1 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 \(l \ge n\) scalar fields. This solution is governed by a certain non-degenerate (quasi-Cartan) matrix \((A_{s s'})\), \(s, s' = 1, \dots , n\). It is a special case of the so-called generalized fluxbrane solutions from Ref. [3]. For fluxbrane solutions see Refs. [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28] and the references therein. The appearance of fluxbrane solutions was motivated by superstring/M theory.

The generalized fluxbrane solutions from Ref. [3] are governed by moduli functions, \(H_s(z) > 0\), defined on the interval \((0, +\infty )\), where \(z = \rho ^2\) and \(\rho \) is a radial variable. These functions obey a set of n non-linear differential master equations governed by the matrix \((A_{s s'})\), equivalent to Toda-like equations, with the following boundary conditions imposed: \(H_{s}(+ 0) = 1\), \(s = 1,\ldots ,n\).

In this paper we assume that \((A_{s s'})\) is a Cartan matrix for some simple finite-dimensional Lie algebra \(\mathcal G\) 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:

$$\begin{aligned} H_{s}(z) = 1 + \sum _{k = 1}^{n_s} P_s^{(k)} z^k, \end{aligned}$$
(1.1)

where the \(P_s^{(k)}\) are constants. Here \(P_s^{(n_s)} \ne 0\) and

$$\begin{aligned} n_s = 2 \sum _{s' =1}^{n} A^{s s'}, \end{aligned}$$
(1.2)

where we denote \((A^{s s'}) = (A_{s s'})^{-1}\). 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 \(\mathcal G\) [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 \(\mathcal G\). Here we calculate 2-form flux integrals \(\Phi ^s = \int _{M_{*}} F^s\), 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 \(A_1 + A_1\).

2 The solutions

We consider a model governed by the action

$$\begin{aligned} S= & {} \int d^Dx \sqrt{|g|} \biggl \{R[g]- h_{\alpha \beta }g^{MN}\partial _M\varphi ^{\alpha }\partial _N\varphi ^{\beta }\nonumber \\&-\frac{1}{2} \sum _{s =1}^{n}\exp [2\lambda _s(\varphi )](F^s)^2 \biggr \} \end{aligned}$$
(2.1)

where \(g=g_{MN}(x)\mathrm{d}x^M\otimes \mathrm{d}x^N\) is a metric, \(\varphi =(\varphi ^\alpha )\in {\mathbb R}^l\) is a set of scalar fields, \((h_{\alpha \beta })\) is a constant symmetric non-degenerate \(l\times l\) matrix \((l\in {\mathbb N})\), \( F^s = dA^s = \frac{1}{2} F^s_{M N} \mathrm{d}x^{M} \wedge \mathrm{d}x^{N}\) is a 2-form, \(\lambda _s\) is a 1-form on \({\mathbb R}^l\): \(\lambda _s(\varphi )=\lambda _{s \alpha }\varphi ^\alpha \), \(s = 1,\ldots , n\); \(\alpha =1,\dots ,l\). Here \((\lambda _{s \alpha })\), \(s =1,\dots , n\), are dilatonic coupling vectors. In (2.1) we denote \(|g| = |\det (g_{MN})|\), \((F^s)^2 = F^s_{M_1 M_{2}} F^s_{N_1 N_{2}} g^{M_1 N_1} g^{M_{2} N_{2}}\), \(s = 1,\dots , n\).

Here we start with a family of exact solutions to field equations corresponding to the action (2.1) and depending on one variable \(\rho \). The solutions are defined on the manifold

$$\begin{aligned} M = (0, + \infty ) \times M_1 \times M_2, \end{aligned}$$
(2.2)

where \(M_1\) is a one-dimensional manifold (say \(S^1\) or \({\mathbb R}\)) and \(M_2\) is a (D-2)-dimensional Ricci-flat manifold. The solution reads [2]

$$\begin{aligned} g= & {} \Bigl (\prod _{s = 1}^{n} H_s^{2 h_s /(D-2)} \Bigr ) \biggl \{ w \mathrm{d}\rho \otimes \mathrm{d}\rho \nonumber \\&+ \Bigl (\prod _{s = 1}^{n} H_s^{-2 h_s} \Bigr ) \rho ^2 \mathrm{d}\phi \otimes \mathrm{d}\phi + g^2 \biggr \}, \end{aligned}$$
(2.3)
$$\begin{aligned} \exp (\varphi ^\alpha )= & {} \prod _{s = 1}^{n} H_s^{h_s \lambda _{s}^\alpha }, \end{aligned}$$
(2.4)
$$\begin{aligned} F^s= & {} q_s \left( \prod _{s' = 1}^{n} H_{s'}^{- A_{s s'}} \right) \rho \mathrm{d}\rho \wedge \mathrm{d}\phi , \end{aligned}$$
(2.5)

