We associate to outer automorphisms of generalized Kac-Moody algebras
generalized character-valued indices, the twining characters. A character
formula for twining characters is derived which shows that they coincide with
the ordinary characters of some other generalized Kac-Moody algebra, the
so-called orbit Lie algebra. Some applications to problems in conformal field
theory, algebraic geometry and the theory of sporadic simple groups are
sketched.Comment: 6 pages, LaTeX, Talk given by C. Schweigert at the XXI international
  colloquium on group theoretical methods in physics, July 1996, Goslar,
  German

Fuchs, J.

Ray, U.

Schellekens, A. N.

Schweigert, C.

English

arXiv

We associate to outer automorphisms of generalized Kac-Moody algebras generalized character-valued indices, the twining characters. A character formula for twining characters is derived which shows that they coincide with the ordinary characters of some other generalized Kac-Moody algebra, the so-called orbit Lie algebra. Some applications to problems in conformal field theory, algebraic geometry and the theory of sporadic simple groups are sketched

Fuchs, Jurgen

Ray, Urmie

Schellekens, Bert

Schweigert, Christoph

UNT Digital Library

    LBNL-39653      TWINING CHARACTERS AND ORBIT LIE ALGEBRAS  Jürgen Fuchs 1 , Urmie Ray 2 , Bert Schellekens 3 and Christoph Schweigert 4  1 DESY, Notkestraße 85, D – 22603 Hamburg 2 IHES, 35, route de Chartres, F – 91440 Bures-sur-Yvette 3 NIKHEF/FOM, Kruislaan 409, NL – 1098 SJ Amsterdam 4 Department of Physics, University of California, Berkeley, CA 94720, USA, and Theoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA                   This work was supported in part by the Director, Office of Science, Office of High Energy Physics, of the U.S. Department of Energy under Contract No. DE-AC03-76SF00098.  DISCLAIMER  This document was prepared as an account of work sponsored by the United States Government. While this document is believed to contain correct information, neither the United States Government nor any agency thereof, nor The Regents of the University of California, nor any of their employees, makes any warranty, express or implied, or assumes any legal responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by its trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof, or The Regents of the University of California. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof or The Regents of the University of California. arXiv:hep-th/9612060v1  5 Dec 1996hep-th/9612060UCB-PTH-96/56, LBNL-39653NIKHEF 96-030, IHES/M/96/79November 1996TWINING CHARACTERS ANDORBIT LIE ALGEBRAS †Jürgen Fuchs 1 , Urmie Ray 2 , Bert Schellekens 3 and Christoph Schweigert 41 DESY, Notkestraße 85, D – 22603 Hamburg2 IHES, 35, route de Chartres, F – 91440 Bures-sur-Yvette3 NIKHEF/FOM, Kruislaan 409, NL – 1098 SJ Amsterdam4 Department of Physics, University of California, Berkeley, CA 94720, USA, andTheoretical Physics Group, Lawrence Berkeley National Laboratory,Berkeley, CA 94720, USAAbstract.We associate to outer automorphisms of generalized Kac–Moody algebras gener-alized character-valued indices, the twining characters. A character formula fortwining characters is derived which shows that they coincide with the ordinarycharacters of some other generalized Kac–Moody algebra, the so-called orbit Lie al-gebra. Some applications to problems in conformal field theory, algebraic geometryand the theory of sporadic simple groups are sketched.