# BRST, Generalized Maurer-Cartan Equations and CFT

###### Abstract

The paper is devoted to the study of BRST charge in perturbed two dimensional conformal field theory. The main goal is to write the operator equation expressing the conservation law of BRST charge in perturbed theory in terms of purely algebraic operations on the corresponding operator algebra, which are defined via the OPE. The corresponding equations are constructed and their symmetries are studied up to the second order in formal coupling constant. It appears that the obtained equations can be interpreted as generalized Maurer-Cartan ones. We study two concrete examples in detail: the bosonic nonlinear sigma model and perturbed first order theory. In particular, we show that the Einstein equations, which are the conformal invariance conditions for both these perturbed theories, expanded up to the second order, can be rewritten in such generalized Maurer-Cartan form.

###### keywords:

string theory, conformal field theory, sigma model###### Pacs:

11.25.Hf, 11.25.Wx, 02.40.Ky, 02.40.Tt, 04.20.Jb^{†}

^{†}journal: Nuclear Physics B

## 1 Introduction

The study of perturbed Conformal Field Theories and String Theory in nontrivial backgrounds is a very complicated task to which many papers were addressed during more than twenty years. One of the most important questions, which emerges in the study of nontrivial backgrounds for String Theory, is: whether this background (e.g. nontrivial metric, dilaton, etc) gives a conformal invariant quantum field theory. The answer to this question was given in 80-s by means of calculation of the -function of the associated nonlinear sigma-model. For the bosonic sigma models with the action

(1) |

considered on a flat worldsheet, the conditions of conformal invariance at zero order in appeared to be the Einstein equations (see e.g. [1]-[6]):

(2) | |||

(3) |

where is a dilaton field. Usually these equations are accompanied with another “dilatonic” equation

(4) |

which is the consequence of (2), such that constant in the r.h.s. of (4) is identified
with central charge of the corresponding CFT [6].
However, this approach gave only superficial
relations with traditional operator approach of String theory.
The question is how these conditions of conformal
invariance and their (target-space) symmetries could be reformulated in this natural operator approach.

In this paper, we try to make first steps in this direction.
As a key tool we consider the accurately defined
deformed BRST operator in the perturbed theory. For both of our examples,
the first order [10]-[14] and the usual second order sigma models,
the equation of BRST charge conservation,
up to the second order in the (formal) coupling constant , leads to the Einstein equations expanded
up to the second order in this formal parameter.
We also show that these operator equations can be interpreted as generalized Maurer-Cartan ones, and
find the operator formulation of associated symmetries.

The outline of the paper is as follows.
In section 2, motivated by the simple analogy with Quantum Mechanics
we postulate the equation expressing the (regularized) conservation law of current in the background of the perturbation
2-form expanded with respect to the formal coupling constant :

(5) |

where is a small circle of radius around and
is a deformation of a 1-form current , such that
is conserved in the non perturbed theory; the charge in the equations
above is a conserved charge associated in the usual way with current .

Since we have an -dependent operation
in the second equation from (1) it is reasonable to make and also -dependent
(the renormalization).

Then we reformulate these two equations in the Maurer-Cartan-like form:

(6) |

where , , is some Grassmann number anticommuting with , and is equal to zero when it is applied to the two form . We also discuss the symmetries of these equations, which are of the form:

(7) |

where and is some bilinear operation properly defined.

In section 3, we specify the type of current/charge, that we deform, and the type of the perturbation to
get more close to
the sigma models (see sections 4,5). The deforming charge we take is a BRST operator,
which can be constructed for any CFT and we put some conditions on the OPEs of perturbing operator.
We choose some ansatz for , motivated by the properties of the BRST operator and
dimensional reasons.
Deriving the concrete equations
relating various operator products, we find some ambiguity in the solutions of (1).
The 1-form is defined up to the total derivative (since it enters the equations
with either a total derivative or with integration over the closed contour) and it is usual for the current,
but studying the first equation from (1) we find that
the solution for is defined up to the closed 1-form, which together with constraints
given by our ansatz has the following explicit form:

(8) |

where , are correspondingly holomorphic and antiholomorphic operators, are usual ghost fields entering the BRST operator. This can lead to the ambiguity in the second equation of (1) since it depends on . However, there is a way to avoid the appearance of such -terms. We put the following constraint

(9) |

for some of ghost number 2. This will automatically lead to the absence of the terms (8). Then, motivated by the properties of BRST operator, one may reexpress the equations (1) in terms of -operators:

(10) |

where are the modes of -ghost fields entering the BRST-operator and
is some operator of ghost number 2 such that
=.

