Mini-BFSS in Silico

1.2 MB·

by  Tarek Anous

MIT-CTP-4877
Mini-BFSS in Silico
Tarek Anous1,2 and Cameron Cogburn3
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0 1Department of Physics and Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver,
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B.C. V6T 1Z1, Canada

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2Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
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J 3Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

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[email protected], [email protected]

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[ Abstract

1 Westudyamass-deformedN =4versionoftheBFSSmatrixmodelwiththreematricesandgaugegroup
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SU(2). ThismodelhaszeroWittenindex. Despitethis,wegivenumericalevidencefortheexistenceoffour
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1 supersymmetric ground states, two bosonic and two fermionic, in the limit where the mass deformation is
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tuned to zero.
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Contents
1 Introduction 2
2 Setup 3
2.1 Supercharges and Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Symmetry algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 Interpretation as D-particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3 Quantizing the SU(2) theory 6
3.1 Polar representation of the matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.2 Gauge-invariant fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4 Numerical results 9
5 Effective theory on the moduli space of the SU(2) model 12
6 Discussion 14
A Reduced Schr¨odinger equation 15
A.1 R=0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
A.2 R=1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
A.3 R=2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
A.4 R=3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
B Metric on the moduli space 21
1 Introduction
This paper concerns itself with the supersymmetric quantum mechanics of three bosonic SU(N) matrices and
their fermionic superpartners. The model in question, introduced in [1–3], has four supercharges and describes
the low energy effective dynamics of a stack of N wrapped D-branes in a string compactification down to
3+1 dimensions. When the compactification manifold has curvature and carries magnetic fluxes, the bosonic
matrices obtain masses [3]. When the compact manifold is Calabi-Yau and carries no fluxes, the matrices are
massless.
This theory has flat directions whenever the matrices are massless, and hence is a simplified version of the
BFSSmatrixmodel[4],which,forthesakeofcomparison,hasninebosonicSU(N)matricesand16supercharges
and describes the non-Abelian geometry felt by D-particles in a non-compact 9+1 dimensional spacetime. We
hencedubthemodelstudiedhere: mini-BFSS(ormini-BMN[5]inthemassivecase). TheWittenindexW has
I
beencomputedformini-BFSS[6–10]andvanishes,meaningthattheexistenceofsupersymmetricgroundstates
is still an open question. Even the refined index, twisted by a combination of global symmetries and calculated
in [9], gives us little information about the set of ground states due to the subtleties associated with computing
suchindicesinthepresenceofflatdirectionsinthepotential. ThisisinstarkcontrastwiththefullBFSSmodel,
whoseWittenindexW =1, implyingbeyonddoubttheexistenceofatleastonesupersymmetricgroundstate.
I
The zero index result for mini-BFSS has led to the interpretation that it may not have any zero energy ground
2
states[6,10],andhencenoholographicinterpretation. Thelogicbeingthat,withoutarichlowenergyspectrum,
scattering in mini-BFSS would not mimic supergraviton scattering in a putative supersymmetric holographic
dual [11]. Of course a vanishing W does not confirm the absence of supersymmetric ground states—as there
I
may potentially be an exact degeneracy between the bosonic and fermionic states at zero energy.
We weigh in on the existence of supersymmetric states in mini-BFSS by solving the Schr¨odinger equation
numerically for the low-lying spectrum of the N = 2 model, in the in silico spirit of [12]. To deal with the
flat directions we numerically diagonalize the Hamiltonian of the mass-deformed mini-BMN matrix model, for
which the flat directions are absent, and study the bound state energies as a function of the mass. A numerical
analysis of mini-BFSS can also be found in [13,14] which use different methods.
What we uncover is quite surprising. As we tune the mass parameter m to zero, we find evidence for four
supersymmetricgroundstates,twobosonicandtwofermionic,whichcancelintheevaluationofW . Thisresult
I
seemstoagreewithplotsfoundin[13,14]. Itmustbesaidthatourresultdoesnotconstituteanexistenceproof
for supersymmetric threshold bound states in the massless limit, but certainly motivates a further study of the
low-lying spectrum of these theories.
The organization of the paper is as follows: in section 2 we present the supercharges, Hamiltonian and
symmetry generators of the mini-BMN model for arbitrary N. In section 3 we restrict to N = 2 and give
coordinatesinwhichtheSchr¨odingerequationbecomesseparable. Insection4weprovideournumericalresults
andinsection5wederivetheone-loopeffectivetheoryonthemodulispaceinthemasslesstheory. Weconclude
with implications for the large-N mini-BFSS model in section 6. We collect formulae for the Schr¨odinger
operators maximally reduced via symmetries in appendix A and compute the one-loop metric on the Coulomb
branch moduli space in appendix B.
