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E-strings and $$N=4$$ topological Yang-Mills theories. (English) Zbl 0951.81025
Summary: We study certain properties of six-dimensional tensionless E-strings (arising from zero size $$E_8$$ instantons). In particular we show that $$n$$ E-strings form a bound string which carries an $$E_8$$ level-$$n$$ current algebra as well as a left-over conformal system with $$c=12n-4-(248n/n+30)$$, whose characters can be computed. Moreover we show that the characters of the $$n$$-string bound state are captured by $$N=4$$ $$U(n)$$ topological Yang-Mills theory on $$\frac 12 K3$$. This relation not only illuminates certain aspects of E-strings but can also be used to shed light on the properties of $$N=4$$ topological Yang-Mills theories on manifolds with $$b_2^+=1$$. In particular the E-string partition functions, which can be computed using local mirror symmetry on a Calabi-Yau threefold, give the Euler characteristics of the Yang-Mills instanton moduli space on $$\frac 12 K3$$. Moreover, the partition functions are determined by a gap condition combined with a simple recurrence relation which has its origins in a holomorphic anomaly that has been conjectured to exist for $$N=4$$ topological Yang-Mills on manifolds with $$b_2^+=1$$ and is also related to the holomorphic anomaly for higher genus topological strings on Calabi-Yau threefolds.

##### MSC:
 81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory 32J81 Applications of compact analytic spaces to the sciences 14J32 Calabi-Yau manifolds (algebro-geometric aspects) 58D27 Moduli problems for differential geometric structures 81T60 Supersymmetric field theories in quantum mechanics
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##### References:
 [1] Strominger, A.; Vafa, C., Phys. lett. B, 379, 99, (1996), hep-th/9601029 [2] J. Maldacena, A. Strominger, E. Witten, Black hole entropy in M-theory, hep-th/9711053. · Zbl 0951.83034 [3] C. Vafa, Black holes and Calabi-Yau three-folds, HUTP-97-A066, hep-th/9711067. [4] E. Witten, Some comments on string dynamics, IASSNS-HEP-95-63, hep-th/9507121; [5] Ganor, O.; Hanany, A., Nucl. phys. B, 474, 122, (1996), hep-th/9602120 [6] Seiberg, N.; Witten, E., Nucl. phys. B, 471, 121, (1996), hep-th/9603003 [7] Morrison, D.; Vafa, C.; Morrison, D.; Vafa, C., Compactifications of F-theory on Calabi-Yau threefolds II, Nucl. phys. B, Nucl. phys. B, 476, 437, (1996), hep-th/9603161 · Zbl 0925.14007 [8] Witten, E., Phase transitions in M-theory and F-theory, Nucl. phys. B, 471, 195, (1996), hep-th/9603150 · Zbl 1003.81537 [9] A. Klemm, P. Mayr and C. Vafa, BPS states of exceptional non-critical strings, to appear in Proc. Conf. Advanced Quantum Field Theory (in memory of Claude [tzykson), CERN-TH-96-184, hep-th/9607139 · Zbl 0976.81503 [10] Minahan, J.; Nemeschansky, D.; Warner, N.P., Investigating the BPS spectrum of non-critical En strings, Nucl. phys. B, 508, 64, (1997), hep-th/9705237 · Zbl 0925.81282 [11] J. Minahan, D. Nemeschansky and N.P. Warner, Partition functions for the BPS states of the E8 non-critical string, USC-97/009, NSF-ITP-97-091, hep-th/9707149. · Zbl 0898.32013 [12] Vafa, C.; Witten, E., A strong coupling test of S-duality, Nucl. phys. B, 431, 3, (1994), hep-th/9408074 · Zbl 0964.81522 [13] Bershadsky, M.; Cecotti, S.; Ooguri, H.; Vafa, C.; Bershadsky, M.; Cecotti, S.; Ooguri, H.; Vafa, C., Nucl. phys. B, Comm. math. phys., 165, 311, (1994), hep-th/9309140 [14] Vafa, C., Evidence for F-theory, Nucl. phys. B, 469, 403, (1996), hep-th/9602022 · Zbl 1003.81531 [15] Ooguri, H.; Vafa, C., Two-dimensional black hole and singularities of CY manifolds, Nucl. phys. B, 463, 55, (1996), hep-th/9511164 · Zbl 1003.83511 [16] Strominger, A., Open p-branes, Phys. lett. B, 383, 44, (1996), hep-th/9512059 · Zbl 0903.53053 [17] Bershadsky, M.; Sadov, V.; Vafa, C., D-branes and topological field