UAHEP 968

August 1996

Bosonization and Current Algebra of Spinning Strings

A. Stern

Dept. of Physics and Astronomy, Univ. of Alabama, Tuscaloosa, Al 35487, U.S.A.

ABSTRACT

We write down a general geometric action principle for spinning strings in -dimensional Minkowski space, which is formulated without the use of Grassmann coordinates. Instead, it is constructed in terms of the pull-back of a left invariant Maurer-Cartan form on the -dimensional Poincaré group to the world sheet. The system contains some interesting special cases. Among them are the Nambu string (as well as, null and tachyionic strings) where the spin vanishes, and also the case of a string with a spin current - but no momentum current. We find the general form for the Virasoro generators, and show that they are first class constraints in the Hamiltonian formulation of the theory. The current algebra associated with the momentum and angular momentum densities are shown, in general, to contain rather complicated anomaly terms which obstruct quantization. As expected, the anomalies vanish when one specializes to the case of the Nambu string, and there one simply recovers the algebra associated with the Poincaré loop group. We speculate that there exist other cases where the anomalies vanish, and that these cases give the bosonization of the known pseudoclassical formulations of spinning strings.

## 1 Introduction

The classical spin of relativistic particles can be described using either classical or pseudoclasssical variables.[1] The same result also holds for the classical spin of relativistic strings. Of course, the pseudoclassical description of spinning strings is well known. Descriptions of spinning strings in terms of classical variables were examined recently in ref. [2]. There we looked at the case of space-time only, and the appropriate classical variables took values in the Poincaré group . The string action was constructed in terms of the pull-back of a left invariant Maurer-Cartan form on to the world sheet. Although, it has a particularly elegant form in dimensions due to the existence of a nondegenerate scalar product on the Poincaré algebra [3], the action can be generalized to the case of an arbitrary number of space-time dimensions, as well as to the case of membranes and p-branes. In this article, we shall examine such generalizations.

As well as generalizing the system of ref. [2] to higher
dimensions, the approach described here generalizes that
developed by Balachandran, Lizzi and Sparano[4],
which in turn gives a unifying description
of the Nambu, null and tachyonic strings.
We are able to recover the dynamics of ref. [4] when we
specialize to the case of spinless strings.
^{1}^{1}1Spinning strings were
also considered in [4] using a Wess-Zumino term. Here we
shall show that there are more possibilities for introducing spin.
Our system also contains the case of pure spin,
where there is a
nonvanishing spin current, but no momentum current.
The existence of different cases is due to the different
choices available for the various constants present in the action.
These constants are the analogues of the mass and spin for the
relativistic particle.

We write down the spinning string action in Sec 2. Our criteria is that it be invariant under Poincaré transformations, as well as, under diffeomorphisms. The result is a straightforward generalization of the spinning particle action described in [1]. The resulting classical dynamics is obtained in Sec. 3 for four separate cases. These cases include the spinless string of ref. [4], the case of pure spin, as well as the most general spinning string.

As is usual, the Hamiltonian description of the string system contains constraints. We proceed with the constrained Hamiltonian dynamics in Sec. 4. There we write down the general expression for the diffeomorphism generators on a fixed time slice of the string world sheet, and we show that it satisfies the Virasoro algebra. We then compute the current algebra for the momentum and angular momentum densities for the four cases mentioned above. For the case of the spinless string corresponding to ref. [4], we recover the algebra associated with the Poincaré loop group. On the other hand, for the case of pure spin, we get an extension of the Lorentz loop group algebra. The extension consists of complicated anomalous terms which are obstructions to the quantization. Similar results are obtained for the most general spinning string. There we, instead, get an extension of the Poincaré loop group algebra.

If we demand that the above mentioned anomalous terms vanish for quantization, we arrive at a set of equations for the constants defining the system. These equations are quite complicated and we have not yet found their solutions. We speculate that solutions exist and they correspond to the bosonic formulation of known pseudoclassical descriptions of spinning strings. We note that the vanishing of the anomalous terms of the classical Poincaré loop group algebra is a necessary but not sufficient condition for a consistent quantization, as new anomalies are expected to arise at the quantum level. This indeed is known to be the case for spinless strings. We also note that a BRST approach to the quantization of this system does not appear to be straightforward due to the appearance of second class as well as first class constraints in the Hamiltonian formalism.

