IFT-P.005/2000

Super-Poincaré Covariant Quantization of the Superstring

Nathan Berkovits^{†}^{†} e-mail:

Instituto de Física Teórica, Universidade Estadual Paulista

Rua Pamplona 145, 01405-900, São Paulo, SP, Brasil

Using pure spinors, the superstring is covariantly quantized. For the first time, massless vertex operators are constructed and scattering amplitudes are computed in a manifestly ten-dimensional super-Poincaré covariant manner. Quantizable non-linear sigma model actions are constructed for the superstring in curved backgrounds, including the background with Ramond-Ramond flux.

January 2000

1. Introduction

There are many motivations for covariantly quantizing the superstring. As in any theory, it is desirable to make all physical symmetries manifest in order to reduce the amount of calculations and simplify any cancellations coming from the symmetry. Recently, an additional motivation has come from the desire to construct a quantizable sigma model action for the superstring in curved backgrounds with Ramond-Ramond flux.

Most attempts
to covariantly quantize the superstring have started from
the classical super-Poincaré invariant
version of the Green-Schwarz (GS) action [1].
One quantization approach is based on gauge-fixing
the fermionic symmetries to get to
“semi-light-cone” gauge where and
[2].
In this gauge, the covariant Green-Schwarz action simplifies to
However, even this simplified action cannot be easily quantized since
the propagator for involves which is not
well-defined.^{†}^{†} On a genus
worldsheet with punctures, vanishes at points
on the worldsheet.
This fact is related to the need for interaction-point operators in
the light-cone GS superstring.
For this reason, it has not
yet been possible to use this approach to construct
physical vertex operators or compute scattering
amplitudes, except in the limit that reproduces
the light-cone gauge computations [3].
Another approach to quantizing the covariant
Green-Schwarz action is based on
replacing the fermionic second-class constraints with
an appropriate set of first-class constraints
[4], sometimes using
SO(9,1)/SO(8) harmonic variables [5]
which covariantize the semi-light-cone gauge choice.
However, despite numerous attempts [6], noone was able to find an
appropriate set of first-class constraints which allows
the covariant computation of scattering amplitudes.

In the absence of Ramond states, it is possible to quantize the superstring in a manifestly Lorentz-covariant manner using the standard Ramond-Neveu-Schwarz (RNS) formalism. However, none of the spacetime supersymmetries are manifest in the RNS formalism and, in order to explicitly construct the spin field for Ramond states, manifest SO(9,1) Lorentz invariance must be broken (after Wick-rotation) to a U(5) subgroup [7]. Recently, an alternative formalism for the superstring was constructed which manifestly preserves this same U(5) subgroup in addition to manifestly preserving six of the sixteen spacetime supersymmetries [8]. The worldsheet variables of this supersymmetric U(5) formalism are related to those of the RNS formalism by a field redefinition, allowing one to prove that physical vertex operators and scattering amplitudes in the two formalisms are equivalent. However, the lack of manifest Lorentz invariance makes it difficult to use this formalism to describe the superstring in curved (Wick-rotated) backgrounds which do not preserve U(5) holonomy.

In this paper, a new formalism for the superstring will be presented which can be quantized in a manifestly super-Poincaré covariant manner. The worldsheet variables of this formalism will consist of the usual ten-dimensional superspace variables in addition to a bosonic spacetime spinor satisfying the ‘pure’ spinor condition

for to 9. must be complex to satisfy (1.1) and, after Wick-rotating SO(9,1) to SO(10), can be parameterized by eleven complex variables. One of these eleven variables is an overall scale factor, and the other ten parameterize the coset space SO(10)/U(5). So this new formalism is probably related to a covariantization of the U(5) formalism of [8]. Although the precise relation between the two formalisms is still unclear, it will be argued in section 2 that pure spinor variables are necessary for equating RNS vertex operators with the GS vertex operators proposed in [4].

In section 3, physical states will be defined as elements in the cohomology of the BRST-like operator

where is the generator of supersymmetric derivatives as defined in [4]. Since where is the supersymmetric translation generator, (1.1) implies that . Note that the operator of (1.2) was used in [9] by Howe to show that the constraints of ten-dimensional super-Yang-Mills and supergravity can be understood as integrability conditions on pure spinor lines.

Using this definition of physical states, one can easily construct the physical massless vertex operators. For the open superstring, the massless vertex operator in unintegrated form is and in integrated form is

where are the super-Yang-Mills prepotentials, and are the gauge-invariant superfields whose lowest components are the gluino and the gluon field strength, and is the pure spinor contribution to the Lorentz generator. Except for the term, the vertex operator of (1.3) is that proposed by Siegel in [4]. As will be shown in section 4, these vertex operators can be used to compute scattering amplitudes in a manifestly super-Poincaré covariant manner.

