Emil J. Martinec

Conformal invariance, supersymmetry and string theory

Covariant quantization of string theories is developed in the context of conformal field theory and the BRST quantization procedure. The BRST method is used to covariantly quantize superstrings, and in particular to construct the vertex operators for string emission as well as the supersymmetry charge. The calculation of string loop diagrams is sketched. We discuss how conformal methods can be used to study string compactification and dynamics.

Strings in background fields

Abstract We formulate the propagation of strings in background fields, including the effects of metric, antisymmetric tensor, and dilaton expectation values, as well as gauge field backgrounds in the case of heterotic strings. The inclusion of background fermion fields is sketched. The equations of motion of all these fields are shown to be the consequence of (super) conformal invariance of the string.

Heterotic string theory (I). The free heterotic string

A new theory of closed orientable superstrings is constructed as a chiral combination of the closed D = 26 bosonic and D = 10 fermionic strings. The resulting ten-dimensional theory is N = 1 supersymmetric, Lorentz invariant, and free of tachyons. An alternate formulation where sixteen of the bosonic coordinates are replaced by fermionic coordinates is presented and used to construct a manifestly Lorentz covariant and reparametrization invariant action for the theory. The theory is consistent only for gauge group Spin(32)/Z 2 or E 8 × E 8 .

The Conformal Field Theory of Orbifolds

A prescription for the calculation of any correlation function in orbifold conformal field theory is given. The method is applied to the scattering of four twisted string states, which allows the extraction of operator product coefficients of conformal twist fields. We derive Yukawa couplings in the effective field theory for fermionic strings on orbifolds.

Heterotic string theory: (II). The interacting heterotic string

Abstract The theory of interacting heterotic strings is presented. Vertex operators are derived in both the bosonic and fermionic formulations of the theory and are shown to be consistent with gauge invariance, Lorentz invariance, and supersymmetry. Three- and four-point amplitudes for the scattering of massless string states are calculated and used to derive the low-energy field theory limit of the heterotic string. Divergences in string theories are discussed and it is shown that one-loop heterotic string amplitudes are finite and modular invariant only for gauge group E 8 ×E 8 or spin (32)/Z 2 .

Integrable systems and supersymmetric gauge theory

Abstract After the work of Seiberg and Witten, it has been seen that the dynamics of N = 2 Yang-Mills theory is governed by a Riemann surface ϵ In particular, the integral of a special differential ωSW over (a subset of) the periods of ϵ gives the mass formula for BPS-saturated states. We show that, for each simple group G, the Riemann surface is a spectral curve of the periodic Toda lattice for the dual group, GV whose affine Dynkin diagram is the dual of that of G. This curve is not unique, rather it depends on the choice of a representation ϱ of GV; however, different choices of ϱ lead to equivalent constructions. The Seiberg-Witten differential ωSW is naturally expressed in Toda variables, and the N = 2Yang-Mills pre-potential is the free energy of a topological field theory defined by the data eg, π and ωSW.

Algebraic Geometry and Effective Lagrangians

Abstract N=2 supersymmetric Landau-Ginsburg fixed points describe nonlinear models whose target spaces are algebraic varieties in certain generalized projective spaces; the defining equation is precisely the zero set of the superpotential, considered as a condition in the projective space. The ADE classification of modular invariants arises as the classification of projective descriptions of P 1; in general, the hierarchy of fixed points is conjectured to be isomorphic to the classification of quasihomogeneous singularities. The condition of vanishing first Chern class is an integrality condition on the Virasoro central charge; the central charge is determined by the superpotential. The operator algebra is given by the algebra of Wick contractions of perturbations of the superpotential.

Covariant Quantization of Superstrings

Abstract We present a manifestly Lorentz covariant formulation of supersymmetric string theory. In particular, we construct the fermion vertex and the supersymmetry generators in a BRST quantization using the techniques of superconformal field theory.

Phenomenology and conformal field theory or can string theory predict the weak mixing angle

Abstract We show that the weak mixing angle θ w is the same for continuously connected classical vacua of the heterotic string which have chiral fermions in their massless spectra. We also show that the world-sheet quantum field theory for any classical vacuum with spacetime supersymmetry possesses an N = 2 superconformal invariance.

D-branes and strings as non-commutative solitons

The non-commutative geometry of a large auxiliary B-field simplifies the construction of D-branes as solitons in open string field theory. Similarly, fundamental strings are constructed as localized flux tubes in the string field theory. Tensions are determined exactly using general properties of non-BPS branes, and the non-abelian structure of gauge fields on coincident D-branes is recovered.