It is straightforward to generalize our action for spinning strings to higher dimensional spinning objects, like membranes. We show how to do this in Sec. 5.

## 2 Classical Actions

Before we write down the general expression for the spinning string action, we give a discussion of our mathematical conventions and a brief review of the classical description for the relativistic spinning particle.

### 2.1 Mathematical Conventions

As stated above, the target space for the spinning string will be taken to be the Poincaré group. We denote an element of the Poincaré group in space-time dimensions by , where is a Lorentz matrix and a Lorentz vector, . will also serve to denote the Minkowski coordinates of the string. We will only consider closed strings so and are functions on , being associated with the time.

Under the left action of the Poincaré group by , transforms according to the semidirect product rule:

(1) |

Similarly, under the right action of the Poincaré group by , we have that

(2) |

Let and denote a basis for the corresponding Poincaré algebra. For their commutation relations we have

(3) | |||||

(4) | |||||

(5) |

where diag is the Minkowski metric.

A left invariant Maurer-Cartan form associated with the Poincaré group can be written as follows:

(6) |

where the components and are one forms given by

(7) |

It is easy to check that is invariant under left transformations (1). Under right transformations (2), the Maurer-Cartan form transforms as follows:

(8) |

where

(9) | |||||

(10) |

The transformation from and to and by defines the adjoint action of the Poincaré group on the basis vectors.

We can now construct geometric actions which are invariant under Poincaré transformations. For this, we let the action depend only on the components and of . It will then automatically be invariant under (left) Poincaré transformations (1). With this prescription, we will arrive at a general description for relativistic objects with spin. To illustrate this we first review the case of relativistic particles.[1]

### 2.2 Spinning Particle Action

The particle action is constructed in terms of the pull-back of to the world line. The most general geometric particle action which we can write in this way is

(11) |

where

(12) | |||||

(13) |

and and are constants, the latter being antisymmetric in the indices and . In space-time dimensions there are then a total of constants. These constants determine the particle dynamics. Actually for that purpose, we only need to specify certain ‘orbits’ of and . These orbits are induced by the action of the Poincaré group. We define this action by the following set of transformations :

(14) | |||||

(15) |

where

(16) | |||||

(17) |

and thus transform under the Poincaré group like momentum and angular momentum. The orbits can be classified by their invariants which are the usual ones for the Poincaré algebra. In four space-time dimensions they are and , where .

Using (15) it can be shown that

(18) |

Thus, replacing and in the action by and is equivalent to transforming the variables and by the right action (2) of the Poincaré group. The equations of motion obtained by varying and in (18) are identical to those resulting from variations of and in (11). Therefore (15) define maps between equivalent dynamical systems.

The term in the action (11) by itself describes a spinless particle, while (which is a Wess-Zumino term for this system) gives rise to spin. This is easily seen from the equations of motion, which we can obtain by extremizing the action with respect to Poincaré transformations. The equations of motion state that there are two sets of constants of the motion. From infinitesimal translations, and , we get the constants of motion associated with the momentum,

(19) |

From infinitesimal Lorentz transformations, , , for infinitesimal , we get constants of motion associated with angular momentum,

(20) |

then gives the spin contribution of the particle to the total angular momentum.

As usual, and are not all independent. For example, for a massive spinning particle they are subject to two conditions:

(21) |

where is the Pauli-Lubanski vector . The quantum analogues of and generate the Poincaré algebra in the quantum theory. Their representations must be irreducible, the particular representation being determined by the orbit of via conditions like (21) on the quantum operators.

### 2.3 Spinning String Action

We now adapt a similar approach to the description of spinning strings.

A geometric action for strings can be expressed as a wedge products of the one forms and defined in eq. (7). There are thus three possible terms:

(22) |

where

(23) | |||||

(24) | |||||

(25) |

denotes a set of constants and they are the analogues of the constants and appearing in the particle action. They satisfy the following symmetry properties:

(26) | |||||

(27) | |||||

(28) |

In space-time dimensions there are then a total of constants .

