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Wall Crossing, Discrete Attractor Flow and Borcherds Algebra
Symmetry, integrability and geometry: methods and applications, Tome 4 (2008) Cet article a éte moissonné depuis la source Math-Net.Ru

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The appearance of a generalized (or Borcherds–) Kac–Moody algebra in the spectrum of BPS dyons in $\mathcal N=4$, $d=4$ string theory is elucidated. From the low-energy supergravity analysis, we identify its root lattice as the lattice of the $T$-duality invariants of the dyonic charges, the symmetry group of the root system as the extended $S$-duality group $PGL(2,\mathbb Z)$ of the theory, and the walls of Weyl chambers as the walls of marginal stability for the relevant two-centered solutions. This leads to an interpretation for the Weyl group as the group of wall-crossing, or the group of discrete attractor flows. Furthermore we propose an equivalence between a “second-quantized multiplicity” of a charge- and moduli-dependent highest weight vector and the dyon degeneracy, and show that the wall-crossing formula following from our proposal agrees with the wall-crossing formula obtained from the supergravity analysis. This can be thought of as providing a microscopic derivation of the wall-crossing formula of this theory.
Keywords: generalized Kac–Moody algebra; black hole; dyons. 
@article{SIGMA_2008_4_a67,
     author = {Miranda C. N. Cheng and Erik P. Verlinde},
     title = {Wall {Crossing,} {Discrete} {Attractor} {Flow} and {Borcherds} {Algebra}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2008},
     volume = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2008_4_a67/}
}
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Miranda C. N. Cheng; Erik P. Verlinde. Wall Crossing, Discrete Attractor Flow and Borcherds Algebra. Symmetry, integrability and geometry: methods and applications, Tome 4 (2008). http://geodesic.mathdoc.fr/item/SIGMA_2008_4_a67/

[1] Harvey J. A., Moore G. W., “Algebras, BPS states, and strings”, Nuclear Phys. B, 463 (1996), 315–368 ; hep-th/9510182 | DOI | MR | Zbl

[2] Harvey J. A., Moore G. W., “On the algebras of BPS states”, Comm. Math. Phys., 197 (1998), 489–519 ; hep-th/9609017 | DOI | MR | Zbl

[3] Denef F., Moore G. W., Split states, entropy enigmas, holes and halos, hep-th/0702146

[4] Diaconescu E., Moore G. W., Crossing the wall: branes vs. bundles, arXiv:0706.3193

[5] Kontsevich M., Talk in Paris workshop on Black Holes, Black Rings and Modular Forms 17 (August 2007)

[6] Dijkgraaf R., Verlinde E. P., Verlinde H. L., “Counting dyons in $N=4$ string theory”, Nuclear Phys. B, 484 (1997), 543–561 ; hep-th/9607026 | DOI | MR | Zbl

[7] Kawai T., “$N=2$ heterotic string threshold correction, $K3$ surface and generalized Kac–Moody superalgebra”, Phys. Lett. B, 372 (1996), 59–64 ; hep-th/9512046 | DOI | MR | Zbl

[8] Gritsenko V., Nikulin V., “Siegel automorphic form correction of some Lorentzian Kac–Moody Lie algebras”, Amer. J. Math., 119 (1997), 181–224 ; alg-geom/9504006 | DOI | MR | Zbl

[9] Sen A., “Walls of marginal stability and dyon spectrum in $N=4$ supersymmetric string theories”, J. High Energy Phys., 2007:5 (2007), 039, 33 pp., ages ; hep-th/0702141 | DOI | MR

[10] Dabholkar A., Gaiotto D., Nampuri S., “Comments on the spectrum of CHL dyons”, J. High Energy Phys., 2008:1 (2008), 023, 23 pp., ages ; hep-th/0702150 | DOI | MR

[11] Cheng M. C. N., Verlinde E., “Dying dyons don't count”, J. High Energy Phys., 2007:9 (2007), 070, 21 pp., ages ; arXiv:0706.2363 | DOI | MR

[12] Sen A., “Rare decay modes of quarter BPS dyons”, J. High Energy Phys., 2007:10 (2007), 059, 8 pp., ages ; arXiv:0707.1563 | DOI | MR

[13] Dabholkar A., Gomes J., Murthy S., Counting all dyons in $N=4$ string theory, arXiv:0803.2692

[14] Sen A., “Two centered black holes and $N=4$ dyon spectrum”, J. High Energy Phys., 2007:9 (2007), 045, 11 pp., ages ; ; Sen A., Wall crossing formula for $N=4$ dyons: a macroscopic derivation, arXiv:0705.3874arXiv:0803.3857 | DOI | MR | MR

[15] Shih D., Strominger A., Yin X., “Recounting dyons in $N=4$ string theory”, J. High Energy Phys., 2006:10 (2006), 087, 5 pp., ages ; hep-th/0505094 | DOI | MR

[16] David J. R., Sen A., “CHL dyons and statistical entropy function from D1-D5 system”, J. High Energy Phys., 2006:11 (2006), 072, 40 pp., ages ; hep-th/0605210 | DOI | MR

[17] Cvetic M., Tseytlin A. A., “Solitonic strings and BPS saturated dyonic black holes”, Phys. Rev. D, 53 (1996), 5619–5633 ; Erratum Phys. Rev. D, 55 (1997), 3907 ; ; Cvetic M., Youm D., “Dyonic BPS saturated black holes of heterotic string on a six torus”, Phys. Rev. D, 53 (1996), R584–R588 ; hep-th/9512031hep-th/9507090 | DOI | MR | DOI | MR | DOI | MR

