Advances in High Energy Physics
Volume 2017 (2017), Article ID 6124189, 11 pages
https://doi.org/10.1155/2017/6124189

Research Article

Double Hodge Theory for a Particle on Torus

Vipul Kumar Pandey and Bhabani Prasad Mandal

Department of Physics, Banaras Hindu University, Varanasi 221005, India

Correspondence should be addressed to Vipul Kumar Pandey; vipulvaranasi@gmail.com

Received 1 July 2017; Accepted 31 October 2017; Published 27 November 2017

Academic Editor: Edward Sarkisyan-Grinbaum

Copyright © 2017 Vipul Kumar Pandey and Bhabani Prasad Mandal. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP³.

Abstract.

We investigate all possible nilpotent symmetries for a particle on torus. We explicitly construct four independent nilpotent BRST symmetries for such systems and derive the algebra between the generators of such symmetries. We show that such a system has rich mathematical properties and behaves as double Hodge theory. We further construct the finite field dependent BRST transformation for such systems by integrating the infinitesimal BRST transformation systematically. Such a finite transformation is useful in realizing the various theories with toric geometry.

1. Introduction

The formulation based on BRST symmetry [1–4] plays a crucial role in the discussion of quantization, renormalization, and unitarity and other aspects of gauge theories. The nilpotency nature of BRST transformation is mainly responsible for simplified treatment in all these discussions. Thus it is extremely important to find more and more nilpotent symmetry associated with any system to study, particularly the systems with constraints. Toric geometry which is generalization of the projective identification that defines corresponding to the most general linear sigma model provides a scheme for constructing Calabi-Yau manifolds and their mirrors [5]. Recently, on the basis of boundary string field theory [6], the brane-antibrane system was exploited [7] in the toroidal background to investigate its thermodynamic properties associated with the Hagedorn temperature [8, 9]. The Nahm transform and moduli spaces of models were also studied on the toric geometry [10]. In a four-dimensional, toroidally compactified heterotic string, the electrically charged BPS-saturated states were shown to become massless along the hyper surfaces of enhanced gauge symmetry of a two-torus moduli subspace [11].

In the present work we investigate various possible nilpotent symmetries for a particle on torus. Usual BRST symmetry for a particle on torus has already been constructed [12]. In this work we construct four different nilpotent symmetries associated with this system, namely, BRST symmetry, anti-BRST symmetry, dual BRST (also known as co-BRST) symmetry, and anti-dual BRST (also known as anti-co-BRST) symmetry [13–15]. We further construct two different bosonic symmetries using these nilpotent BRST symmetries and some discrete symmetries associated with ghost number are also written for such systems. Complete algebra satisfied by charges, which generate these symmetries, is derived. Deep mathematical connections of such system with Hodge theory [16–19] are established in this work. We found that the system of particle on a torus is realized as Hodge theory with respect to two different sets of operators. The generators for BRST, dual BRST symmetries, and generator for corresponding bosonic symmetries constructed out of BRST and dual BRST symmetries are analogous to exterior derivative, coexterior derivative, and Laplace operator in Hodge theory [20–28]. On the other hand the charges corresponding to anti-BRST symmetry, anti-dual BRST symmetry, and bosonic symmetry constructed out of these two BRST symmetries are also from set of de Rham cohomological operators. This indicates that the mathematical foundation of the theory of a particle on a torus is extremely rich.

We further extend the BRST transformation for this system by considering the BRST parameter as finite and field dependent. More than two decades ago Joglekar and Mandal introduced for the first time the concept of finite field dependent BRST (FFBRST) transformation [29], which had similar structure and properties of usual BRST transformation. However the path integral measure is not invariant due to finite nature of such transformation. It has been shown that by constructing suitable finite parameter one can calculate desirable Jacobian factor which under certain condition is added to the effective action of the theory. Thus FFBRST is capable of connecting generating functionals of two different effective theories. Because of these remarkable properties, FFBRST has become a useful tool of studying various field theoretic systems with BRST symmetry and it has found many applications [30–43]. We have constructed FFBRST transformation for the system of particle on torus to show the connection between two theories on torus with different gauge fixing. Now we present the plan of this manuscript.

