Papers by Gordon Chalmers
The solution term by term to the scattering of all consistent string theories is given. The modul... more The solution term by term to the scattering of all consistent string theories is given. The moduli space of M-theory is derived and connects the various string theories. The solutions contain both the perturbative and non-perturbative sectors of the string. Modular forms found by differential equations on subspaces of the M-theory moduli space and transfinite algebras play an essential role in deriving the coefficients. Various results and identities in algebra are found from the explicit solution. Archetypes and models are presented in accord with phenomenology and cosmology.
The moduli space describing the low-energy dynamics of BPS multi-monopoles for several charge con... more The moduli space describing the low-energy dynamics of BPS multi-monopoles for several charge configurations is presented. We first prove the conjectured form of the moduli space of n − 1 distinct monopoles in a spontaneously broken SU (n) gauge theory. We further propose the solution where one of the charge components has two units, hence asymptotically corresponds to embeddings of two monopoles of one charge type and the rest different. The latter hyperkähler metrics possess features of the two-monopole Atiyah-Hitchin metric. We also conjecture classes of solutions to multi-monopole moduli spaces with arbitrary charge and no more than two units in each component, which models the gluing together of Atiyah-Hitchin metrics. Our approach here uses the generalized Legendre transform technique to find the new hyperkähler manifolds and rederive previously conjectured ones.
The quantum correlations of scalar fields are examined as a power series in derivatives. Recursiv... more The quantum correlations of scalar fields are examined as a power series in derivatives. Recursive algebraic equations are derived and determine the amplitudes; all loop integrations are performed. This recursion contains the same information as the usual loop expansion. The approach is pragmatic and generalizable to most quantum field theories.
Aspects of Gauge and Scalar Field Theory: Factorization, Non-Polynomial Interactions, and Renormalization
Eprint Arxiv Physics 0510233, Oct 1, 2005
The graviton scattering in IIB superstring theory is examined in the context of S-duality and sym... more The graviton scattering in IIB superstring theory is examined in the context of S-duality and symmetry. There is an algebra that generates all of the terms in the four-point function to any order in derivatives. A map from the algebra to the scattering is given; it suggests the correctness of the full four-point function with the S-duality. The higher point functions are expected to follow a similar pattern.
Solitons, 2000
We summarize our recent work on low energy quantum corrections to monopoles in N = 2 supersymmetr... more We summarize our recent work on low energy quantum corrections to monopoles in N = 2 supersymmetric Yang-Mills theory. The details maybe found in [1]. Most of our understanding of solitons in quantum field theory comes from semiclassical expansion about classical solutions. However, it was realized long ago [2] that in principle there is an alternative: One can study "classical" solutions of the quantum effective action. Of course, in practice, one rarely has access to the full effective action. Nevertheless, one can study the solutions to various approximations and get approximate information. An example of this procedure is the Skyrme model where solitons of the QCD low energy effective action (depending on a mesonic field) are considered to be baryons. The key difference between that case and our problem is, as explained below, that we know the full non-perturbative as well as perturbative low-energy effective action; this should of course make our predictions more reliable. Three years ago, Seiberg and Witten [3] started a revolution when they found the full nonperturbative low-energy effective action of the massless states in N = 2 supersymmetric SU(2) Yang-Mills theory. This effective action can be gauge covariantized with repect to the spontaneously broken SU(2) gauge symmetry (as described below) to describe the massive charged vector multiplets of the theory. Evidence for the correctness of this procedure is provided by an interesting strong-coupling phenomenon: the massive charged vector multiplets destabilize for certain values of the Higgs vacuum expectation value, and precisely at these
A number theoretic algorithm is given for writing gauge theory amplitudes in a compact manner. It... more A number theoretic algorithm is given for writing gauge theory amplitudes in a compact manner. It is possible to write down all details of the complete $L$ loop amplitude with two integers, or a complex integer. However, a more symmetric notation requires more integers, five or seven, depending on the type of theory. It is possible that in the symmetric form (or in the non-symmetric form) that a direct (or less direct) recursive algorithm or generating function can be developed to compute these numbers at arbitrary loop order. The existence of this function is implied by the recursive structure of loop amplitudes and their analyticity, i.e. multi-particle poles; a function requiring a finite number of computations such as a polynomial with derivable coefficients is desired.
