Tao probing the end of the world

Abstract

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.

Brane webs, 5d gauge theories and 6d N = 1 , 0 $$ \mathcal{N}=\left(1,\;0\right) $$ SCFT’s

“…Specifically, we add three flavors as shown in figure 21 (a), corresponding to the gauge theory description in figure 21 (b). This is expected to lift to 6d, as first pointed out in [28]. As a cross check, one can use the methods of [29] to show that this theory has an enhancement to an affine A (1) 3N −1 symmetry suggesting the 6d theory should have an SU(3N ) global symmetry.…”

“…The starting point is to generalize the discussion of [26] to the class of 5d gauge theories of the form (N + 2)F + SU 0 (N ) k + (N + 2)F . These were recently conjectured to lift to 6d SCFT [28]. Furthermore, in [29] a conjecture for this 6d SCFT appeared.…”

“…The analysis of [29] suggests it should have a D (1) 6 global symmetry and so is expected to lift to 6d. Indeed, as figure 25 (a) shows, it has a spiraling tau diagram which are characteristic of theories lifting to 6d [28].…”

Type IIB 7-branes in warped AdS6: partition functions, brane webs and probe limit

“…The space of 5d SCFTs that can be realized in Type IIB string theory can be extended substantially by adding additional 7-branes into 5-brane webs [15], and many insights have been obtained through the inclusion of 7-branes and in particular their associated branch cuts [16][17][18][19][20][21]. This motivates a corresponding extension of the construction of supergravity solutions.…”

Topological vertex for Higgsed 5d T N theories

“…Finally, there has recently been a lot of progress in computing the elliptic genera of strings in six-dimensional superconformal field theories [46][47][48][49][50][51][52][53] and also the elliptic genera of strings in five-dimensional superconformal field theories [54,55]. It would be interesting to see if our prescription can be applied to such computation.…”