Solutions of D=2 supersymmetric Yang-Mills quantum mechanics with SU(N) gauge group

0.2 MB·English

by  Piotr Korcyl

D
Solutions of = 2 supersymmetric Yang-Mills
SU N
quantum mechanics with ( ) gauge group
1
1
0
2
Piotr Korcyl

n
a
J M. Smoluchowski Institute of Physics, Jagiellonian University,
3 Reymonta 4, 30-059 Krak´ow, Poland

]
h
Abstract
p
-
We describe the generalization of the recently derived solutions of
h
t D = 2 supersymmetric Yang-Mills quantum mechanics with SU(3)
a
gauge group to the generic case of SU(N) gauge group. We discuss
m
the spectra and eigensolutions in bosonic as well as fermionic sectors.
[

1
1 Introduction
v
1
9
Supersymmetric Yang-Mills quantum mechanics [1, 2] attract a lot of at-
5
tention since the works Ref.[3, 4]. Among many variants of such quantum
0
. mechanics, the twodimensional systems are the simplest ones. Although
1
0 proposed nearly 30 years ago by Claudson and Halpern [2], their solutions
1 for gauge groups other than SU(2) are poorly known. Recently, complete
1
solutions were derived for the model with the SU(3) gauge group [5]. In
:
v
this work we present a generalization of such solutions to the case of models
i
X with any SU(N) gauge group. We discuss their derivation as well as their
r properties. We use the framework of cut Fock basis described in details in
a
Ref.[6], which allows a systematic analysis of the SYMQM systems by both
numerical and analytic methods.
The cut Fock space approach basically consists in introducing the Fock
basis in the Hilbert space and then in considering only a finite subset of
basis states. This subset is composed of states with less than N quanta
cut
and N is usually called the cut-off.
cut
The main results presented in this paper are the closed formulae for
the eigenenergies and corresponding eigenstates of SYMQM in any, bosonic
or fermionic, sector, valid for any N . Their infinite cut-off limit is also
cut
discussed.
1
Such results are especially interesting since the enable one to study the
large-N behavior of the wavefunctions of SYMQM as well as the supersym-
metricstructureofthelimitingmodel. Theymaybeusedtoconstructhigher
dimensional wave-functions which can have an interpretation in the context
of supermembrane theory [7]. One may also try to build a perturbation
theory around their finite cut-off versions [6].
This paper is composed in the following way. We start by briefly pre-
senting the D = 2 supersymmetric Yang-Mills quantum mechanics and the
approach of cut Fock basis. Then, we translate the eigenequation for the
Hamiltonian to a recursion relation for the coefficients describing the de-
composition of the eigenstate in the Fock basis. The recursion relation is
valid for any N. Subsequently, we discuss the implications of this recur-
rence relation. Specifically, we argue that a closed formula for the spectra
can be deduced in all, bosonic as well as fermionic, sectors of the SYMQM
models. We explicitly present it for the SU(4) and SU(5) models. We also
mention the form of the spectra in the infinite cut-off limit. In section 7
expressions for the eigenstates are presented. The emphasis is put on the
wave-functions of supersymmetric vacua in a generic SU(N) model. Next,
the completeness, orthogonality, normalization and infinite cut-off limit of
our eigensolutions are discussed. Finally, we end with some conclusions.
2 Supersymmetric Yang-Mills Quantum Mechan-
ics
Although supersymmetric Yang-Mills quantum mechanics were described
already many times [2, 8, 9, 6], let us briefly remind the main statements
in order to keep this work self-contained. SYMQM can be obtained by a
dimensional reduction of a supersymmetric, D = d+1 dimensional Yang-
Mills quantum field theory to D = 0 + 1, i.e. to a single point in space.