\(s = 1,\dots , n\); \(\alpha = 1,\dots , l\), where \(w = \pm 1\), \(g^1 = \mathrm{d}\phi \otimes \mathrm{d}\phi \) is a metric on \(M_1\) and \(g^2\) is a Ricci-flat metric on \(M_{2}\). Here \(q_s \ne 0\) are integration constants, \(q_s = - Q_s\) in the notations of Ref. [2], \(s = 1,\dots , n\).

The functions \(H_s(z) > 0\), \(z = \rho ^2\), obey the master equations

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}z} \left( \frac{ z}{H_s} \frac{\mathrm{d}}{\mathrm{d}z} H_s \right) = P_s \prod _{s' = 1}^{n} H_{s'}^{- A_{s s'}}, \end{aligned}$$
(2.6)

with the following boundary conditions:

$$\begin{aligned} H_{s}(+ 0) = 1, \end{aligned}$$
(2.7)

where

$$\begin{aligned} P_s = \frac{1}{4} K_s q_s^2, \end{aligned}$$
(2.8)

\(s = 1,\dots ,n\). The boundary condition (2.7) guarantees the absence of a conic singularity [in the metric (2.3)] for \(\rho = +0\).

The parameters \(h_s\) satisfy the relations

$$\begin{aligned} h_s = K_s^{-1}, \quad K_s = B_{s s} > 0, \end{aligned}$$
(2.9)

where

$$\begin{aligned} B_{ss'} \equiv 1 +\frac{1}{2-D}+ \lambda _{s \alpha } \lambda _{s' \beta } h^{\alpha \beta }, \end{aligned}$$
(2.10)

\(s, s' = 1,\ldots , n\), with \((h^{\alpha \beta })=(h_{\alpha \beta })^{-1}\). In the relations above we denote \(\lambda _{s}^{\alpha } = h^{\alpha \beta } \lambda _{s \beta }\) and

$$\begin{aligned} (A_{ss'}) = \left( 2 B_{s s'}/B_{s' s'} \right) . \end{aligned}$$
(2.11)

The latter is the so-called quasi-Cartan matrix.

We note that the constants \(B_{s s'}\) and \(K_s = B_{s s}\) 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. \(B_{s s'} =(U^s,U^{s'})\) 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 \((h_{\alpha \beta })\) has an Euclidean signature and \(l \ge n\), and \((A_{ss'})\) is a Cartan matrix for a simple Lie algebra \(\mathcal G\) of rank n, there exists a set of co-vectors \(\lambda _1, \dots , \lambda _n\) obeying (2.11) (for \(l = n\) see Remark 1 in the next section). Thus the solution is valid at least when \(l \ge n\) and the matrix \((h_{\alpha \beta })\) is positive-definite.

The solution under consideration is a special case of the fluxbrane (for \(w = +1\), \(M_1 = S^1\)) and S-brane (\(w = -1\)) solutions from [3] and [25], respectively.

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 \(D = 4\), \(n = 1\), \(l =0\), \(M_1 = S^1\) (\(0< \phi < 2 \pi \)), \(M_2 = {\mathbb R}^2\), \(g^2 = - \mathrm{d}t \otimes \mathrm{d}t + \mathrm{d}x \otimes \mathrm{d}x\) and \(\mathcal{G} = A_1\).

For \(w = -1\) and \(g^2\) of Euclidean signature we obtain a cosmological solution with a horizon (as \(\rho = + 0\)) if \(M_1 = {\mathbb R}\) (\( - \infty< \phi < + \infty \)).

