Hyperbolic Kac-Moody superalgebras

We present a classification of the hyperbolic Kac-Moody (HKM) superalgebras. The HKM superalgebras of rank larger or equal than 3 are finite in number (213) and limited in rank (6). The Dynkin-Kac diagrams and the corresponding simple root systems are determined. We also discuss a class of singular sub(super)algebras obtained by a folding procedure

Missing Modules, the Gnome Lie Algebra, and E10E_{10}

We study the embedding of Kac-Moody algebras into Borcherds (or generalized Kac-Moody) algebras which can be explicitly realized as Lie algebras of physical states of some completely compactified bosonic string. The extra ``missing states'' can be decomposed into irreducible highest or lowest weight ``missing modules'' w.r.t. the relevant Kac-Moody subalgebra; the corresponding lowest weights are associated with imaginary simple roots whose multiplicities can be simply understood in terms of certain polarization states of the associated string. We analyse in detail two examples where the momentum lattice of the string is given by the unique even unimodular Lorentzian lattice $II_{1,1}$ or $II_{9,1}$, respectively. The former leads to the Borcherds algebra $g_{1,1}$, which we call ``gnome Lie algebra", with maximal Kac-Moody subalgebra $A_1$. By the use of the denominator formula a complete set of imaginary simple roots can be exhibited, whereas the DDF construction provides an explicit Lie algebra basis in terms of purely longitudinal states of the compactified string in two dimensions. The second example is the Borcherds algebra $g_{9,1}$, whose maximal Kac-Moody subalgebra is the hyperbolic algebra $E_{10}$. The imaginary simple roots at level 1, which give rise to irreducible lowest weight modules for $E_{10}$, can be completely characterized; furthermore, our explicit analysis of two non-trivial level-2 root spaces leads us to conjecture that these are in fact the only imaginary simple roots for $g_{9,1}$.Comment: 31 pages, LaTeX2e, AMS packages, PSTRICK

On the Imaginary Simple Roots of the Borcherds Algebra gII9,1g_{II_{9,1}}

In a recent paper (hep-th/9703084) it was conjectured that the imaginary simple roots of the Borcherds algebra $g_{II_{9,1}}$ at level 1 are its only ones. We here propose an independent test of this conjecture, establishing its validity for all roots of norm $\geq -8$. However, the conjecture fails for roots of norm -10 and beyond, as we show by computing the simple multiplicities down to norm -24, which turn out to be remakably small in comparison with the corresponding $E_{10}$ multiplicities. Our derivation is based on a modified denominator formula combining the denominator formulas for $E_{10}$ and $g_{II_{9,1}}$, and provides an efficient method for determining the imaginary simple roots. In addition, we compute the $E_{10}$ multiplicities of all roots up to height 231, including levels up to $\ell =6$ and norms -42.Comment: 14 pages, LaTeX2e, packages amsmath, amsfonts, amssymb, amsthm, xspace, pstricks, longtable; substantially extended, appendix with new $E_{10}$ root multiplicities adde

Algebras, BPS States, and Strings

We clarify the role played by BPS states in the calculation of threshold corrections of D=4, N=2 heterotic string compactifications. We evaluate these corrections for some classes of compactifications and show that they are sums of logarithmic functions over the positive roots of generalized Kac-Moody algebras. Moreover, a certain limit of the formulae suggests a reformulation of heterotic string in terms of a gauge theory based on hyperbolic algebras such as $E_{10}$. We define a generalized Kac-Moody Lie superalgebra associated to the BPS states. Finally we discuss the relation of our results with string duality.Comment: 64 pages, harvmac (b), Discussion of BRST improved, typos fixed, two references adde

The classification of almost affine (hyperbolic) Lie superalgebras

We say that an indecomposable Cartan matrix A with entries in the ground field of characteristic 0 is almost affine if the Lie sub(super)algebra determined by it is not finite dimensional or affine but the Lie (super)algebra determined by any submatrix of A, obtained by striking out any row and any column intersecting on the main diagonal, is the sum of finite dimensional or affine Lie (super)algebras. A Lie (super)algebra with Cartan matrix is said to be almost affine if it is not finite dimensional or affine, and all of its Cartan matrices are almost affine. We list all almost affine Lie superalgebras over complex numbers correcting two earlier claims of classification and make available the list of almost affine Lie algebras obtained by Li Wang Lai.Comment: 92 page