2 Setup
2.1 Supercharges and Hamiltonian
Let us consider a supersymmetric quantum mechanics of SU(N) bosonic matrices Xi and their superpartners
A
λ . The quantum mechanics we have in mind has four supercharges:1
Aα
(cid:16) (cid:17) (cid:16) (cid:17)
Q = −i∂ −imXi −iWi σi γλ , Q¯β =λ¯ γσi β −i∂ +imXi +iWi . (2.2)
α Xi A A α Aγ A γ Xi A A
A A
The parameter m is simply the mass of Xi. The massless version of this model was introduced in [1] and
A
can be derived by dimensionally reducing N = 1, d = 4 super Yang-Mills to the quantum mechanics of its
zero-modes. The mass deformation was introduced in [3], and can be obtained from a dimensional reduction
of the same gauge theory on R×S3. We direct the reader to [2,3] for an introduction to these models. This
quantum mechanics should be thought of as a simplified version of the BMN matrix model [5] (mini-BMN for
brevity). The massless limit should then be thought of as a mini-BFSS matrix model [4]. The lowercase index
i=1,...,3 runs over the spatial dimensions (in the language of the original gauge theory), and the uppercase
index A = 1,...,N2−1, runs over the generators of the gauge group SU(N). The σi are the Pauli matrices
1Spinorsandtheirconjugatestransformrespectivelyinthe2and2¯ofofSO(3). Spinorindicesareraisedandloweredusingthe
Levi-Civitasymbol(cid:15)αβ =−(cid:15) with(cid:15)12=1. Thusinourconventions:
αβ
(cid:0)ψ¯(cid:15)(cid:1)α=ψ¯γ(cid:15)γα , ((cid:15)ψ)α=(cid:15)αγψγ , (cid:15)αω(cid:15)ωβ =δαβ . (2.1)
3
and greek indices run over α=1,2. In keeping with [1], we have defined Wi ≡∂W/∂Xi where
A A
g
W ≡ f (cid:15) Xi Xj Xk, (2.3)
6 ABC ijk A B C
and f are the structure constants of SU(N). The gauginos obey the canonical fermionic commutation
ABC
(cid:110) (cid:111)
relations λ ,λ¯β =δ δ β, and hence the algebra generated by these supercharges is [3]
Aα B AB α
(cid:8)Q ,Q¯β(cid:9)=2(cid:0)δ βH −gσk βXk G + mσk βJk(cid:1) , {Q¯α,Q¯β}={Q ,Q }=0 , (2.4)
α α α A A α α β
with Hamiltonian:
H ≡−1∂ ∂ +1m2(cid:0)Xi(cid:1)2+mXi Wi +g2 (cid:16)f Xi Xj(cid:17)2−3m[λ¯ ,λ ]+igf λ¯ Xk σkλ . (2.5)
2 XAi XAi 2 A A A 4 ABC B C 4 A A ABC A B C
The operators G and Jk appearing in the algebra are, respectively, the generators of gauge transformations
A
and SO(3) rotations. These are given by:
(cid:16) (cid:17) 1
G ≡−if Xi ∂ +λ¯ λ , Ji ≡−i(cid:15) Xj ∂ + λ¯ σiλ . (2.6)
A ABC B XCi B C ijk A XAk 2 A A
In solving for the spectrum of this theory, we must impose the constraint G |ψ(cid:105) = 0 ,∀ A. In the above
A
expressions, whenever fermionic indices are suppressed, it implies that they are being summed over.
Let us briefly note the dimensions of the fields and parameters in units of the energy [E] = 1. These are
[X] = −1/2, [λ] = 0, [g] = 3/2 and [m] = 1. Therefore, an important role will be played by the dimensionless
quantity
m
ν ≡ . (2.7)
g2/3
Weconsiderherethemassdeformedgaugequantummechanicsbecause,intheabsenceofthemassparameter
m, the classical potential has flat directions (see figure 1). Turning on this mass deformation gives us a
dimensionless parameter ν, to tune in studying the spectrum of this theory, and allows us to approach the
massless limit from above.