[18] Damour T., Henneaux M., Nicolai H., “Cosmological billiards”, Classical Quantum Gravity, 20 (2003), R145–R200 ; hep-th/0212256 | DOI | MR | Zbl

[19] Henneaux M., Persson D., Spindel P., “Spacelike singularities and hidden symmetries of gravity”, Living Rev. Relativity, 11 (2008), 232 pages, lrr-2008-1; arXiv:0710.1818

[20] Ferrara S., Kallosh R., Strominger A., “$N=2$ extremal black holes”, Phys. Rev. D, 52 (1995), R5412–R5416 ; hep-th/9508072 | DOI | MR

[21] Mohaupt T., “Black hole entropy, special geometry and strings”, Fortsch. Phys., 49 (2001), 3–161 ; hep-th/0007195 | 3.0.CO;2-%23 class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[22] Maldacena J. M., Moore G. W., Strominger A., Counting BPS black holes in toroidal type II string theory, hep-th/9903163

[23] Banerjee S., Sen A., “$S$-duality action on discrete $T$-duality invariants”, J. High Energy Phys., 2008:4 (2008), 012, 8 pp., ages ; ; Banerjee S., Sen A., Srivastava Y. K., “Generalities of quarter BPS dyon partition function and dyons of torsion two”, J. High Energy Phys., 2008:5 (2008), 101, 54 pp., ages ; ; Banerjee S., Sen A., Srivastava Y. K., “Partition functions of torsion $>1$ dyons in heterotic string theory on $T^6$”, J. High Energy Phys., 2008:5 (2008), 098, 14 pp., ages ; arXiv:0801.0149arXiv:0802.0544arXiv:0802.1556 | DOI | MR | DOI | MR | DOI | MR

[24] Dabholkar A., Harvey J. A., “Nonrenormalization of the superstring tension”, Phys. Rev. Lett., 63 (1989), 478–481 | DOI

[25] Dabholkar A., Kallosh R., Maloney A., “A stringy cloak for a classical singularity”, J. High Energy Phys., 2004:12 (2004), 059, 11 pp., ages ; hep-th/0410076 | DOI | MR

[26] Denef F., “Quantum quivers and Hall/hole halos”, J. High Energy Phys., 2002:10 (2002), 023, 42 pp., ages ; hep-th/0206072 | DOI | MR

[27] Fré P., Sorin A. S., The arrow of time and the Weyl group: all supergravity billiards are integrable, arXiv:0710.1059

[28] Denef F., Greene B. R., Raugas M., “Split attractor flows and the spectrum of BPS D-branes on the quintic”, J. High Energy Phys., 2001:5 (2001), 012, 47 pp., ages ; hep-th/0101135 | DOI | MR

[29] Feingold A. J., Frenkel I. B., “A hyperbolic Kac–Moody Algebra and the theory of Siegel modular forms of genus 2”, Math. Ann., 263 (1983), 87–144 | DOI | MR | Zbl

[30] Feingold A. J., Nicolai H., “Subalgebras of hyperbolic Kac–Moody algebras”, Kac-Moody Lie algebras and related topics, Contemp. Math., 343, 2004, 97–114 ; math.QA/0303179 | MR | Zbl

[31] Ray U., Automorphic forms and Lie superalgebra, Springer, Dordrecht, 2006 | MR

[32] Eguchi T., Ooguri H., Taormina A., Yang S. K., “Superconformal algebras and string compactification on manifolds with $SU(N)$ holonomy”, Nuclear Phys. B, 315 (1989), 193–221 | DOI | MR

[33] Borcherds R. E., “Automorphic forms with singularities on Grassmannian”, Invent. Math., 132 (1998), 491–562 ; alg-geom/9609022 | DOI | MR | Zbl

[34] Göttsche L., “The Betti numbers of the Hilbert scheme of points on a smooth projective surface”, Math. Ann., 286 (1990), 193–207 | DOI | MR | Zbl

[35] Dijkgraaf R., Moore G. W., Verlinde E. P., Verlinde H. L., “Elliptic genera of symmetric products and second quantized strings”, Comm. Math. Phys., 185 (1997), 197–209 ; hep-th/9608096 | DOI | MR | Zbl

[36] David J. R., Jatkar D. P., Sen A., “Dyon spectrum in generic $N=4$ supersymmetric $Z^N$ orbifolds”, J. High Energy Phys., 2007:1 (2007), 016, 31 pp., ages ; hep-th/0609109 | DOI | MR

[37] Yau S. T., Zaslow E., “BPS states, string duality, and nodal curves on $K3$”, Nuclear Phys. B, 471 (1996), 503–512 ; hep-th/9512121 | DOI | MR | Zbl

[38] Cachazo F., Fiol B., Intriligator K. A., Katz S., Vafa C., “A geometric unification of dualities”, Nuclear Phys. B, 628 (2002), 3–78 ; hep-th/0110028 | DOI | MR | Zbl

[39] Gaiotto D., Re-recounting dyons in $N=4$ string theory, hep-th/0506249

[40] Sen A., “Three string junction and $N=4$ dyon spectrum”, J. High Energy Phys., 2007:12 (2007), 019, 12 pp., ages ; arXiv:0708.3715 | DOI | MR

[41] Cheng M. C. N., Dabholkar A., Work in progress

[42] Humphreys J. E., Reflection groups and Coxeter groups, Cambridge University Press, Cambridge, 1990 | MR

[43] Björner A., Brenti F., Combinatorics of Coxeter groups, Springer, New York, 2005 | MR

[44] Davis M. W., The geometry and topology of Coxeter groups, Princeton University Press, Princeton, NJ, 2008 | MR | Zbl