def tf_A123(self, t_T, want_A1=True, want_A2=True, want_A3=True):
"""See base class."""
t_A1 = t_A2 = t_A3 = None
if want_A1:
t_A1 = mu.tfc128(-4 / 21) * tf.einsum('mijm->ij', t_T)
if want_A2 or want_A3:
t_A2 = mu.tfc128(-4 / (3 * 3)) * (
# Antisymmetrize in last 3 indices, but using antisymmetry in last 2.
# Note factor 1/3 above (in -4/(3*3) rather than -4/3).
t_T + tf.einsum('lijk->ljki', t_T) + tf.einsum('lijk->lkij', t_T))
if want_A3:
t_A3 = a123.tf_A3_from_A2(t_A2)
return t_A1, t_A2, t_A3
def tf_A123(self, t_T, want_A1=True, want_A2=True, want_A3=True):
t_A1, t_A2, t_A3 = None, None, None
if want_A1:
t_A1 = tf.einsum('mijm->ij', t_T) * mu.tfc128(-4.0 / 21)
# TODO(tfish): Do check if the SO(8)-origin-A1 is actually diag([1.0]*8)
# or perhaps now has an extra factor 1j.
if want_A2 or want_A3:
# Over-satisfying is OK: When asked for A3 only, we also compute A2.
t_A2 = mu.tfc128(-4.0 /
(3 * 3)) * (t_T + tf.einsum('lijk->ljki', t_T) +
tf.einsum('lijk->lkij', t_T))
if want_A3:
t_A3 = a123.tf_A3_from_A2(t_A2)
return t_A1, t_A2, t_A3
def __init__(
self,
Theta,
e7=e7,
verbose=False,
check_gaugeability=True,
gaugeability_atol=1e-8,
# Either `None`, or ('SUSY', None),
# or ('M2G', {[8]-sequence of masses}).
stationarity_tweak=None):
super().__init__(e7.t56r, verbose=verbose)
if check_gaugeability:
if get_gaugeability_condition_violations(Theta,
e7=e7,
atol=gaugeability_atol):
raise ValueError('Non-gaugeable Theta-tensor provided.')
self._stationarity_tweak = stationarity_tweak
self._opt_tc_stationarity_tweak = (
None if stationarity_tweak is None or stationarity_tweak[1] is None
else mu.tff64(stationarity_tweak[1]))
self._tc_X = tc_X = mu.tff64(get_X(Theta, e7=e7))
self._tc_XX = tf.einsum('MNQ,PQN->MP', tc_X, tc_X)
self._tc_e7_S_rc = tf.constant(e7.S_rc, dtype=tf.complex128)
self._tc_1j = mu.tfc128(0, 1)
self._tc_28_8_8 = tf.constant(algebra.g.su8.m_28_8_8,
dtype=tf.complex128)
self._tc_56_888 = tf.constant(algebra.g.su8.m_56_8_8_8,
dtype=tf.complex128)
self._tc_eps_56_56_8_8 = tf.constant(algebra.g.su8.eps_56_56_8_8,
dtype=tf.complex128)
self._tc_omega = tf.constant(e7.omega, dtype=tf.complex128)
def tf_fermion_massmatrix(t_A3, t_potential, tc_masses_factor):
"""Computes the spin-1/2 mass matrix from the A3-tensor."""
# The extra factor 2.0 relative to https://arxiv.org/abs/1906.00207
# makes the fermion masses align with the way particle states are
# grouped into SUSY multiplets in appendix (B.2) of:
# https://arxiv.org/abs/1909.10969
return mu.tfc128(2.0) * tf.einsum('ij,ik->jk', t_A3, tf.math.conj(
t_A3)) * (tc_masses_factor / tf.cast(t_potential, tf.complex128))
def tf_T(self, t_vielbein):
# Here, we do not actually compute the full T-tensor, but the T_i^jkl part.
# This is all that we ever need here.
V56_RC = tf.einsum('RS,SC->RC', tf.cast(t_vielbein, tf.complex128),
self._tc_e7_S_rc)
t_Q_Mijkl = self._tc_1j * tf.einsum(
'NP,NI,Iij,MPQ,QK,Kkl->Mijkl', self._tc_omega, V56_RC[:, 28:],
self._tc_28_8_8, tf.cast(
self._tc_X, tf.complex128), V56_RC[:, :28], self._tc_28_8_8)
t_Q_Mij = tf.einsum('Mijil->Mjl', t_Q_Mijkl) * mu.tfc128(-2 / 3.0)
return mu.tfc128(0.0, 0.75 / 2**.5) * tf.einsum(
# Note fudge-factor sqrt(2) above.
# This makes the T-tensor match the colab notebook on SO(8) gauging.
'MN,Mkl,NI,Iij->klij',
self._tc_omega,
t_Q_Mij,
V56_RC[:, :28],
self._tc_28_8_8)
def tf_A3_from_A2(t_A2):
"""Computes the A3-tensor from the A2-tensor."""
