# Duality Identities for Moduli Functions of Generalized Melvin Solutions Related to Classical Lie Algebras of Rank 4

Authors: S. V. Bolokhov; V. D. Ivashchuk
Journal: Advances in Mathematical Physics
 (2018)

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## Abstract

We consider generalized Melvin-like solutions associated with nonexceptional Lie algebras of rank4 (namely, A4, B4, C4, and D4) corresponding to certain internal symmetries of the solutions. The system under consideration is a static cylindrically symmetric gravitational configuration in D dimensions in presence of four Abelian 2-forms and four scalar fields. The solution is governed by four moduli functions Hs(z) (s=1,…,4) of squared radial coordinate z=ρ2 obeying four differential equations of the Toda chain type. These functions turn out to be polynomials of powers (n1,n2,n3,n4)=(4,6,6,4),(8,14,18,10),(7,12,15,16),(6,10,6,6) for Lie algebras A4, B4, C4, and D4, respectively. The asymptotic behaviour for the polynomials at large distances is governed by some integer-valued 4×4 matrix ν connected in a certain way with the inverse Cartan matrix of the Lie algebra and (in A4 case) the matrix representing a generator of the Z2-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 2-dimensional discs and corresponding Wilson loop factors over their boundaries.


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## Body

## 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, n Abelian 2-forms and l≥n scalar fields. Here we consider a special solutions with n=l=4, governed by a 4 × 4 Cartan matrix (Aij) for simple nonexceptional Lie algebras of rank 4: A4, B4, C4, and D4. 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 andM-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 Hs(z)>0 defined on the interval (0,+∞), where z=ρ2 and ρ is a radial coordinate. Functions Hs(z) obey n nonlinear differential master equations of Toda-like type governed by some matrix (Ass′), and the following boundary conditions are imposed: Hs(+0)=1, s=1,…,n.Here, as in [2], we assume that the matrix (Ass′) is a Cartan matrix for some simple finite-dimensional Lie algebra G of rank n (Ass=2 for all s). 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: (1)Hsz=1+∑k=1nsPskzk,where Ps(k) are constants. Here Ps(ns)≠0 and (2)ns=2∑s′=1nAss′,where we denote (Ass′)=Ass′-1. Integers ns are components of the twice dual Weyl vector in the basis of simple (co)roots [22].Therefore, the functionsHs (which may be called “fluxbrane polynomials”) define a special solution to open Toda chain equations [23, 24] corresponding to simple finite-dimensional Lie algebra G [25]. In [2, 26] a program (in Maple) for calculation of these polynomials for classical series of Lie algebras (A-, B-, C-, and D-series) was suggested. It was pointed out in [3] that the conjecture on polynomial structure of Hs(z) is valid for Lie algebras of A- and C- 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 transformationρ→1/ρ, which makes the model in tune with T-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 B4 and C4) or isomorphic to the group Z2 (for Lie algebra A4) or isomorphic to the group S3 which is the group of permutation of 3 elements (for Lie algebra D4). These duality identities may be used in deriving a 1/ρ-expansion for solutions at large distances ρ. The corresponding asymptotic behaviour of the solutions is studied.The analogous analysis was performed recently for the case of rank-2 Lie algebras:A2, B2=C2, G2 in [27], and for rank-3 algebras A3, B3, and C3 in [28]. Also, in [29] the conjecture from [3] was verified for the Lie algebra E6 and certain duality relations for six E6-polynomials were found.The paper is organized as follows. In Section2 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 A4, B4, C4, and D4. We find symmetry properties and duality relations for polynomials and present asymptotic relations for the solutions. We also calculate 2-form flux integrals Φs(R)=∫DRFs and corresponding Wilson loop factors, where Fs are 2-forms and DR is 2-dimensional disc of radius R. The flux integrals converge, i.e., have finite limits for R=+∞ [30].