Hagedorn String Thermodynamics in Curved Spacetimes and near Black Hole Horizons

This thesis concerns the study of high-temperature string theory on curved backgrounds, generalizing the notions of Hagedorn temperature and thermal scalar to general backgrounds. Chapter 2 contains a review on string thermodynamics in flat space, setting the stage. Chapters 3 and 4 contain the detailed study of the random walk picture in a general curved background. Chapters 5 and 6 then apply this to Rindler space, the near-horizon approximation of a generic (uncharged) black hole. Chapters 7 and 8 contain a study of the AdS3 and BTZ WZW models where we study the thermal spectrum and the resulting random walk picture that emerges. Chapters 9 and 10 attempt to draw general conclusions from the study of the two specific examples earlier: we draw lessons on string thermodynamics in general and on (perturbative) string thermodynamics around black hole horizons. For the latter, we point out a possible link to the firewall paradox. Finally, chapter 11 contains a detailed discussion on the near-Hagedorn (and high-energy) stress tensor in a generic spacetime, the results of which are applied to provide a description of the Bekenstein-Hawking entropy in terms of long string equilibration.Comment: PhD Thesis, based on arXiv:1305.7443, arXiv:1307.3491, arXiv:1402.2808, arXiv:1408.6999, arXiv:1408.7012, arXiv:1410.8009 and arXiv:1505.04025. v2: corrected typos and added reference

Multistring vertices and hyperbolic Kac Moody algebras

Multistring vertices and the overlap identities which they satisfy are exploited to understand properties of hyperbolic Kac Moody algebras, and E_1_0 in particular. Since any such algebra can be embedded in the larger Lie algebra of physical states of an associated completely compactified subcritical bosonic string, one can in principle determine the root spaces by analyzing which (positive norm) physical states decouple from the N-string vertex. Consequently, the Lie algebra of physical states decomposes into a direct sum of the hyperbolic algebra and the space of decoupled states. Both these spaces contain transversal and longitudinal states. Longitudinal decoupling holds generally, and may also be valid for uncompacitified strings, with possible consequences for Liouville theory; the identification of the decoupled states simply amounts to finding the zeroes of certain ''decoupling polynomials''. This is not the case for transversal decoupling, which crucially depends on special properties of the root lattice, as we explicitly demonstrate for a non-trivial root space of E_1_0. Because the N-vertices of the compactified string contain the complete information about decoupling, all the properties of the hyperbolic algebra are encoded into them. In view of the integer grading of hyperbolic algebras such as E_1_0 by the level, these algebras can be interpreted as interacting strings moving on the respective group manifolds associated with the underlying finite-dimensional Lie algebras. (orig.)64 refs.Available from TIB Hannover: RA 2999(95-092) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

Multistring vertices and hyperbolic Kac-Moody algebras

Multistring vertices and the overlap identities which they satisfy are exploited to understand properties of hyperbolic Kac Moody algebras, and $E_{10}$ in particular. Since any such algebra can be embedded in the larger Lie algebra of physical states of an associated completely compactified subcritical bosonic string, one can in principle determine the root spaces by analyzing which (positive norm) physical states decouple from the $N$-string vertex. Consequently, the Lie algebra of physical states decomposes into a direct sum of the hyperbolic algebra and the space of decoupled states. Both these spaces contain transversal and longitudinal states. Longitudinal decoupling holds generally, and may also be valid for uncompactified strings, with possible consequences for Liouville theory; the identification of the decoupled states simply amounts to finding the zeroes of certain ``decoupling polynomials. This is not the case for transversal decoupling, which crucially depends on special properties of the root lattice, as we explicitly demonstrate for a non-trivial root space of $E_{10}$. Because the $N$-vertices of the compactified string contain the complete information about decoupling, all the properties of the hyperbolic algebra are encoded into them. In view of the integer grading of hyperbolic algebras such as $E_{10}$ by the level, these algebras can be interpreted as interacting strings moving on the respective group manifolds associated with the underlying finite-dimensional Lie algebras