The remaining degrees of freedom are those of internal symmetries of the
field theory. Consequently, the initial local gauge symmetry is reduced to a
global symmetry of the quantum mechanical system. In this work we will
be interested in systems obtained from = 1 Yang-Mills field theory in
N
two dimensions with different SU(N) gauge symmetries [2]. The degrees of
freedom are described by a scalar field φ and a complex fermion λ , where
A A
A labels the generators of the gauge group, hence φ and λ transform in the
adjoint representation of SU(N). The Hamiltonian of the reduced system
2
reads 1
1
H = π π +igf λ¯ φ λ . (1)
A A ABC A B C
2
H is supersymmetric since we can define the supercharges Q and Q¯,
Q = λ π , Q¯ = λ¯ π , (2)
A A A A
such that
Q,Q = Q¯,Q¯ = 0, and Q,Q¯ = π π 2gφ G , (3)
A A A A
{ } { } { } −
where
G = f φ π iλ¯ λ , (4)
A ABC B C B C
−
is the generator of the gauge transfo(cid:0)rmations. Upon(cid:1)canonical quantization,
one defines operators satisfying canonical commutation/anticommutation
relations,
[φ ,π ] = iδ , λ ,λ¯ = δ . (5)
A B A,B A B A,B
{ }
Thequantization procedurerequires the imposition of the Gauss’ constraint
on physical states. The dimensionally reduced Gauss’ law translates to a
requirement of invariance of physical states under gauge transformations,
G physical state = 0. (6)
A
| i
Thus, the fermionic term of the Hamiltonian, being proportional to the
Gauss’ constraint, vanishes on any physical state. Therefore,
1
H = Q,Q = π π in the physical Hilbert space. (7)
† A A
2
(cid:8) (cid:9)
We end this section by rewriting the above Hamiltonian in terms of creation
and annihilation operators, defined as
1 1
aA = √2 φA+iπA , a†A = √2 φA −iπA . (8)
(cid:0) (cid:1) (cid:0) (cid:1)
Thus,
N2 1 1 1
H = tr(a†a)+ − tr(a†a†) tr(aa), (9)
4 − 2 − 2
where we used the matrix notation, in which every operator transforming
in the adjoint representation is summed with the generators of the SU(N)
group in the fundamental representation, giving an operator valued N N
×
matrix. In what follows we will usea simplified notation for the trace of any
such matrix, namely, tr(O) (O).
≡
1Thesummation of doubled indices is understood, i.e. φAφA≡PAN=2−11φAφA.
3
3 Gauge invariant Fock basis for SU(N) SYMQM
A systematic construction of the Fock basis for the D = 2 SYMQM models
was proposed in Ref.[8] and developed in Ref.[9]. We use the notation in-
troduced in Ref.[6] where the recursive construction of the basis using the
notion of elementary bosonic and fermionic bricks was described in details.
We recall here very briefly the most important conclusions concerning the
Fock basis of the SYMQM Hilbert space.
An operator is called an elementary bosonic brick if it is a single trace
operator composed exclusively of creation operators. For a given N we have
N 1linearly independentelementary bosonicbricks, which we labelby C .
†
−
They are
CN† (2) ≡ (a†2), CN† (3) ≡ (a†3), ..., CN† (N−1)≡ (a†N−1), CN† (N) ≡(a†N).
A generic basis state can be written as
|p2,p3,...,pNi =CN† (2)p2CN† (3)p3...CN† (N −1)pN−1CN† (N)pN|0i. (10)
Additionally, in the fermionic sector with n fermionic quanta there
F
are dnF(N) fermionic bricks. We label them by CN† (nαB,nF,α), where nαB
denotes the number of bosonic creation operators and n the number of
F
fermionic creation operators incorporated in CN† (nαB,nF,α). α is an addi-
tional index, since nα andn donotspecify unambiguouslytheoperator. α
B F
runs from 1 to dnF(N) in each fermionic sector. Fermionic basis states can
be obtained by the application of the fermionic bricks to the bosonic basis
states eq.(10). Hence, we define
|α,nF;p2,p3,...,pNi = CN† (nαB,nF,α)|p2,p3,...,pNi. (11)
Although the sets of fermionic bricks are not explicitly known for N > 4,
it turns out that they are not necessary for the derivation of spectra of the
SU(N) SYMQM models.
A generic state from the bosonic sector with up to N bosonic quanta
cut
can be decomposed as
E = a p ,p ,...,p , (12)
| i p2,p3,...,pN| 2 3 Ni
2p2+3p3+·X··+NpN≤Ncut
whereas a generic state from the sector with n fermionic quanta can bede-
F
composedinthesoconstructedbasiswithunknownamplitudesaα (E),
p2,p3,...,pN
4
where the index α describes which one of the fermionic bricks was used. For
the cut-off N we get
cut
dnF(N)
E = aα α,n ;p ,p ,...,p . (13)
| i p2,p3,...,pN| F 2 3 Ni
αX=1 PNk=2kpXk≤Ncut−nαB
The physical results correspond to the limit of N . Such limit is
cut
→ ∞
nontrivial in the case of systems with continuous spectra and was discussed
in details in the case of a system with one degree of freedom [10] as well as
systems with a SO(d) gauge symmetry [11]. In this paper we will describe
the N limit, however, the study of the scaling law in the spirit of
cut
→ ∞
Refs.[10, 11] will be discussed elsewhere.
4 Recurrence relations
Inthissectionwederivetherecurrencerelationforthecoefficientsa
p2,p3,...,pN
and aα . To this goal we follow the derivation of a similar recurrence
p2,p3,...,pN
relations for the SU(3) model [12].