——————† Slightly extended version of a talk given by C. Schweigert at the XXI International Colloquium onGroup Theoretical Methods in Physics (Goslar, Germany, July 1996)11 Generalized Kac–Moody algebrasGeneralized Kac--Moody algebras constitute a class of Lie algebras which comprise many Lie algebrasthat describe symmetries in physical systems. In particular, they include the finite-dimensional simpleLie algebras (i.e. the four series of classical Lie algebras and the five exceptional simple Lie algebras),twisted and untwisted affine Lie algebras (i.e. centrally extended loop algebras), hyperbolic Lie algebras(such as E10), as well as the Monster Lie algebra and its various relatives. Moreover, any chiral algebraof a conformal field theory, i.e. any vertex operator algebra, gives rise to a generalized Kac--Moodyalgebra.Any finite-dimensional simple Lie algebra is generated (as a Lie algebra) by several copies of sl(2),one copy for each simple root. This is also true for ordinary Kac--Moody algebras; a generalizedKac--Moody algebra, however, is generated by copies of sl(2) and of the Heisenberg algebra whichhas a basis {e, f,K} with K a central element and non-trivial Lie bracket [e, f ] = K. Nonetheless,generalized Kac--Moody algebras can still be characterized by a square matrix, the Cartan matrixA = (aij)i,j∈I . The index set I can either be finite, I = {1, 2, ... , n}, or countably infinite, I = Z+.In case of finite-dimensional simple Lie algebras the entries of A are integers; here we allow for realentries, but still we keep the following properties of A:(i) aij ≤ 0 if i 6= j; (ii)2aijaii∈ Z if aii > 0;(iii) if aij = 0, then aji = 0;(iv) there exists a diagonal matrix D = diag(ǫ1, ..., ǫn), with ǫi a positivereal number for all i, such that DA is symmetric.(1.1)From these data the generalized Kac--Moody algebra is constructed by the same procedure that isused to construct a finite-dimensional simple Lie algebra from its Cartan matrix. One starts with anabelian Lie algebra g0 of dimension greater or equal to n and fixes n linearly independent elementsh1, ... , hn of g0 and n linear forms αj on g0, the simple roots, such that αj(hi) = aij. The generalizedKac--Moody algebra g = g(A) with Cartan matrix A and Cartan subalgebra g0 is then the Lie algebragenerated by ei, fi , i∈I, and g0, modulo the relations[ei, fj ] = δijhi , [h, ei] = αi(h)ei , [h, fi] = −αi(h)fi ,(ad ei)1−2aij/aiiej = 0 = (ad fi)1−2aij/aiifj if aii > 0 ,[ei, ej ] = 0 = [fi, fj ] if aij = 0 .(1.2)2 Automorphisms of generalized Kac–Moody algebrasWe now turn to the main object of our interest: a class of outer automorphisms of generalizedKac--Moody algebras and some new structures associated to them. Such automorphisms occur natu-rally in many physical applications; for some examples see the last section and [5]. We start with apermutation ω̇ of finite order of the index set I which leaves the Cartan matrix A invariant:aω̇i,ω̇j = ai,j . (2.3)If the generalized Kac--Moody algebra has a Dynkin diagram, ω̇ corresponds to a symmetry of theDynkin diagram. Such a permutation ω̇ induces an automorphism ω : g → g of the Lie algebra g,2which is defined by its action on the generators ei, fi and hi:1ω(ei) := eω̇i , ω(fi) := fω̇i , ω(hi) := hω̇i . (2.4)The automorphism ω gives rise to a map τω : HΛ → Hω⋆Λ of integrable highest weight modules(and analogously also for Verma modules) which ‘ω-twines’ the action of g:τω(RΛ(x) · v) = Rω⋆Λ(ω(x)) · τω(v) , (2.5)and maps a fixed highest weight vector vΛ of HΛ to a highest weight vector vω⋆Λ of Hω⋆Λ. Analogousmaps can also be defined for Verma modules; they are compatible with the submodule structure ofthe Verma modules and hence also with their null vector structures. Pictorially, the map τω can berepresented as follows:HΛ Hω⋆Λv = xvΛ τωv = ω(x)vω⋆Λτωx∈U(g−) ω(x)∈U(g−)vΛ vω⋆Λτω3 Fixed pointsNote that τω is generically a linear map from a module HΛ to a different module Hω⋆Λ. A particularlyinteresting situation occurs when the highest weight Λ is symmetric, i.e. satisfies ω⋆(Λ) = Λ. In thiscase the heighest weight module HΛ is called a fixed point of ω and τω is an endomorphism of thevector space underlying the module HΛ. To keep track of some properties of the endomorphism τω,we introduce a character-like object, the twining character χ[ω]Λ :χ[ω]Λ (h) = trHΛτω e2πiRΛ(h) for h∈g0 . (3.6)Like ordinary characters the twining characters are (formal) functions on the Cartan subalgebra. Theyare the generating functions of the trace of τω on the weight spaces, i.e. they are generalized (since ω1 In general, one also must