The constants determine the string dynamics. In analogy with the particle description, we only need to specify the ‘orbits’ on which lie. These orbits are again induced by the action of the Poincaré group. We define this action by the following set of transformations :

(30) | |||||

(32) | |||||

(34) | |||||

where

(35) | |||||

(36) | |||||

(37) |

Using these definitions it can be shown that

(38) |

in analogy to (18). Thus, replacing in the action by is equivalent to transforming the variables and by the right action (2) of the Poincaré group. The equations of motion obtained by varying and in (38) are identical to those resulting from variations of and in (22). Therefore (34) define maps between equivalent dynamical systems.

The orbits induced by the action (34) of the Poincaré group on can be labeled by their invariants. One simple quadratic invariant is of course

In general, the expression for the invariant depends on the number of space-time dimensions. For the example of three space-time dimensions, we found the following additional quadratic invariant: [2]

In four space-time dimensions, one instead has the quadratic invariant:[4]

## 3 Equations of Motion

We next examine the classical string dynamics following from the actions i) , ii) , iii) and iv) .

i) The action , which can be expressed by

(39) |

was discussed in ref. [4] and for certain orbits of is known to be equivalent to the Nambu action. Here we define The standard form of the Nambu action is obtained upon eliminating (which in this case play the role of auxiliary variables) from . For other orbits of , the action can yield either tachyonic or null strings[5].

For all choices of the action alone describes a spinless string. This will be evident from the form of the conserved momenta and angular momenta. These conserved currents are found by extremizing the action with respect to Poincaré transformations. From infinitesimal translations and , we get the equations of motion corresponding to momentum current conservation,

(40) |

where denote world sheet indices and . From infinitesimal Lorentz transformations, , , for infinitesimal , we get angular momentum current conservation,

(41) |

Consequently, the angular momentum current consists of only an orbital term, and therefore the string defined by the action is spinless.

ii) A spin contribution to the angular momentum current is present for strings described by the action , which can also be written as

(42) |

where . Now from infinitesimal translations and , we get the equations of motion

(43) |

From infinitesimal Lorentz transformations, , , we get that

(44) |

denotes the spin contribution to the angular momentum current. It is given by

(45) |

iii) We next consider the case of strings described by the action , which can be written

(46) |

where Now the angular momentum current consists solely of a spin term. This is evident because the momentum current vanishes, , and hence so does the orbital angular momentum. Here the spin current has the following form:

(47) |

which is seen by extremizing with respect to variations . In space-time dimensions, was seen to be the integral of an exact two form, and as a result there was identically conserved.[2]

iv) For the most general action (22) consisting of all three terms and the conserved momentum and angular momentum currents are given by the sum of the individual currents:

(48) | |||||

(49) |

The angular momentum current can be expressed as a sum of an orbital part and a spin current. The latter is given by

(50) |

We note that the tensors , and which appear in the string equations of motion satisfy the same symmetry properties as the constant tensors , and , i.e., we can replace by in (28).

## 4 Constraint Analysis and Current Algebra

We next proceed with the Hamiltonian formulation of the system. Constraints are present in the Hamiltonian description since all terms in the action (22) are first order in world sheet time (-) derivatives. Furthermore, the constraints can be either first or second class. As a result of this, the analysis of the current algebra is quite involved.

Below we shall handle the cases i)-iv) separately. Before doing so, however, we shall guess the expression for the diffeomorphism generators on a fixed time slice of the string world sheet, and show that it satisfies the Virasoro algebra.

We first introduce the momentum variables and , which along with and span the phase space. The variables are defined to be canonically conjugate to , while generate left transformations on . This can be expressed in terms of the following equal (world sheet) time Poisson brackets:

(51) | |||||

(52) | |||||

(53) |

where we write the phase space variables on a world sheet time slice, the phase space variable being periodic functions of the spatial coordinate of the world sheet. All other Poisson brackets between the phase space variables are zero. Below we shall utilize the following matrix representation for :

(54) |

As mentioned above, the constraints can be classified as both first and second class. In this regard, we know from the reparametization symmetry of the action, that there exist some combinations of the constraints which are first class. Those which generate diffeomorphisms along a surface should satisfy the Virasoro algebra. It is easy to construct such generators. They are:

(55) |

where are periodic functions on a fixed slice. From (53), it can be verified that indeed do satisfy the Virasoro algebra,

(56) |

In the subsections that follow, we shall show that are first class constraints. For this we shall need the specific form for the action in order to obtain the explicit expressions for the constraints, which look different in the four different cases mentioned earlier. With this in mind we now specialize to the cases i)-iv).