2.2 Symmetry algebra
Let us now give the symmetry algebra of the theory. The components of J(cid:126) satisfy:
(cid:2)Ji,Jj(cid:3)=i(cid:15) Jk , (cid:2)Ji,Q (cid:3)=−1σi γQ ,
ijk α 2 α γ
(cid:104)J(cid:126)2,Ji(cid:105)=0 , (cid:2)Ji,Q¯α(cid:3)= 1Q¯βσi α . (2.8)
2 β
There is an additional U(1) generator R≡λ¯ λ which counts the number of fermions. It satisfies
R A A
[R,Q ]=−Q , (cid:2)R,Q¯α(cid:3)=+Q¯α , (cid:2)R,Ji(cid:3)=0 . (2.9)
α α
The Hamiltonian also has a particle-hole symmetry:
λ¯α →(cid:15)αγλ , λ →λ¯γ(cid:15) , (cid:15)12 =−(cid:15) =1 , (2.10)
A Aγ Aα A γα 12
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where (cid:15)αβ is the Levi-Civita symbol. This transformation leaves the Hamiltonian invariant but takes R →
2(N2−1)−R and effectively cuts our problem in half.
One peculiar feature of the mass deformed theory is that the supercharges do not commute with the Hamil-
tonian as a result of the vector J(cid:126) appearing in (2.4). It is easy to show that
(cid:2)H,Q (cid:3)= mQ , (cid:2)H,Q¯β(cid:3)=−mQ¯β . (2.11)
α 2 α 2
Thus,actingwithasuperchargeincreases/decreasestheenergyofastateby±m. ThisisaquestionofR-frames,
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as discussed in [3]. Essentially we can choose to measure energies with respect to the shifted Hamiltonian
H ≡H + mR, which commutes with the supercharges, and write the algebra as:
m 2
(cid:8)Q ,Q¯β(cid:9)=2(cid:110)δ β (cid:16)H − mR(cid:17)−gσk βXk G + mσk βJk(cid:111) . (2.12)
α α m 2 α A A α
2.3 Interpretation as D-particles
Theν →0limitofthismodelcanbethoughtofastheworldvolumetheoryofastackofN D-branescompactified
along a special Lagrangian cycle of a Calabi-Yau three-fold [2]. The Xi then parametrize the non-Abelian
A
geometry felt by the compactified D-particles in the remaining non-compact 3+1 dimensional asymptotically
flatspacetime. Theadditionofthemassparametercorrespondstoaddingcurvatureandmagneticfluxestothe
compactmanifold[3]changingtheasymptoticsofthenon-compactspacetimetoAdS . Thisinterpretationwas
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argued in [3,15] and passes several consistency checks. Hence we should think of the mass deformed theory as
describing the non-relativistic dynamics of D-particles in an asymptotically AdS spacetime and the massless
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limit as taking the AdS radius to infinity in units of the string length.
To be more specific, it will be useful to translate between our conventions and the conventions of [3]. One
identifiesm=Ω,g2 =1/m ,{X,λ} =m1/2{X,λ} inunitswherethestringlengthl =1. Reintroducing
v us v them s
√
l , this dictionary implies that g2 = g /l3 2π, with g the string coupling, gets set by a combination of the
s s s s
magnetic fluxes threading the compact manifold and similarly (cid:96) ≡ 1/m gets set by a combination of these
AdS
magnetic fluxes and the string length. For AdS ×CP3 compactifications dual to ABJM this was worked out
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in detail in [3] and they identify
(cid:18)32π2N(cid:19)14 (cid:18) N (cid:19)41
g = , (cid:96) = l , (2.13)
s k5 AdS 8π2k s
where k and N are, respectively, integrally quantized magnetic 2-form and 6-form flux. In this example taking
√
ν = 2π(cid:0)k2/N(cid:1)1/3 →0 while keeping g fixed takes the AdS radius to infinity in units of l .
s s
The main focus of the next sections is on whether this stack of D-particles forms a supersymmetric bound
state, particularly in the ν → 0 limit. There the Witten index W ≡ Tr (cid:8)(−1)Re−βH(cid:9) has been computed
I H
[6–10] and evaluates to zero. This is in contrast with the full BFSS matrix model, whose index is W = 1,
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confirming the existence of a supersymmetric ground state. We will use the numerical approach of [12] and
verify if supersymmetry is preserved or broken in the SU(2) case. We find evidence that supersymmetry is
preserved in the ν →0 limit, and that there are precisely 4 ground states contributing to the vanishing Witten
index.
