# Tpju-9/04, Ifup-Th 2004/24 High precision study of the structure of supersymmetric Yang-Mills quantum mechanics

###### Abstract

The spectrum of supersymmetric Yang-Mills quantum mechanics is computed with high accuracy in all channels of angular momentum and fermion number. Localized and non-localized states coexists in certain channels as a consequence of the supersymmetric interactions with flat valleys. All states fall into well identifiable supermultiplets providing an explicit realization of supersymmetry on the spectroscopic level. An accidental degeneracy among some supermultiplets has been found. Regularized Witten index converges to a time-independent constant which agrees with earlier calculations.

PACS: 11.10.Kk, 04.60.Kz

Keywords: M-theory, matrix
model, quantum mechanics, supersymmetry

## 1 Introduction

In the present paper we report detailed studies of the supersymmetric Yang-Mills quantum mechanics (SYMQM) [1],[2]. The particular model addressed here results from the dimensional reduction of the supersymmetric Yang-Mills field theory, with the SU(2) gauge group, from a four-dimensional space-time () to a single point in space. It is a member of a family of quantum mechanical systems with the famous , SU, SYMQM at its upper end. The latter, considered as a model of an M-theory [3], attracted a lot of attention in recent years, see [4] and [5] for reviews and further references. For that reason we have launched a systematic, nonperturbative study of the whole family (varying and ) in an attempt to understand their global properties, and to develop adequate techniques while moving gradually to more complex models [6]. The detailed motivation and an account of the relations to the M-theory can be found there.

One of the characteristic property of the supersymmetric quantum mechanics with the Yang-Mills potential is appearance of the continuous spectrum of non-localized states together with discrete, localized bound states [7]. The supersymmetric vacuum is believed to be in the continuous sector, with discrete spectrum beginning at some nonzero energy in general. Interestingly, the (and not less than ) quantum mechanics has also a threshold bound state at zero energy, which agrees with the M-theory correspondence. Apart of the [2],[8], these systems are not soluble and the overall picture just outlined has accumulated over the years of intense studies of particular issues [9]-[21].

In Ref. [6] we proposed to use the standard Hamiltonian formulation of quantum mechanics. To this end we have constructed explicitly the (finite) basis of gauge invariant states and calculated algebraically matrix representations of a Hamiltonian and other relevant observables (e.g., supersymmetry generators). This done, we proceeded to compute numerically the complete spectrum, the energy eigenstates, identified their supersymmetric partners, computed Witten index, etc. The method has an intrinsic cutoff - the total number of allowed bosonic quanta . Since our basis is the eigenbasis of the occupation number operators, the cutoff is easy to implement. It is also gauge and rotationally invariant, hence it preserves these important symmetries. Since the size of the basis, i.e., the dimension of the cut Hilbert space, grows rapidly with , convergence with the cutoff is the crucial question for this approach. It turns out that in all cases studied there (i.e., the Wess-Zumino quantum mechanics, and and , SU(2), SYMQM) many nontrivial results were reliably obtained before the number of basis vectors grew out of control [22]. The approach applies as well to bosons and fermions being entirely insensitive to the notorious sign problem which plagues any Monte Carlo attempts to attack these systems. Later on the new, recursive method of calculating matrix elements significantly improved the precision of the solution of the SYMQM and eventually inspired the exact, analytic calculation of the restricted Witten index for this model [23]. To make further progress one has to deal with the rapidly growing number of states. Of course this problem is most severe in the model where some preliminary results for pure Yang-Mills system were nevertheless already obtained confirming for example the SO(9) invariance [24].

It this paper we have abandoned the brute force construction and diagonalization of the Hamiltonian in the whole (cut) Hilbert space. Instead, we have exploited fully the rotational invariance solving the problem separately for each angular momentum. Second, the recursive construction of matrix elements of Ref. [23] was generalized and adapted to the fixed angular momentum channels (Section III). The two tricks coupled together led to the quantitative improvement of the precision and allowed to uncover a rich structure of the system to a much more complete level (Section IV).

Finally, for the scalar () sector, one can push the cutoff even higher performing complete analytic separation of variables in this case [25, 26]. Using the method adapted by van Baal for the noncompact system considered here, one can reach yet higher cutoffs in the two () channels. Results of this approach will be briefly discussed in the next Section.

Recently, a new possibility to optimize numerical solutions for the lowest state of the system has been investigated [27].

Effective Lagrangians for various dimensionally reduced supersymmetric Yang-Mills theories, including SYMQM, have been very recently derived in Ref. [28].

Supersymmetric Yang-Mills theories in extended space have been studied for some time with the aid of the Hamiltonian approach on the light cone [29].