In the following, we will refer to the system (1) as Master Equation since we suppose that
this system can be unified into one equation:

(11) |

where dots mean the terms of higher order in and its descendants
and , such that .
In the next section, we apply the obtained results to the concrete examples.

Section 4 is devoted to study of Master Equation in the case of first order field theory
with the action

(12) |

perturbed by the 2-form

(13) |

where ( is even); () are (1,0) ((0,1)) primary fields and ,
are also primary of zero conformal weights.
This theory (with and without the perturbation above) has many interesting properties and applications (see e.g.
[9]-[14]).

We find that in this case the Master Equation leads to the equations (2) and (3),
where metric and B-field are expressed as follows:

(14) |

expanded up to the second order in the formal parameter.
This is what we expected, since after (functional) integration over -variables
we arrive to the usual sigma model with metric and the B-field given as above.
In this case Einstein equations are quadratic in the tensor field, so our
second order approximation is exact.
In the end of this section we study the unexpected (target-space) algebraic structure of these equations.

In section 5, the usual sigma model with the action

(15) |

where ( is a dimension of the target space), is discussed. Expanding we find that the perturbation operators naively have the following expressions:

(16) | |||

However, analyzing the symmetries and the Master Equation itself, we find that the proper expression for should include the bivertex operator:

(17) | |||

The Master Equation applied to the leads to the equation (2) (with )
expanded up to the second order in .

Section 6 outlines the possible way of construction of the complete Master equation and other directions of
further study.

## 2 Basic Equations and their Symmetries.

### 2.1 Motivation: Deformed currents and charges

Let’s consider the charge in the euclidean quantum mechanics with the hamiltonian . If it is conserved, it commutes with the hamiltonian: . Let’s suppose there is a perturbation , where is expanded with respect to some formal coupling constant : . The charge usually does not commute with and therefore it is not conserved in the perturbed theory. However, let’s try to deform the charge () in such a way that is conserved, i.e. . This leads to the following relation between and at the second order in :

(18) |

where we have included time dependence with respect to the hamiltonian (recall that in euclidean Quantum Mechanics, the evolution is given as follows ). However, to be well defined, the commutator between , should be regularized. One of possible ways to do this is to include a compact operator of type between operators in the commutator, i.e. it is reasonable to consider the following pair of equations with shifted time variables inside the commutator:

(19) |

Now let’s consider the 2d CFT on the complex plane and a conserved current . This means that the following relation holds under the correlator

(20) |

for any closed contour , with the condition that no other operators are inserted inside . The commutator of the associated conserved charge with any operator can be defined in the following way:

(21) |

where is a circle contour of radius around ( is supposed to be small enough such that no other operators are inserted inside ). Suppose we have perturbed our CFT by a 2-form such that it is expanded with respect to the formal coupling constant . Generically the current is no longer conserved in such a background. So, we need to build something similar to the relations (2.1) in the field theory case. We propose the following equations providing the conservation of the deformed current () in the background of a perturbation up to the second order in :

(22) | |||

(23) |

where is some finite regularization parameter ^{1}^{1}1The equations (22)-(23) also can be
substantiated by consideration of the deformed current in the background of the perturbation
series regularized by the point-splitting with splitting parameter . This problem will be studied
elsewhere..

### 2.2 Hidden Maurer-Cartan Structures

Before we proceed to the study the equations (22) and (23), we need to define the
structure of , properly. We make two assumptions, motivated by our basic examples.

Assumption 1. Any two operators are supposed to have the OPE of such a type:

(24) |

for some ,
where and are some formal parameters, ; if
the operators don’t depend on on and , then
polynomially depend on and don’t depend on .

