201 research outputs found
Electric-magnetic Duality of Abelian Gauge Theory on the Four-torus, from the Fivebrane on T2 x T4, via their Partition Functions
We compute the partition function of four-dimensional abelian gauge theory on
a general four-torus T4 with flat metric using Dirac quantization. In addition
to an SL(4, Z) symmetry, it possesses SL(2,Z) symmetry that is electromagnetic
S-duality. We show explicitly how this SL(2, Z) S-duality of the 4d abelian
gauge theory has its origin in symmetries of the 6d (2,0) tensor theory, by
computing the partition function of a single fivebrane compactified on T2 x T4,
which has SL(2,Z) x SL(4,Z) symmetry. If we identify the couplings of the
abelian gauge theory \tau = {\theta\over 2\pi} + i{4\pi\over e^2} with the
complex modulus of the T2 torus, \tau = \beta^2 + i {R_1\over R_2}, then in the
small T2 limit, the partition function of the fivebrane tensor field can be
factorized, and contains the partition function of the 4d gauge theory. In this
way the SL(2,Z) symmetry of the 6d tensor partition function is identified with
the S-duality symmetry of the 4d gauge partition function. Each partition
function is the product of zero mode and oscillator contributions, where the
SL(2,Z) acts suitably. For the 4d gauge theory, which has a Lagrangian, this
product redistributes when using path integral quantization.Comment: 41 pages, published versio
Proof of the Formula of Cachazo, He and Yuan for Yang-Mills Tree Amplitudes in Arbitrary Dimension
A proof is given of the formula, recently proposed by Cachazo, He and Yuan
(CHY) for gluon tree amplitudes in pure Yang-Mills theory in arbitrary
dimension. The approach is to first establish the corresponding result for
massless theory using the BCFW recurrence relation and then to extend
this to the gauge theory case. Additionally, it is shown that the scattering
equations introduced by CHY can be generalized to massive particles, enabling
the description of tree amplitudes for massive theory.Comment: 27 p
General Solution of the Scattering Equations
The scattering equations, originally introduced by Fairlie and Roberts in
1972 and more recently shown by Cachazo, He and Yuan to provide a kinematic
basis for describing tree amplitudes for massless particles in arbitrary
space-time dimension, have been reformulated in polynomial form. The scattering
equations for N particles are equivalent to N-3 polynomial equations h_m=0,
m=1,...,N-3, in N-3 variables, where h_m has degree m and is linear in the
individual variables. Facilitated by this linearity, elimination theory is used
to construct a single variable polynomial equation of degree (N-3)! determining
the solutions. \Delta_N is the sparse resultant of the system of polynomial
scattering equations and it can be identified as the hyperdeterminant of a
multidimensional matrix of border format within the terminology of Gel'fand,
Kapranov and Zelevinsky. Macaulay's Unmixedness Theorem is used to show that
the polynomials of the scattering equations constitute a regular sequence,
enabling the Hilbert series of the variety determined by the scattering
equations to be calculated, independently showing that they have (N-3)!
solutions.Comment: v2 completes the proof that the construction yields \Delta_N for all
N, identifies it as the hyperdeterminant of a multidimensional matrix, and
proves that the polynomial scattering equations constitute a regular
sequence, enabling the Hilbert series of the associated variety to be
calculated, 26 page
The Polynomial Form of the Scattering Equations
The scattering equations, recently proposed by Cachazo, He and Yuan as
providing a kinematic basis for describing tree amplitudes for massless
particles in arbitrary space-time dimension (including scalars, gauge bosons
and gravitons), are reformulated in polynomial form. The scattering equations
for particles are shown to be equivalent to a Moebius invariant system of
equations, , , in variables, where
is a homogeneous polynomial of degree m, with the exceptional
property of being linear in each variable taken separately. Fixing the Moebius
invariance appropriately, yields polynomial equations , , in variables, where has degree . The linearity of the
equations in the individual variables facilitates computation, e.g the
elimination of variables to obtain single variable equations determining the
solutions. Expressions are given for the tree amplitudes in terms of the
and . The extension to the massive case for scalar particles
is described and the special case of four dimensional space-time is discussed.Comment: 24 page
Vertex Operators for AdS3 Background With Ramond Ramond Flux
In order to study vertex operators for the Type IIB superstring on AdS space,
we derive supersymmetric constraint equations for the vertex operators in
AdS3xS3 backgrounds with Ramond-Ramond flux, using Berkovits-Vafa-Witten
variables. These constraints are solved to compute the vertex operators and
show that they satisfy the linearized D=6, N=(2,0) equations of motion for a
supergravity and tensor multiplet expanded around the AdS3xS3 spacetime.Comment: harvmac, 23 page
Conformal Operators for Partially Massless States
The AdS/CFT correspondence is explored for ``partially massless'' fields in
AdS space (which have fewer helicity states than a massive field but more than
a conventional massless field). Such fields correspond in the boundary
conformal field theory to fields obeying a certain conformally-invariant
differential equation that has been described by Eastwood et al. The first
descendant of such a field is a conformal field of negative norm. Hence,
partially massless fields may make more physical sense in de Sitter as opposed
to Anti de Sitter space.Comment: 14 page
Current Algebra on the Torus
We derive the N-point one-loop correlation functions for the currents of an arbitrary affine Kac-Moody algebra. The one-loop amplitudes, which are elliptic functions defined on the torus Riemann surface, are specified by group invariant tensors and certain constant taudependent functions. We compute the elliptic functions via a generating function, and explicitly construct the invariant tensor functions recursively in terms of Young tableaux. The lowest tensors are related to the character formula of the representation of the affine algebra. These general current algebra loop amplitudes provide a building block for open twistor string theory, among other applications
Conformal Supergravity Tree Amplitudes from Open Twistor String Theory
We display the vertex operators for all states in the conformal supergravity
sector of the twistor string, as outlined by Berkovits and Witten. These
include `dipole' states, which are pairs of supergravitons that do not
diagonalize the translation generators. We use canonical quantization of the
open string version of Berkovits, and compute N-point tree level scattering
amplitudes for gravitons, gluons and scalars. We reproduce the Berkovits-Witten
formula for maximal helicity violating (MHV) amplitudes (which they derived
using path integrals), and extend their results to the dipole pairs. We compare
these trees with those of Einstein gravity field theory.Comment: 31 pages, expanded version, references adde
The Ramond-Ramond self-dual Five-form's Partition Function on the Ten Torus
In view of the recent interest in formulating a quantum theory of
Ramond-Ramond p-forms, we exhibit an SL(10,Z) invariant partition function for
the chiral four-form of Type IIB string theory on the ten-torus. We follow the
strategy used to derive a modular invariant partition function for the chiral
two-form of the M-theory fivebrane. We also generalize the calculation to
self-dual quantum fields in spacetime dimension 2p=2+4k, and display the
SL(2p,Z) automorphic forms for odd p>1. We relate our explicit calculation to a
computation of the B-cycle periods, which are discussed in the work of Witten.Comment: 18 page
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