T-duality is used to extract information on an instanton of zero size in the E 8 × E 8 heterotic string. We discuss the possibility of the appearance of a tensionless antiself-dual non-critical string through an implementation of the mechanism suggested by Strominger of two coincident 5-branes. It is argued that when an instanton shrinks to zero size a tensionless non-critical string appears at the core of the instanton. It is further conjectured that appearance of tensionless strings in the spectrum leads to new phase transitions in six dimensions in much the same way as massless particles do in four dimensions.Recently Witten [1] showed what happens when instantons on the SO(32) heterotic string shrink to zero size. The low energy six dimensional effective theory has an extra Sp (1) gauge symmetry which is supported at the core of the instanton. The finite size instanton is obtained from the zero size instanton through a Higgs mechanism. In six dimensions there is no vector multiplet moduli space and therefore the Higgs mechanism is presumed to be the only relevant low energy dynamics.In this paper we will explore some features in the problem of small instantons on the E 8 ⊗E 8 heterotic strings. A straight forward generalization of Witten's construction does not seem to work because the dimensions of the E 8 representations are too big. It was suggested by Witten [2] that the solution involves a new type of physics unknown before. In [3] Duff and Lu constructed a supersymmetric solitonic one-brane solution of N = 2 D = 6 supergravity which is self-dual. This soliton was realized in [4] as a self-dual three brane of the type IIB string compactified on K3 wrapped on a self-dual two-cycle of the K3. When the K3 degenerates in such a way that the pullback of the complexified Kähler form on the two cycle is very small the tension of the one-brane (which is proportional to the area of the two cycle) is much smaller than any other scale in the theory. Another realization of this one-brane was discussed in [5]. A membrane stretched between two five-branes in the M-theory has the one-brane solution at the boundaries of the membrane. When the two five-branes coincide the tension of the one-brane vanishes. The constructions of [5] and of [4] are related as explained in [6].Returning to low energy in six dimensions we see that a new type of dynamics may appear through the emergence of a tensionless self-dual string.We asked ourselves the following questions:• What can we learn about the E 8 ⊗E 8 small instanton from T-duality between SO(32) and E 8 ⊗E 8 ?• Is there any relation between tensionless strings in 6D and an E 8 ⊗E 8 small instanton?• What happens when a 5-brane of the M-theory on S 1 /Z 2 (which describes the strong coupling limit of the E 8 ⊗E 8 heterotic string) approaches one of the fixed points?

We consider compactifications of the N = 1, d = 6, E 8 theory on tori to five, four, and three dimensions and learn about some properties of this theory. As a by-product we derive the SL(2, Z) duality of the N = 2, d = 4, SU (2) theory with N f = 4. Using this theory on a D-brane probe we shed new light on the singularities of F-theory compactifications to eight dimensions. As another application we consider compactifications of F-theory, M-theory and the IIA string on (singular) Calabi-Yau spaces where our theory appears in spacetime.Our viewpoint leads to a new perspective on the nature of the singularities in the moduli space and their spacetime interpretations. In particular, we have a universal understanding of how the singularities in the classical moduli space of Calabi-Yau spaces are modified by worldsheet instantons to singularities in the moduli space of the corresponding conformal field theories.

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T-duality of gauge theories on a noncommutative T d can be extended to include fields with twisted boundary conditions. The resulting T-dual theories contain novel nonlocal fields. These fields represent dipoles of constant magnitude. Several unique properties of field theories on noncommutative spaces have simpler counterparts in the dipole-theories.