3 Flux integrals for a simple finite-dimensional Lie algebra

Here we deal with the solution which corresponds to a simple finite-dimensional Lie algebra \(\mathcal{G}\), 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 \(M_1 = S^1\), \(h_{\alpha \beta } = \delta _{\alpha \beta }\) and denote \((\lambda _{s a }) = (\lambda _{s}^{a}) = \mathbf {\lambda }_{s}\), \(s = 1, \dots , n\).

Due to (2.9)–(2.11) we get

$$\begin{aligned} K_s = \frac{D - 3}{D -2} + \mathbf {\lambda }_{s}^2, \end{aligned}$$
(3.1)

\(h_s = K_s^{-1}\), and

$$\begin{aligned} \mathbf {\lambda }_{s} \mathbf {\lambda }_{l} = \frac{1}{2} K_l A_{sl} - \frac{D - 3}{D -2} \equiv \Gamma _{sl}, \end{aligned}$$
(3.2)

\(s,l = 1, \dots , n\). [Equation (3.1) is a special case of (3.2)].

It follows from (2.9)–(2.11) that

$$\begin{aligned} \frac{h_i}{h_j} = \frac{K_{j}}{K_{i}} = \frac{B_{jj}}{B_{ii}} = \frac{B_{ji}}{B_{ii}} \frac{B_{jj}}{B_{ij}} = \frac{A_{ji}}{A_{ij}} \end{aligned}$$
(3.3)

for any \(i \ne j\) obeying \(A_{ij}, A_{ji} \ne 0\); \(i,j = 1, \dots ,n\). It may be readily shown from (3.3) that the ratios \(\frac{h_i}{h_j} = \frac{K_{j}}{K_{i}}\) are fixed numbers for any given Cartan matrix \(({A_{ij}})\) of a simple (finite-dimensional) Lie algebra \(\mathcal{G}\). (This follows from (3.3) and the connectedness of the Dynkin diagram of a simple Lie algebra.) The ratios (3.3) may be written as follows:

$$\begin{aligned} \frac{h_i}{h_j} = \frac{K_{j}}{K_{i}} = \frac{r_j }{r_i} \end{aligned}$$
(3.4)

\(i \ne j\), where \(r_i = (\alpha _{i}, \alpha _{i})\) is the length squared of a simple root \(\alpha _{i}\) corresponding to the Lie algebra \(\mathcal{G}\). Here we use the notations \(A_{ij} = 2 (\alpha _{i}, \alpha _{j})/(\alpha _{j}, \alpha _{j})\); \(i,j = 1, \dots ,n\). Equation (3.4) implies

$$\begin{aligned} K_{i} = \frac{1}{2} K r_i, \end{aligned}$$
(3.5)

\(i = 1, \dots ,n\), where \(K > 0\). (For simply laced (ADE) Lie algebras all \(r_i\) are equal.)

Remark 1

For large enough K in (3.5) there exist vectors \(\mathbf {\lambda }_s\) obeying (3.2) [and hence (3.1)]. Indeed, the matrix \((\Gamma _{sl})\) is positive-definite if \(K > K_{*}\), where \(K_{*}\) is some positive number. Hence there exists a matrix \(\Lambda \), such that \(\Lambda ^{T}\Lambda = \Gamma \). We put \((\Lambda _{as}) = (\lambda _{s}^a)\) and get the set of vectors obeying (3.2).

Now let us consider the oriented 2-dimensional manifold \(M_{*} =(0, + \infty ) \times S^1\). The flux integrals

$$\begin{aligned} \Phi ^s= & {} \int _{M_{*}} F^s = \int _{0}^{+ \infty } \mathrm{d}\rho \int _{0}^{2 \pi } \mathrm{d}\phi \ \rho \mathcal{B}^s(\rho ^2) \nonumber \\= & {} 2 \pi \int _{0}^{+ \infty } \mathrm{d}\rho \ \rho \mathcal{B}^s(\rho ^2) , \end{aligned}$$
(3.6)

where

$$\begin{aligned} \mathcal{B}^s(\rho ^2) = q_s \prod _{l = 1}^{n} (H_{l}(\rho ^2))^{- A_{s l}}, \end{aligned}$$
(3.7)

are convergent for all s, if the conjecture for the Lie algebra \(\mathcal{G}\) (on polynomial structure of moduli functions \(H_s\)) is obeyed for the Lie algebra \(\mathcal{G}\) under consideration.

