Abstract
We consider generalized Melvin-like solutions associated with nonexceptional Lie algebras of rank (namely, , , , and ) corresponding to certain internal symmetries of the solutions. The system under consideration is a static cylindrically symmetric gravitational configuration in dimensions in presence of four Abelian 2-forms and four scalar fields. The solution is governed by four moduli functions () of squared radial coordinate obeying four differential equations of the Toda chain type. These functions turn out to be polynomials of powers for Lie algebras , , , and , respectively. The asymptotic behaviour for the polynomials at large distances is governed by some integer-valued matrix connected in a certain way with the inverse Cartan matrix of the Lie algebra and (in case) the matrix representing a generator of the -group of symmetry of the Dynkin diagram. The symmetry properties and duality identities for polynomials are obtained, as well as asymptotic relations for solutions at large distances. We also calculate 2-form flux integrals over -dimensional discs and corresponding Wilson loop factors over their boundaries.
1. Introduction
In this paper, we investigate properties of multidimensional generalization of Melvin’s solution [1], which was presented earlier in [2]. Originally, model from [2] contains metric, Abelian 2-forms and scalar fields. Here we consider a special solutions with , governed by a 4 × 4 Cartan matrix for simple nonexceptional Lie algebras of rank 4: , , , and . The solutions from [2] are special case of the so-called generalized fluxbrane solutions from [3].
It is well known that the original Melvin’s solution in four dimensions describes the gravitational field of a magnetic flux tube. The multidimensional analog of such a flux tube, supported by a certain configuration of form fields, is referred to as a fluxbrane (a “thickened brane” of magnetic flux). The appearance of fluxbrane solutions was originally motivated by superstring/brane models and -theory. For generalizations of the Melvin solution and fluxbrane solutions see [4–21] and references therein.
In [3] there were considered the generalized fluxbrane solutions which are described in terms of moduli functions defined on the interval , where and is a radial coordinate. Functions obey nonlinear differential master equations of Toda-like type governed by some matrix , and the following boundary conditions are imposed: , .
Here, as in [2], we assume that the matrix is a Cartan matrix for some simple finite-dimensional Lie algebra of rank ( for all ). A conjecture was suggested in [3] that in this case the solutions to master equations with the above boundary conditions are polynomials of the form: where are constants. Here and where we denote . Integers are components of the twice dual Weyl vector in the basis of simple (co)roots [22].
Therefore, the functions (which may be called “fluxbrane polynomials”) define a special solution to open Toda chain equations [23, 24] corresponding to simple finite-dimensional Lie algebra [25]. In [2, 26] a program (in Maple) for calculation of these polynomials for classical series of Lie algebras (-, -, -, and -series) was suggested. It was pointed out in [3] that the conjecture on polynomial structure of is valid for Lie algebras of - and - series.
One of the goals of this paper is to study interesting geometric properties of the solution considered for case of nonexceptional Lie algebras of rank 4. In particular, we prove some symmetry properties, as well as the so-called duality relations of fluxbrane polynomials which establishes a behaviour of the solutions under the inversion transformation , which makes the model in tune with -duality in string models and also can be mathematically understood in terms of the groups of symmetry of Dynkin diagrams for the corresponding Lie algebras. In our case these groups of symmetry are either identical ones (for Lie algebras and ) or isomorphic to the group (for Lie algebra ) or isomorphic to the group which is the group of permutation of 3 elements (for Lie algebra ). These duality identities may be used in deriving a -expansion for solutions at large distances . The corresponding asymptotic behaviour of the solutions is studied.
The analogous analysis was performed recently for the case of rank-2 Lie algebras: , , in [27], and for rank-3 algebras , , and in [28]. Also, in [29] the conjecture from [3] was verified for the Lie algebra and certain duality relations for six -polynomials were found.
The paper is organized as follows. In Section 2 we present a generalized Melvin solutions from [2] for the case of four scalar fields and four 2-forms. In Section 3 we deal with the solutions for the Lie algebras , , , and . We find symmetry properties and duality relations for polynomials and present asymptotic relations for the solutions. We also calculate 2-form flux integrals and corresponding Wilson loop factors, where are 2-forms and is 2-dimensional disc of radius . The flux integrals converge, i.e., have finite limits for [30].
