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hep-th/9907191 HUTP-99/A045

The Geometry of (Super) Conformal Quantum Mechanics

Jeremy Michelson^{†}^{†} Current address: New High
Energy Theory Center; Rutgers University; 126 Frelinghuysen Road;
Piscataway, NJ 08854. and Andrew Strominger

Department of Physics

Harvard University

Cambridge, MA 02138

Department of Physics

University of California

Santa Barbara, CA 93106

Abstract

-particle quantum mechanics described by a sigma model with an -dimensional target space with torsion is considered. It is shown that an conformal symmetry exists if and only if the geometry admits a homothetic Killing vector whose associated one-form is closed. Further, the can always be extended to superconformal symmetry, with a suitable choice of torsion, by the addition of real fermions. Extension to requires a complex structure and a holomorphic isometry . Conditions for extension to the superconformal group , which involve a triplet of complex structures and isometries, are derived. Examples are given.

1. Introduction

Conformal and superconformal field theories in various dimensions have played a central role in our understanding of modern field theory and string theory. Oddly, the subject of this paper—one dimension—is one of the least well understood cases. The simplest example of conformally invariant single-particle quantum mechanics was pioneered in [1], following the general analysis of [2–4]. Supersymmetric generalizations were discussed in [5–11]. The quantum mechanics case has taken on renewed interest because superconformal quantum mechanics may provide a dual description of string theory on [12].

Most of the discussions so far have concerned relatively simple systems either with small numbers of particles or exact integrability. In this paper we consider a more general class of models with particles.

We begin in section 2 with a bosonic sigma model with an -dimensional target space. It is shown that the model has a nonlinearly-realized conformal symmetry if and only if the target space metric has a vector field whose Lie derivative obeys

and whose associated one-form is closed

Given (1.1) and (1.2) it is shown that, in a Hamiltonian formalism, the dilations are (roughly) generated by while the special conformal transformations are generated by

The conformal symmetry persists in the presence of a potential obeying A general class of examples is given.

In section 3 we turn to the supersymmetric case. The geometry of Poincaré-supersymmetric quantum mechanics with a variety of supermultiplets was discussed by Coles and Papadopoulos [13]. We restrict our attention to the case for which the multiplet structure with respect to the Poincaré super-subgroup consists of bosons with real superpartners . Such multiplets arise in the reduction of two-dimensional chiral multiplets, where is the number of supersymmetries, to one dimension, and give rise to what is sometimes referred to as “type ” models (most of the literature concerns “type ” multiplets). In section 3.1 we show that every bosonic conformal model can be extended to an theory with superconformal symmetry provided the torsion obeys certain constraints. In section 3.2 we consider and find that the extension to requires a complex structure with respect to which must be holomorphic. is found to generate a isometry. In section 3.3 we first derive a simplified version of the conditions for Poincaré supersymmetry with an -symmetry as first-order differential relations between the triplet of complex structures . We further show that an model has a superconformal symmetry if the vector fields generate an isometry group and obey generalizations of the identities required for . The parameter is determined by the constant in the Lie bracket algebra. In section 3.4 we construct a large class of theories in terms of an unconstrained potential . superconformal symmetry then follows if is a homogeneous and rotationally-invariant function of the coordinates. Related results in four dimensions were recently discussed in [14].

Throughout the paper we use a Hamiltonian formalism. In appendix A we give a Lagrangian derivation of the supercharges used in the text. We use real coordinates throughout the body of the text, but appendix B gives various useful formulae for the geometry and supercharges in complex coordinates. In appendix C we discuss the conditions under which an geometry can be written in terms of a potential .

A primary motivation for this work is the expectation that quantum mechanics on the five-dimensional multi-black hole moduli space is an theory with a superconformal symmetry at low energies [15].