## 2 The system and early results

### 2.1 Definitions

The reduced quantum-mechanical Yang-Mills system is described by nine canonically conjugate pairs of bosonic coordinates and momenta , , , and six independent fermionic coordinates composing a Majorana spinor , , satisfying canonical anticommutation relations. In , it is equally possible to impose the Weyl condition and work with Weyl spinors. The Hamiltonian reads [30]

(1) | |||||

in , are the standard Dirac matrices. In all explicit calculations we use the Majorana representation of Ref. [31].

Even though three-dimensional space was reduced to a single point, the system still has an internal Spin(3) rotational symmetry, inherited from the original theory, and generated by the angular momentum

(2) |

with

(3) |

Furthermore, the model posesses the residual of the local gauge transformation generated by

(4) |

and it is invariant under the supersymmetry transformations with the generators

(5) |

The bosonic potential in Eq. (1), when written in a vector notation in the color space, has a form

(6) |

which exhibits the famous flat directions responsible for a rich structure of the spectrum.

### 2.2 Creation and annihilation operators

The Hamiltonian (1) is polynomial in momenta and coordinates. Therefore it is convenient to employ the eigenbasis of the occupation number operators associated with all degrees of freedom. To this end we rewrite bosonic and fermionic variables in terms of creation and annihilation operators of simple, normalized harmonic oscillators

(7) |

obeying the canonical (anti)commutation relations

(8) |

As usual bosonic variables are given by

(9) |

For fermionic variables we use the following representation for a quantum Hermitian Majorana spinor

(10) |

which is easily shown to satisfy Eq. (8), with the help of Eq. (7). Other choices of fermionic creation and annihilation operators are possible [13, 30, 15].

### 2.3 The cutoff

For completeness, we shortly review the practical construction of the cut Fock space used in Ref. [6]. The entire Hilbert space is generated by all independent polynomials of the elementary creation operators and acting on the empty state, i.e., the state with zero occupation number for all of the above-defined oscillators. In practical applications we shall work in the finite-dimensional Hilbert space of states containing a total of at most bosonic quanta, i.e.,

(11) |

There is no need to cut the fermionic number, which is limited to 6 by construction.

The physical Hilbert space is restricted to gauge-invariant states only. It can be conveniently generated by all independent polynomials of gauge-invariant creators – bilinear or trilinear combinations of ’s and ’s (the explicit form will be given later).

Finally, since elementary creation and annihilation operators have a straightforward action in the occupation-number basis, one can readily calculate all matrix elements of the Hamiltonian and other observables.

All these steps can be implemented automatically in a computer algebra system. The matrix elements of any operator are calculated by writing the operator in terms of creation and annihilation operators. Finally, the complete spectrum and eigenstates of the cut Hamiltonian (1) are obtained by numerical diagonalization.

This approach has proved reasonably successful. But of curse there is a limit to it. It is possible to improve the results considerably by exploiting fully the symmetries of the cut system, and by foregoing an explicit construction of states using only matrix elements (cf. Ref. [23]).

### 2.4 The symmetries

Some of the symmetries were already exploited earlier to reduce the size of the bases. We now discuss shortly their significance.

The fermion number is conserved:

(12) |

This is best seen in the Weyl representation of Dirac matrices, where the Majorana spinor 10 assumes the simple form [32]

(13) |

Since the Dirac matrices are block-diagonal in this representation, the fermionic Hamiltonian contains only bilinears of the type . Therefore it cannot change . Because the Pauli principle allows only six colored Majorana fermions in this system, the whole Hilbert space splits into seven sectors, . The cutoff on the bosonic quanta preserves , and consequently the diagonalization described above can be carried out independently in each fermionic sector for finite .

The system is invariant under the particle-hole symmetry

(14) |

therefore it suffices to find the spectrum only in the first four sectors, , with the sector being selfdual with respect to Eq. (14).

The local gauge invariance of the full (non-reduced) theory turns into a global constraint of the reduced quantum mechanics. Namely, the physical Hilbert space consists of the gauge-invariant states, which in this case are invariant under the global SU(2) rotations in the color space. This is taken care of by using the gauge invariant combinations of the creation operators. This symmetry is preserved by the gauge invariant cutoff (11), and was already maximally exploited by working exclusively in the color-singlet sector.

On the other hand, rotational invariance had not been fully used until now. Again, the cutoff is rotationally invariant and, accordingly, only exactly degenerate SO(3) multiplets with well-defined angular momentum were observed in the spectrum. However no attempt was made to generate separate bases in each angular momentum channel. This is the main source of improvement in the present work and will be discussed in detail in the next Section.

The system is also invariant under parity. In the sector, it is equivalent to bosonic parity, , and states can be classified according to the parity of .