2. The Setup and Generalized Melvin Solutions
Let us consider the following product manifold: where and is a -dimensional Ricci-flat manifold.
On this manifold, we define the action where is a metric on , is vector of scalar fields, is a 2-form, and is dilatonic coupling vector, ; . Here we use the notations ; .
There is a family of exact cylindrically symmetric solutions to the field equations corresponding for the action (4) and depending on the radial coordinate . The solution has the form [2], where is a metric on and is a Ricci-flat metric of signature on . Here are integration constants ( in notations of [2]).
For further convenience, let us denote . As it was shown in earlier works, the functions obey the set of master equations with the boundary conditions where . The boundary condition (9) guarantees the absence of a conic singularity (for the metric (5)) for .
There are some relations for the parameters : where . In these relations, we have denoted The latter matrix is the so-called “quasi-Cartan” matrix. One can prove that if is a Cartan matrix for a certain simple Lie algebra of rank 4 then there exists a set of vectors obeying (13). See also Remark 1 in the next section.
The solution considered can be understood as a special case of the fluxbrane solutions from [3, 19].
Therefore, here we investigate a multidimensional generalization of Melvin’s solution [1] for the case of four scalar fields and four 2-forms. Note that the original Melvin’s solution without scalar field would correspond to , one (electromagnetic) 2-form, (), , and .
3. Solutions Related to Simple Classical Rank-4 Lie Algebras
In this section we consider the solutions associated with the simple nonexceptional Lie algebras of rank 4. This means than the matrix coincides with one of the Cartan matrices
for , respectively.
Each of these matrices can be graphically described by drawing the Dynkin diagrams pictured on Figure 1 for these four Lie algebras.
Using (11)-(13) we can getwhere and; (15) is a special case of (16).
From (11) and (13) it also follows that for any obeying . This implies or () for , respectively.
Remark 1. For large enough in (18) there exist vectors obeying (16) (and hence (15)). Indeed, the matrix is positive definite if , where is some positive number. Hence there exists a matrix , such that . We put and get the set of vectors obeying (16).
Polynomials. According to the polynomial conjecture, the set of moduli functions , obeying (8) and (9) with the Cartan matrix from (14) are polynomials with powers , , , calculated by using (2) for Lie algebras , , , and , respectively.
One can prove this conjecture by solving the system of nonlinear algebraic equations for the coefficients of these polynomials following from master equations (8). Below we present a list of the polynomials obtained by using appropriate MATHEMATICA algorithm. For convenience, we use the rescaled variables (as in [25]):
-Case. For the Lie algebra we have
-Case. For the Lie algebra the fluxbrane polynomials are
-Case. For the Lie algebra we get the following polynomials:
-Case. For the Lie algebra we obtain the polynomials
Let us denote
One can easily write down the asymptotic behaviour of the polynomials obtained:where we introduced the integer-valued matrix having the formfor Lie algebras , respectively. In these four cases there is a simple property
Note that for Lie algebras , , and we havewhere is inverse Cartan matrix, whereas in the -case the matrix is related to the inverse Cartan matrix as follows:Here is identity matrix andis a permutation matrix corresponding to the permutation ( is symmetric group)by the following relation . Here is the generator of the group which is the group of symmetry of the Dynkin diagram for . is isomorphic to the group .
In case of the group of symmetry of the Dynkin diagram is isomorphic to the symmetric group acting on the set of three vertices of the Dynkin diagram via their permutations. The existence of the above symmetry groups and implies certain identity properties for the fluxbrane polynomials .
Let us denote for the case and for , , and cases (). We call the ordered set as dual one to the ordered set . It corresponds to the action (trivial or nontrivial) of the group on vertices of the Dynkin diagrams for above algebras.
Then we obtain the following identities which were directly verified by using MATHEMATICA algorithms.