## 2. The Setup and Generalized Melvin Solutions

Let us consider the following product manifold:(3)M=0,+∞×M1×M2,where M1=S1 and M2 is a (D-2)-dimensional Ricci-flat manifold.On this manifold, we define the action(4)S=∫dDxgRg-δabgMN∂Mφa∂Nφb-12∑s=14exp2λ→sφ→Fs2,where g=gMN(x)dxM⊗dxN is a metric on M, φ→=(φa)∈R4 is vector of scalar fields, Fs=dAs=1/2FMNsdxM∧dxN is a 2-form, and λ→s=(λsa)∈R4 is dilatonic coupling vector, s=1,…,4; a=1,…,4. Here we use the notations g≡detgMN; Fs2≡FM1M2sFN1N2sgM1N1gM2N2.There is a family of exact cylindrically symmetric solutions to the field equations corresponding for the action (4) and depending on the radial coordinate ρ. The solution has the form [2](5)g=∏s=14Hs2hs/D-2dρ⊗dρ+∏s=14Hs-2hsρ2dϕ⊗dϕ+g2,(6)expφa=∏s=14Hshsλsa,(7)Fs=qs∏l=14Hl-Aslρdρ∧dϕ,s,a=1,…,4, where g1=dϕ⊗dϕ is a metric on M1=S1 and g2 is a Ricci-flat metric of signature (-,+,…,+) on M2. Here qs≠0 are integration constants (qs=-Qs in notations of [2]).For further convenience, let us denotez=ρ2. As it was shown in earlier works, the functions Hs(z)>0 obey the set of master equations (8)ddzzHsddzHs=Ps∏l=14Hl-Asl,with the boundary conditions (9)Hs+0=1,where (10)Ps=14Ksqs2,s=1,…,4. The boundary condition (9) guarantees the absence of a conic singularity (for the metric (5)) for ρ=+0.There are some relations for the parametershs: (11)hs=Ks-1,Ks=Bss>0,where (12)Bsl≡1+12-D+λ→sλ→l,s,l=1,…,4. In these relations, we have denoted (13)Asl=2BslBll.The latter matrix is the so-called “quasi-Cartan” matrix. One can prove that if (Asl) is a Cartan matrix for a certain simple Lie algebra G of rank 4 then there exists a set of vectors λ→1,…,λ→4 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 D=4, one (electromagnetic) 2-form, M1=S1 (0<ϕ<2π), M2=R2, and g2=-dt⊗dt+dx⊗dx.


## 3. Solutions Related to Simple Classical Rank-4 Lie Algebras

In this section we consider the solutions associated with the simple nonexceptional Lie algebrasG of rank 4. This means than the matrix A=(Asl) coincides with one of the Cartan matrices (14)Ass′=2-100-12-100-12-100-12,2-100-12-100-12-200-12,2-100-12-100-12-100-22,2-100-12-1-10-1200-102forG=A4,B4,C4,D4, respectively.Each of these matrices can be graphically described by drawing the Dynkin diagrams pictured on Figure1 for these four Lie algebras.Figure 1
The Dynkin diagrams for the Lie algebrasA4, B4, C4, and D4, respectively.Using (11)-(13) we can get(15)Ks=D-3D-2+λ→s2,where hs=Ks-1 and(16)λ→sλ→l=12KlAsl-D-3D-2≡Gsl,s,l=1,2,3,4; (15) is a special case of (16).From (11) and (13) it also follows that (17)hshl=KlKs=BllBss=BlsBssBllBsl=AlsAslfor any s≠l obeying Asl,Als≠0. This implies (18)K1=K2=K3=K,K4=K,12K,2K,K,or (19)h1=h2=h3=h,h4=h,2h,12h,h(h=K-1) for G=A4,B4,C4,D4, respectively.Remark 1.