2. Conformally Invariant -Particle Quantum Mechanics

In this section we find the conditions under which a general
-particle quantum mechanics admits an symmetry.
We will adopt a Hamiltonian formalism,
and derive the conditions for the existence of appropriate
operators generating the symmetries.
The general Hamiltonian is^{†}^{†} The canonical momentum
obeys and (for the norm ). In this and all subsequent expressions, the operator
ordering is as indicated.

where . We now determine the conditions under which the theory, defined by equation (2.1), admits an symmetry.

We first look for a dilational symmetry of the general form

This is generated by an operator

which should obey

From the definitions (2.3) and (2.1) one finds

where is the usual Lie derivative obeying:

Therefore, given a metric and potential a dilational symmetry exists if and only if there exists a conformal killing vector obeying

and

Note that (2.7) implies the vanishing of the last term of equation (2.5). A vector field obeying (2.7) is known as a vector field, and the action of is known as a .

Next we look for a special conformal symmetry generated by an operator obeying

and

Equations (2.9) and (2.10) together with (2.5) is an algebra. Equation (2.9) is equivalent to

while (2.10) can be written

Hence the one-form is exact. One can solve for as

We shall adopt the phrase “closed homothety” to refer to a homothety whose associated one-form is closed and exact.

An alternate basis of generators is

In this basis the generators obey the standard commutation relations

The nature of these geometries can be illuminated by choosing coordinates such that and for . This is always locally possible away from the zeros of . One then finds

Hence, given metric in dimensions, one can construct a geometry with a closed homothety in dimensions by dressing it with an extra radial dimension. Similar comments pertain to the potential .

An alternate useful choice is dilational coordinates, in which

where is an arbitrary constant. These are related to the coordinates in (2.16) by . In such dilational coordinates one finds

Hence in dilational gauge the metric components are homogeneous functions of degree . (It is not, however, the case that every homogeneous metric admits an symmetry.) At this point can be changed by transformations which take the coordinates to powers of themselves, and so has no coordinate independent meaning. However, it turns out that for supersymmetry, a preferred value of is obtained in quaternionic coordinates, when such coordinates exist and coincide with dilational gauge, as in the class of examples considered in section 3.4.

In conclusion, the Hamiltonian (2.1) describes an invariant quantum mechanics if and only if the metric admits a closed homothety

under which the potential transforms according to (2.8).

3. The Supersymmetric Case

In the following we supersymmetrize the bosonic sigma model by extending the boson to the supermultiplet with . A number of other multiplets exist [13] which will not be considered in the following. Furthermore we will set the potential . An operator approach to a similar system can be found in [16].

3.1. Poincaré supersymmetry and superconformal symmetry

Let us supersymmetrize the bosonic sigma-model (2.1) for with fermions where is a tangent space index. These obey the standard anticommutation relations

and of course commute with and . It is convenient to make the field redefinitions

where is related to
the usual spin
connection by
.^{†}^{†} We note that where is constructed
from the connection ; and , where is the
Christoffel connection. The Hilbert space can be viewed as a
spinor (as is seen by identifying equation (3.1) with the
-matrix algebra) and as the covariant derivative (with
torsion )
on Hilbert space states.

A supercharge can then be constructed as^{†}^{†} Despite the
non-hermiticity of , this expression is
hermitian with the indicated operator ordering.

where is a 3-form, which at this point is arbitrary. A derivation of the supercharge from a supersymmetric Lagrangian is given in appendix A. The supercharge obeys

where the bosonic part of agrees with (2.1) for .

We wish to extend this Poincaré-superalgebra to the superconformal algebra whose non-vanishing commutation relations are

As before, the bosonic subalgebra requires a closed homothety. The new supercharge can then be constructed as

with given by (2.13). The anticommutator is then used to find

Then, is satisfied, but the commutator is

Agreement with (3.5) then requires to be orthogonal to :

Given (3.9), the full commutator becomes

We therefore demand that transform under dilations as

The remaining commutators in (3.5) then follow from the Jacobi identities, with no further constraints on the geometry.