describe the action of ω on the derivations of the generalized Kac–Moody algebra.For more details see section 3 of [2].3does not necessarily have order two) character-valued indices. Clearly, for the trivial automorphismω = 1 we recover the ordinary character of the module HΛ.It is a rather surprising result [2, 1] that the twining character is essentially identical to thecharacter of some other generalized Kac--Moody algebra ğ, called the orbit Lie algebra. This resultmakes the twining characters explicitly computable. In particular, it implies that the coefficients inthe expansion of the twining character are not arbitrary complex numbers, but non-negative integers,and hence justifies a posteriori the name twining character.The orbit Lie algebra corresponding to g and its diagram automorphism ω is obtained by a simpleprescription which corresponds to folding the Dynkin diagram of g according to the action of ω̇.Pictorially we have e.g.:ğ = G(1)2 ğ = B̃(2)5g = E(1)6 g = B(1)6In formulæ, the orbit Lie algebra is described as follows. Denote by Î a set of representatives in Ifor each ω̇-orbit. The Cartan matrix of the orbit Lie algebra is then labelled by the subsetĬ := {i∈ Î |Ni−1∑l=0ai,ω̇li ≤ 0 =⇒Ni−1∑l=0ai,ω̇li = aii} (3.7)of Î. For each i∈ Î we define the numbersi :={aii/∑Ni−1l=0 ai,ω̇li if i∈ Ĭ and aii 6= 0 ,1 otherwise ,(3.8)which is either 1 or 2. The Cartan matrix Â = (âij)i,j∈Ĭ of the orbit Lie algebra is then defined bysumming over one index of the Cartan matrix of g:âij := sjNj−1∑l=0ai,ω̇lj . (3.9)We emphasize that ğ is not constructed as a subalgebra of g; in particular the orbit Lie algebra isin general not isomorphic to the subalgebra of g that stays fixed under ω. One can also show thatthe algebra g and the orbit Lie algebra ğ are 2 of the same type (i.e. simple, affine, indefinite); theorbit Lie algebra of hyperbolic Lie algebras is hyperbolic as well. A particularly interesting situation2 except for the order-N automorphisms of sl(N)4is illustrated on the right hand side of the figure: for certain orbit Lie algebras, one obtains thetwisted affine Lie algebras B̃(2)n which are the only twisted affine Lie algebras for which the charactershave good modular transformation properties. This had been predicted [6] on the basis of level-rankdualities of N = 2 superconformal coset theories.We conjecture that when the original Lie algebra g is an untwisted affine Lie algebra and thesymmetry of the Dynkin diagram corresponds to the action of a simple current, 3 then the modularmatrix S of the orbit Lie algebra ğ describes the modular transformation properties of the chiralblocks for the one-point function of the relevant simple current on the torus. This would provide aconceptual explanation of the observation that in these cases the twining characters of g span again aunitary representation of SL(2,Z). For some further evidence for this conjecture, see [5].4 Sketch of the proofWe will now outline the main ideas which enter in the proof of a character formula for the twiningcharacter; for the full details we refer the reader to the original publications [2, 1]. The first step isto show that the subgroup Ŵ of the Weyl group W of g that consists of those elements of W whoseaction on the weight space of g commutes with ω⋆ is isomorphic to the Weyl group W̆ of the orbit Liealgebra ğ. This part of the proof also provides new insight in the structure of Coxeter groups.The idea is to show that Ŵ plays for the twining characters the role the full Weyl group plays forthe ordinary characters: First, the twining characters of Verma modules are Ŵ -odd. More precisely,let V(ω)Λ denote the twining character of the Verma module; then V(ω) := e−ρ−Λ V(ω)Λ (which does notdepend on Λ) is antisymmetric under Ŵ , ŵ(V(ω)) = ǫ̂(ŵ)V(ω). Here ǫ̂ is the sign function associatedto Ŵ as a Coxeter group, and not the restriction of the sign function of W . The twining charactersχ[ω]Λ of the irreducible modules, on the other hand, are symmetric under the action of Ŵ :ŵ(χ[ω]Λ ) = χ[ω]Λ . (4.10)We can now