Indeed, due to the polynomial assumption (1.1) we have

$$\begin{aligned} H_s(\rho ^2) \sim C_s \rho ^{2n_s}, \quad C_s = P_s^{(n_s)}, \end{aligned}$$
(3.8)

as \(\rho \rightarrow + \infty \); \(s =1, \dots , n\). From (3.7), (3.8) and the equality \(\sum _{1}^{n} A_{s l} n_l = 2\), following from (1.2), we get

$$\begin{aligned} \mathcal{B}^s(\rho ^2) \sim q_s C^s \rho ^{-4}, \quad C^s = \prod _{l = 1}^{n} C_l^{-A_{sl}}, \end{aligned}$$
(3.9)

and hence the integral (3.6) is convergent for any \(s =1, \dots , n\).

By using the master equations (2.6) we obtain

$$\begin{aligned} \int _{0}^{+ \infty } \mathrm{d}\rho \rho \mathcal{B}^s(\rho ^2)= & {} q_s P_s^{-1} \frac{1}{2} \int _{0}^{+ \infty } \mathrm{d}z \frac{\mathrm{d}}{\mathrm{d}z} \left( \frac{ z}{H_s} \frac{\mathrm{d}}{\mathrm{d}z} H_s \right) \nonumber \\= & {} \frac{1}{2} q_s P_s^{-1} \lim _{z \rightarrow + \infty } \left( \frac{ z}{H_s} \frac{\mathrm{d}}{\mathrm{d}z} H_s \right) \nonumber \\= & {} \frac{1}{2} n_s q_s P_s^{-1}, \end{aligned}$$
(3.10)

which implies [see (2.8)]

$$\begin{aligned} \Phi ^s = 4 \pi n_s q_s^{-1} h_s, \end{aligned}$$
(3.11)

\(s =1, \dots , n \).

Thus, any flux \(\Phi ^s\) depends upon one integration constant \(q_s \ne 0\), while the integrand form \(F^s\) depends upon all constants: \(q_1, \dots , q_n\).

We note that for \(D =4\) and \(g^2 = - \mathrm{d}t \otimes \mathrm{d}t + \mathrm{d}x \otimes d x\), \(q_s\) 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 \(\mathcal{G} = A_1\) [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 \(\mathcal{G} = A_1\)) see also Ref. [14].

Due to (3.4) the ratios

$$\begin{aligned} \frac{q_i \Phi ^i}{q_j \Phi ^j} = \frac{n_i h_i}{n_j h_j} = \frac{n_i r_{j} }{n_j r_{i}} \end{aligned}$$
(3.12)

are fixed numbers depending upon the Cartan matrix \(({A_{ij}})\) of a simple finite-dimensional Lie algebra \(\mathcal{G}\).

Remark 2

The relation for flux integrals (3.11) is also valid when the matrix \((A_{ss'})\) is a Cartan matrix of a finite-dimensional semi-simple Lie algebra \(\mathcal{G} = \mathcal{G}_1 \oplus \cdots \oplus \mathcal{G}_k\), where \(\mathcal{G}_1, \dots , \mathcal{G}_k\) are simple Lie (sub)algebras. In this case the Cartan matrix \(({A_{ij}})\) has a block-diagonal form, i.e. \(({A_{ij}}) = \mathrm{diag} \left( \left( {A^{(1)}_{i_1 j_1}}\right) , \ldots , \left( {A^{(k)}_{i_k j_k}}\right) \right) \), where \(\left( {A^{(a)}_{i_a j_a}}\right) \) is the Cartan matrix of the Lie algebra \(\mathcal{G}_a\), \(a = 1, \dots , k\). The set of polynomials in this case splits in a direct union of sets of polynomials corresponding to the Lie algebras \(\mathcal{G}_1, \dots , \mathcal{G}_k\). Equations (3.4) and (3.12) are valid, when the indices ij correspond to one ath block, \(a = 1, \dots , k\). The quantities \(q_i \Phi ^i\) and \(q_j \Phi ^j\) corresponding to different blocks are independent. Equation (3.5) should be replaced by

$$\begin{aligned} K_{i_a} = \frac{1}{2} K^{(a)} r_{i_a}, \quad K^{(a)} > 0, \end{aligned}$$
(3.13)

for any index \(i_a\) corresponding to the ath block; \(a = 1, \dots , k\). The existence of dilatonic coupling vectors \(\mathbf {\lambda }_s\) obeying (3.2) [(and (3.1)] just follows from the arguments of Remark 1, if we put all \(K^{(a)} = K > 0\).

