We study extremal black hole solutions of the S3 model (obtained by setting S=T=U in the STU model) using group theoretical methods. Upon dimensional reduction over time, the S3 model exhibits the pseudo-Riemannian coset structure G/K with G=G2(2) and K=SO(2,2). We study nilpotent K-orbits of G2(2) corresponding to non-rotating single-center extremal solutions. We find six such distinct K-orbits. Three of these orbits are supersymmetric, one is non-supersymmetric, and two are unphysical. We write general solutions and discuss examples in all four physical orbits. We show that all solutions in supersymmetric orbits when uplifted to five-dimensional minimal supergravity have single-center Gibbons-Hawking space as their four-dimensional Euclidean hyper-K\"ahler base space. We construct hitherto unknown extremal (supersymmetric as well as non-supersymmetric) pressureless black strings of minimal five-dimensional supergravity and briefly discuss their relation to black rings.Comment: 45 pages, 2 figures. v2: minor changes. Published versio

The symmetries of the N = 4 SuperYang-Mills theory on the light-cone are discussed, solely in terms of its physical degrees of freedom. We derive explicit expressions for the generators of the P SU (2, 2 | 4) superalgebra, both in the free theory, and to all orders in the gauge coupling of the classical theory. We use these symmetries to construct its Hamiltonian, and show that it can be written as a quadratic form of a fermionic superfield.

We introduce a new type IIB 5-brane description for the E-string theory which is the world-volume theory on the M5-brane probing the end of the world M9-brane.The E-string in the new realization is depicted as spiral 5-branes web equipped with the cyclic structure which is key to uplifting to six dimensions. Utilizing the topological vertex to the 5-brane web configuration enables us to write down a combinatorial formula for the generating function of the E-string elliptic genera, namely the full partition function of topological strings on the local 1 2 K3 surface. * In this paper we propose a different description of the E-string theory based on the familiar type IIB 5-brane setup. In the case of 5d N = 1 gauge theories, the worldvolume theory of the corresponding (p, q) 5-brane web configuration leads to the desired gauge theory [41][42][43][44]. The same web diagram specifies the toric Calabi-Yau 3-fold of the compactified M-theory dual to the 5-brane web, and then we calculate the 5d Nekrasov partition function [45] using the topological string method [46-51] known as the topological vertex [52-56] through geometric engineering [57,58]. If these techniques are also applicable to the 6d E-string theory, they will be a very powerful method conceptually and computationally to investigate 6d dynamics. To this end, we utilize the key fact:the E-string theory of interest, which is a 6d N = (1, 0) SCFT, appears as the UV fixed point of 5d N = 1 SU (2) gauge theory with eight flavors. Indeed, the agreement of the corresponding partition functions is checked in [39]. This means that once one finds a 5-brane web for SU (2) gauge theory with eight flavors, one can apply all the established techniques to study the E-strings. In this paper, we discuss such (p, q) 5-brane web description of the E-string theory. The (p, q) 5-brane web that we found is of a spiral shape with a cyclic structure associated with the spiral direction, which is the source of the hidden 6d direction accounting for the Kaluza-Klein (KK) direction. By implementing the topological vertex to this spiral web, we can write down a combinatorial expression of the full-order partition function, which is the generating function of the elliptic genera of the E-strings. This generating function is precisely the full partition function of topological strings on the local 1 2 K3 surface. Our analysis may provide a stepping stone allowing one to look for a similar description for other 6d theories as well as a new class of 5-brane web.The paper is organized as follows. In Sect. 2, we review type IIB (p, q) 5-brane web construction for SU (2) gauge theory with eight flavors based on 7-brane monodromies, and find a spiral structure of the web diagram, which we call the "Tao diagram." In Sects. 3 and 4, applying the topological vertex method to this Tao diagram, we compute the partition function and compare the obtained result to the elliptic genera computed in [39]. We discuss various Tao diagrams as well as other future work in Sect. 5.

No abstract