D. H. Phong

Multiloop amplitudes for the bosonic Polyakov string

Abstract We provide a simple formula for multiloop amplitudes of the bosonic, closed oriented Polyakov string (in d = 26) as integrals over moduli space with respect to the Weil-Petersson measure. The integrand consists of Green functions and the determinants of laplacians acting on functions and vectors. We compute these determinants in terms of the lengths of the closed geodesics on the surface. They are finite and different from zero. The one on functions equals the derivative at 1 of the Selberg zeta function. A discussion of lengths of closed geodesics and coordinates for Teichmueller space is given.

On determinants of Laplacians on Riemann surfaces

Determinants of Laplacians on tensors and spinors of arbitrary weights on compact hyperbolic Riemann surfaces are computed in terms of values of Selberg zeta functions at half integer points.

Two-loop superstrings.: I. Main formulas☆

Abstract An unambiguous and slice-independent formula for the two-loop superstring measure on moduli space for even spin structure is constructed from first principles. The construction uses the super-period matrix as moduli invariant under worldsheet supersymmetry. This produces new subtle contributions to the gauge-fixing process, which eliminate all the ambiguities plaguing earlier gauge-fixed formulas. The superstring measure can be computed explicitly and a simple expression in terms of modular forms is obtained. For fixed spin structure, the measure exhibits the expected behavior under degenerations of the surface. The measure allows for a unique modular covariant GSO projection. Under this GSO projection, the cosmological constant, the 1-, 2- and 3-point functions of massless supergravitons all vanish pointwise on moduli space without the appearance of boundary terms. A certain disconnected part of the 4-point function is shown to be given by a convergent, finite integral on moduli space. A general slice-independent formula is given for the two-loop cosmological constant in compactifications with central charge c =15 and N =1 worldsheet supersymmetry in terms of the data of the compactification conformal field theory. In this Letter, a summary of the above results is presented with detailed constructions, derivations and proofs to be provided in a series of subsequent publications.

Two-loop superstrings IV: The cosmological constant and modular forms☆

Abstract The slice-independent gauge-fixed superstring chiral measure in genus 2 derived in the earlier papers of this series for each spin structure is evaluated explicitly in terms of theta-constants. The slice-independence allows an arbitrary choice of superghost insertion points q 1 , q 2 in the explicit evaluation, and the most effective one turns out to be the split gauge defined by S δ ( q 1 , q 2 )=0. This results in expressions involving bilinear theta-constants M . The final formula in terms of only theta-constants follows from new identities between M and theta-constants which may be interesting in their own right. The action of the modular group Sp(4, Z ) is worked out explicitly for the contribution of each spin structure to the superstring chiral measure. It is found that there is a unique choice of relative phases which insures the modular invariance of the full chiral superstring measure, and hence a unique way of implementing the GSO projection for even spin structure. The resulting cosmological constant vanishes, not by a Riemann identity, but rather by the genus 2 identity expressing any modular form of weight 8 as the square of a modular form of weight 4. The degeneration limits for the contribution of each spin structure are determined, and the divergences, before the GSO projection, are found to be the ones expected on physical grounds.

Two-loop superstrings II: The chiral measure on moduli space☆

Abstract A detailed derivation from first principles is given for the unambiguous and slice-independent formula for the two-loop superstring chiral measure which was announced in the first paper of this series. Supergeometries are projected onto their superperiod matrices, and the integration over odd supermoduli is performed by integrating over the fibers of this projection. The subtleties associated with this procedure are identified. They require the inclusion of some new finite-dimensional Jacobian superdeterminants, a deformation of the worldsheet correlation functions using the stress tensor, and perhaps paradoxically, another additional gauge choice, “slice μ choice”, whose independence also has to be established. This is done using an important correspondence between superholomorphic notions with respect to a supergeometry and holomorphic notions with respect to its superperiod matrix. Altogether, the subtleties produce precisely the corrective terms which restore the independence of the resulting gauge-fixed formula under infinitesimal changes of gauge-slice. This independence is a key criterion for any gauge-fixed formula and hence is verified in detail.