The manifold \(M_{*} =(0, + \infty ) \times S^1\) is isomorphic to the manifold \({\mathbb R}^2_{*} = {\mathbb R}^2 \setminus \{ 0 \}\). The solution (2.3)–(2.5) may be understood (or rewritten by pull-backs) as defined on the manifold \({\mathbb R}^2_{*} \times M_2\), where the coordinates \(\rho \), \(\phi \) are understood as coordinates on \({\mathbb R}^2_{*}\). They are not globally defined. One should consider two charts with coordinates \(\rho \), \(\phi = \phi _1\) and \(\rho \), \(\phi = \phi _2\), where \(\rho > 0\), \(0< \phi _1 < 2 \pi \) and \(- \pi< \phi _2 < \pi \). Here \(\exp (i \phi _1 ) = \exp (i \phi _2)\). In both cases we have \(x = \rho \cos \phi \) and \(y = \rho \sin \phi \), where xy are standard coordinates of \({\mathbb R}^2\). Using the identity \( \rho \mathrm{d}\rho \wedge \mathrm{d}\phi = \mathrm{d}x \wedge \mathrm{d}y\) we get

$$\begin{aligned} F^s= q_s \prod _{s' = 1}^{n} (H_{s'}(x^2 + y^2))^{- A_{s s'}} \mathrm{d}x \wedge \mathrm{d}y, \end{aligned}$$
(3.14)

\(s =1, \dots , n \). The 2-forms (3.14) are well defined on \({\mathbb R}^2\). Indeed, due to the conjecture from Ref. [3] any polynomial \(H_{s}(z)\) is a smooth function on \({\mathbb R}= (- \infty , + \infty )\) which obeys \(H_{s}(z) > 0\) for \(z \in (- \varepsilon _s, + \infty )\), where \(\varepsilon _s > 0\). This is valid due to the conjecture from Ref. [3] \(H_{s}(z) > 0\) for \(z > 0\) and \(H_{s}(+0) = 1\). Thus, \(\left( \prod _{s' = 1}^{n} \left( H_{s'}\left( x^2 + y^2\right) \right) ^{- A_{s s'}} \right) \) is a smooth function since it is a composition of two well-defined smooth functions \(\left( \prod _{s' = 1}^{n} (H_{s'}(z))^{- A_{s s'}} \right) \) and \(z = x^2 + y^2\).

Now we show that there exist 1-forms \(A^s\) obeying \(F^s = dA^s\) which are globally defined on \({\mathbb R}^2\). We start with the open submanifold \({\mathbb R}^2_{*}\). The 1-forms

$$\begin{aligned} A^s = \left( \int _{0}^{\rho } \mathrm{d}\bar{\rho } \bar{\rho } \mathcal{B}^s(\bar{\rho }^2) \right) \mathrm{d}\phi = \frac{1}{2} \left( \int _{0}^{\rho ^2} \mathrm{d}\bar{z} \mathcal{B}^s(\bar{z}) \right) \mathrm{d}\phi \end{aligned}$$
(3.15)

are well defined on \({\mathbb R}^2_{*}\) (here \(\mathrm{d}\phi = (x^2 + y^2)^{-1} ( - y \mathrm{d}x + x \mathrm{d}y)\)) and obey \(F^s = dA^s\), \(s = 1, \dots , n \). Using the master equation (2.6) we obtain

$$\begin{aligned} A^s= & {} \frac{q_s}{2 P_s} \left( \int _{0}^{\rho ^2} \mathrm{d}\bar{z} \frac{\mathrm{d}}{\mathrm{d}\bar{z}} \left( \frac{ \bar{z}}{H_s(\bar{z})} \frac{\mathrm{d}}{\mathrm{d}\bar{z}} H_s ( \bar{z}) \right) \right) \mathrm{d}\phi \nonumber \\= & {} \frac{2 h_s}{q_s} \frac{H^{'}_{s} (\rho ^2)}{H_s(\rho ^2)} \rho ^2 d \phi , \end{aligned}$$
(3.16)

\(s = 1, \dots , n \). Here \(H{'}_s = \frac{\mathrm{d}}{\mathrm{d}z} H_s\). Due to the relation \(\rho ^2 \mathrm{d}\phi = - y \mathrm{d}x + x \mathrm{d}y \), we obtain

$$\begin{aligned} A^s = \frac{2 h_s}{q_s} \frac{H^{'}_{s} (x^2 + y^2)}{H_s(x^2 + y^2)} (- y \mathrm{d}x + x \mathrm{d}y), \end{aligned}$$
(3.17)

\(s = 1, \dots , n \). The 1-forms (3.17) are well-defined smooth 1-forms on \({\mathbb R}^2\).