tc_56_8_8_8 = tf.constant(algebra.g.su8.m_56_8_8_8.astype(
numpy.complex128),
dtype=tf.complex128)
tc_eps_56_56_8_8 = tf.constant(algebra.g.su8.eps_56_56_8_8.astype(
numpy.complex128),
dtype=tf.complex128)
t_A2_nP = tf.einsum('nijk,Pijk->nP', tf.math.conj(t_A2), tc_56_8_8_8)
return tf.einsum('nP,APlm,Blmn->AB', t_A2_nP, tc_eps_56_56_8_8,
tc_56_8_8_8) * mu.tfc128(2**.5 / 24.0)
def tf_vector_massmatrix(t_A2, t_potential, tc_masses_factor):
"""Computes the spin-1 mass matrix from the A2-tensor."""
# This is most readily available in https://arxiv.org/pdf/1103.2785.pdf,
# Eq. (5.27) and (5.28), setting the scaling-symmetry ('trombone')
# contributions B to zero.
# With the conventions from arXiv:1103.2785, we reproduce
# Table B.3 of arXiv:1909.10969.
# Note that we get 56 masses, 28 of which are spurious and will always be zero
# in any valid gauging.
tc_56_8_8_8 = tf.constant(algebra.g.su8.m_56_8_8_8.astype(
numpy.complex128),
dtype=tf.complex128)
tc_28_8_8 = tf.constant(algebra.g.su8.m_28_8_8.astype(numpy.complex128),
dtype=tf.complex128)
tc_eps_56_56_8_8 = tf.constant(algebra.g.su8.eps_56_56_8_8.astype(
numpy.complex128),
dtype=tf.complex128)
t_A2_8_56 = mu.tfc128(1 / 6.0) * tf.einsum('ipqr,Xpqr->iX', t_A2,
tc_56_8_8_8)
t_A2c_8_56 = tf.math.conj(t_A2_8_56)
M_ij_KL = (
mu.tfc128(-1 / 4.0) *
tf.einsum('iX,lX,jk,Iij,Kkl->IK', t_A2_8_56, t_A2c_8_56,
tf.eye(8, dtype=tf.complex128), tc_28_8_8, tc_28_8_8) +
mu.tfc128(1 / 8.0) *
tf.einsum('ipqk,ljpq,Iij,Kkl->IK', t_A2, tf.math.conj(t_A2), tc_28_8_8,
tc_28_8_8))
M_ijkl = mu.tfc128(
1 / 4.0) * tf.einsum('iX,lY,XYjk,Iij,Kkl->IK', t_A2_8_56, t_A2_8_56,
tc_eps_56_56_8_8, tc_28_8_8, tc_28_8_8)
M_vec = tf.reshape(
tf.einsum(
'abAB->aAbB',
tf.reshape(
tf.stack([
M_ij_KL, M_ijkl,
tf.math.conj(M_ijkl),
tf.math.conj(M_ij_KL)
]), [2, 2, 28, 28])), [56, 56])
return (M_vec * tc_masses_factor / tf.cast(t_potential, tf.complex128))
def tf_dwn_stationarity_vec(t_A1, t_A2):
"""Computes stationarity-violation 70-vector 'in the local frame'."""
# Formula:
# arXiv: https://arxiv.org/pdf/1302.6219.pdf, formula (3.2);
# originally: https://inspirehep.net/literature/191530
# (2.20) - (2.22).
_m_8888sd_35ortho, _m_8888asd_35ortho = _get_35sd_asd_bases()
t_x0 = (tf.einsum('mi,mjkl->ijkl', t_A1, t_A2) +
mu.tfc128(-0.75, 0.0) * tf.einsum('mnij,nklm->ijkl', t_A2, t_A2))
t_x0_real = tf.math.real(t_x0)
t_x0_imag = tf.math.imag(t_x0)
# The self-dual part must be zero.
t_x0_re_sd = tf.einsum('ijkl,ijklX->X', t_x0_real,
mu.tff64(_m_8888sd_35ortho))
t_x0_im_asd = tf.einsum('ijkl,ijklX->X', t_x0_imag,
mu.tff64(_m_8888asd_35ortho))
return tf.concat([t_x0_re_sd, t_x0_im_asd], axis=0)
def tf_vector_massmatrix(self, ts_A123, t_potential):
"""See base class."""
_, t_A2, _ = ts_A123
return a123.tf_vector_massmatrix(
t_A2, t_potential, mu.tfc128(-self.signature.vector_masses_factor))
def tf_fermion_massmatrix(self, ts_A123, t_potential):
"""See base class."""
*_, t_A3 = ts_A123
return a123.tf_fermion_massmatrix(
t_A3, t_potential,
mu.tfc128(-self.signature.fermion_masses_factor))