The physical vertex operators for the closed superstring can be obtained by taking the ‘left-right’ product of two open superstring vertex operators. In section 5, the integrated form of the closed superstring massless vertex operator will be used to construct a quantizable sigma model action for the superstring in a curved superspace background. As a special case, a quantizable sigma model action will be constructed for the superstring in an background with Ramond-Ramond flux. This action differs from that of Metsaev and Tseytlin [10] in containing a kinetic term for the fermions which allows quantization.

In section 6, further evidence will be given for equivalence with the RNS formalism and some possible applications of the new formalism will be discussed.

2. Pure Spinors and Lorentz Currents

In conformal gauge, the left-moving contribution to the covariant Green-Schwarz superstring action can be written as

where is related to and by the constraint with[4]

Since where , involves first and second-class constraints. The idea of [4] is to find an appropriate set of first-class constraints constructed from which can replace the second-class constraints. In such a framework, is treated as an independent field and physical vertex operators are annihilated by the first-class constraints. Although an appropriate set of first-class constraints were not found in [4], Siegel used supersymmetry arguments to conjecture that the massless open superstring vertex operator should have the form

where are the super-Yang-Mills prepotentials and is the super-Yang-Mills spinor field strength.

For a gluon, the vertex operator
of (2.2) reduces to
where and are the ordinary -independent
gluon gauge field and field strength,
which closely resembles the gluon vertex operator in the RNS formalism
. However, there is
a crucial difference between the OPE’s of the SO(9,1) Lorentz currents
and
which will
force the introduction of pure spinors. Namely, the
OPE of
with has a
double pole proportional to where the factor of 16 comes from the spinor dimension.
However, the double pole in the OPE of with
is proportional to without the factor of .
So the vertex operator of [4] can only be equivalent at the
quantum level to the RNS vertex operator if one adds a new term to the
Lorentz current where satisfies the OPE
^{†}^{†} In four dimensions,
has a double pole proportional to
, so there is no need to
add new Lorentz degrees of freedom when quantizing the four-dimensional
superstring [11].
In six dimensions, (where to 2 is an internal
SU(2) index)
has a double pole proportional to
, so one needs to add degrees of freedom
whose Lorentz current has a double pole with itself
proportional to
. These degrees of freedom are a bosonic spinor
and its conjugate momentum for to 4.
They are the ghosts for the
‘harmonic’ constraints
of [12]
whose contribution was incorrectly ignored in [12]. The
correct massless six-dimensional open superstring vertex operator
is
where is a superfield whose lowest
component is the gluon field strength. Note that this vertex operator is
annihilated on-shell by the ‘harmonic’ BRST-like operator
and the central charge contribution from
cancels the contribution from in the stress tensor
to give a vanishing conformal anomaly.

As will now be shown, such a Lorentz current can be explicitly constructed from a pure spinor , i.e. a complex bosonic spinor satisfying (1.1). To parameterize the eleven independent complex degrees of freedom of , it is convenient to Wick-rotate and temporarily break SO(10) to SU(5) U(1) as in [8]. The sixteen complex components of split into for to 5, which transform respectively as

satisfies the pure spinor condition of (1.1). Note that -matrices in U(5) notation satisfy and where the SO(10) vector has been split into a and representation.

In conformal gauge, the worldsheet action for the left-moving variables will be defined as

with the left-moving stress tensor

where are the conjugate momenta for . As desired, has no conformal anomaly since the central contribution for the new degrees of freedom is , which cancels the central charge contribution from the and variables.

In U(5) notation, the SO(10) Lorentz currents split into which transform respectively as representations. After fermionizing and as in [7], will be defined as

Using the free-field OPE’s,

one can check that

which correctly reproduces the OPE of (2.3).

Furthermore, as can
be easily shown
by noting that
where is a spinor of the opposite chirality to with
components^{†}^{†} Terms coming from normal-ordering ambiguities in
can be ignored since they only involve
and , which have no singularities with .

Note that

where the in signifies the spinor component in the SU(5) notation of (2.4). The second term in the OPE of (2.11) is necessary for to have no singularity with , however, it does not contribute to the commutator since

So after introducing pure spinors, it is possible to obtain vanishing conformal anomaly and to relate the RNS gluon vertex operator with the proposal of Siegel in [4]. It will now be shown how these pure spinors can be used to define physical vertex operators and compute scattering amplitudes in a super-Poincaré covariant manner.