In summary any conformal quantum mechanics can be promoted to , but the torsion appearing in the supercharges must obey

3.2. Poincaré supersymmetry and superconformal symmetry

supersymmetry requires a complex structure and a hermitian metric on the target space [13]. The relevant formulae are simplest in complex coordinates. However complex coordinates are less useful in the extension to the case (which has an triplet of complex structures) considered in the next subsection. Accordingly we continue with real coordinates, but give the complex version in appendix B.

The second supercharge is given by

A derivation is given in appendix B. Whereas is unconstrained for , for the vanishing of requires [13]

where the torsion connection involves the Christoffel connection plus the torsion as In complex coordinates (3.14) can be solved for the part of as

with

The part of must be closed under but is otherwise unconstrained, and the and parts are obtained by complex conjugation.

We wish to promote the algebra to . This involves an additional bosonic generator which is the generator of the symmetry group of the subalgebra. The non-vanishing commutation relations are given by (3.5), an identical set of relations with both and replaced by and , together with

As before closure of the algebra requires that the geometry must admit a closed homothety, as well as the constraints (3.12) on . Commutation of the supercharges with leads to the superconformal charges

Obtaining the correct commutator requires that the action of preserves the complex structure:

This is equivalent to the statement that acts holomorphically. Alternate forms of (3.19) are

It follows from (3.20), together with (3.9) and (3.15) that generates a holomorphic isometry

as expected from . Moreover the part of the torsion is annihilated by while the part has weight .

is determined from the commutator of and as

where we used equation (3.20). One finds

In complex coordinates and dilational gauge , when such coordinates exist, this reduces to

Notice that commutes with in complex coordinates with .

All the remaining commutators (3.17) and (3.5) are satisfied without any additional constraints.

In summary, there is an symmetry if and only if, in addition to the constraints (2.19) and (3.12), and the constraints, preserves the complex structure:

It further follows that generates a holomorphic isometry.

3.3. Poincaré supersymmetry and superconformal symmetry

Remarks on Poincaré supersymmetry

Extending the algebra to include 4 supersymmetries requires 3 complex structures , . With each one can associate a generalized exterior derivative

where the connection appearing in is^{†}^{†} defined in this way gives a connection acting on
forms as described but not on general tensors.

One of the conditions for supersymmetry found in [13,17] can be expressed

These are the vanishings of the Nijenhuis tensors
and concomitants.^{†}^{†} So, Theorem 3.9 of [18] implies that the vanishing of any two
of these equations
yields the vanishing of all six.
In complex coordinates adapted to ,
vanishes and .
Equation (3.28) further implies

Additional requirements for supersymmetry discussed in [13,17] are

In this last equation, we used the covariant derivative with torsion defined just below equation (3.14).

The commutators of are related to the -symmetry group.
We shall consider the case^{†}^{†} We have employed an obvious
summation convention in this equation. We hope that it will be clear from
the context when repeated indices should or should not
be summed over.

This case is sometimes referred to as supersymmetry, and arises in the reduction of supersymmetry from two dimensions.

We now show, defining the two-forms

that the necessary and sufficient conditions for supersymmetry can be recast in the simpler form

Note that the last two conditions (3.32) and (3.33) which involve the torsion have been replaced by the condition (3.39) which is independent of . Let us write the torsion appearing in (3.33) as

for some three-form . It can be checked that the torsion connection with set to zero is the unique such connection annihilating , and therefore has holonomy contained in . It follows that, in complex coordinates adapted to , the condition (3.33) for reduces to

(This is the argument that led to equation (3.15).) On the other hand, adding the plus or minus times the component of (3.33) yields

We conclude that and . By symmetry we must also have and , from which (3.39) follows. Conversely given (3.39), adding the torsion to the Christoffel connection implies (3.33). It can be further checked that this choice of satisfies (3.32).

This single choice of torsion connection annihilates all three complex structures

In fact the condition (3.43) is equivalent to (3.39). It differs from (3.33) by the absence of symmetrization but is nevertheless equivalent for . Equation (3.43) is referred to in [17] as the weak HKT (hyperkähler with torsion) condition. We have shown that (which includes the condition (3.38)) implies weak HKT.