We start with the brief introduction about the free particle on the surface of torus in Section 2. Hamiltonian formulation for this theory is presented in Section 3. In Section 4 the BFV formulation for this model has been discussed and BRST symmetry for such model has been constructed. In Section 5 the other nilpotent symmetry transformations for same system have been constructed. Co-BRST and anti-co-BRST have been discussed in Section 6. Other symmetries have been discussed in Section 7. The connection between algebra satisfied by the nilpotent charges and de Rham cohomological operators of differential geometry is shown in Section 8. In Section 9 we introduce FFBRST transformation and in the next section we connect theory in different gauges using FFBRST transformations. We conclude our results in Section 10.

2. Free Particle on Surface of Torus

BRST Symmetry for free particle system on toric geometry has already been studied using BFV formalism in [12]. Here we review some parts of the work in [12] relevant to our calculation in later chapters. A particle moving freely on the surface of a torus is described by Lagrangian [12]:

where are toroidal coordinates related to Cartesian coordinates as

Here we have considered a torus with axial circle in the plane centered at the origin, of radius , having a circular cross section of radius . The angle ranges from 0 to and the angle from 0 to . Since the particle moves on the surface of torus of radius , it is constrained to satisfy

The canonical Hamiltonian corresponding to the Lagrangian in (1) with the above constraint is then written as

where , , and are the canonical momenta conjugate to the coordinate , , and , respectively, given by

The time evolution of the constraint yields the secondary constraint as

3. Wess-Zumino Term and Hamiltonian Formulation

To construct a gauge invariant theory corresponding to the gauge noninvariant model in (4), we introduce the Wess-Zumino term [28] in the Lagrangian density . For this purpose we enlarge the Hilbert space of the theory by introducing a new quantum field , called Wess-Zumino field, through the redefinition of fields and in the original Lagrangian density as follows:

With this redefinition of the fields, the modified Lagrangian density becomes

Canonical momenta corresponding to this modified Lagrangian density are then given by

The primary constraint for this extended theory is

The Hamiltonian density corresponding to is written as

The total Hamiltonian density after the introduction of a Lagrange multiplier field corresponding to the primary constraint is then obtained as

Following Dirac’s method of constraint analysis [44–47], we obtain secondary constraint

In next two sections, we extend this constrained theory to study the nilpotent symmetries associated with this theory.

4. BFV Formulation for Free Particle on the Surface of Torus

To discuss all possible nilpotent symmetries we further extend the theory using BFV formalism [48–53]. In the BFV formulation associated with this system, we introduce a pair of canonically conjugate ghost fields with ghost numbers 1 and −1, respectively, for the primary constraint and another pair of ghost fields with ghost numbers −1 and 1, respectively, for the secondary constraint, . The effective action for a particle on surface of the torus in extended phase space is then written as

where is the BRST charge and is the gauge fixed fermion. This effective action is invariant under BRST transformation generated by which is constructed by using constraints in the theory as

The canonical brackets for all dynamical variables are written as

where rest of the brackets are zero. Now, the nilpotent BRST transformation, using the relation ( sign represents the fermionic and bosonic nature of the fields ), is explicitly written as

In BFV formulation the generating functional is independent of gauge fixed fermion [48–53]; hence we have liberty to choose it in the convenient form as

Putting the value of in (14) and using (15) and (16), we obtain

and the generating functional for this effective theory is represented as

Now integrating this generating functional over and , we get

where is the path integral measure for effective theory when integrations over fields and are carried out. Further integrating over field we obtain an effective generating functional as

where is the path integral measure corresponding to all the dynamical variables involved in the effective action. The BRST symmetry transformation for this effective theory is written as

These transformations are on shell nilpotent.

5. Nilpotent Symmetries

In this section we will study various other nilpotent symmetries of this model with particle on a torus [54]. For this purpose it is convenient to work using Nakanishi-Lautrup type auxiliary field which linearizes the gauge fixing part of the effective action in (22). The first-order effective action is then given by

We can easily show that this action is invariant under the following off-shell nilpotent BRST transformation:

Corresponding anti-BRST transformation for this theory is then written by interchanging the role of ghost and anti-ghost field as

The conserved BRST and anti-BRST charges and which generate above BRST and anti-BRST transformations are written for this effective theory as

Further by using the following equation of motion

it is shown that these charges are constants of motion, that is, , and satisfy following relations:

To arrive to these relations, the canonical brackets (16) of the fields and the definition of canonical momenta have been used

The physical states of theory are annihilated by the BRST and anti-BRST charges, leading to

This implies that the operator form of the first class constraint and annihilates the physical state of the theory. Thus the physicality criteria are consistent with Dirac’s method of quantization.

