Seiberg-Witten prepotential for E-string theory and random partitions

We find a Nekrasov-type expression for the Seiberg-Witten prepotential for the six-dimensional non-critical E_8 string theory toroidally compactified down to four dimensions. The prepotential represents the BPS partition function of the E_8 strings wound around one of the circles of the toroidal compactification with general winding numbers and momenta. We show that our expression exhibits expected modular properties. In particular, we prove that it obeys the modular anomaly equation known to be satisfied by the prepotential.Comment: 14 page

Exceptional String: Instanton Expansions and Seiberg-Witten Curve

We investigate instanton expansions of partition functions of several toric E-string models using local mirror symmetry and elliptic modular forms. We also develop a method to obtain the Seiberg--Witten curve of E-string with arbitrary Wilson lines with the help of elliptic functions.Comment: 71 pages, three Wilson line

Folds in 2D String Theories

We study maps from a 2D world-sheet to a 2D target space which include folds. The geometry of folds is discussed and a metric on the space of folded maps is written down. We show that the latter is not invariant under area preserving diffeomorphisms of the target space. The contribution to the partition function of maps associated with a given fold configuration is computed. We derive a description of folds in terms of Feynman diagrams. A scheme to sum up the contributions of folds to the partition function in a special case is suggested and is shown to be related to the Baxter-Wu lattice model. An interpretation of folds as trajectories of particles in the adjoint representation of $SU(N)$ gauge group in the large $N$ limit which interact in an unusual way with the gauge fields is discussed.Comment: 56 pages, latex, followed by epsf, 13 uuencoded epsf figure

Seiberg-Witten prepotential for E-string theory and global symmetries

We obtain Nekrasov-type expressions for the Seiberg-Witten prepotential for the six-dimensional (1,0) supersymmetric E-string theory compactified on T^2 with nontrivial Wilson lines. We consider compactification with four general Wilson line parameters, which partially break the E_8 global symmetry. In particular, we investigate in detail the cases where the Lie algebra of the unbroken global symmetry is E_n + A_{8-n} with n=8,7,6,5 or D_8. All our Nekrasov-type expressions can be viewed as special cases of the elliptic analogue of the Nekrasov partition function for the SU(N) gauge theory with N_f=2N flavors. We also present a new expression for the Seiberg-Witten curve for the E-string theory with four Wilson line parameters, clarifying the connection between the E-string theory and the SU(2) Seiberg-Witten theory with N_f=4 flavors.Comment: 22 pages. v2: comments and a reference added, version to appear in JHE

Beta-functions in Yang-Mills Theory from Non-critical String

The renormalization group equations of the Yang-Mills theory are examined in the non-critical string theory according to the framework of the holography. Under a simple ansatz for the tachyon, we could find several interesting solutions which are classified by the value of the tachyon potential at the vacuum. We show two typical, asymptotic-free solutions which are different in their infrared behaviors. For both types of solutions, we could obtain quark-confinement potential from the Wilson-loop. The stability of the tachyon and the ZigZag symmetry are also discussed for these solutions.Comment: 16 pages, 5 figure

Characterization of solar-grade silicon produced by the SiF4-Na process

A process was developed for producing low cost solar grade silicon by the reaction between SiF4 gas and sodium metal. The results of the characterization of the silicon are presented. These results include impurity levels, electronic properties of the silicon after crystal growth, and the performance of solar photovoltaic cells fabricated from wafers of the single crystals. The efficiency of the solar cells fabricated from semiconductor silicon and SiF4-Na silicon was the same

Yang-Mills theory from non-critical string

The correspondence of the non-critical string theory and the Yang-Mills theory is examined according to the recent Polyakov's proposal, and two possible solutions of the bulk equations are addressed near the fixed points of the pure Yang-Mills theory: (i) One solution asymptotically approaches to the AdS space at the ultraviolet limit where the conformally invariant field theory is realized. (ii) The second one approaches to the flat space in both the infrared and the ultraviolet limits. The area law of the Wilson-loop and the asymptotic freedom with logarithmic behaviour are seen in the respective limit.Comment: 17 pages, no figure, Late

Plane wave limit of local conserved charges

We study the plane wave limit of the Backlund transformations for the classical string in AdS space times a sphere and obtain an explicit expression for the local conserved charges. We show that the Pohlmeyer charges become in the plane wave limit the local integrals of motion of the free massive field. This fixes the coefficients in the expansion of the anomalous dimension as the sum of the Pohlmeyer charges.Comment: v2: added explanation

Exceptional Indices

Recently a prescription to compute the superconformal index for all theories of class S was proposed. In this paper we discuss some of the physical information which can be extracted from this index. We derive a simple criterion for the given theory of class S to have a decoupled free component and for it to have enhanced flavor symmetry. Furthermore, we establish a criterion for the "good", the "bad", and the "ugly" trichotomy of the theories. After interpreting the prescription to compute the index with non-maximal flavor symmetry as a residue calculus we address the computation of the index of the bad theories. In particular we suggest explicit expressions for the superconformal index of higher rank theories with E_n flavor symmetry, i.e. for the Hilbert series of the multi-instanton moduli space of E_n.Comment: 33 pages, 11 figures, v2: minor correction

The String Theory Approach to Generalized 2D Yang-Mills Theory

We calculate the partition function of the $SU(N)$ ( and $U(N)$) generalized $YM_2$ theory defined on an arbitrary Riemann surface. The result which is expressed as a sum over irreducible representations generalizes the Rusakov formula for ordinary YM_2 theory. A diagrammatic expansion of the formula enables us to derive a Gross-Taylor like stringy description of the model. A sum of 2D string maps is shown to reproduce the gauge theory results. Maps with branch points of degree higher than one, as well as ``microscopic surfaces'' play an important role in the sum. We discuss the underlying string theory.Comment: TAUP-2182-94, 53 pages of LaTeX and 5 uuencoded eps figure