Extension to superconformal symmetry

We now turn to superconformal symmetry. It turns out that the relevant supergroup is , where the parameter will be determined by the geometry. In order to write down the commutators, it is convenient to define the four-component supercharges and for ; these transform in the of the -symmetry group of . Operators , , , , and (to be described) then comprise the algebra. The non-vanishing commutators are

The matrices defined by

obey

Notice that when or , one of the two s can be decoupled, and there is an subalgebra.

Since has three and one subalgebra, the (previously discussed) conditions on the geometry for the existence of those subgroups can all be assumed. In particular, must now be holomorphic with respect to all three complex structures

Expressions
for and are then of the forms (3.13) and
(3.18) with replaced by . Somewhat lengthy expressions for
as a
function of then follow from linear combinations of
anticommutators as determined by (3.44).^{†}^{†} In principle, we should treat or as special
cases. In fact, cannot be realized with the supermultiplet
we are considering.
For , the logic is slightly different but
the results are the same.
Obtaining
properly normalized algebras for the operators
so determined requires

with and

Equation (3.48) can be
taken as the definition of the constant .^{†}^{†} Note that the two excluded values and
, correspond respectively to and , for which
the algebra (3.48) is clearly singular.
Since the
normalization of is fixed in terms of , is a
coordinate-invariant parameter associated to the geometry.

Reproducing the proper commutators leads to the stronger requirement

In fact (3.50) (including ) implies both (3.48) and (3.47). Using (3.50) one then finds are given by

The torsion can be eliminated from (3.51) using the identity .

Using the Jacobi identity, the remaining commutators follow with no further constraints on the geometry.

We note that equations (3.50) and (3.43) imply

Properties of and then imply equation (3.39), which thus needs not be taken as a further condition.

We also find, in quaternionic coordinates and dilational gauge, when such coordinates exist, that

In summary, a quantum mechanical theory has supersymmetry if and only if the complex structure and metric obey equations (3.36)–(3.39). The torsion is then uniquely determined as

A symmetry arises if and only if in addition there is a vector field obeying

where and is a constant characterizing the geometry. The parameter in the superconformal algebra is related to the constant in (3.56) by

3.4. Examples of Quantum Mechanics

In this subsection, we show that a large class of examples of quantum mechanical systems with symmetry (and an integrable quaternionic structure) can be constructed from a potential . In an superspace formalism (not described here, but similar to the ones in [19,20]) turns out to be the superspace integrand.

has an obvious triplet of complex structures associated to self-dual two-forms obeying (3.38). Let be the generalizations to . We may then define a triplet of fundamental two-forms by

It follows immediately from this definition and that the obey (3.39). Moreover the associated metric can be written (in a coordinate system in which the are constant)

This expression is
manifestly hermitian. In other words for any we can construct
an quantum mechanics.^{†}^{†} Although one may wish in
addition to impose positivity of the metric , which further
constrains . It is natural to ask whether or not every weak
HKT geometry is described by some potential . This is related
to the integrability of the quaternionic structure, as discussed
in appendix C.

The full symmetry follows by imposing

where is an arbitrary constant and

The first condition implies that is a homogeneous function of degree on , while the second states that it is invariant under -symmetry rotations. These conditions manifestly ensure the existence of the required homothety

as well as the isometries. Remarkably, it follows from (3.60) and (3.61) with a little algebra that is automatically a closed homothety,

As discussed in section 2 this implies the existence of special conformal transformations generated by

In fact, all the requirements of (3.56) are automatically satisfied with these conditions, and so indeed the full algebra is obtained.

The conditions (3.60) and (3.61) are sufficient but not necessary to insure invariance. More generally one could add to the right hand side anything which is in the kernel of the second-order differential operator in (3.58). This is especially relevant for the interesting case , for which equations (3.63) and (3.64) show that the metric is otherwise degenerate. An example of this will appear in [15].

The simplest case is

where . This has . The metric is then simply the flat metric on

while the torsion vanishes. The generators of are then