Journal of High Energy Physics, 1999
We rewrite the SL(2, Z) covariant worldsheet action for the IIB string proposed by Townsend in a ... more We rewrite the SL(2, Z) covariant worldsheet action for the IIB string proposed by Townsend in a Polyakov form. In a flat background the formalism yields separate (p, q) sectors. In each one the action is that of the IIB string action with the string slope parameter α ′ replaced by its SL(2, Z) analogue α ′ pq. SL(2, Z) invariant graviton scattering amplitudes are obtained from those of the fundamental (1, 0) string by summing over the different sectors. The tree-level four-graviton amplitude in this formalism differs from a previously conjectured non-perturbative form; both yield the same expansion in order α ′3 .
A scheme is presented based on numbers that represent a manifold in $d$ dimensions for generaliza... more A scheme is presented based on numbers that represent a manifold in $d$ dimensions for generalizations of textbook cryptosystems. The interlocking or intersection of geometries, requiring the addition of a series of integers $q_j$, can be used to enhance a cryptosystem via algorithms based on the form of the geometry. Further operations besides addition of these numbers, e.g. sewing of the geometries, can be given such as rotations or contractions.
Physical Review Letters, 1994
We present a conjecture for the leading color part of the n-gluon one-loop amplitudes with maxima... more We present a conjecture for the leading color part of the n-gluon one-loop amplitudes with maximal helicity violation. The conjecture emerges from the powerful requirement that the amplitudes have the correct behavior in the collinear limits of external momenta. One implication is that the corresponding amplitudes where three or more gluon legs are replaced by photons vanish for n > 4.
Physical Review D, 1996
We show that general cutoff scalar field theories in four dimensions are perturbatively renormali... more We show that general cutoff scalar field theories in four dimensions are perturbatively renormalizable through the use of diagrammatic techniques and an adapted BPH renormalization method. Weinberg's convergence theorem is used to show that operators in the Lagrangian with dimension greater than four, which are divided by powers of the cutoff, produce perturbatively only local divergences in the two-, three-, and four-point correlation functions. We also show that the renormalized Green's functions are the same as in ordinary Φ 4 theory up to corrections suppressed by inverse powers of the cutoff. These conclusions are consistent with those of existing proofs based on the renormalization group.
Physical Review D, 1998
Using twistor methods we derive a generating function which leads to the hyperkähler metric on a ... more Using twistor methods we derive a generating function which leads to the hyperkähler metric on a deformation of the Atiyah-Hitchin monopole moduli space. This deformation was first considered by Dancer through the quotient construction and is related to a charge two monopole configuration in a completely broken SU (3) gauge theory. The manifold and metric are the first members of a family of hyperkähler manifolds which are deformations of the D k rational singularities of C 2 .
Physical Review D, 2001
We show the existence of a global anomaly in the one-loop graphs of N = 2 string theory, defined ... more We show the existence of a global anomaly in the one-loop graphs of N = 2 string theory, defined by sewing tree amplitudes, unless spacetime supersymmetry is imposed. The anomaly is responsible for the non-vanishing maximally helicity violating amplitudes. The supersymmetric completion of the N = 2 string spectrum is formulated by extending the previous cohomological analysis with an external spin factor; the target space-time spin-statistics of these individual fields in a selfdual background are compatible with previous cohomological analysis as fields of arbitrary spin may be bosonized into one another. We further analyze duality relations between the open and closed string amplitudes and demonstrate this in the supersymmetric extension of the target space-time theory through the insertion of zero-momentum operators.
Physical Review D, 2000
The general correlator of composite operators of N=4 supersymmetric gauge field theory is diverge... more The general correlator of composite operators of N=4 supersymmetric gauge field theory is divergent. We introduce a means for renormalizing these correlators by adding a boundary theory on the AdS space correcting for the divergences. Such renormalizations are not equivalent to the standard normal ordering of current algebras in two dimensions. The correlators contain contact terms that contribute to the OPE; we relate them diagrammatically to correlation functions of compound composite operators dual to multi-particle states.
Physical Review D, 2001
The strongly coupled limit of string scattering and the automorphic construction of the graviton ... more The strongly coupled limit of string scattering and the automorphic construction of the graviton S-matrix is compared with the eleven dimensional formulation of Mtheory. In a particular scaling limit at strong string coupling, M-theory is described by eleven-dimensional supergravity which does not possess a dilaton, but rather a perturbative expansion in the gravitational coupling and derivatives. The latter theory provides an off-shell description of the string, upon dimensional reduction.
