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We introduce \texttt{cymyc}, a high-performance Python library for numerical investigation of the geometry of a large class of string compactification manifolds and their associated moduli spaces. We develop a well-defined geometric ansatz to numerically model tensor fields of arbitrary degree on a large class of Calabi-Yau manifolds. \texttt{cymyc} includes a machine learning component which incorporates this ansatz to model tensor fields of interest on these spaces by finding an approximate solution to the system of partial differential equations they should satisfy., Comment: 35 pages, 12 figures

Calabi--Yau compactifications of the $E_8\times E_8$ heterotic string provide a promising route to recovering the four-dimensional particle physics described by the Standard Model. While the topology of the Calabi--Yau space determines the overall matter content in the low-energy effective field theory, further details of the compactification geometry are needed to calculate the normalized physical couplings and masses of elementary particles. In this work, we present numerical computations of physical Yukawa couplings in a number of heterotic models in the standard embedding and demonstrate the existence of natural hierarchies, a coveted feature in string model building., Comment: 7 pages, 3 figures

We apply reinforcement learning (RL) to generate fine regular star triangulations of reflexive polytopes, that give rise to smooth Calabi-Yau (CY) hypersurfaces. We demonstrate that, by simple modifications to the data encoding and reward function, one can search for CYs that satisfy a set of desirable string compactification conditions. For instance, we show that our RL algorithm can generate triangulations together with holomorphic vector bundles that satisfy anomaly cancellation and poly-stability conditions in heterotic compactification. Furthermore, we show that our algorithm can be used to search for reflexive subpolytopes together with compatible triangulations that define fibration structures of the CYs.

One of the challenges of heterotic compactification on a Calabi-Yau threefold is to determine the physical $(\mathbf{27})^3$ Yukawa couplings of the resulting four-dimensional $\mathcal{N}=1$ theory. In general, the calculation necessitates knowledge of the Ricci-flat metric. However, in the standard embedding, which references the tangent bundle, we can compute normalized Yukawa couplings from the Weil-Petersson metric on the moduli space of complex structure deformations of the Calabi-Yau manifold. In various examples (the Fermat quintic, the intersection of two cubics in $\mathbb{P}^5$, and the Tian-Yau manifold), we calculate the normalized Yukawa couplings for $(2,1)$-forms using the Weil-Petersson metric obtained from the Kodaira-Spencer map. In cases where $h^{1,1}=1$, this is compared to a complementary calculation based on performing period integrals. A third expression for the normalized Yukawa couplings is obtained from a machine learned approximate Ricci-flat metric making use of explicit harmonic representatives. The excellent agreement between the different approaches opens the door to precision string phenomenology., Comment: 33 pages, 11 figures, 2 tables, 3 lemmas, 1 theorem. v2: Minor edits

We apply machine learning to understand fundamental aspects of holographic duality, specifically the entropies obtained from the apparent and event horizon areas. We show that simple features of only the time series of the pressure anisotropy, namely the values and half-widths of the maxima and minima, the times these are attained, and the times of the first zeroes can predict the areas of the apparent and event horizons in the dual bulk geometry at all times with a fixed maximum length (10) of the input vector. Given that simple Vaidya-type metrics constructed just from the apparent and event horizon areas can be used to approximately obtain unequal time correlation functions, we argue that the corresponding entropy functions are the measures of information that need to be extracted from simple one-point functions to reconstruct specific aspects of correlation functions of the dual state with the best possible approximations., Comment: 16 pages, 1 Figure; results tested with initial conditions given by 153 parameters

In this work we employ machine learning to understand structured mathematical data involving finite groups and derive a theorem about necessary properties of generators of finite simple groups. We create a database of all 2-generated subgroups of the symmetric group on n-objects and conduct a classification of finite simple groups among them using shallow feed-forward neural networks. We show that this neural network classifier can decipher the property of simplicity with varying accuracies depending on the features. Our neural network model leads to a natural conjecture concerning the generators of a finite simple group. We subsequently prove this conjecture. This new toy theorem comments on the necessary properties of generators of finite simple groups. We show this explicitly for a class of sporadic groups for which the result holds. Our work further makes the case for a machine motivated study of algebraic structures in pure mathematics and highlights the possibility of generating new conjectures and theorems in mathematics with the aid of machine learning., Comment: 25 pages, 6 figures and 5 tables

Calabi-Yau manifolds can be obtained as hypersurfaces in toric varieties built from reflexive polytopes. We generate reflexive polytopes in various dimensions using a genetic algorithm. As a proof of principle, we demonstrate that our algorithm reproduces the full set of reflexive polytopes in two and three dimensions, and in four dimensions with a small number of vertices and points. Motivated by this result, we construct five-dimensional reflexive polytopes with the lowest number of vertices and points. By calculating the normal form of the polytopes, we establish that many of these are not in existing datasets and therefore give rise to new Calabi-Yau four-folds. In some instances, the Hodge numbers we compute are new as well., Comment: 24 pages, 2 figures. Accepted version