Symmetry Relations
Proposition 2. The fluxbrane polynomials obey for all and the identities for any , . We call relations (45) as symmetry ones.
Duality Relations
Proposition 3. The fluxbrane polynomials corresponding to Lie algebras , , , and obey for all and the identities.
We call relations (46) as duality ones.
Fluxes. Here we deal with an oriented 2-dimensional manifold , . One can define the flux integrals over this manifold:where we denotedIt can be easily understood that total flux integrals are convergent. Indeed, due to polynomial assumption (1) we have as . From (48), (49), and the equality , following from (2), we get and hence the integral (47) is convergent for any .
Using (42) and (50) we have for the -case Similarly, due to (41) and (50) we get for Lie algebras , , and After that, we can calculate the fluxes . Using the master equations (8) one can write where . Thus, using (47) we easily obtain Note that the manifold is isomorphic to the manifold . Therefore, one can understand the family of solutions under consideration as defined on the manifold , where coordinates , are polar coordinates in a domain of : and , where are standard coordinates of . It was shown in [30] that there exist forms globally defined on and obeying the relation .
Now let us we consider a 2-dimensional oriented manifold (disk) . Its boundary is a circle of radius , i.e., 1-dimensional oriented manifold with the orientation inherited from that of obeying the relation .
The Stokes theorem yields in this case According to the definition of Abelian Wilson loop (factor), we have Relations (1) and (54) imply (see (10)) . Any (total) flux depends upon one integration constant , while the integrand form depends upon all constants: . As a consequence, we obtain finite limits
In the -case we have where .
In the -case we find where , .
In the -case we obtain where , .
In the -case we are led to relations where . (In all examples relations (19) are used.)
Note that, for and , coincides with the value of the -component of the -th magnetic field on the axis of symmetry, .
Asymptotic Relations. Here we can write down the asymptotic relations for the solution under consideration as : , where and in (65) we put for and for . In derivation of asymptotic relations, (40), (49), and (51) were used. We note that for the asymptotic value of form depends upon , , while in the -case depends upon and for , and depends upon for .
We note also that by putting we get the Melvin-type solutions corresponding to Lie algebras , , , and , respectively, which were analyzed in [28]. (The case of the rank 2 Lie algebra [27] may be obtained for the case when .)
Dilatonic Black Holes. Relations (constraints) on dilatonic coupling vectors (12), (13) appear also for dilatonic black hole (DBH) solutions which are defined on the manifold where and is a Ricci-flat manifold. These DBH solutions on from (67) for the model under consideration may be extracted from general black brane solutions; see [21, 25, 31]. They read , where , is the standard metric on , and is a Ricci-flat metric of signature on . Here are integration constants (charges).
The functions obey the master equations with the following boundary conditions on the horizon and at infinity imposed: where . Here relations (11) are also valid. For Lie algebras of rank 4 the functions are polynomials with respect to , which may be obtained (at least for small enough ) from fluxbrane polynomials presented in this paper. See [25].
4. Conclusions
In this paper, the generalized multidimensional family of Melvin-type solutions was considered corresponding to simple nonexceptional finite-dimensional Lie algebras of rank 4: . Each solution of that family is governed by a set of 4 fluxbrane polynomials , . These polynomials define special solutions to open Toda chain equations corresponding to the Lie algebra .
The polynomials depend also upon parameters , which coincides for (up to a sign) with the values of colored magnetic fields on the axis of symmetry.
We have found the symmetry relations and the duality identities for polynomials. These identities may be used in deriving -expansion for solutions at large distances , e.g., for asymptotic relations which are presented in the paper.
There were also calculated two-dimensional flux integrals () over a disc of radius and a corresponding Wilson loop factors over a circle . It turns out that each total flux depends only upon one corresponding parameter , whereas the integrand depends on all parameters , .
The case of exceptional Lie algebra will be considered in a separate publication.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The publication has been prepared with the support of the “RUDN University Program 5-100”. It was also partially supported by the Russian Foundation for Basic Research, Grant no. 16-02-00602.