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3 Quantizing the SU(2) theory
3.1 Polar representation of the matrices
We are aiming to solve the Schr¨odinger problem H |ψ(cid:105) = E |ψ(cid:105). We will not be able to do this for arbitrary
m m
N and from here on we will restrict to gauge group SU(2) for which the structure constants f =(cid:15) . In
ABC ABC
this case the wavefunctions depend on 9 bosonic degrees of freedom tensored into a 64-dimensional fermionic
Hilbert space. It is thus incumbent upon us to reduce this problem maximally via symmetry. In order to do so,
we exploit the fact that the matrices Xi admit a polar decomposition as follows
A
Xi =L Λj MTji (3.1)
A AB B
with
L≡e−iϕ1L3e−iϕ2L2e−iϕ3L3 , M ≡e−iϑ1L3e−iϑ2L2e−iϑ3L3 , (3.2)
and (cid:2)Li(cid:3) ≡−i(cid:15) are the generators of SO(3). The diagonal matrix
jk ijk
Λ≡diag(x ,x ,x ) (3.3)
1 2 3
represents the spatial separation between the pair of D-branes in the stack. The ϕ and ϑ represent the
i i
(respectively gauge-dependent and gauge-independent) Euler-angle rigid body rotations of the configuration
space. This parametrization is useful because the Schr¨odinger equation is separable in these variables, as we
show in appendix A.
The metric on configuration space can be re-expressed as:
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(cid:88)dXi dXi =(cid:88)dx2 +I (cid:0)dΩ2 +dω2(cid:1)−2K dΩ dω , (3.4)
A A a a a a a a a
A,i a=1
I ≡x x −x2 , K ≡|(cid:15) |x x . (3.5)
a b b a a abc b c
The angular differentials are the usual SU(2) Cartan-Maurer differential forms defined as follows:
dω =−1(cid:15) (cid:2)LT ·dL(cid:3) , dΩ =−1(cid:15) (cid:2)MT ·dM(cid:3) . (3.6)
a 2 abc bc a 2 abc bc
The volume element used to compute the norm of the wavefunction is
3 3 3
(cid:89) (cid:89) (cid:89) (cid:89)
dXidXi =∆(x ) dx sinϕ dϕ sinϑ dϑ , (3.7)
A A a i 2 j 2 k
i,A i=1 j=1 k=1
where ∆(x ) ≡ (cid:0)x2−x2(cid:1)(cid:0)x2−x2(cid:1)(cid:0)x2−x2(cid:1) is the Vandermonde determinant with squared eigenvalues. To
a 1 2 3 2 3 1
cover the configuration space correctly, we take the new coordinates to lie in the range [16]:
x ≥x ≥|x |≥0 , π ≥ϕ ,ϑ ≥0 , 2π ≥ϕ ,ϑ ≥0 . (3.8)
3 1 2 2 2 i(cid:54)=2 i(cid:54)=2
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The generators of gauge-transformations G and rotations Ji are given in (2.6). These satisfy
A
(cid:2)Ji,Jj(cid:3)=i(cid:15) Jk , [G ,G ]=i(cid:15) G ,
ijk A B ABC C
(cid:104)J(cid:126)2,Ji(cid:105)=0 , (cid:2)Ji,G (cid:3)=0 . (3.9)
A
To label the SU(2) ×SO(3) representations of the wavefunctions, it is useful to define the “body fixed”
gauge J
angular momentum and gauge operators P(cid:126) ≡M−1·J(cid:126) and S(cid:126) ≡L−1·G(cid:126), which satisfy
P(cid:126)2 =J(cid:126)2 , S(cid:126)2 =G(cid:126)2 ,
(cid:2)Pi,Pj(cid:3)=−i(cid:15) Pk , [S ,S ]=−i(cid:15) S ,
ijk A B ABC C
(cid:2)Pi,Jj(cid:3)=0 , [S ,G ]=0 . (3.10)
A B
Unlike the generators of angular momentum, P(cid:126) is not conserved. However, as we explain in appendix A, it is
still useful for separating variables.