Loop Amplitudes for the Fermionic String

Abstract We obtain simple formulae for loop amplitudes of the fermionic closed oriented Polyakov string in d = 10, as integrals over moduli and supermoduli space. The integrals over supermoduli may be carried out, leaving an integral over moduli with respect to the Weil-Petersson measure. The integrand consists of Green functions and determinants of laplacians on tensors and spinors.

THE EFFECTIVE PREPOTENTIAL OF N=2 SUPERSYMMETRIC SU(NC) GAUGE THEORIES

We determine the effective prepotential for N = 2 supersymmetric SU(Nc) gauge theories with an arbitrary number of flavors Nf < 2Nc from the exact solution constructed out of spectral curves. The prepotential is the same for the several models of spectral curves proposed in the literature. It has to all orders the logarithmic singularities of the one-loop perturbative corrections, thus confirming the non-renormalization theorems from supersymmetry. In particular, the renormalized order parameters and their duals have all the correct monodromy transformations prescribed at weak coupling. We evaluate explicitly the contributions of 1- and 2-instanton processes.

Lectures on Supersymmetric Yang-Mills Theory and Integrable Systems

We present a series of four self-contained lectures on the following topics; (I) An introduction to 4-dimensional 1 ≤ N ≤ 4 supersymmetric Yang-Mills theory, including particle and field contents, N = 1 and N = 2 superfield methods and the construction of general invariant Lagrangians; (II) A review of holomorphicity and duality in N = 2 super-Yang-Mills, of Seiberg-Witten theory and its formulation in terms of Riemann surfaces; (III) An introduction to mechanical Hamiltonian integrable systems; such as the Toda and Calogero-Moser systems associated with general Lie algebras; a review of the recently constructed Lax pairs with spectral parameter for twisted and untwisted elliptic Calogero-Moser systems; (IV) A review of recent solutions of the Seiberg-Witten theory for general gauge algebra and adjoint hypermultiplet content in terms of the elliptic Calogero-Moser integrable systems.

Calogero-Moser systems in SU(N) Seiberg-Witten theory

Abstract The Seiberg-Witten curve and differential for N = 2 supersymmetric SU(N) gauge theory, with a massive hypermultiplet in the adjoint representation of the gauge group, are analyzed in terms of the elliptic Calogero-Moser integrable system. A new parametrization for the Calogero-Moser spectral curves is found, which exhibits the classical vacuum expectation values of the scalar field of the gauge multiplet. The one-loop perturbative correction to the effective prepotential is evaluated explicitly, and found to agree with quantum field theory predictions. A renormalization group equation for the variation with respect to the coupling is derived for the effective prepotential, and may be evaluated in a weak-coupling series using residue methods only. This gives a simple and efficient algorithm for the instanton corrections to the effective prepotential to any order. The one- and two-instanton corrections are derived explicitly. Finally, it is shown that certain decoupling limits yield N = 2 supersymmetric theories for simple gauge groups SU(N1) with hypermultiplets in the fundamental representation, while others yield theories for product gauge groups SU(N1) ×…× SU(Np), with hypermultiplets in fundamental and bi-fundamental representations. The spectral curves obtained this way for these models agree with the ones proposed by Witten using D-branes and M-theory.

Calogero-Moser Lax pairs with spectral parameter for general Lie algebras

We construct a Lax pair with spectral parameter for the elliptic Calogero-Moser Hamiltonian systems associated with each of the finite-dimensional Lie algebras, of the classical and of the exceptional type. When the spectral parameter equals one of the three half periods of the elliptic curve, our result for the classical Lie algebras reduces to one of the Lax pairs without spectral parameter that were known previously. These Calogero-Moser systems are invariant under the Weyl group of the associated untwisted affine Lie algebra. For non-simply laced Lie algebras, we introduce new integrable systems, naturally associated with twisted affine Lie algebras, and construct their Lax operators with spectral parameter (except in the case of G2).