For large enoughK1 in (18) there exist vectors λ→s obeying (16) (and hence (15)). Indeed, the matrix (Gsl) is positive definite if K1>K∗, where K∗ is some positive number. Hence there exists a matrix Λ, such that ΛTΛ=G. We put (Λas)=(λsa) and get the set of vectors obeying (16).Polynomials.According to the polynomial conjecture, the set of moduli functions H1(z),…,H4(z), obeying (8) and (9) with the Cartan matrix A=(Asl) from (14) are polynomials with powers (n1,n2,n3,n4)=(4,6,6,4), (8,14,18,10), (7,12,15,16), (6,10,6,6) calculated by using (2) for Lie algebras A4, B4, C4, and D4, 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]): (20)ps=Psns.A 4-Case.For the Lie algebra A4≅sl(5) we have (21)H1=1+4p1z+6p1p2z2+4p1p2p3z3+p1p2p3p4z4,(22)H2=1+6p2z+6p1p2+9p2p3z2+16p1p2p3+4p2p3p4z3+6p1p22p3+9p1p2p3p4z4+6p1p22p3p4z5+p1p22p32p4z6,(23)H3=1+6p3z+9p2p3+6p3p4z2+4p1p2p3+16p2p3p4z3+9p1p2p3p4+6p2p32p4z4+6p1p2p32p4z5+p1p22p32p4z6,(24)H4=1+4p4z+6p3p4z2+4p2p3p4z3+p1p2p3p4z4.B 4-Case.For the Lie algebra B4≅so(9) the fluxbrane polynomials are (25)H1=1+8p1z+28p1p2z2+56p1p2p3z3+70p1p2p3p4z4+56p1p2p3p42z5+28p1p2p32p42z6+8p1p22p32p42z7+p12p22p32p42z8,(26)H2=1+14p2z+28p1p2+63p2p3z2+224p1p2p3+140p2p3p4z3+196p1p22p3+630p1p2p3p4+175p2p3p42z4+980p1p22p3p4+896p1p2p3p42+126p2p32p42z5+490p1p22p32p4+1764p1p22p3p42+700p1p2p32p42+49p22p32p42z6+3432p1p22p32p42z7+49p12p22p32p42+700p1p23p32p42+1764p1p22p33p42+490p1p22p32p43z8+126p12p23p32p42+896p1p23p33p42+980p1p22p33p43z9+175p12p23p33p42+630p1p23p33p43+196p1p22p33p44z10+140p12p23p33p43+224p1p23p33p44z11+63p12p23p33p44+28p1p23p34p44z12+14p12p23p34p44z13+p12p24p34p44z14,(27)H3=1+18p3z+63p2p3+90p3p4z2+56p1p2p3+560p2p3p4+200p3p42z3+630p1p2p3p4+630p2p32p4+1575p2p3p42+225p32p42z4+1260p1p2p32p4+2016p1p2p3p42+5292p2p32p42z5+490p1p22p32p4+9996p1p2p32p42+1225p22p32p42+5103p2p33p42+1750p2p32p43z6+5616p1p22p32p42+12600p1p2p33p42+3528p22p33p42+5040p1p2p32p43+5040p2p33p43z7+441p12p22p32p42+17172p1p22p33p42+4410p1p22p32p43+15750p1p2p33p43+4410p22p33p43+1575p2p33p44z8+2450p12p22p33p42+5600p1p23p33p42+32520p1p22p33p43+5600p1p2p33p44+2450p22p33p44z9+1575p12p23p33p42+4410p12p22p33p43+15750p1p23p33p43+4410p1p22p34p43+17172p1p22p33p44+441p22p34p44z10+5040p12p23p33p43+5040p1p23p34p43+3528p12p22p33p44+12600p1p23p33p44+5616p1p22p34p44z11+1750p12p23p34p43+5103p12p23p33p44+1225p12p22p34p44+9996p1p23p34p44+490p1p22p34p45z12+5292p12p23p34p44+2016p1p23p35p44+1260p1p23p34p45z13+225p12p24p34p44+1575p12p23p35p44+630p12p23p34p45+630p1p23p35p45z14+200p12p24p35p44+560p12p23p35p45+56p1p23p35p46z15+90p12p24p35p45+63p12p23p35p46z16+18p12p24p35p46z17+p12p24p36p46z18,(28)H4=1+10p4z+45p3p4z2+70p2p3p4+50p3p42z3+35p1p2p3p4+175p2p3p42z4+126p1p2p3p42+126p2p32p42z5+175p1p2p32p42+35p2p32p43z6+50p1p22p32p42+70p1p2p32p43z7+45p1p22p32p43z8+10p1p22p33p43z9+p1p22p33p44z10.C 4-Case.For the Lie algebra C4≅sp(6) we get the following polynomials:(29)H1=1+7p1z+21p1p2z2+35p1p2p3z3+35p1p2p3p4z4+21p1p2p32p4z5+7p1p22p32p4z6+p12p22p32p4z7,(30)H2=1+12p2z+21p1p2+45p2p3z2+140p1p2p3+80p2p3p4z3+105p1p22p3+315p1p2p3p4+75p2p32p4z4+420p1p22p3p4+336p1p2p32p4+36p22p32p4z5+924p1p22p32p4z6+36p12p22p32p4+336p1p23p32p4+420p1p22p33p4z7+75p12p23p32p4+315p1p23p33p4+105p1p22p33p42z8+80p12p23p33p4+140p1p23p33p42z9+45p12p23p33p42+21p1p23p34p42z10+12p12p23p34p42z11+p12p24p34p42z12,(31)H3=1+15p3z+45p2p3+60p3p4z2+35p1p2p3+320p2p3p4+100p32p4z3+315p1p2p3p4+1050p2p32p4z4+1302p1p2p32p4+576p22p32p4+1