### 4.1 Case i)

As stated before, case i) describes a spinless string. The constraints on the momentum variables are:

(57) | |||||

(58) |

From the first constraint and (40), is thus identified with the time component of the momentum current . In addition to , we can define the variables

(59) |

which are weakly equal (i.e., up to a linear combination of constraints) to the time component of the angular momentum current .

To compute their Poisson brackets, we find it convenient to write the constraints as distributions:

(60) | |||||

(61) |

where and are periodic functions of the spatial coordinate . From the Poisson brackets (53), we obtain the following algebra of the constraints:

(62) | |||||

(63) | |||||

(64) |

Note that the last two relations only hold weakly.

From the reparametization symmetry of the action, we know that there exist linear combinations of and which are first class constraints. Let

(65) |

be a general first class constraint, where the Lagrange multipliers and denote the functions and , respectively. They are solutions to the following equations:

(66) | |||||

(67) |

which follow from demanding that is first class. Since these equations are linear in and , it follows that if and are solutions to (67), then and are also solutions to (67), where is an arbitrary function on the world sheet. The generators of diffeomorphisms of the world sheet then have the general form: .

One solution to equations (67) is and . The resulting first class constraint (corresponding to ),

(68) |

generates translations in :

(69) | |||||

(70) |

When we get the generators of diffeomorphisms

(71) |

on a fixed slice of the world sheet. were given in (55) and satisfy the Virasoro algebra (56). We have therefore shown that the Virasoro generators are first class constraints.

For the case where is the generator of translations, we need that is a time-like vector. By computing the Hamilton equations of motion for and , we can recover the current conservation law for momentum and angular momentum. This follows because the Poisson brackets of and with the constraints are spatial derivatives,

(72) | |||||

(73) | |||||

(74) | |||||

(75) |

The Hamilton equations are then:

(76) | |||||

(77) |

Here we get the identification of with the space component of the momentum current defined in (40). These equations once again show that the case i) string is spinless.

The generators of the Poincaré symmetry are the charges

(78) |

They have zero Poisson brackets with the constraints. This again follows because the Poisson brackets of and with the constraints are spatial derivatives. The charges (78) are thus first class variables as well as Dirac variables.

It remains to construct the Dirac variables associated with current densities and . For this we first define

(79) |

denote the functions . have (weakly) zero Poisson brackets with . For them to have zero Poisson brackets with , we need that the functions satisfy:

(80) |

Similarly, we can define the variables

(81) |

denote the functions . Like , they have (weakly) zero Poisson brackets with . For them to have zero Poisson brackets with , we need that satisfy:

(82) |

and are then Dirac variables. Since they have zero Poisson brackets with first, as well as second class constraints, they are gauge invariant.

We next compute the Poisson bracket (or current) algebra for the momenta and angular momenta. For this purpose it turns out, in this case, not to be necessary to solve (82) for . The current algebra is simply the algebra of the Poincaré loop group:

(83) | |||||

(84) | |||||

(85) |

and are gauge invariant coordinates which label the reduced phase space. Whether or not they form a complete set of variables is not evident. On the other hand, we note that for spinless strings and are not independent on the reduced phase space, since from (58) and (59), are and are subject to:

(86) |

### 4.2 Case ii)

This case is of interest because it yields a nontrivial momentum and angular momentum current, the latter containing a spin contribution . Here we shall derive a current algebra which is an extension of the Poincaré loop group.

In this case the constraints on the momentum variables are:

(87) | |||||

(88) |

From the first constraint and (43), is thus identified with the time component of the momentum current , while from the second constraint and (45) is identified with the time component of the spin current . In addition to and , we can once again define the variables , as was done in (59), which are weakly equal to the angular momentum densities .

For the algebra of the constraints we now get:

(89) | |||||

(90) | |||||

(91) |

where and are once again the distributions defined in (61). As with case i), from the reparametization symmetry of the action, we know that there exist linear combinations of and which are first class constraints. First class constraints may also arise due to additional symmetries associated with some particular choices for . For as defined in (65) to be a first class constraint, we need that the Lagrange multipliers and now satisfy:

(92) | |||

(93) | |||

(94) | |||

(95) |

Since, like (67), these equations are linear in and , it again follows that if