3. Physical Vertex Operators

Since the stress-tensor of (2.6) has vanishing central charge, one can require that physical vertex operators in unintegrated form are primary fields of dimension zero. However, this requirement is clearly insufficient since, for a massless vertex operator depending only on the zero modes of the worldsheet fields, it implies which has far more propagating fields than super-Yang-Mills. One therefore needs a further constraint on physical vertex operators, and using the intuition of [4], this constraint should be constructed from of (2.1).

Using the pure spinor defined in terms of and as in (2.4), one can define a nilpotent BRST-like operator

Defining ghost charge to be , carries
ghost-number one. So it is natural to define physical vertex operators
as states of ghost-number 1 in the cohomology of . Note that
after Wick rotation, and are complex spinors, so
the Hilbert space of states should be restricted to analytic functions
of these variables.^{†}^{†} Although it is difficult to impose reality
conditions on the states in Euclidean space, this is not a problem
for computing scattering amplitudes since it will be trivial to Wick-rotate
the final result back to Minkowski space where the reality conditions
are easily defined.

It will now be shown for the massless sector of the open superstring that this definition of physical states reproduces the desired super-Yang-Mills spectrum. Massless vertex operators of dimension zero can only depend on the worldsheet zero modes, so the most general such vertex operator of ghost number 1 is where is an analytic function of and . Since is Lorentz invariant (after including the contribution of (2.7) in the Lorentz generators), elements in its cohomology must transform Lorentz covariantly. But because of the non-linear nature of the Lorentz transformations generated by , the only finite-dimensional covariantly transforming object which is linear in is . So if the cohomology is restricted to finite-dimensional elements, the most general massless vertex operator of ghost number 1 is

where is a generic spinor function of and .

The constraint implies that where , this implies that , which is the on-shell constraint for the spinor prepotential of super-Yang-Mills. Furthermore, the gauge transformation . Since

reproduces the usual super-Yang-Mills gauge transformation where is a generic scalar superfield. So the ghost number 1 cohomology of for the massless sector reproduces the desired super-Yang-Mills spectrum.

To compute scattering amplitudes, one also needs vertex operators in integrated form, i.e. integrals of dimension 1 primary fields. Normally, these are obtained from the unintegrated vertex operator by anti-commuting with . But in this formalism, there are no ghosts, so it is presently unclear how to relate the two types of vertex operators. Nevertheless, one can define physical integrated vertex operators as elements in the BRST cohomology of ghost-number zero.

In the massless sector, there is an obvious candidate which is the dimension 1 vertex operator of (2.2) suitably modified to include the pure spinor contribution to the Lorentz current, i.e.

where is the superfield whose lowest component is the gluon field strength. To show that , first note that[4]

where Since ,

where is the spinor field strength. But using from (2.10),

since from ten-dimensional -matrix identities. Finally, using the gluino equation of motion,

so describes a physical integrated vertex operator. Note that the super-Yang-Mills gauge transformation transforms by the total derivative , so is manifestly gauge-invariant.

4. Computation of Scattering Amplitudes

In this section, it will be shown how to compute tree-level open superstring scattering amplitudes in a manifestly super-Poincaré covariant manner. To compute -point tree-level scattering amplitudes, one needs three vertex operators in unintegrated form and vertex operators in integrated form. Since only the massless vertex operators are known explicitly, only scattering of massless states will be considered here.

The two-dimensional correlation function which needs to be evaluated for computing tree-level scattering of super-Yang-Mills multiplets is

where is the dimension 0 vertex operator of (3.2), is the dimension 1 vertex operator of (3.4), and the locations of can be chosen arbitrarily because of SL(2,R) invariance.

The functional integral over the non-zero modes of the various worldsheet fields is completely straightforward using the free-field OPE’s. For example, the dimension 1 worldsheet fields can be integrated out by contracting with other dimension 1 fields or with . Note that manifest Lorentz invariance is preserved by the contractions of because its only singular OPE’s are and (2.3).

However, the functional integral prescription for the zero modes of the worldsheet fields needs to be explained. Besides the zero modes of (which are treated in the usual manner using conservation of momentum), there are the eleven bosonic zero modes of and the sixteen fermionic zero modes of . After integrating out the non-zero modes, one gets an expression

where only the zero modes of contribute and is a function which depends on , on the momenta for to , and on the zero modes of .