4.1 Bosonic sectors
We start with the purely bosonic sector. In order to obtain the recurrence
relation for a we must evaluate the action of the Hamiltonian on a
p2,p3,...,pN
generic basis state. We get (see Appendix A for the details of calculations):
N
1
(a a)p ,...,p = kp p ,...,p ,
† 2 N k 2 N
| i 2 | i
(cid:16)Xk=2 (cid:17)
(a a )p ,...,p = p +1,p ,...,p ,
† † 2 N 2 3 N
| i | i
N
1
(aa)p ,...,p = p p + (N2 1) 1 kp p 1,p ,...,p +
2 N 2 2 k 2 3 N
| i 2 − − | − i
(cid:16) (cid:0) Xk=3 (cid:1)(cid:17)
+ N j2pj(pj −1) (a†2j−2) 1 (a†j−1)2 + jpj j−4(a†t)(a†j)(a†j−2−t)+
4 − N 4

j=3 t=2
X (cid:0) (cid:1) X
Njp j 1 N jp sp (a j+s 2)(a j)
+ 4 j 1− N−2 (a†j)(a†j−2)+ j2 s † (a−s) † +
(cid:0) (cid:1) s=Xj+1 (cid:16) †
1 (a j 1)(a j)(a s 1)
† − † † −
p ,...,p 2,...,p . (14)
− N (a s) | 2 j − Ni
† !
(cid:17)
5
Eqs.(14) lead to the general recursion relation
N
1
a kp + (N2 1) 2E a +
p2−1,...,pN − k 2 − − p2,...,pN
(cid:16)Xk=2 (cid:17)
N
1
+ (p +1) p + (N2 1)+ kp a +
2 2 2 − k p2+1,...,pN
(cid:16) (cid:0) Xk=3 (cid:1)(cid:17)
N
p (p 1) 1
+ j j − j2 a a +
4 p2,...,pj+2,...,p2j−2−1,...,pN − N p2,...,pj−1−2,pj+2,...,pN
Xj=3 (cid:16) (cid:17)
N j 1
+p j 1 − a +
j 4 − N2 p2,...,pj−2−1,pj−1,pj+1,...,pN
(cid:0)j 4 (cid:1)
pjj −
+ a +
4 p2,...,pt−1,...,pj−2−t−1,...,pj+1,...,pN
t=2
X
N
p s
s
+p j a +
j 2 p2,...,pj+1,...,ps+1,...,pj+s−2−1,...,pN
s=Xj+1 (cid:16)
1
a = 0. (15)
− N p2,...,pj−1−1,pj+1,...,ps−1−1,ps+1,...,pN !
(cid:17)
An important remark concerns the Cayley-Hamilton theorem. In de-
riving eq.(15) we did not simplified the operators with powers of bosonic
creation operators bigger than N. Such operators can appear in two places
ineq.(14),namelyfortheoperators(a 2j 2)and(a j+s 2),where3 j N
† − † −
≤ ≤
and j+1 N. They must be reduced once the final form of the recurrence
≤
relation for a given N is obtained. Note, however, that when examining the
large-N limit, the Cayley-Hamilton theorem does not apply, therefore, our
recurrence relation is a good starting point for such investigation.
With N = 3 and after simplifying the operator (a 4) with the Cayley-
†
Hamilton theorem, we recover the recurrence relation discussed in Refs.[5,
12], namely,
a 2p +3p +4 2E a +(p +1)(p +3p +4)a +
p2−1,p3 − 2 3 − p2,p3 2 2 3 p2+1,p3
3
(cid:0) (cid:1) + (p +1)(p +2)a = 0. (16)
8 3 3 p2−2,p3+2
The first three terms does not involve any change of the p index. The
3
mixing of amplitudes with different values of the p index is described only
3
bythefourthterm. Itisinducedbythenon-orthogonalstates 0,23,0 = 0.