We note that in the case of the Gibbons–Maeda solution [5] corresponding to \(D = 4\), \(n = l= 1\) and \(\mathcal{G} = A_1\) 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) \(D_R = \{ (x,y): x^2 + y^2 \le R^2 \}\) with the boundary \(\partial D_R = C_R = \{ (x,y): x^2 + y^2 = R^2 \}\). \(C_R\) is a circle of radius R. It is an 1d oriented manifold with the orientation (inherited from that of \(D_R\)) obeying the relation \(\int _{C_R} \mathrm{d}\phi = 2 \pi \). Using the Stokes–Cartan theorem we get

$$\begin{aligned} \int _{D_{R}} F^s= \int _{D_{R}} \mathrm{d} A^s= \int _{C_{R}} A^s = \frac{4 \pi h_s}{q_s} \frac{H^{'}_{s} (R^2)}{H_s(R^2)} R^2, \end{aligned}$$
(3.18)

\(s = 1, \dots , n \). By using the asymptotic relation (3.8) we find

$$\begin{aligned} \lim _{R \rightarrow + \infty } \int _{D_{R}} F^s = \frac{4 \pi h_s n_s}{q_s}, \end{aligned}$$
(3.19)

\(s = 1, \dots , n \), in agreement with (3.11).

Remark 3

We note (for completeness) that the metric and scalar fields for our solution with \(w = +1\) and \(l = n\) can be extended to the manifold \({\mathbb R}^2 \times M_2\). Indeed, in the coordinates xy the metric (2.3) and scalar fields (2.4) read as follows:

$$\begin{aligned} g= & {} \Bigl (\prod _{s = 1}^{n} H_s^{2 h_s /(D-2)} \Bigr ) \biggl \{ \mathrm{d}x \otimes \mathrm{d}x + \mathrm{d}y \otimes \mathrm{d}y \nonumber \\&+ f (- y \mathrm{d}x + x \mathrm{d}y)^2 + g^2 \biggr \}, \end{aligned}$$
(3.20)
$$\begin{aligned} \varphi ^{a}= & {} \sum _{s = 1}^{n} h_s \lambda _{s}^{a} \ln H_s, \end{aligned}$$
(3.21)

\(a =1, \dots , l \). Here \(H_s = H_s(x^2 + y^2)\), \(s = 1, \dots , n \), and \(f = f(x^2 + y^2)\), where

$$\begin{aligned} f(z) = \left( \Bigl (\prod _{s = 1}^{n} (H_s(z))^{-2 h_s} \Bigr ) - 1 \right) z^{-1}, \end{aligned}$$
(3.22)

for \(z \ne 0\) and \(f(0)= \lim _{z \rightarrow 0} f(z)\) (the limit does exist). The function f(z) is smooth in the interval \( (- \varepsilon , + \infty )\) for some \(\varepsilon > 0\). Indeed, it is smooth in the interval \( (0, + \infty )\) and holomorphic in the domain \(\{z | 0< |z| < \varepsilon \}\) for a small enough \(\varepsilon > 0\). Since the limit \(\lim _{z \rightarrow 0} f(z)\) does exist the function f(z) is holomorphic in the disc \(\{z | |z| < \varepsilon \}\) and hence it is smooth in the interval \( (- \varepsilon , + \infty )\). This implies that the metric is smooth on the manifold \({\mathbb R}^2 \times M_2\). (See the text after Eq. (3.14).) The scalar fields are also smooth on \({\mathbb R}^2 \times M_2\).