Nuclear Physics B, 2000
S and U-duality dictate that graviton scattering amplitudes in IIB superstring theory be automorp... more S and U-duality dictate that graviton scattering amplitudes in IIB superstring theory be automorphic functions on the appropriate fundamental domain which describe the inequivalent vacua of (compactified) theories. A constrained functional form of graviton scattering is proposed using Eisenstein series and their generalizations compatible with: a) two-loop supergravity, b) genus one superstring theory, c) the perturbative coupling dependence of the superstring, and d) with the unitarity structure of the massless modes. The form has a perturbative truncation in the genus expansion at a given order in the derivative expansion. Comparisons between graviton scattering S-matrices and effective actions for the first quantized superstring are made at the quantum level. Possible extended finiteness properties of maximally extended quantum supergravity theories in different dimensions is implied by the perturbative truncation of the functional form of graviton scattering in IIB superstring theory.
Nuclear Physics B, 2000
We examine the dual correspondence between holographic IIB superstring theory and N = 4 super Yan... more We examine the dual correspondence between holographic IIB superstring theory and N = 4 super Yang-Mills theory at finite values of the coupling constants. In particular we analyze a field theory strong-coupling expansion which is the S-dual of the planar expansion. This expansion arises naturally as the AdS/CFT dual of the IIB superstring scattering amplitudes given a genus truncation property due to modular invariance. The space-time structure of the contributions to the field theory four-point correlation functions obtained from the IIB scattering elements is investigated in the example of the product of four conserved stress tensors, and is expressed as an infinite sum of field theory triangle integrals. The OPE structure of these contributions to the stress tensor four-point function is analyzed and shown not to give rise to any poles. Quantization of the string in the background of a five-form field strength is performed through a covariantized background field approach, and relations to the N = 4 topological string are found.
Nuclear Physics B, 1995
Factorization properties of one-loop gauge theory amplitudes have been used as checks on explicit... more Factorization properties of one-loop gauge theory amplitudes have been used as checks on explicitly computed amplitudes and in the construction of ansätze for higher-point ones. In massless theories, such as QCD at high energies, infrared divergences complicate factorization. Here we prove that factorization in such theories is described by a set of universal functions. In particular, a proof of the universality of one-loop splitting functions as the momenta of two particles become collinear is presented. Factorization in multi-particle channels is also given. The discontinuity functions that appear in the splitting functions may also be used to obtain infrared divergent box integral functions from finite ones.
Nuclear Physics B, 2000
We calculate the genus-one three-and four-point amplitudes in the 2+2 dimensional closed N=(2, 2)... more We calculate the genus-one three-and four-point amplitudes in the 2+2 dimensional closed N=(2, 2) worldsheet supersymmetric string within the RNS formulation. Vertex operators are redefined with the incorporation of spinor helicity techniques, and the quantum scattering is shown to be manifestly gauge and Lorentz invariant after normalizing the string states. The continuous spin structure summation over the monodromies of the worldsheet fermions is carried out explicitly, and the field-theory limit is extracted. The amplitude in this limit is shown to be the maximally helicity violating amplitude in pure gravity evaluated in a twodimensional setting, which vanishes, unlike the four-dimensional result. The vanishing of the genus-one N=2 closed string amplitude is related to the absence of one-loop divergences in dimensionally regulated IIB supergravity. Comparisons and contrasts between self-dual field theory and the N=2 string theory are made at the quantum level; they have different S-matrices. Finally, we point to further relations with self-dual field theory and two-dimensional models. −} are spinor indices of SL(2, R) (or SU (2)).
Classical and Quantum Gravity, 2002
We examine the form of the cosmological constant in the loop expansion of broken maximally supers... more We examine the form of the cosmological constant in the loop expansion of broken maximally supersymmetric supergravity theories, and after embedding, within superstring and M-theory. Supersymmetry breaking at the TeV scale generates values of the cosmological constant that are in agreement with current astrophysical data. The form of perturbative quantum effects in the loop expansion is consistent with this parameter regime.
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Papers by Gordon Chalmers