h | i 6
6
In order to demonstrate the increasing complexity of the recursion rela-
tion with increasing N induced by the increasing number of mixing terms
between states with the same number of bosonic quanta but made with dif-
ferent elementary bricks, let us present the recursion relation for the SU(4)
model, which reads
15
a 2p +3p +4p + 2E a +
p2−1,p3,p4 − 2 3 4 2 − p2,p3,p4
(cid:0) (cid:1) 15
+(p +1)(p +3p +4p + )a
2 2 3 4 2 p2+1,p3,p4
3 13
+ 5p (p +1) +3p (p +1)+ (p +1) a +
3 4 − 2 4 4 4 4 p2−1,p3,p4+1
1 (cid:0) 1 (cid:1)
+ (p +1)(p +2)a (p +1)(p +2)a +
3 4 4 p2,p3−2,p4+2− 2 4 4 p2−3,p3,p4+2
9 9
+ (p +1)(p +2)a (p +1)(p +2)a . (17)
4 3 3 p2,p3+2,p4−1− 16 3 3 p2−2,p3+2,p4
The first line again contains terms where only the p index vary (p and p
2 3 4
remain fixed in these expressions). The remaining terms are mixing terms,
which are induced by the nonvanishing scalar products: 2,0,00,0,1 = 0,
h | i 6
1,2,00,0,2 = 0, 1,0,10,2,0 = 0 and 3,0,00,2,0 = 0.
h | i 6 h | i 6 h | i 6
Summarizing, a general feature of the recursion relation eq.(15) is that
only the first three terms describe a change in the p index of the a
2 p2,p3...,pN
coefficients. Moreover, these terms have the structure of the Laguerre re-
cursion relation 2 with 2E being the argument of the polynomials and
N kp + 1(N2 1) 1 playing the role of their index. The additional
k=3 k 2 − −
terms areresponsibleforthemixingbetween states with anequalnumberof
P
quanta but constructed with different elementary bricks. However, keeping
these indices as external parameters, one can solve eq.(17).
In sections 6 and 7 we will use these observations and the general theo-
rems developed in Ref.[12] for the SU(3) model, to discuss the eigenvalues
and eigenstates of H which solves the above recursion relations.
2WedefinetheLaguerrepolynomialsLα(x)asthesolutionsofthedifferentialequation
m
Txyh′e′+p(oαly+n1o−mxia)lys′+Lαmny(x=) a0raenrdeltahteedorttohoLgαmon(axl)itvyiareLlaαmti(oxn)R=0∞ΓL(Lmαmαm+(x(αx)+)L1αn),(xw)hxeαree−Γx(dmx)=isδmthne.
Euler gamma function, Γ(m+1)=m! for m integer.
7
4.2 Fermionic sectors
Before we discuss the solutions of eq.(15) let us generalize the above treat-
ment for the case of fermionic sectors. Since we have
1 1
H,C (nα,n ,α) = nαC (nα,n ,α) (aa),C (nα,n ,α) (18)
† B F 2 B † B F − 2 † B F
(cid:2) (cid:3) (cid:2) (cid:3)
we can rewrite the eigenequation of H as
dnF nα
aα C (nα,n ,α) H + B p ,p ,...,p
p2,p3,...,pN † B F 2 | 2 3 Ni
αX=1PNk=2kpXk≤Ncut−nαB (cid:16) (cid:17)
dnF
1
aα (aa),C (nα,n ,α) p ,p ,...,p = 0.
−2 p2,p3,...,pN † B F | 2 3 Ni
αX=1PNk=2kpXk≤Ncut−nαB (cid:2) (cid:3)
(19)
The action of the Hamiltonian on the basis states has been derived in the
previous section. In order to obtain the recurrence relation we have to
compute the commutators (aa),C (nα,n ,α) for any fermionic brick and
† B F
any N. We proceed in the same manneras was did for theSU(3) model[12].
(cid:2) (cid:3)
For any fermionic brick C (nα,n ,α), the commutator with (aa) will be
† B F
equal to a sum of nα terms, each of them being equal to the fermionic brick
B
C (nα,n ,α) with one of the bosonic creation operators substituted by an
† B F
annihilation operator. We will write them as Gt, where the index t goes
α
from 1 to nα,
B
nα
B
(aa),C (nα,n ,α) = Gt. (20)
† B F α
t=1
(cid:2) (cid:3) X
The operators Gt should be now pushed on the right through the creation
i
operators Nk=2(a† k)pk. We get,
QN N j 1 N
−
t Gtα, (a† k)pk = (a†k)pk Gtα,(a†j)pj (a†k)pk (21)
∀
(cid:2) kY=2 (cid:3) Xj=2(cid:16)kY=2 (cid:17)h i(cid:16)k=Yj+1 (cid:17)
Gt contains exactly one bosonic annihilation operators, therefore we can
α
write
Gt,(a j)pj = p (a j)pj 1 Gt,(a j) . (22)
t α † j † − α †
∀
(cid:2) (cid:3) (cid:2) (cid:3)
8
Forj = 2,wecanreplace Gt,(a a ) intheaboveexpressionbyC (nα,n ,α).
