On flux integrals for generalized Melvin solution related to simple finite-dimensional Lie algebra

A generalized Melvin solution for an arbitrary simple finite-dimensional Lie algebra G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal G$$\end{document} is considered. The solution contains a metric, n Abelian 2-forms and n scalar fields, where n is the rank of G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal G$$\end{document}. It is governed by a set of n moduli functions Hs(z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_s(z)$$\end{document} obeying n ordinary differential equations with certain boundary conditions imposed. It was conjectured earlier that these functions should be polynomials—the so-called fluxbrane polynomials. These polynomials depend upon integration constants qs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_s$$\end{document}, s=1,⋯,n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s = 1,\dots ,n$$\end{document}. In the case when the conjecture on the polynomial structure for the Lie algebra G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal G$$\end{document} is satisfied, it is proved that 2-form flux integrals Φs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi ^s$$\end{document} over a proper 2d submanifold are finite and obey the relations qsΦs=4πnshs\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_s \Phi ^s = 4 \pi n_s h_s$$\end{document}, where the hs>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_s > 0$$\end{document} are certain constants (related to dilatonic coupling vectors) and the ns\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_s$$\end{document} are powers of the polynomials, which are components of a twice dual Weyl vector in the basis of simple (co-)roots, s=1,⋯,n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s = 1,\dots ,n$$\end{document}. The main relations of the paper are valid for a solution corresponding to a finite-dimensional semi-simple Lie algebra G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal G$$\end{document}. Examples of polynomials and fluxes for the Lie algebras A1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_1$$\end{document}, A2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_2$$\end{document}, A3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_3$$\end{document}, C2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_2$$\end{document}, G2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_2$$\end{document} and A1+A1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_1 + A_1$$\end{document} are presented.


Introduction
In this paper we start with a generalization of a Melvin solution [1], which was presented earlier in Ref. [2]. It appears in the model which contains a metric, n Abelian 2-forms and l ≥ n scalar fields. This solution is governed by a certain nondegenerate (quasi-Cartan) matrix (A ss ), s, s = 1, . . . , n. It is a special case of the so-called generalized fluxbrane solutions from Ref. [3]. For fluxbrane solutions see Refs.  and the references therein. The appearance of fluxbrane solutions was motivated by superstring/M theory. a e-mail: ivashchuk@mail.ru The generalized fluxbrane solutions from Ref. [3] are governed by moduli functions, H s (z) > 0, defined on the interval (0, +∞), where z = ρ 2 and ρ is a radial variable. These functions obey a set of n non-linear differential master equations governed by the matrix (A ss ), equivalent to Toda-like equations, with the following boundary conditions imposed: H s (+0) = 1, s = 1, . . . , n.
In this paper we assume that (A ss ) is a Cartan matrix for some simple finite-dimensional Lie algebra G of rank n (A ss = 2 for all s). According to a conjecture suggested in Ref. [3], the solutions to the master equations with the boundary conditions imposed are polynomials: where we denote (A ss ) = (A ss ) −1 . The integers n s are components of a twice dual Weyl vector in the basis of simple (co-)roots [29]. The set of fluxbrane polynomials H s defines a special solution to open Toda chain equations [30,31] corresponding to a simple finite-dimensional Lie algebra G [32]. In Refs. [2,33] a program (in Maple) for the calculation of these polynomials for the classical series of Lie algebras ( A-, B-, C-and D-series) was suggested. It was pointed out in Ref. [3] that the conjecture on the polynomial structure of H s (z) is valid for Lie algebras of the A-and C-series. In Ref. [34] the conjecture from Ref. [3] was verified for the Lie algebra E 6 and certain duality relations for six E 6 -polynomials were proved. In Sect. 2 we present the generalized Melvin solution from Ref. [2]. In Sect. 3 we deal with the generalized Melvin solution for an arbitrary simple finite-dimensional Lie algebra G. Here we calculate 2-form flux integrals s = M * F s , where F s are 2-forms and M * is a certain 2d submanifold. These integrals (fluxes) are finite when moduli functions are polynomials. In Sect. 3 we consider examples of fluxbrane polynomials and fluxes for the Lie algebras: A 1 , A 2 , A 3 , C 2 , G 2 and A 1 + A 1 .

The solutions
We consider a model governed by the action Here we start with a family of exact solutions to field equations corresponding to the action (2.1) and depending on one variable ρ. The solutions are defined on the manifold where M 1 is a one-dimensional manifold (say S 1 or R) and M 2 is a (D-2)-dimensional Ricci-flat manifold. The solution reads [2] The latter is the so-called quasi-Cartan matrix. We note that the constants B ss and K s = B ss have a certain mathematical sense. They are related to scalar products of certain vectors U s (brane vectors, or U -vectors), which belong to a certain linear space ("truncated target space", for our problem it has dimension l + 2), i.e. B ss = (U s , U s ) and K s = (U s , U s ) [35][36][37]. The scalar products of such a type are of physical significance, since they appear for various solutions with branes, e.g. black branes, S-branes, fluxbranes etc. Several physical parameters in multidimensional models with branes, e.g. the Hawking-like temperatures and the entropies of black holes and branes, PPN parameters, Hubble-like parameters, fluxes etc., contain such scalar products; see [36,37] and Sect. 3 of this paper. The relation (2.11) defines generalized intersection rules for branes which were suggested in [35]. The constants K s are invariants of dimensional reduction. It is well known, see [37] and the references therein, that K s = 2 for branes in numerous supergravity models, e.g. in dimensions D = 10, 11.
It may be shown that if the matrix (h αβ ) has an Euclidean signature and l ≥ n, and (A ss ) is a Cartan matrix for a simple Lie algebra G of rank n, there exists a set of co-vectors λ 1 , . . . , λ n obeying (2.11) (for l = n see Remark 1 in the next section). Thus the solution is valid at least when l ≥ n and the matrix (h αβ ) is positive-definite.
If w = +1 and the (Ricci-flat) metric g 2 has a pseudo-Euclidean signature, we get a multidimensional generalization of Melvin's solution [1].
In our notations Melvin's solution (without scalar field) corresponds to D = 4, n = 1, l = 0, For w = −1 and g 2 of Euclidean signature we obtain a cosmological solution with a horizon (as