Let us give expressions for the bosonic parts of J(cid:126) and P(cid:126), which we call J(cid:126) and P(cid:126) respectively, in terms of
the angular coordinates. These are:
(cid:18) (cid:19)
cosϑ
J1 =−i −cosϑ cotϑ ∂ −sinϑ ∂ + 1 ∂ , (3.11)
1 2 ϑ1 1 ϑ2 sinϑ ϑ3
2
(cid:18) (cid:19)
sinϑ
J2 =−i −sinϑ cotϑ ∂ +cosϑ ∂ + 1 ∂ , (3.12)
1 2 ϑ1 1 ϑ2 sinϑ ϑ3
2
J3 =−i∂ , (3.13)
ϑ1
and
(cid:18) (cid:19)
cosϑ
P1 =−i − 3 ∂ +sinϑ ∂ +cotϑ cosϑ ∂ , (3.14)
sinϑ ϑ1 3 ϑ2 2 3 ϑ3
2
(cid:18) (cid:19)
sinϑ
P2 =−i 3 ∂ +cosϑ ∂ −cotϑ sinϑ ∂ , (3.15)
sinϑ ϑ1 3 ϑ2 2 3 ϑ3
2
P3 =−i∂ . (3.16)
ϑ3
Similarly let us define G and S as the bosonic parts of the the G and S operators. The G are related to
A A A A A
theJi byreplacingϑ →ϕ . ItiseasytoguessthattheS arethenrelatedtothePi viathesamereplacement.
i i A
We are now ready to give expressions for the momentum operators and the kinetic energy operator in terms
of the new variables. These are [17]:
(cid:26) (cid:27)
(cid:15)
−i∂ =−iL Mib δ ∂ +i abc (x Pc+x S ) , (3.17)
XAi Aa ab xa x2 −x2 a b c
a b
−1∂ ∂ =− 1 ∂ ∆∂ + 1(cid:88)3 Ia(Pa2+Sa2)+2KaPaSa . (3.18)
2 XAi XAi 2∆ xa xa 2 I2−K2
a=1 a a
It is also straightforward to write down the bosonic potential V in terms of the new variables:
V = 1m2x x +3gmx x x + g2 (cid:0)x2x2+x2x2+x2x2(cid:1) . (3.19)
2 a a 1 2 3 2 1 2 1 3 2 3
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Figure 1: Contours of constant potential energy V =2 in units where g =1 as a function of x . The left hand
a
figure is evaluated at m = 0 whereas the right hand figure is evaluated at m = 1. The long spikes in the left
figure are indicative of the flat directions along the moduli space. These flat directions get lifted for any finite
m.
As expected it is independent of the angular variables. We have depicted constant potential surfaces in figure
1.
Apartfromthecoordinatesx thefollowingnon-linearcoordinateswilloftenappearintheequationsbelow:
a
I 1 x2+x2 K x x
y ≡ a = |(cid:15) | b c , z ≡ a =|(cid:15) | b c . (3.20)
a Ia2−Ka2 2 abc (x2b −x2c)2 a Ia2−Ka2 abc (x2b −x2c)2
With these definitions the kinetic term can be written as:
− 1∂ ∂ =− 1 ∂ ∆∂ + 1(cid:2)y (Pa2+S2)+2z PaS (cid:3) .
2 XAi XAi 2∆ xa xa 2 a a a a
Notice that the term (cid:80)3 y Pa2 is the kinetic energy of a rigid rotor with principal moments of inertial y−1.
a=1 a a
Unlike the c = 1 matrix model, the angular-independent piece of the kinetic term can not be trivialized by
√
absorbing a factor of ∆ into the wavefunction [18]. Instead we have:
1 1(cid:18) 1 √ (cid:19)
− ∂ ∆∂ =− √ ∂2 ∆+T , (3.21)
2∆ xa xa 2 ∆ xa
where
(cid:88)3 x2+x2 x2+x2 x2+x2
T ≡ y = 1 2 + 1 3 + 2 3 , (3.22)
a (x2−x2)2 (x2−x2)2 (x2−x2)2
a=1 1 2 1 3 3 3
and its appearance in the Schr¨odinger equation acts as an attractive effective potential between the x .