6. Co-BRST and Anti Co-BRST Symmetries

In this section, we investigate two other nilpotent transformations, namely, co-BRST and anti-co-BRST transformation, which are also the symmetry of the effective action in (24). Further these transformations leave the gauge fixing term of the action invariant independently and the kinetic energy term (which remains invariant under BRST and anti-BRST transformations) transforms under it to compensate for the transformation of the ghost terms. These transformations are also called dual and anti-dual BRST transformation [13–15].

The nilpotent co-BRST transformation and anti-co-BRST transformation , which leave the effective action [in (24)] for a particle on torus invariant, are given by

These transformations are absolutely anticommuting as . The conserved charges for above symmetries are found using Noether’s theorem and are written as

which generate the symmetry transformations in (32) and (33), respectively. It is easy to verify the following relations:

which reflect the nilpotency and anti-commutativity property of and (i.e., and ).

7. Other Symmetries

In this section, we construct other symmetries related to this system. Two different bosonic symmetries are constructed out of four nilpotent symmetries. Discrete symmetry related to ghost number is also constructed.

7.1. Bosonic Symmetry

In this part we construct the bosonic symmetry out of these nilpotent BRST symmetries of the theory using [50–52]. The BRST , anti-BRST , co-BRST , and anti-co-BRST symmetry operators satisfy the following algebra:

and we define bosonic symmetries and as

The fields variables transform under bosonic symmetry as

On the other hand transformation generated by is

However the transformation generated by and is not independent as it is easy to see from (38) and (39) that the operators and satisfy the relation . This implies from (37) that

It is clear from above algebra that the operator analogous of the Laplacian operator in the language of differential geometry and the conserved charge for the above symmetry transformation is calculated as

which generates the transformation in (38).

Using equation of motion, it can readily be checked that

Hence is the constant of motion for this theory.

7.2. Ghost Symmetry and Discrete Symmetry

Now we consider yet another kind of symmetry of this system called ghost symmetry discussed in [50]. The ghost numbers of the ghost and anti-ghost fields are 1 and −1, respectively. Rest of the variables in the action of this theory have ghost number zero. Keeping this fact in mind we can introduce a scale transformation of the ghost field, under which the effective action is invariant, as

where and is a global scale parameter. The infinitesimal version of the ghost scale transformation can be written as

Noether’s conserved charge for above symmetry transformation is calculated as

In addition to above continuous symmetry transformation, the ghost sector respects the following discrete symmetry transformations:

8. Geometric Cohomology and Double Hodge Theory

In this section we study the de Rham cohomological operators [17–19] and their realization in terms of conserved charges which generate the nilpotent symmetries for the theory of a particle on the surface of torus. In particular we point out the similarities between the algebra obeyed by de Rham cohomological operators and that by different BRST conserved charges.

Before we proceed to discuss the analogy, we briefly review the essential features of Hodge theory [50–52]. The de Rham cohomological operators in differential geometry obey the following algebra:

where , , and are exterior, coexterior, and Laplace-Beltrami operator, respectively. The operators and are adjoint or dual to each other and is self-adjoint operator. It is well known that the exterior derivative raises the degree of form by one when it operates on forms , whereas the dual-exterior derivative lowers the degree of a form by one when it operates on forms . However does not change the degree of form . denotes an arbitrary -form object.

The Hodge-de Rham decomposition theorem can be stated as follows.

A regular differential form of degree may be uniquely decomposed into a sum of the harmonic form , exact form , and coexact form ; that is,

where and

The generators of all the nilpotent symmetry transformations satisfy the following algebra [50–52]:

Here the relations between the conserved charges and as well as and can be found using equation of motions only. This algebra is similar to the algebra satisfied by de Rham cohomological operators of differential geometry given in (50). Comparing (50) and (55) we obtain following analogies

Let be the ghost number associated with a given state defined in the total Hilbert space of states; that is,

Then it is easy to verify the following relations:

which imply that the ghost numbers of the states and are , and , respectively. The states and have ghost numbers and , respectively. The properties of sets and are same as of operators and . It is evident from (55) that the set raises the ghost number of a state by one and the set lowers the ghost number of the same state by one. Keeping the analogy between charges of different nilpotent symmetries and Hodge-de Rham differential operators, we express any arbitrary state in terms of the sets and as

where the most symmetric state is the harmonic state that satisfies

analogous to (50). Therefore the BRST charges for a particle on a torus form two separate sets of de Rham cohomological operator, namely, and . Thus we call the theory of a particle on torus as double Hodge theory. Fermionic charges and follow the following physicality criteria:

which lead to

This is the operator form of the first class constraint which annihilates the physical state as a consequence of physical criteria, which further is consistent with Dirac’s method of quantization of a system with first class constraints.

