161 research outputs found
Non-Perturbative Quantum Geometry III
The Nekrasov-Shatashvili limit of the refined topological string on toric
Calabi-Yau manifolds and the resulting quantum geometry is studied from a
non-perturbative perspective. The quantum differential and thus the quantum
periods exhibit Stokes phenomena over the combined string coupling and
quantized Kaehler moduli space. We outline that the underlying formalism of
exact quantization is generally applicable to points in moduli space featuring
massless hypermultiplets, leading to non-perturbative band splitting. Our prime
example is local P1xP1 near a conifold point in moduli space. In particular, we
will present numerical evidence that in a Stokes chamber of interest the string
based quantum geometry reproduces the non-perturbative corrections for the
Nekrasov-Shatashvili limit of 4d supersymmetric SU(2) gauge theory at strong
coupling found in the previous part of this series. A preliminary discussion of
local P2 near the conifold point in moduli space is also provided.Comment: 34 pages; v2: Minor correction and refs added; v3: Table 2 modified,
clarifying comment and footnote adde
Non-Perturbative Quantum Geometry
The beta-ensemble with cubic potential can be used to study a quantum
particle in a double-well potential with symmetry breaking term. The quantum
mechanical perturbative energy arises from the ensemble free energy in a novel
large N limit. A relation between the generating functions of the exact
non-perturbative energy, similar in spirit to the one of Dunne-Unsal, is found.
The exact quantization condition of Zinn-Justin and Jentschura is equivalent to
the Nekrasov-Shatashvili quantization condition on the level of the ensemble.
Refined topological string theory in the Nekrasov-Shatashvili limit arises as a
large N limit of quantum mechanics.Comment: 20 page
Modelling conditional probabilities with Riemann-Theta Boltzmann Machines
The probability density function for the visible sector of a Riemann-Theta
Boltzmann machine can be taken conditional on a subset of the visible units. We
derive that the corresponding conditional density function is given by a
reparameterization of the Riemann-Theta Boltzmann machine modelling the
original probability density function. Therefore the conditional densities can
be directly inferred from the Riemann-Theta Boltzmann machine.Comment: 7 pages, 3 figures, in proceedings of the 19th International Workshop
on Advanced Computing and Analysis Techniques in Physics Research (ACAT 2019
Exact Chern-Simons / Topological String duality
We invoke universal Chern-Simons theory to analytically calculate the exact
free energy of the refined topological string on the resolved conifold. In the
unrefined limit we reproduce non-perturbative corrections for the resolved
conifold found elsewhere in the literature, thereby providing strong evidence
that the Chern-Simons / topological string duality is exact, and in particular
holds at arbitrary N as well. In the refined case, the non-perturbative
corrections we find are novel and appear to be non-trivial. We show that
non-perturbatively special treatment is needed for rational valued deformation
parameter. Above results are also extend to refined Chern-Simons with
orthogonal groups.Comment: 32 page
Riemann-Theta Boltzmann Machine
A general Boltzmann machine with continuous visible and discrete integer
valued hidden states is introduced. Under mild assumptions about the connection
matrices, the probability density function of the visible units can be solved
for analytically, yielding a novel parametric density function involving a
ratio of Riemann-Theta functions. The conditional expectation of a hidden state
for given visible states can also be calculated analytically, yielding a
derivative of the logarithmic Riemann-Theta function. The conditional
expectation can be used as activation function in a feedforward neural network,
thereby increasing the modelling capacity of the network. Both the Boltzmann
machine and the derived feedforward neural network can be successfully trained
via standard gradient- and non-gradient-based optimization techniques.Comment: 29 pages, 11 figures, final version published in Neurocomputin
A gauge theory analog of some "stringy" D-instantons
We argue that one can see a specific class of "stringy" D-instantons in the
underlying 4D gauge theory as the UV completion of an ordinary gauge instanton
of a completely broken gauge group corresponding to the "empty" cycle the
D-instanton is located on. In this sense, the D-instanton induced
non-perturbative superpotential can be qualitatively inferred directly from
pure field-theory considerations.Comment: 4 pages, 3 figures; typos corrected; discussion of dynamical scale
extende
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