Flux integrals for a simple finite-dimensional Lie algebra
Here we deal with the solution which corresponds to a simple finite-dimensional Lie algebra G, i.e. the matrix A = (A ss ) is coinciding with the Cartan matrix of this Lie algebra. We put also n = l, w = +1 and M 1 = S 1 , h αβ = δ αβ and denote is the length squared of a simple root α i corresponding to the Lie algebra G. Here we use the notations A i j = 2(α i , α j )/(α j , α j ); i, j = 1, . . . , n. where

Equation (3.4) implies
are convergent for all s, if the conjecture for the Lie algebra G (on polynomial structure of moduli functions H s ) is obeyed for the Lie algebra G under consideration. Indeed, due to the polynomial assumption (1.1) we have  Thus, any flux s depends upon one integration constant q s = 0, while the integrand form F s depends upon all constants: q 1 , . . . , q n . We note that for D = 4 and g 2 = −dt ⊗ dt + dx ⊗ dx, q s is coinciding with the value of the x-component of the sth magnetic field on the axis of symmetry.
In the case of the Gibbons-Maeda dilatonic generalization of the Melvin solution, corresponding to D = 4, n = l = 1 and G = A 1 [5], the flux from (3.11) (s = 1) is in agreement with that considered in Ref. [26]. For Melvin's case and some higher dimensional extensions (with G = A 1 ) see also Ref. [14].
Due to (3.4) the ratios q i i q j j = n i h i n j h j = n i r j n j r i (3.12) are fixed numbers depending upon the Cartan matrix (A i j ) of a simple finite-dimensional Lie algebra G.

Remark 2
The relation for flux integrals (3.11) is also valid when the matrix (A ss ) is a Cartan matrix of a finitedimensional semi-simple Lie algebra G = G 1 ⊕ · · · ⊕ G k , where G 1 , . . . , G k are simple Lie (sub)algebras. In this case the Cartan matrix (A i j ) has a block-diagonal form, i.e. The manifold M * = (0, +∞) × S 1 is isomorphic to the manifold R 2 * = R 2 \ {0}. The solution (2.3)-(2.5) may be understood (or rewritten by pull-backs) as defined on the manifold R 2 * × M 2 , where the coordinates ρ, φ are understood as coordinates on R 2 * . They are not globally defined. One should consider two charts with coordinates ρ, φ = φ 1 and ρ, φ = φ 2 , where ρ > 0, 0 < φ 1 < 2π and −π < φ 2 < π . Here exp(iφ 1 ) = exp(iφ 2 ). In both cases we have x = ρ cos φ and y = ρ sin φ, where x, y are standard coordinates of R 2 . Using the identity ρdρ ∧ dφ = dx ∧ dy we get (H s (x 2 + y 2 )) −A ss dx ∧ dy, (3.14) s = 1, . . . , n. The 2-forms (3.14) are well defined on R 2 . Indeed, due to the conjecture from Ref. [3] any polynomial H s (z) is a smooth function on R = (−∞, +∞) which obeys H s (z) > 0 for z ∈ (−ε s , +∞), where ε s > 0. This is valid due to the conjecture from Ref. [3] H s (z) > 0 for z > 0 and H s (+0) = 1. Thus, n s =1 H s x 2 + y 2 −A ss is a smooth function since it is a composition of two well-defined smooth functions n s =1 (H s (z)) −A ss and z = x 2 + y 2 . Now we show that there exist 1-forms A s obeying F s = d A s which are globally defined on R 2 . We start with the open submanifold R 2 * . The 1-forms are well defined on R 2 * (here dφ = (x 2 + y 2 ) −1 (−ydx + xdy)) and obey F s = d A s , s = 1, . . . , n. Using the master equation (2.6) we obtain We note that in the case of the Gibbons-Maeda solution [5] corresponding to D = 4, n = l = 1 and G = A 1 the gauge potential from (3.16) coincides (up to notations) with that considered in Ref. [7]. Now we verify our result (3.11) for flux integrals by using the relations for the 1-forms A s . Let us consider a 2d oriented manifold (disk) D R = {(x, y) : It is an 1d oriented manifold with the orientation (inherited from that of D R ) obeying the relation C R dφ = 2π . Using the Stokes-Cartan theorem we get for z = 0 and f (0) = lim z→0 f (z) (the limit does exist). The function f (z) is smooth in the interval (−ε, +∞) for some ε > 0. Indeed, it is smooth in the interval (0, +∞) and holomorphic in the domain {z|0 < |z| < ε} for a small enough ε > 0. Since the limit lim z→0 f (z) does exist the function f (z) is holomorphic in the disc {z||z| < ε} and hence it is smooth in the interval (−ε, +∞). This implies that the metric is smooth on the manifold R 2 × M 2 . (See the text after Eq. (3.14).) The scalar fields are also smooth on R 2 × M 2 .