Assumption 2. The regularization parameter is supposed to be small enough in order to the operation in
(23) could be calculated via the OPE. We consider as an -independent operator
and we assume that and
can depend on and in the following way:

and
for some finite .

The reason for -independence of is quite obvious,
since there is no need in renormalization at the first order in the coupling constant, while at the
second order we have -dependent operation and there inclusion of -terms is necessary.

After these preparations we switch to the algebraic analysis of the equations (22) and (23).
Let’s define the differential

(25) |

where is a charge associated with some conserved current , is a de Rham differential, and is a Grassmann variable anticommuting with d. Then the equation (22) has the following form:

(26) |

where .

To describe (23) in a similar way, we introduce an operation on and
as follows:

(27) |

where integral over is equal to zero, acting on 2-forms and . In other words, we have:

(28) |

Then the equation (23) allows the following representation:

(29) |

where .

Thus the generalized equation describing the conservation of current should be of the following form:

(30) |

where and dots mean the higher operations.

### 2.3 Symmetries

The equation (26) has the obvious symmetry:

(31) |

where is infinitesimal. In components, this looks as follows:

(32) |

The transformations

(33) |

where is infinitesimal and is some bilinear operation on and , are the symmetries of equations (29) iff

(34) |

One can easily show that is always closed with respect to ;
we need now to find such
that is exact. To show that such exists we
need the following Proposition.

Proposition 2.1. The expressions

(35) | |||

(36) |

can be represented in the following form:

(37) |

for some operators and , built from and their derivatives
(here

The proof can be easily obtained using the Assumption 1 and comparing the coefficients
).
and for (35), and the coefficients
and for (36).

Proposition 2.2.
The expression

(38) |

is always exact with respect to the de Rham differential for any bosonic operator valued 1-forms and .

Proof. Let’s denote and
. Then, showing that

(39) |

reduces to sum for some operators and , we prove the Proposition 2.2.. Let’s consider the first line in (39). Recalling that the action of and is equivalent to the action of Virasoro generators and correspondingly, the first term of (39) can be rewritten as follows:

(40) |

We can see that the first line in the formula above is represented in the needed form, while the second and the third lines can be reexpressed:

(41) |

using the fact that the integral of the total derivative vanishes. Let’s compare this with the second line in (39):

(42) |

In order to see that the sum of (2.3) and (2.3) is equal to the sum for some
, one needs to use Proposition 2.1.

Let’s consider the following type of transformation of :

(43) |

and :

(44) |

where can be obtained from the following relation:

(45) |

and are some bilinear operations on operator-valued 1-forms and 2-forms. It is easy to see that equation (23) is invariant under such type of a transformation and thus the expression for is:

(46) |

However, the concrete choice of the bilinear operations , and (it is defined up to some closed 1-form) is unclear in the general situation. So, here we have only the class of transformations under which (26) and (29) are invariant; in order to pick out a symmetry, which should be a part of a symmetry of more general equation (30), one should find the expressions for higher operations.

## 3 Deformation of BRST operator

### 3.1 OPE and Operator Equations

Let’s consider the following expressions for the components of the perturbation 2-forms from equations (22) and (23):

(47) |

By we mean the possible -dependent terms from . We say that the pair and is of type (1,1) if the following relations take place:

(48) | |||

(49) | |||

(50) | |||

(51) |

where are Virasoro operators.

Proposition 3.1. The coefficients in the OPE (3.1) are not independent and the following relations
take place:

(52) | |||

To prove this one should expand the OPE around and and compare the coefficients.

Now let’s recall the definition of the BRST operator in CFT [7]:

(53) | |||

where and and are the primary operators with conformal weights (2,0) and (-1,0)
((0,2) and (0,-1)) correspondingly with the operator
product ().

The BRST current obeys the following relation:

(54) |

where and .

This motivates us to formulate the constraints we need to put on a deformation of BRST current.

We say that the operator-valued 1-form is a deformation of the
BRST operator if:

1). is of ghost number 1 and of the first order in and and their derivatives.

2). obeys the projection relation:

(55) |

These constraints allow to write a (1,1) ansatz for and corresponding to the (1,1) type of the perturbation:

(56) |

and

(57) |

such that the OPEs between “dilatonic” terms , , and have the following form:

(58) | |||