The prescription for integration over the remaining worldsheet zero modes will be

where is the complex conjugate of in Euclidean space and is an integration over the different possible orientions of . Although this prescription is defined in Euclidean space, it is trivial to Wick-rotate the result back to Minkowski space using the fact that

Equation (4.4) can be derived using the fact that there is a unique covariantly transforming tensor which is symmetrized with respect to its upper and lower indices and which satisfies and

So the amplitude of (4.2) can be written in SO(9,1) Lorentz-covariant notation as

where is defined in (4.4). By expanding as a power series in , one can check that the prescription of (4.3) selects out the term

in the power series, i.e. .

This prescription for integrating out the zero modes is reasonable since it is Lorentz invariant and since the eleven bosonic zero mode integrations are expected to cancel eleven of the sixteen fermionic zero mode integrations, leaving five zero modes of which are removed with five ’s. Further evidence for this zero-mode prescription comes from the fact that it is gauge invariant and spacetime supersymmetric, as will now be shown.

To show that is invariant under a gauge transformation , note that commutes with and , so

for some after integrating out the non-zero modes. Using the zero-mode prescription of (4.5),

where conservation of momentum has been used to replace with in (4.8). But using anti-symmetry properties of , one can show that

so .

It will now be shown that the prescription of (4.5) is invariant under spacetime supersymmetry transformations, implying that the amplitudes are SO(9,1) super-Poincaré invariant. Under a spacetime supersymmetry transformation with global parameter , the term of (4.6) transforms as where appears in the power series for as

But comes from vertex operators which commute with , so it must satisfy the constraint

for any pure spinor . Plugging (4.9) into (4.10) and using , one finds that

must vanish for any pure spinor . But this is only possible if , implying that

so the amplitude prescription of (4.5) is spacetime supersymmetric.

5. Superstring Action in a Curved Background

In this section, the massless integrated vertex operator for the closed superstring will be used to construct a quantizable action for the superstring in a curved background. As a special case, a quantizable action will be constructed for the Type IIB superstring in an background with Ramond-Ramond flux.

In bosonic string theory and in the Neveu-Schwarz sector of superstring theory, the action in a curved background (ignoring the Fradkin-Tseytlin term for dilaton coupling) can be constructed by ‘covariantizing’ the massless closed string vertex operator with respect to target-space reparameterization invariance. As in [13] and [12], this procedure can also be used here after constructing the massless closed string vertex operator from the ‘left-right’ product of two massless open string vertex operators of (3.4).

To do this, one first needs to introduce right-moving analogs of the worldsheet fields described in (2.5). The complete worldsheet action for the Type II superstring in a flat background in conformal gauge is

where is constructed from and in a manner similar to (2.4). Note that and are independent of and , and are not related by complex conjugation. For the Type IIA superstring, the hatted spinor index has the opposite chirality to the unhatted spinor index while, for the Type IIB superstring, the hatted spinor index has the same chirality as the unhatted spinor index.

The action for the Type II superstring in a curved background obtained
by
‘covariantizing’ the massless closed superstring vertex operator
with respect to target-space super-reparameterization invariance is^{†}^{†} In the action for the
superstring in a curved six-dimensional background, there are terms
coming from the bosonic ghosts described in footnote 3 which were
incorrectly omitted from the action of [12].
The correct action should have terms of the type
,
as well as a kinetic action for the
and ghosts.

where parameterizes the curved superspace background, and are the spinor parts of the super-vierbein , and with being defined by (2.7) and being defined similarly in terms of the hatted variables, and is the superfield whose lowest components are the bispinor Ramond-Ramond field strengths. The operators and in (5.2) are understood to act on the superfield to their left, e.g. on , or .

Note that the first line of (5.2) is identical to the Green-Schwarz action in a curved background, however, the second and third lines are crucial for quantization since they provide an invertible propagator for and . Furthermore, since there is no fermionic -symmetry which needs to be preserved, there is no problem with adding a Fradkin-Tseytlin term to (5.2) of the type where is the worldsheet curvature and is a scalar superfield whose lowest component is the dilaton.

When the background superfields satisfy their effective low-energy equations of motion, the action of (5.2) together with the Fradkin-Tseytlin term is expected to be conformally invariant where the left-moving stress tensor is

Furthermore, the current is expected to be holomorphic and nilpotent when the background superfields are on-shell.

5.1. Superstring Action in background

In this subsection, the action of (5.2) will be explicitly constructed for the special case of the Type IIB superstring in an background with Ramond-Ramond flux. As discussed in [10], this background can be conveniently described by a coset supergroup taking values in where the super-vierbein satisfies

and ranges over the 10 bosonic and 32 fermionic entries in the Lie-algebra valued matrix . Furthermore, as discussed in [14], the only non-zero components of and are

where is the value of the Ramond-Ramond flux, is the string coupling constant, and with being the directions of .

Plugging these background superfields into the action of (5.2), one finds