Finally, supersymmetry is broken by limiting , since the generators (5) do change the number of bosonic quanta. It is therefore interesting to look for the restoration of supersymmetry with the increase of the cutoff. Indeed this was qualitatively observed earlier. Present improvements reveal the supersymmetric spectrum with much better precision.

### 2.5 Early results

The structure of the fermionic Hamiltonian has an instructive consequence. The interaction term vanishes in purely bosonic sector , which means that the effective Hamiltonian in this sector is just the pure Yang-Mills, zero-volume Hamiltonian which provides the starting point of the small volume expansion [33]. Indeed the lowest eigenenergies found in this sector agree very well with well established results of Ref. [34]. Later on, this test was extended to higher states crosschecking with recent results by van Baal to 15-digit precision [35, 36].

Sizes of bases which were reached in Refs. [6, 22] are quoted in Table 1. They contain all angular momenta up to , also shown in the Table. Due to the particle-hole symmetry the structure in the sectors is identical with that in respectively. It will be interesting to compare Table 1 with our new results displayed in Table 2.

0 | 1 1 | - - | 1 1 | 4 4 | 0 |
---|---|---|---|---|---|

1 | - 1 | 6 6 | 9 10 | 6 10 | 0 |

2 | 6 7 | 6 12 | 21 31 | 42 52 | 0 |

3 | 1 8 | 36 48 | 63 94 | 56 108 | 0 |

4 | 21 29 | 36 84 | 111 205 | 192 300 | 0 |

5 | 6 35 | 126 210 | 240 445 | 240 540 | 0 |

6 | 56 91 | 126 336 | 370 815 | 600 1140 | 0 |

7 | 21 112 | 336 672 | 675 1490 | 720 1860 | 0 |

8 | 126 238 | 336 1008 | 960 2450 | 1500 3360 | 0 |

8 | 17/2 | 9 | 19/2 |

In Fig. 1 we display the lowest eigenenergies as a function of the cutoff in all fermionic sectors. Clearly the cutoff dependence is different in the than in and sectors. Based on the experience with simpler models, where the correlation between the nature of the spectrum and the rate of convergence with was established, it was claimed that the spectrum in the sectors is discrete, while it is continuous in the “fermion rich” sectors with and . Recent analytic solutions of a sample of quantum mechanical problems in a cut Hilbert space have proven that indeed continuous spectra are characterized by the slow, power-like dependence on the cutoff [37]. All these early results provided an evidence that sizes of the bases displayed in Table 1 were sufficient to calculate lowest localized states with a reasonable precision.

By computing directly supersymmetric images of lowest eigenstates it was found that SUSY in the cutoff system was broken on the level of 10 – 20 % . This was also confirmed by the Witten index calculation.

### 2.6 Separation of variables

The above conclusions, about the signature and coexistence of the discrete and continuous spectra, have been dramatically confirmed recently by van Baal [36]. Decomposing the solutions of the nine-dimensional Schrodinger equation, in the and channels, into covariant tensors, the problem was reduced to a numerically affordable set of coupled ordinary differential equations. As a consequence, van Baal was able to push a cutoff up to in these two channels, as shown in Fig. 2. The discrete, localized states with can be clearly seen with a very high precision. Moreover, the intricate nature of the sector is also evident. As expected, the localized states have quickly convergent eigenenergies while the continuous spectrum manifests itself as a family of levels which slowly fall with the cutoff. We postpone the detailed discussion of this beautiful result until the global picture of the solutions in all channels becomes clear.

Let us move now to the main subject of this paper which extends the results just presented. The new method allows to reach cutoffs in the range in all fermionic sectors and for all angular momenta, providing at the same time detailed information on the supersymmetric interrelations between eigenstates.

## 3 The new approach

We first present the basic features of the new algorithm, which allows us to push the computation much further.

Rotational symmetry is exploited fully: all the objects in the computation, beside being gauge singlets, belong to irreducible representation of the rotation group Spin(3); this allows heavy use of the traditional machinery of Clebsch-Gordan coefficients and and symbols. (In the following, several formulae will be used; they are reported in Appendix A). In addition, parity symmetry is used whenever possible.

Vectors are never constructed explicitly; we build instead a recursive chain of identities between matrix elements of operators; this follows closely our algorithm for the case [23].

### 3.1 Gauge-invariant operators with definite angular momentum for the bosonic sector

To avoid possible confusion, let us rename the bosonic creation and annihilation operators defined in Eq. (7) and respectively. In order to create states of a definite total angular momentum and , take the combinations

(15) |

so that is a state of angular momentum ; now define : the new creation and annihilation operators satisfy the canonical commutation rules

(16) |

and transform as spin-1 triplets under rotations; they have odd parity . (Here and in the following, denotes the usual Hermitian conjugation applied to a single component of an operator; e.g., is the Hermitian conjugate of .)