125p2p33p4z5+1050p1p22p32p4+2240p1p2p33p4+1215p22p33p4+500p2p33p42z6+225p12p22p32p4+3990p1p22p33p4+1260p1p2p33p42+960p22p33p42z7+960p12p22p33p4+1260p1p23p33p4+3990p1p22p33p42+225p22p34p42z8+500p12p23p33p4+1215p12p22p33p42+2240p1p23p33p42+1050p1p22p34p42z9+1125p12p23p33p42+576p12p22p34p42+1302p1p23p34p42z10+1050p12p23p34p42+315p1p23p35p42z11+100p12p24p34p42+320p12p23p35p42+35p1p23p35p43z12+60p12p24p35p42+45p12p23p35p43z13+15p12p24p35p43z14+p12p24p36p43z15,(32)H4=1+16p4z+120p3p4z2+160p2p3p4+400p32p4z3+70p1p2p3p4+1350p2p32p4+400p32p42z4+672p1p2p32p4+1296p22p32p4+2400p2p32p42z5+1400p1p22p32p4+1512p1p2p32p42+4096p22p32p42+1000p2p33p42z6+400p12p22p32p4+5600p1p22p32p42+1120p1p2p33p42+4320p22p33p42z7+2025p12p22p32p42+8820p1p22p33p42+2025p22p34p42z8+4320p12p22p33p42+1120p1p23p33p42+5600p1p22p34p42+400p22p34p43z9+1000p12p23p33p42+4096p12p22p34p42+1512p1p23p34p42+1400p1p22p34p43z10+2400p12p23p34p42+1296p12p22p34p43+672p1p23p34p43z11+400p12p24p34p42+1350p12p23p34p43+70p1p23p35p43z12+400p12p24p34p43+160p12p23p35p43z13+120p12p24p35p43z14+16p12p24p36p43z15+p12p24p36p44z16.D 4-Case.For the Lie algebra D4≅so(8) we obtain the polynomials(33)H1=1+6p1z+15p1p2z2+10p1p2p3+10p1p2p4z3+15p1p2p3p4z4+6p1p22p3p4z5+p12p22p3p4z6,(34)H2=1+10p2z+15p1p2+15p2p3+15p2p4z2+40p1p2p3+40p1p2p4+40p2p3p4z3+25p1p22p3+25p1p22p4+135p1p2p3p4+25p22p3p4z4+252p1p22p3p4z5+25p12p22p3p4+135p1p23p3p4+25p1p22p32p4+25p1p22p3p42z6+40p12p23p3p4+40p1p23p32p4+40p1p23p3p42z7+15p12p23p32p4+15p12p23p3p42+15p1p23p32p42z8+10p12p23p32p42z9+p12p24p32p42z10,(35)H3=1+6p3z+15p2p3z2+10p1p2p3+10p2p3p4z3+15p1p2p3p4z4+6p1p22p3p4z5+p1p22p32p4z6,(36)H4=1+6p4z+15p2p4z2+10p1p2p4+10p2p3p4z3+15p1p2p3p4z4+6p1p22p3p4z5+p1p22p3p42z6.Let us denote(37)Hs≡Hsz=Hsz,pi,pi≡p1,p2,p3,p4.One can easily write down the asymptotic behaviour of the polynomials obtained:(38)Hs=Hsz,pi~∏l=14plνslzns≡Hsasz,pi,asz→∞,where we introduced the integer-valued matrix ν=(νsl) having the form(39)ν=1111122112211111,2222244424661234,2221244224632464,2211242212211212for Lie algebras A4,B4,C4,D4, respectively. In these four cases there is a simple property(40)∑l=14νsl=ns,s=1,2,3,4.Note that for Lie algebrasB4, C4, and D4 we have(41)ν=2A-1,G=B4,C4,D4where A-1 is inverse Cartan matrix, whereas in the A4-case the matrix ν is related to the inverse Cartan matrix as follows:(42)ν=A-1I+P,G=A4.Here I is 4×4 identity matrix and(43)P=0001001001001000is a permutation matrix corresponding to the permutation σ∈S4 (S4 is symmetric group)(44)σ:1,2,3,4↦4,3,2,1,by the following relation P=(Pji)=(δσ(j)i). Here σ is the generator of the group G={σ,id} which is the group of symmetry of the Dynkin diagram for A4. G is isomorphic to the group Z2.In case ofD4 the group of symmetry of the Dynkin diagram G′ is isomorphic to the symmetric group S3 acting on the set of three vertices {1,3,4} of the Dynkin diagram via their permutations. The existence of the above symmetry groups G≅Z2 and G′≅S3 implies certain identity properties for the fluxbrane polynomials Hs(z).Let us denotep^i=pσ(i) for the A4 case and p^i=pi for B4, C4, and D4 cases (i=1,2,3,4). We call the ordered set (p^i) asdual one to the ordered set (pi). It corresponds to the action (trivial or nontrivial) of the group Z2 on vertices of the Dynkin diagrams for above algebras.Then we obtain the following identities which were directly verified by using MATHEMATICA algorithms.Symmetry RelationsProposition 2.