4 Examples

Here we present fluxbrane polynomials corresponding to the Lie algebras \(A_1\), \(A_2\), \(A_3\), \(C_2\), \(G_2\), \(A_1 + A_1\) and related fluxes. Here as in [32] we use other parameters \(p_s\) instead of \(P_s\):

$$\begin{aligned} p_s = P_s/n_s, \end{aligned}$$
(4.1)

\(s = 1, \ldots , n\).

\(A_1\) -case. The simplest example occurs in the case of the Lie algebra \(A_1 = sl(2)\). Here \(n_1 = 1\). We get [3]

$$\begin{aligned} H_{1} = 1 + p_1 z \end{aligned}$$
(4.2)

and

$$\begin{aligned} \Phi ^1 = 4 \pi q_1^{-1} h_1, \end{aligned}$$
(4.3)

which is also valid for Melvin’s solution with \(D = 4\) and \(h_1 = 2\).

\(A_2\) -case. For the Lie algebra \(A_2 = sl(3)\) with the Cartan matrix

$$\begin{aligned} \left( A_{ss'}\right) = \left( \begin{array}{*{6}{c}} 2 &{}\quad -1\\ -1&{}\quad 2\\ \end{array} \right) \quad \end{aligned}$$
(4.4)

we have [3, 25, 32] \(n_1 = n_2 =2\) and

$$\begin{aligned} H_{1}= & {} 1 + 2 p_1 z + p_1 p_2 z^{2}, \end{aligned}$$
(4.5)
$$\begin{aligned} H_{2}= & {} 1 + 2 p_2 z + p_1 p_2 z^{2}. \end{aligned}$$
(4.6)

We get in this case

$$\begin{aligned} (\Phi ^1, \Phi ^2) = 8 \pi h (q_1^{-1},q_2^{-1}), \end{aligned}$$
(4.7)

where \(h_1 = h_2 = h\).

\(A_3\) -case. The polynomials for the \(A_3\)-case read as follows [32, 33]:

$$\begin{aligned} H_{1}= & {} 1 + 3 p_1 z + 3 p_1 p_2 z^{2} + p_1 p_2 p_3 z^{3}, \end{aligned}$$
(4.8)
$$\begin{aligned} H_{2}= & {} 1 + 4 p_2 z + 3 \Bigl ( p_1 p_2 + p_2 p_3 \Bigr ) z^{2} \nonumber \\&+\,4 p_1 p_2 p_3 z^{3} + p_1 p_2^{2} p_3 z^{4}, \end{aligned}$$
(4.9)
$$\begin{aligned} H_{3}= & {} 1 + 3 p_3 z + 3 p_2 p_3 z^{2} + p_1 p_2 p_3 z^{3}. \end{aligned}$$
(4.10)

Here we have \((n_1, n_2, n_3) = (3,4,3)\) and

$$\begin{aligned} (\Phi ^1, \Phi ^2,\Phi ^3) = 4 \pi h (3q_1^{-1},4q_2^{-1}, 3q_3^{-1}) \end{aligned}$$
(4.11)

with \(h_1 = h_2 = h_3 = h\).

\(C_2\) -case. For the Lie algebra \(C_2 = so(5)\) with the Cartan matrix

$$\begin{aligned} \left( A_{ss'}\right) = \left( \begin{array}{*{6}{c}} 2 &{} \quad -1\\ -2&{} \quad 2\\ \end{array} \right) \quad \end{aligned}$$
(4.12)

we get \(n_1 = 3\) and \(n_2 = 4\). For \(C_2\)-polynomials we obtain [25, 32]

$$\begin{aligned} H_1= & {} 1+ 3 p_1 z+ 3 p_1 p_2 z^2 + p_1^2 p_2 z^3, \end{aligned}$$
(4.13)
$$\begin{aligned} H_2= & {} 1+ 4 p_2 z+ 6 p_1 p_2 z^2 + 4 p_1^2 p_2 z^3 + p_1^2 p_2^2 z^4. \end{aligned}$$
(4.14)

In this case we find

$$\begin{aligned} (\Phi ^1, \Phi ^2) = 4 \pi (3 h_1 q_1^{-1}, 4 h_2 q_2^{-1}) \end{aligned}$$
(4.15)

where \(h_1 = 2 h_2\).