From it is possible to build the bilinear gauge-invariant operators , which are then decomposed in components of given angular momentum ; let us introduce the notation

(17) |

where and are arbitrary operators with definite rotational properties; Eq. (17) implies

(18) |

We can now define

(19) |

where is the Hermitian conjugate of . Since is a symmetric combination of ’s, it has no components, but only 1 component and 5 components; and transform as spin-2 quintets under rotations.

In order to express the commutation rules between and , it is necessary to introduce the gauge-invariant “mixed” operators

(20) |

in addition to 1 component and 5 components, has also 3 components. We can now write

(21) | |||||

(22) | |||||

(23) |

It would be pointless to write the detailed form of the coefficients , , and ; their computation will be discussed in Appendix C.

We also introduce the trilinear gauge-invariant creation and annihilation operators

(24) |

(the notation follows from applying Eq. (17) twice), which have only the scalar (i.e., spin-0) component, and the “mixed” trilinear operators

(25) |

The above-defined operators form a complete set of gauge-invariant bosonic operators, in the sense that any gauge-invariant bosonic operator can be written as a polynomial in these operators. In particular, we can write and in terms of , , and as

(26) | |||||

(27) | |||||

### 3.2 Fermionic operators with definite angular momentum

To identify fermionic creation operators with definite angular momentum, recall the origin of the parametrization (10). It represents a Majorana fermion in Majorana representation of Dirac matrices and was obtained by a unitary transformation of a Majorana fermion in the Weyl representation (13) [6]. Therefore creates in fact a fermion in the Weyl representation and as such carries definite angular momentum. This follows from the explicit form of the spin operator defined in Eq. (2):

(28) |

which can be obtained in either Weyl or Majorana representations of Dirac matrices. Therefore, , the fermionic creation and annihilation operators defined in Eq. (7), are already the desired operators, and we set

and are , operators; the anticommutation relations are

### 3.3 Gauge-invariant operators involving fermions

Let us complete the set of gauge-invariant operators, bilinear or trilinear , with definite , , , and . In the bilinear case, we complement the bosonic operators , , and with

( vanishes identically); note that , , , and give zero when applied to a bosonic state.

In the trilinear case, we complement the bosonic creation and annihilation operators and with

the antisymmetrized product of two ’s only produces , and likewise for ’s; the factor in , , and is required to have real matrix elements of between a even, and a odd, state. We also define the “mixed” operators , , and

note that , , , , , , and give zero when applied to a bosonic state.

We can establish (anti)commutation relations between pairs of gauge-invariant operators, similar to Eqs. (21) and (22); it would be pointless to present here their explicit form; their computation will be discussed in Appendix C.

The above-defined operators form a complete set of gauge-invariant operators, in the sense that any gauge-invariant operator can be written as a polynomial in these operators. In particular, we show by explicit computation that

(29) |

must be decomposed in components with definite and ; since in our Majorana representation is real and is purely imaginary, is Hermitian. Denoting the and doublets by and respectively, we have

(30) |

where is an arbitrary phase; the anticommutation relations are

where is defined by the analogous of Eq. (15); gives zero when applied to a gauge-invariant state; the only nontrivial anticommutator can be rewritten as

(31) |

we choose ; an explicit computation gives

(32) |

Note that, with the present conventions, all matrix elements of interest are real.

### 3.4 Construction and orthonormalization of states with definite angular momentum

All states are classified into even and odd states, according to the parity of . (This label coincides with parity only for states.)

It is useful to set up a common naming scheme for all our creation operators: is the creation operator with and ; i.e., , , , , , , and ; is identically zero.

We build our states recursively, applying to a state of an orthonormal basis with definite and , and taking linear combinations to produce again an orthonormal basis.

It is important to note that, given the contraction rule

(33) |

the product of two trilinear operators can always be decomposed into a sum of products of three bilinear operators; therefore, even states can be built by applying any number of even () creation operators to the vacuum; correspondingly, odd states can be built by applying one odd () and any number of even creation operators to the vacuum. is never needed in combination with fermionic operators, since for can be written as a linear combination of terms of the form .

Creation operators (which (anti)commute between themselves) can be ordered to get every fermionic operator to the left of every bosonic operator and every trilinear operator to the left of every bilinear operator. Therefore, using the notation for our states, we can build all bosonic states from even bosonic states as , and all fermionic states of parity from even states of lower as , .

In order to create a fermionic state with , at least one must be used; therefore, such states can also be built as ; this second recipe turns out to be much more efficient, both in generating and orthonormalizing the states and in computing matrix elements of operators between them.

A basis for the sector with given and is contained in the set

(34) |

where

(35) | |||||

The scalar product of two such states can be written as

By Gram-Schmidt orthonormalization we obtain the orthonormal basis