The fluxbrane polynomials obey for allpi and z>0 the identities (45)Hσsz,pi=Hsz,p^iforA4case,Hσ′sz,pi=Hs(z,pσ′iforD4case,for any σ′∈S3, s=1,…,4. We call relations (45) as symmetry ones.Duality RelationsProposition 3.
The fluxbrane polynomials corresponding to Lie algebrasA4, B4, C4, and D4 obey for all pi>0 and z>0 the identities(46)Hsz,pi=Hsasz,piHsz-1,p^i-1,s=1,2,3,4.We call relations (46) as duality ones.Fluxes.Here we deal with an oriented 2-dimensional manifold MR=(0,R)×S1, R>0. One can define the flux integrals over this manifold:(47)ΦsR=∫MRFs=2π∫0RdρρBs,where we denoted(48)Bs=qs∏l=14Hl-Asl.It can be easily understood that total flux integrals Φs=Φs(+∞) are convergent. Indeed, due to polynomial assumption (1) we have (49)Hs~Csρ2ns,Cs=∏l=14plνsl,as ρ→+∞. From (48), (49), and the equality ∑1nAslnl=2, following from (2), we get (50)Bs~qsCsρ-4,Cs=∏l=14pl-Aνsl,and hence the integral (47) is convergent for any s=1,2,3,4.Using (42) and (50) we have for the A4-case (51)Cs=∏l=14pl-I+Psl=∏l=14pl-δsl-δσsl=ps-1pσs-1.Similarly, due to (41) and (50) we get for Lie algebras B4, C4, and D4(52)Cs=ps-2.After that, we can calculate the fluxes Φs(R). Using the master equations (8) one can write (53)∫0RdρρBs=qsPs-112∫0R2dzddzzHsddzHs=12qsPs-1R2Hs′R2HsR2,where Hs′=dHs/dz. Thus, using (47) we easily obtain (54)ΦsR=4πqs-1hsR2Hs′R2HsR2.Note that the manifold M∗=(0,+∞)×S1 is isomorphic to the manifold R∗2=R2∖{0}. Therefore, one can understand the family of solutions under consideration as defined on the manifold R∗2×M2, where coordinates ρ, ϕ are polar coordinates in a domain of R∗2: x=ρcosϕ and y=ρsinϕ, where x,y are standard coordinates of R2. It was shown in [30] that there exist forms As globally defined on R2 and obeying the relation Fs=dAs.Now let us we consider a 2-dimensional oriented manifold (disk)DR=x,y:x2+y2≤R2. Its boundary ∂DR≡CR={(x,y):x2+y2=R2} is a circle of radius R, i.e., 1-dimensional oriented manifold with the orientation inherited from that of DR obeying the relation ∫CRdϕ=2π.The Stokes theorem yields in this case(55)ΦsR=∫MRFs=∫DRdAs=∫CRAs.According to the definition of Abelian Wilson loop (factor), we have (56)WsCR=expi∫CRAs=expiΦsR.Relations (1) and (54) imply (see (10)) (57)Φs=Φs+∞=4πnsqs-1hs,s=1,2,3,4. Any (total) flux Φs depends upon one integration constant qs≠0, while the integrand form Fs depends upon all constants: q1,q2,q3,q4. As a consequence, we obtain finite limits (58)limR→+∞WsCR=expiΦs.In theA4-case we have (59)q1Φ1,q2Φ2,q3Φ3,q4Φ4=4πh4,6,6,4,where h1=h2=h3=h4=h.In theB4-case we find (60)q1Φ1,q2Φ2,q3Φ3,q4Φ4=4π8h1,14h2,18h3,10h4=4πh8,14,18,20,where h1=h2=h3=h, h4=2h.In theC4-case we obtain (61)q1Φ1,q2Φ2,q3Φ3,q4Φ4=4π7h1,12h2,15h3,16h4=4πh7,12,15,8,where h1=h2=h3=h, h4=1/2h.In theD4-case we are led to relations (62)q1Φ1,q2Φ2,q3Φ3,q4Φ4=4πh6,10,6,6,where h1=h2=h3=h4=h. (In all examples relations (19) are used.)Note that, forD=4 and g2=-dt⊗dt+dx⊗dx, qs coincides