generalize the arguments used in the proof of the Weyl-Kac-Borcherds character formulaof ordinary characters and implement the symmetry properties under Ŵ to derive an explicit characterformula for the twining characters:χ[ω]Λ =∑ŵ∈Ŵ ǫ̂(ŵ) ŵ(SωΛ)∑ŵ∈Ŵ ǫ̂(ŵ) ŵ(Sω0 ), (4.11)whereSωΛ = eΛ+ρ∑ǫ̂(β) e−β (4.12)and ǫ̂(β) = (−1)n if β is the symmetric sum of n pairwise orthogonal imaginary simple roots, allorthogonal to Λ, and ǫ̂(β) = 0 otherwise. Comparison with the character formula for the orbit Liealgebra then proves the claim.3 Dynkin diagram symmetries coming from simple currents [7] are elements of the unique maximal abeliannormal subgroup of the diagram automorphisms. This subgroup is isomorphic to the center of the universalcovering Lie group that has the horizontal subalgebra ḡ ⊂ g as its Lie algebra.55 Applications and conclusionsOrbit Lie algebras and twining characters are novel structures in the representation theory of general-ized Kac--Moody algebras. They already found several applications in two-dimensional conformal fieldtheory, e.g. in the solution of the problem of field identification fixed points in coset conformal fieldtheories [3] and in fixed point resolution for integer spin simple current (‘D-type’) modular invariantsof conformal field theories [4]. More details can be found in the original publications and in [5].In the case of WZW theories, integer spin modular invariants describe the WZW theory on anon-simply connected group manifold. In general, twining characters and orbit Lie algebras play arole as soon as one considers non-simply connected Lie groups. In particular, they allow to describeChern-Simons theories based on such groups [8] and they give a Verlinde formula also for non-simplyconnected groups [4]. Other applications concern the biggest sporadic simple group, the monster group,which acts on a generalized Kac--Moody algebra, the monster Lie algebra, by outer automorphisms(recently, this group has been proposed as the symmetry underlying N = 2 string theory). We expectthat orbit Lie algebras and twining characters will be a useful tool in this situation as well. Finally,twining characters of finite-dimensional simple Lie algebras are closely related to the characters ofnon-connected finite-dimensional simple Lie groups (the latter have e.g. been studied in [9]).Acknowledgement. J. Fuchs was supported by a Heisenberg fellowship. C. Schweigert was sup-ported in part by the Director, Office of Energy Research, Office of Basic Energy Sciences, of theU.S. Department of Energy under Contract DE-AC03-76F00098 and in part by the National ScienceFoundation under grant PHY95-14797.References[1] J. Fuchs, U. Ray, and C. Schweigert, Some automorphisms of Generalized Kac–Moody algebras,preprint q-alg/9605046, to appear in J. of Algebra[2] J. Fuchs, A.N. Schellekens, and C. Schweigert, From Dynkin diagram symmetries to fixed pointstructures, Commun.Math. Phys. 180 (1996) 39[3] J. Fuchs, A.N. Schellekens, and C. Schweigert, The resolution of field identification fixed pointsin diagonal coset theories, Nucl. Phys. B 461 (1996) 371[4] J. Fuchs, A.N. Schellekens, and C. Schweigert, A matrix S for all simple current extensions,Nucl. Phys. B 473 (1996) 323[5] J. Fuchs, A.N. Schellekens, and C. Schweigert, Fixed point resolution in conformal field theory,contribution to these proceedings.[6] J. Fuchs and C. Schweigert, Level-rank duality of WZW theories and isomorphisms of N = 2coset models, Ann.Phys. 234 (1994) 102[7] A.N. Schellekens and S. Yankielowicz, Simple currents, modular invariants, and fixed points,Int. J.Mod.Phys. A 5 (1990) 2903[8] C. Schweigert, On moduli spaces of flat connections with non-simply connected structure group,preprint hep-th/9611092[9] T. Shintani, On irreducible unitary characters of a certain group extension of GL(2,C), J.Math.Soc. Japan 29 (1977) 1656

Twining characters and orbit Lie algebras

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Twining characters and orbit Lie algebras

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