9. Finite Field BRST Transformations

In this section we show that these nilpotent symmetries can be generalized by making the parameter finite and field dependent following the work of Joglekar and Mandal [29]. The BRST transformations can be generated from BRST charge using relation where is infinitesimal anticommuting BRST parameter under which effective action remains invariant. Joglekar and Mandal generalized the anticommuting BRST parameter to be finite field dependent but space-time independent parameter . Under this generalization the path integral measure varies nontrivially. The Jacobian for these transformations for certain can be calculated by the following way:

where is a numerical parameter whose value lies between 0 and 1 (). Here all the fields are taken to be dependent. For a field and .

The invariance of the under is a BRST transformation given by

can be replaced by for a certain functional which can be determined in each individual case using following condition

where is a total derivative of with respect to in which dependence on is also differentiated and the Jacobian can be expressed as where is local functional of fields which satisfies (60) where change in Jacobian is calculated as

is sign for bosonic and fermionic fields , respectively.

10. FFBRST for Free Particle on Surface of Torus

The effective action for the free particle on surface of torus using BFV formulation is written in (19) and its BRST transformation is given by (23). In BRST transformation given by (23), is global, infinitesimal, and anticommuting parameter. FFBRST transformation corresponding to this BRST transformation is written as

where is finite field dependent, global, and anticommuting parameter. Under this transformation too, effective action is invariant.

Generating functional for this effective theory can be written as

where

where is the path integral measure integrated over total phase space. The finite BRST transformation given above leaves the effective action invariant but path integral measure in generating functional is not invariant under this transformation. It gives rise to a Jacobian in the extended phase space which can be calculated as

Write it in compact form as

where , which can be written as

Now we consider an example to illustrate the FFBRST formulation. For that purpose we construct finite BRST parameter obtained

through

The Jacobian change is calculated

We make an ansatz for as

where is a dependent arbitrary parameter. Now,

Using condition in (62), we will get . Now the modified generating functional can be written as

Here generating functional at is the theory for a free particle on a surface of torus with a gauge parameter and at the generating functional for same theory with a different gauge parameter . These two effective theories with two different gauge parameters on the surface of a torus are related through the FFBRST transformation with finite parameter given in (70). FFBRST transformation is thus helpful in showing the gauge independence of physical quantities.

11. Conclusions

BFVsystem: We have used this technique to study all the symmetries of a free particle on the surface of torus. We have constructed nilpotent BRST, dual BRST, anti-BRST, and anti-dual BRST transformations for this system. Dual BRST transformations are also the symmetry of effective action and leave gauge fixing part of the effective action invariant. Interchanging the role of ghost and anti-ghost fields the anti-BRST and anti-dual BRST symmetry transformations is constructed. We have shown that the nilpotent BRST and anti-dual BRST charges are analogous to the exterior derivative operators as the ghost number of the state on the total Hilbert space is increased by one when these charges operate on this state and algebra followed by these operators is the same as the algebra obeyed by the de Rham cohomological operators. Similarly the dual BRST and anti-BRST charges are analogous to coexterior derivative. The anticommutators of BRST and dual BRST and anti-BRST and anti-dual BRST charges lead to bosonic symmetry. The corresponding charges are analogous to Laplacian operator. Further, this theory has another nilpotent symmetry called ghost symmetry under which the ghost term of the effective action is invariant. We further have shown that this theory behaves as double Hodge theory as the charges for BRST and dual BRST and the charges for the bosonic symmetry generated out of these two symmetries form the algebra for Hodge theory. On the other hand charges for anti-BRST , anti-dual BRST , and , charge for bosonic symmetry generalized out of these nilpotent symmetries, also satisfy the Hodge algebra. Thus a particle on the surface of the torus has very rich mathematical structure.

We further constructed the FFBRST transformation for this system. By constructing appropriate field dependent parameter we have explicitly shown that such generalized BRST transformations are capable of connecting different theories on torus. It will be interesting to construct finite version dual BRST transformations and study its consequences in studying system with constraints.

Conflicts of Interest

There are no conflicts of interest regarding the publication of this paper.

Acknowledgments

Vipul Kumar Pandey acknowledges University Grant Commission (UGC), India, for its financial assistance under CSIR-UGC JRF/SRF Scheme.

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