with the value of the x-component of the s-th magnetic field on the axis of symmetry, s=1,2,3,4.Asymptotic Relations.Here we can write down the asymptotic relations for the solution under consideration as ρ→+∞: (63)gas=∏l=14plal2/D-2ρ2Adρ⊗dρ+∏l=14plal-2ρ2-2AD-2dϕ⊗dϕ+g2,(64)φasa=∑s=14hsλsa∑l=14νsllnpl+2nslnρ,(65)Fass=qsps-1pθs-1ρ-3dρ∧dϕ,a,s=1,2,3,4, where (66)al=∑s=14hsνsl,A=2D-2-1∑s=14nshs,and in (65) we put θ=σ for G=A4 and θ=id for G=B4,C4,D4. In derivation of asymptotic relations, (40), (49), and (51) were used. We note that for G=B4,C4,D4 the asymptotic value of form Fass depends upon qs, s=1,2,3,4, while in the A4-case Fass depends upon q1 and q4 for s=1,4, and Fass depends upon q2,q3 for s=2,3.We note also that by puttingq1=0 we get the Melvin-type solutions corresponding to Lie algebras A3, B3, C3, and A3, respectively, which were analyzed in [28]. (The case of the rank 2 Lie algebra G2 [27] may be obtained for the D4 case when q1=q3=q4.)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 (67)M=R0,+∞×M0=S2×M1=R×M2,where R0=2μ>0 and M2 is a Ricci-flat manifold. These DBH solutions on M from (67) for the model under consideration may be extracted from general black brane solutions; see [21, 25, 31]. They read (68)g=∏s=14Hs2hs/D-2f-1dR⊗dR+R2g0-∏s=14H-2hsfdt⊗dt+g2,(69)expφa=∏s=14Hhsλsa,(70)Fs=-QsR-2∏l=14Hl-AsldR∧dt,s,a=1,2,3,4, where f=1-2μR-1, g0 is the standard metric on M0=S2, and g2 is a Ricci-flat metric of signature (+,…,+) on M2. Here Qs≠0 are integration constants (charges).The functionsHs=Hs(R)>0 obey the master equations (71)R2ddRfR2HsddRHs=Bs∏l=14Hl-Asl,with the following boundary conditions on the horizon and at infinity imposed: (72)HsR0+0=Hs0>0,Hs+∞=1,where (73)Bs=-KsQs2,s=1,2,3,4. Here relations (11) are also valid. For Lie algebras of rank 4 the functions Hs are polynomials with respect to R-1, which may be obtained (at least for small enough qs) from fluxbrane polynomials Hs(z) 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:G=A4,B4,C4,D4. Each solution of that family is governed by a set of 4 fluxbrane polynomials Hs(z), s=1,2,3,4. These polynomials define special solutions to open Toda chain equations corresponding to the Lie algebra G.The polynomialsHs(z) depend also upon parameters qs, which coincides for D=4 (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 deriving1/ρ-expansion for solutions at large distances ρ, e.g., for asymptotic relations which are presented in the paper.There were also calculated two-dimensional flux integralsΦs(R)=∫DRFs (s=1,2,3,4) over a disc DR of radius R and a corresponding Wilson loop factors Ws(CR) over a circle CR. It turns out that each total flux Φs(∞) depends only upon one corresponding parameter qs, whereas the integrand Fs depends on all parameters qs, s=1,2,3,4.The case of exceptional Lie algebraF4 will be considered in a separate publication.



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