of first kind and one q = of second kind. Our solutions are regular
∞
at q =0. Because the singular point q =0 is first kind, we can consider their indicial equations.
ν(ν 1)+ν =0 , µ(µ 1)+µ=0 . (17)
− −
This means that general solutions of these equations might have log divergence at q =0. Polynomial approximation
is not so bad because the convergent radii are infinite. However if we consider their logarithm, sometimes we are led
8
to wrong direction. In order to avoid it, we use differential equation though its validity in q >> 1 region is not so
clear.
Connected generating function is log of grand partition function Θ; = lnΘ in Euclidean model. The
W W −
expectation value of the matrix size N are defined as the derivative of [O;β] := lnΘ[O;q] where the symbol O
W −
represents the type of sequence O = A,C; ϕ = N = (∂/∂β) [O;β]. The effective action Γ[O;ϕ] is defined by
O
h i W
the Legendre transformation; Γ[O;ϕ]:= [O;β] βϕ. We will show their graph later. In the evaluation of matrix
W −
102 103
Θ[A]+ hNiA+
ΘΘ[[CA]]+− 102 hh2NNiiAC−+
Θ[C]− h2NiC−
101
101 Ni 100
Θ h
10-1
10-2
100 10-3
-3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3
β β
FIG. 2: Grand partition functions and the expectation valuesof thematrix size N and 2N for A and C, respectively.
400 0.5
350 Γ[A]+ Γ[A]−
0.4
300
0.3
250
200 0.2
Γ 150 Γ 0.1
100
0
50
-0.1
0
-50 -0.2
0 10 20 30 40 50 60 70 80 90 100 0 0.2 0.4 0.6 0.8 1 1.2
hNi hNi
4500 0
4000 Γ[C]+ Γ[C]−
-0.1
3500
3000 -0.2
2500
-0.3
2000
Γ Γ -0.4
1500
1000 -0.5
500
-0.6
0
-500 -0.7
0 200 400 600 800 1000 1200 1400 1600 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
hNi hNi
FIG. 3: Effective potentials Γ[A]±, Γ[A]± as functions of theexpectation valueof matrix size, hNi.
integrals, we omit the discussion on sign. Let us consider grand partition functions with sign;
∞ ∞
Θ[A;q] :=1+ [A ]( q)N Θ[C;q] :=1+ [B ]( )N−1q2N . (18)
− N−1 − N
Z − Z −
N=2 N=1
X X
9
We use the subscript in order to explain this difference. They follow to the following differential equations;
±
d2 d d2 d
+(1+q−1) Θ[A;q] =1 , +(q+q−1) +1 Θ[C;q] =2 . (19)
dq2 dq − dq2 dq −
(cid:20) (cid:21) (cid:20) (cid:21)
Their graphs are included to the previous figure and we show their effective potential. The graphs for Θ have cusp
−
singularities. Theseeffectivepotentialswhicharefunctionsoftheexpectationvalueofthesizeofmatrixarenotsingle
valued. There are two branches in SU and USp cases. These results imply the phase transition of these models with
respect to the variation of the expectation value of the size of matrix.
IV. CONCLUSION
We reviewed non-Abelian Berry’s phases in the USp matrix model as an effect of fermionic integrationagainst the
classicalbackground. We encounteredgeneralizedmonopoles. As suchamonopole,westudiedaboutthe Tchrakian’s
five dimensional monopole. The charges of the Tchrakian’s monopole and the Yang monopole are instanton number.
The effect of Berry’s phase is considered in the T-dualized model. In order to interpret them in terms of the original
model, weperformedmatrix integrals. We obtainfull resultforallofclassicalgaugegroupsandtheir grandpartition
functions are computed. The grand partition functions are written as an exponential integral function and confluent
hypergeometric functions. We showed graphs of grand partition functions of A and C series. In other words, matrix
integral for the little IIB and little USp matrix model were exhibited. In these models, the expectation values of
numbers N and their effective potentials were considered. There exist a problem of choice of sign. In the case of
minus sign, we obtain cusp singularities. It shows that there are another phases of these model. From the figure the
start points which correspond with q = 0 are not stable. If we assume that these models have sense only when the
matrix sizes are large enough, the reliable regions are around q =1 and they are neighbor of those cusps. The graph
for SU(N) show that the potential fall into the expectation value < N >= 1, while for USp(2N) the stable point
looks like < N >= 0. We would like to compute matrix integral for full Itoyama-Tokura model. There are several
interesting papers which might be related to our work [30].
acknowledgment
This work has been supported by Osaka University, JSPS(01J00801), COE program in Osaka City Univer-
sity(Constitution of wide-angle mathematical basis focused on knots), Korea Institute for Advanced Study. I would
like to thank to Hiroshi Itoyama, Asato Tsuchiya, Reiji Yoshioka, Leonard Susskind, Ki-Myeong Lee, Piljin Yi for
various advices. I would like to thank to Jun Nishimura for his seminar and advices. I would like to thank to all of
my colleague in Osaka University, Osaka City University, KIAS and all visitors.
APPENDIX A: BASIC TABLES ON CLASSICAL LIE GROUPS
Let us list linear groups. We omit the Lorentz groups.
GL(N,K):= g M(N,K) detg =0 , SL(N,K):= g M(N,K) detg =1 ,
{ ∈ | 6 } { ∈ | }
U(N):= g M(N,C)g† =g−1 , SU(N):=U(N) SL(N,C) ,
{ ∈ | } ∩
SO(N):= g M(N,R)gt =g−1 , USp(2N):= g U(2N)gtJ g =J ,
2N 2N
{ ∈ | } { ∈ | }
0 1
J = N , (A1)
2N 1N 0 !
−
where K = R or C and N 1. Group SO(1) is a discrete group which is isomorphic to 1 . Group SU(1) is a
≥ {± }
unit group 1 . Groups U(1),SO(2) are isomorphic to each other. They are commutative and their fundamental
groups are π{ (}U(1)) π (SO(2)) Z. We will omit these groups. Group SO(N)(N 3) is not simply connected,
1 1
but doubly connected≃; π (SO(N))≃ Z (N 3). The universal covering group of SO≥(N) is denoted by Spin(N).
1 2
≃ ≥
Groups SU(N)(N 2) are simply connected. groupSpin(N) is formed in terms of Clifford algebra. Let us list their
≥
centers. We denote the center of a group G Z(G). The Schur’s lemma show that the center of the general linear
10