α † † † B F
Thus, we get the general form of the recurrence relation,
(cid:2) (cid:3)
dnF
aα +
p2 1,p3,...,pN
( −
αX=1PNk=2kpXk≤Ncut−nαB
N
1
2p + kp + (N2 1)+nα 2E aα
− 2 k 2 − B − p2,p3,...,pN
(cid:16) Xk=3 (cid:17)
N
1
+(p +1) p + kp + (N2 1)+nα aα α,n ;p ,p ,...,p
2 2 k 2 − B p2+1,p3,...,pN | F 2 3 Ni
!
(cid:16) Xk=3 (cid:17)
N
p (p 1) 1
+ j j − j2 aα aα +
4 p2,...,pj+2,...,p2j−2−1,...,pN − N p2,...,pj−1−2,pj+2,...,pN
Xj=3 (cid:16) (cid:17)
N j 1
+p j 1 − aα +
j 4 − N2 p2,...,pj−2−1,pj−1,pj+1,...,pN
(cid:0)j 4 (cid:1)
+ pjj − aα +
4 p2,...,pt−1,...,pj−2−t−1,...,pj+1,...,pN
t=2
X
N
p s
+p j s aα +
j 2 p2,...,pj+1,...,ps+1,...,pj+s−2−1,...,pN
s=Xj+1 (cid:16)
1
aα α,n ;p ,...,p
− N p2,...,pj−1−1,pj+1,...,ps−1−1,ps+1,...,pN !| F 2 Ni
(cid:17)
nαB N
+aα p Gt,(a k) p ,p ,...,p 1,...,p +
p2,p3,...,pN k α † | 2 3 j − Ni

Xt=1 Xk=3 h i
N
+ (a† k)pk Gtα 0 = 0. (23)
| i
!)
(cid:16)kY=2 (cid:17)
This recursion relation can be divided into three parts. The first part (first
three terms contained in a parenthesis) is diagonal in all indices except
p , this includes also the fermionic index α. Again, it has the structure
2
of the Laguerre polynomials recursion relation with 2E playing the role of
the argument of these polynomials. The second part (four terms contained
in a parenthesis) mixes the indices p , 2 j N. However, it is still
j
≤ ≤
diagonal in the fermionic index α. The third part contains terms which mix
different values of the α index. It can be argued that this latter part does
9
not contain terms proportional to the C (nα,n ,α) fermionic brick, hence
† B F
it really corresponds to a mixing part.
5 Grouping into families
One of the most important conclusion of this note is the observation that
the solutions, both bosonic as well as fermionic, group into disjoint sets.
This is true for the models with arbitrary N. Such sets of solutions were
called families in the study of the SU(3) model[12]. In this latter case, each
solution belonged to an unique family. Families were labeled by a single
integer denoting the maximal number of cubic bricks which appeared in the
decomposition of the solutions from this family in the Fock basis. Basing on
theformoftherecursionrelations eqs.(15)and(23)wenowarguethatthese
remarks can be generalized to the case of the SU(N) model with arbitrary
N. This is by no means a strick proof; in order to obtain a specific solution
one should apply a general theorem presented in Ref.[12]. Below, we just
sketch the argument.
We start with the discussion of the purely bosonic recursion relation.
Let us note that at finite cut-off there always exists one set of coefficients
a which recursion relation is not coupled to any other. This is be-
p2,p3,...,pN
cause by cutting the Fock basis we do not consider states with sufficiently
large number of quanta. In order to be more specific let us fix the cut-
off to be equal to N . We choose a set of integers pmax,pmax,...,pmax,
cut 3 4 N
such that N kpmax = N . One of the possible choices is pmax =
k=3 k cut 4
pmax = = pmax = 0 and 3pmax = N . Other choices are indepen-
5 ···P N 3 cut
dent and lead to the same conclusions. Consider now the recursion relation
for ap2,pm3ax,pm4ax,...,pm5ax. There is only one equation which reads
1
N + (N2 1) 2E a = 0. (24)
− cut 2 − − 0,31Ncut,0,...,0
Other coefficients(cid:0)a are absen(cid:1)t because they contain to many
p2,13Ncut,0,...,0
quanta. From eq.(24) follow two possibilities. Either 2E = N +1(N2 1)
cut 2 −
and then a can be arbitrary or 2E = N + 1(N2 1) in which
0,13Ncut,0,...,0 6 cut 2 −
case a must vanish. The condition for E can be also rewritten in
0,13Ncut,0,...,0
the form,
LNcut+21(N2−1)−1(2E) = 0 or LNcut+12(N2−1)−1(2E) = 0. (25)
1 1 6
Now, if the coefficient a vanishes (we assume that so do the co-
efficients corresponding0t,o13Noctuht,e0r,...c,0hoices of pmax,pmax,...,pmax, such that
3 4 N
10