\(G_2\) -case. For the Lie algebra \(G_2\) with the Cartan matrix

$$\begin{aligned} \left( A_{ss'}\right) = \left( \begin{array}{*{6}{c}} 2 &{} \quad -1\\ -3&{} \quad 2\\ \end{array} \right) \quad \end{aligned}$$
(4.16)

we get \(n_1 = 6\) and \(n_2 = 10\). In this case the fluxbrane polynomials read [25, 32]

$$\begin{aligned} H_{1}= & {} 1+ 6 p_1 z+ 15 p_1 p_2 z^2 + 20 p_1^2 p_2 z^3 \nonumber \\&+\,15 p_1^3 p_2 z^4 + 6 p_1^3 p_2^2 z^5 + p_1^4 p_2^2 z^6 , \end{aligned}$$
(4.17)
$$\begin{aligned} H_2= & {} 1+ 10 p_2 z + 45 p_1 p_2 z^2 + 120 p_1^2 p_2 z^3\nonumber \\&+\, p_1^2 p_2( 135 p_1 + 75 p_2) z^4 \nonumber \\&+\,252 p_1^3 p_2^2 z^5 +\,p_1^3 p_2^2 \biggl (75 p_1 + 135 p_2 \biggr )z^6 \nonumber \\&+\,120 p_1^4 p_2^3 z^7\nonumber \\&+\,45 p_1^5 p_2^3 z^8 + 10 p_1^6 p_2^3 z^9 + p_1^{6} p_2^{4} z^{10}. \end{aligned}$$
(4.18)

We are led to the relations

$$\begin{aligned} (\Phi ^1, \Phi ^2) = 4 \pi (6 h_1 q_1^{-1}, 10 h_2 q_2^{-1}) \end{aligned}$$
(4.19)

where \(h_1 = 3 h_2\).

\((A_1 + A_1)\) -case. For the semi-simple Lie algebra \(A_1 + A_1\) we obtain \(n_1 = n_2 = 1\),

$$\begin{aligned} H_{1} = 1 + p_1 z, \quad H_{2} = 1 + p_2 z, \end{aligned}$$
(4.20)

and

$$\begin{aligned} (\Phi ^1, \Phi ^2) = 4 \pi (q_1^{-1} h_1, q_2^{-1} h_2), \end{aligned}$$
(4.21)

where \(h_1\) and \(h_2\) are independent, as well as the quantities \(q_1 \Phi ^1\) and \(q_2 \Phi ^2\).

5 Conclusions

Here we have considered a multidimensional generalization of Melvin’s solution corresponding to a simple finite-dimensional Lie algebra \(\mathcal{G}\). We have assumed that the solution is governed by a set of n fluxbrane polynomials \(H_s(z)\), \(s =1,\dots ,n\). These polynomials define special solutions to open Toda chain equations corresponding to the Lie algebra \(\mathcal{G}\).

The polynomials \(H_s(z)\) depend also upon parameters \(q_s\), which are coinciding for \(D =4\) (up to a sign) with the values of colored magnetic fields on the axis of symmetry.

We have calculated 2d flux integrals \(\Phi ^s = \int F^s\), \(s =1, \dots , n\). Any flux \(\Phi ^s\) depends only upon one parameter \(q_s\), while the integrand \(F^s\) depends upon all parameters \(q_1, \dots , q_n\). The relation for flux integrals (3.11) is also valid when the matrix \((A_{ss'})\) is a Cartan matrix of a finite-dimensional semi-simple Lie algebra \(\mathcal G\).

Here we have considered examples of polynomials and fluxes for the Lie algebras \(A_1\), \(A_2\), \(A_3\), \(C_2\), \(G_2\) and \(A_1 + A_1\). The approach of this paper will be used for a calculation of certain flux integrals for forms \(F^s\) of arbitrary ranks corresponding to certain fluxbrane solutions (of electric type by p-brane notation or magnetic type by fluxbrane classificationFootnote 1) governed by fluxbrane polynomials [38].

An open problem is to find the fluxes for the solutions which are related to infinite-dimensional Lorentzian Kac–Moody algebras, e.g. hyperbolic ones [39, 40]. In this case one should deal with phantom scalar fields in the model (2.1) and non-polynomial solutions to Eqs. (2.6). Another possibility is to study the convergence of flux integrals for non-polynomial solutions for moduli functions corresponding to non-Cartan matrices \((A_{ss'})\) (e.g. for the model with two 2-forms from Ref. [41]).