theories, Nucl. phys. B, 463, 420, (1996), hep-th/9511222 · Zbl 1004.81560 [18] M. Douglas, Branes within branes, hep-th/9512077. [19] Vafa, C., Instantons on D-branes, Nucl. phys. B, 463, 435, (1996), hep-th/9512078 · Zbl 1004.81537 [20] R. Friedman, J.W. Morgan and E. Witten, Vector bundles over elliptic fibrations, alg-geom/9709029. · Zbl 0937.14004 [21] Bershadsky, M.; Johansen, A.; Pantev, T.; Sadov, V., On four-dimensional compactifications of F-theory, Nucl. phys. B, 505, 165, (1997), hep-th/9701165 · Zbl 0925.14019 [22] Lerche, W.; Mayr, P.; Warner, N.P., Non-critical strings, del Pezzo singularities and Seiberg-Witten curves, Nucl. phys. B, 499, 125, (1997), hep-th/9612085 · Zbl 0934.81036 [23] Katz, S.; Klemm, A.; Vafa, C., Geometric engineering of quantum field theories, Nucl. phys. B, 497, 173, (1997), hep-th/9609239 · Zbl 0935.81058 [24] Witten, E., Physical interpretation of certain strong coupling singularities, Mod. phys. let. A, 11, 2649, (1996), hep-th/9609159 · Zbl 1022.81720 [25] M. Bershadsky and C. Vafa, Global anomalies and geometric engineering of critical theories in six dimensions, hep-ttt/9703167. [26] Verlinde, E., Global aspects of electric-magnetic duality, Nucl. phys. B, 455, 211, (1995), hep-th/950601l · Zbl 0925.58107 [27] O.J. Ganor, Compactification of tensionless string theories, hep-th/9607092. [28] Hooft, G.J.; Hooft, G.J., A property of electric and magnetic flux in non-abelian gauge theories, Nucl. phys. B, Nucl. phys. B, 153, 141, (1979) [29] Dijkgraaf, R.; Verlinde, E.; Verlinde, H., Nucl. phys. B, 486, 89, (1997) [30] Dijkgraaf, R.; Moore, G.; Verlinde, E.; Verlinde, H., Elliptic genera of symmetric products and second quantized strings, Commun. math. phys., 185, 197, (1997), hep-th/9608096 · Zbl 0872.32006 [31] Lang, S., Introduction to modular forms, (1976), Springer Berlin [32] G. Moore and E. Witten, Integration over the u-plane in Donaldson theory, hep-th/9709193. · Zbl 0899.57021 [33] A. Lossev, N. Nekrassov and S. Shatashvili, Testing Seiberg-Witten solution, hep-th/9801061. [34] Itzykson, C., Int. J. mod. phys. A B, 8, 1994, (1994) [35] T. Pantev, private communication. [36] S. Katz, P. Mayr and C. Vafa, Mirror symmetry and exact solution of 4D N = 2 gauge theories i, hep-th/9706110. · Zbl 0912.32016 [37] N.C. Leung and C. Vafa, Branes and tonic geometry, hep-th/9711013. · Zbl 0914.14024 [38] Vafa, C.; Witten, E., On orbifolds with discrete torsion, J. geom. phys., 15, 189, (1995), hepth/9409188 · Zbl 0816.53053 [39] J. Minahan. D. Nemeschansky and N.P. Warner, Instanton expansions for mass deformed N = 4 super Yang-Mills theories, USC-97/016, hep-th/9710146. · Zbl 0951.81080 [40] Ganor, O.; Morrison, D.; Seiberg, N., Nucl. phys. B, 487, 93, (1997), hep-th/9610251 [41] Yau, S.-T., Essays on mirror manifolds, (1992), international Press Hong Kong · Zbl 0816.00010 [42] Seiberg, N.; Witten, E.; Seiberg, N.; Witten, E., Nucl. phys. B, Nucl. phys. B, 431, 484, (1994), hep-th/9408099 [43] Ganor, O.J., A test of the chiral E8 current algebra on a 6D non-critical string, Nucl. phys. B, 479, 197, (1996), hep-th/9607020 · Zbl 0925.81202 [44] Cecotti, S.; Fendley, P.; Inniligator, K.; Vafa, C., A new supersymmetric index, Nucl. phys. B, 386, 405, (1992), hep-th/9204102 [45] Douglas, M.R.; Katz, S.; Vafa, C., Nucl. phys. B, 497, 155, (1997), hep-th/9609071 [46] Morrison, D.R.; Seiberg, N., Nucl. phys. B, 483, 229, (1997), hep-th/9609070 [47] K. Yoshioka, The Betti numbers of the moduli space of stable sheaves of rank 2 on p2, preprint, Kyoto University; The Betti numbers of the moduli space of stable sheaves of rank 2 on a ruled surface, preprint, Kyoto University; · Zbl 0731.14009 [48] K. Yoshioka, to appear. [49] L. Gottsche and D. Zagier, Jacobi forms and the structure of Donaldson invariants for 4-manifolds with b+ = 1, alg-geom/9612020. · Zbl 0924.57025
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