58 research outputs found
From strings in 6d to strings in 5d
We show how recent progress in computing elliptic genera of strings in six
dimensions can be used to obtain expressions for elliptic genera of strings in
five-dimensional field theories which have a six-dimensional parent. We further
connect our results to recent mathematical results about sheaf counting on
ruled surfaces.Comment: 17 pages, 4 figures. v3: Definitions clarified, Section "Concluding
Thoughts" adde
M5 branes and Theta Functions
We propose quantum states for Little String Theories (LSTs) arising from M5
branes probing A- and D-type singularities. This extends Witten's picture of M5
brane partition functions as theta functions to this more general setup.
Compactifying the world-volume of the five-branes on a two-torus, we find that
the corresponding theta functions are sections of line bundles over complex
4-tori. This formalism allows us to derive Seiberg-Witten curves for the
resulting four-dimensional theories. Along the way, we prove a duality for LSTs
observed by Iqbal, Hohenegger and Rey.Comment: 27 pages, 1 Figure. v3: included further explanations and corrected
typos. This the version published in JHE
A 5d/2d/4d correspondence
We propose a correspondence between two-dimensional (0,4) sigma models with
target space the moduli spaces of r monopoles, and four-dimensional N=4, U(r)
Yang-Mills theory on del Pezzo surfaces. In particular, the two- and
four-dimensional BPS partition functions are argued to be equal. The
correspondence relies on insights from five-dimensional supersymmetric gauge
theory and its geometric engineering in M-theory, hence the name "5d/2d/4d
correspondence". We provide various tests of our proposal. The most stringent
ones are for r=1, for which we prove the equality of partition functions.Comment: 39 pages, final versio
On Topological String Theory with Calabi-Yau Backgrounds
String theory represents a unifying framework for quantum field theory as well as for general relativity combining them into a theory of quantum gravity. The topological string is a subsector of the full string theory capturing physical amplitudes which only depend on the topology of the compactification manifold. Starting with a review of the physical applications of topological string theory we go on to give a detailed description of its theoretical framework and mathematical principles. Having this way provided the grounding for concrete calculations we proceed to solve the theory on three major types of Calabi-Yau manifolds, namely Grassmannian Calabi-Yau manifolds, local Calabi-Yau manifolds, and K3 fibrations. Our method of solution is the integration of the holomorphic anomaly equations and fixing the holomorphic ambiguity by physical boundary conditions. We determine the correct parameterization of the ambiguity and new boundary conditions at various singularity loci in moduli space. Among the main results of this thesis are the tables of degeneracies of BPS states in the appendices and the verification of the correct microscopic entropy interpretation for five dimensional extremal black holes arising from compactifications on Grassmannian Calabi-Yau manifolds
ADE String Chains and Mirror Symmetry
6d superconformal field theories (SCFTs) are the SCFTs in the highest
possible dimension. They can be geometrically engineered in F-theory by
compactifying on non-compact elliptic Calabi-Yau manifolds. In this paper we
focus on the class of SCFTs whose base geometry is determined by curves
intersecting according to ADE Dynkin diagrams and derive the corresponding
mirror Calabi-Yau manifold. The mirror geometry is uniquely determined in terms
of the mirror curve which has also an interpretation in terms of the
Seiberg-Witten curve of the four-dimensional theory arising from torus
compactification. Adding the affine node of the ADE quiver to the base
geometry, we connect to recent results on SYZ mirror symmetry for the case
and provide a physical interpretation in terms of little string theory. Our
results, however, go beyond this case as our construction naturally covers the
and cases as well.Comment: version 2: typos corrected, 30 pages, 8 figure
Blowup Equations for 6d SCFTs. I
We propose novel functional equations for the BPS partition functions of 6d
(1,0) SCFTs, which can be regarded as an elliptic version of
Gottsche-Nakajima-Yoshioka's K-theoretic blowup equations. From the viewpoint
of geometric engineering, these are the generalized blowup equations for
refined topological strings on certain local elliptic Calabi-Yau threefolds. We
derive recursion formulas for elliptic genera of self-dual strings on the
tensor branch from these functional equations and in this way obtain a
universal approach for determining refined BPS invariants. As examples, we
study in detail the minimal 6d SCFTs with SU(3) and SO(8) gauge symmetry. In
companion papers, we will study the elliptic blowup equations for all other
non-Higgsable clusters.Comment: 52 pages, 3 figure
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
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