a
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3.2 Gauge-invariant fermions
Because the operators G in (2.6) have a nontrivial dependence on the gauginos λ it is not sufficient to
A Aα
suppress the wavefunction’s dependence on gauge angles ϕ entirely. Instead we can write down a set of gauge-
i
invariant fermions that will contain the entire dependence on the gauge angles [19]:
χ ≡L λ , χ¯β ≡L λ¯β . (3.23)
Aα BA Bα A BA B
(cid:110) (cid:111)
Thesesatisfy χ ,χ¯β =δ δ β,butnolongercommutewithbosonicderivatives. Definingσ˜i β ≡Mjiσj β,
Aα B AB α α α
we can now write the supercharges in terms of the new parametrization. These are:
(cid:18) (cid:110) g (cid:111) (cid:15) (cid:19)
Q =−iσ˜b γχ δ ∂ +mx + |(cid:15) |x x +i abc (x Pc+x S ) ,
α α aγ ab xb b 2 bst s t x2 −x2 a b c
a b
(cid:18) (cid:110) g (cid:111) (cid:15) (cid:19)
Q¯β =−iχ¯γσ˜b β δ ∂ −mx − |(cid:15) |x x +i abc (x Pc+x S ) , (3.24)
a γ ab xb b 2 bst s t x2 −x2 a b c
a b
wherewehaveputthegauge-invariantfermionstotheleftsoastoremindthereaderthatthebosonicderivatives
are not meant to act on them in the supercharges. The Hamiltonian H (not H ) in the new parametrization
m
is:
H =− 1 ∂ ∆∂ + 1(cid:2)y (cid:0)Pa2+S2(cid:1)+2z PaS (cid:3)
2∆ xa xa 2 a a a a
+ 1m2x x +3gmx x x + g2 (cid:0)x2x2+x2x2+x2x2(cid:1)− 3m[χ¯ ,χ ]+ig(cid:15) χ¯ x σ˜kχ . (3.25)
2 a a 1 2 3 2 1 2 1 3 2 3 4 A A AkC A k C
4 Numerical results
In order to calculate the spectrum of the Hamiltonian (3.25), we must reduce our problem using symmetry,
that is we should label our states via the maximal commuting set of conserved quantities: H , J3,J(cid:126)2, R.
m
Because of the discrete particle-hole symmetry (2.10) we need only consider R = 0,...,3. In appendix A we
construct gauge-invariant highest-weight representations of SO(3) in each R-charge sector. This means we fix
J
the wavefunctions’ dependence on the angles ϑ and ϕ and provide the reduced Schr¨odinger operators that
i i
depend only on x .2
a
OurnumericalresultsforthelowestenergystatesofH foreachR andj arepresentedinTable1andwere
m
obtained by inputting the restricted Schr¨odinger equations of appendix A into Mathematica’s NDEigenvalues
command, whichusesafiniteelementapproachtosolvefortheeigenfunctionsofacoupleddifferentialoperator
on a restricted domain. We have labeled each row by the fermion number R and each column by the SO(3)
J
highest weight eigenvalue j (i.e. J(cid:126)2|ψ(cid:105)=j(j+1)|ψ(cid:105) and J3|ψ(cid:105)=j|ψ(cid:105)).
A few comments are in order:
1. The most striking feature of these plots is the seeming appearance of zero energy states for (R,j)=(2,0)
and (R,j) = (3,1/2) as ν → 0. Since the Witten index W = 0, and since the states in the (2,0) and
I
(3,1/2)sectorsseemhavenonzeroenergyforanyfiniteν,itmustbethecasethatthesestatesareelements
of the same supersymmetry multiplet. This must be so for the deformation invariance of W .
I
2WeonlyprovideasmallsetofthesereducedSchr¨odingeroperators,astheyincreaseinsizewithincreasingSO(3)J eigenvalue
j.
9
j =0 j =1/2
25 25
20 20
3 3
2/ 15 2/ 15
- -
g g
R=0
× ×
m10 m10
ℰ ℰ
5 5
0 0
0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0
ν ν
20 20
15 15
3 3
2/ 2/
- -
g g
R=1 10 10
× ×
m m
ℰ ℰ
5 5
0 0
0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0
ν ν
14 14
12 12
10 10
3 3
2/ 2/
- 8 - 8
g g
R=2
× ×
6 6
m m
ℰ ℰ
4 4
2 2
0 0
0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0
ν ν
12 12
10 10
3 3
2/ 8 2/ 8
- -
g g
R=3 × 6 × 6
m m
ℰ ℰ
4 4
2 2
0 0
0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0
ν ν
Table1: LowestenergyeigenvalueforR={0,1,2,3}andj ={0,1/2}asafunctionofν. Eachrowcorresponds
to a different value of R up to 3 and the columns are labeled by j = 0 or j = 1/2. Note that for ν = 0 there
are E =0 energy eigenstates in both the R=2 and R=3 sectors of the theory. This implies the existence of 4
supersymmetric ground states at ν =0, a fermionic j =1/2 doublet in the R=3 sector and two bosonic j =0
singlets in the R=2 and R=4 sectors.
10