Finding Ricci-flat (Calabi-Yau) metrics is a long standing problem in geometry with deep implications for string theory and phenomenology. A new attack on this problem uses neural networks to engineer approximations to the Calabi-Yau metric within a given K\"ahler class. In this paper we investigate numerical Ricci-flat metrics over smooth and singular K3 surfaces and Calabi-Yau threefolds. Using these Ricci-flat metric approximations for the Cefal\'u family of quartic twofolds and the Dwork family of quintic threefolds, we study characteristic forms on these geometries. We observe that the numerical stability of the numerically computed topological characteristic is heavily influenced by the choice of the neural network model, in particular, we briefly discuss a different neural network model, namely Spectral networks, which correctly approximate the topological characteristic of a Calabi-Yau. Using persistent homology, we show that high curvature regions of the manifolds form clusters near the singular points. For our neural network approximations, we observe a Bogomolov--Yau type inequality $3c_2 \geq c_1^2$ and observe an identity when our geometries have isolated $A_1$ type singularities. We sketch a proof that $\chi(X~\smallsetminus~\mathrm{Sing}\,{X}) + 2~|\mathrm{Sing}\,{X}| = 24$ also holds for our numerical approximations., Comment: Version accepted for publication: 48 pages, 32 figures, 8 tables, 3 appendices

We automate the process of machine learning correlations between knot invariants. For nearly 200,000 distinct sets of input knot invariants together with an output invariant, we attempt to learn the output invariant by training a neural network on the input invariants. Correlation between invariants is measured by the accuracy of the neural network prediction, and bipartite or tripartite correlations are sequentially filtered from the input invariant sets so that experiments with larger input sets are checking for true multipartite correlation. We rediscover several known relationships between polynomial, homological, and hyperbolic knot invariants, while also finding novel correlations which are not explained by known results in knot theory. These unexplained correlations strengthen previous observations concerning links between Khovanov and knot Floer homology. Our results also point to a new connection between quantum algebraic and hyperbolic invariants, similar to the generalized volume conjecture., Comment: 30 pages, 9 figures

Superconformal indices of four-dimensional $\mathcal{N}=1$ gauge theories factorize into holomorphic blocks. We interpret this as a modular property resulting from the combined action of an $SL(3,\mathbb{Z})$ and $SL(2,\mathbb{Z})\ltimes \mathbb{Z}^2$ transformation. The former corresponds to a gluing transformation and the latter to an overall large diffeomorphism, both associated with a Heegaard splitting of the underlying geometry. The extension to more general transformations leads us to argue that a given index can be factorized in terms of a family of holomorphic blocks parametrized by modular (congruence sub)groups. We find precise agreement between this proposal and new modular properties of the elliptic $\Gamma$ function. This leads to our conjecture for the ``modular factorization'' of superconformal lens indices of general $\mathcal{N}=1$ gauge theories. We provide evidence for the conjecture in the context of the free chiral multiplet and SQED, and sketch the extension of our arguments to more general gauge theories. Based on this result, we systematically prove that a normalized part of superconformal lens indices defines a non-trivial first cohomology class associated with $SL(3,\mathbb{Z})$. Finally, we use this framework to propose a formula for the general lens space index., Comment: 63 pages + appendices. v2: added clarifications in section 3.5. v3: additional clarifications, added new section 3.6.2 proving modular factorization for the SQED index with non-zero FI parameter

Topological data analysis (TDA) is a powerful technique for extracting complex and valuable shape-related summaries of high-dimensional data. However, the computational demands of classical algorithms for computing TDA are exorbitant, and quickly become impractical for high-order characteristics. Quantum computers offer the potential of achieving significant speedup for certain computational problems. Indeed, TDA has been purported to be one such problem, yet, quantum computing algorithms proposed for the problem, such as the original Quantum TDA (QTDA) formulation by Lloyd, Garnerone and Zanardi, require fault-tolerance qualifications that are currently unavailable. In this study, we present NISQ-TDA, a fully implemented end-to-end quantum machine learning algorithm needing only a short circuit-depth, that is applicable to high-dimensional classical data, and with provable asymptotic speedup for certain classes of problems. The algorithm neither suffers from the data-loading problem nor does it need to store the input data on the quantum computer explicitly. The algorithm was successfully executed on quantum computing devices, as well as on noisy quantum simulators, applied to small datasets. Preliminary empirical results suggest that the algorithm is robust to noise., Comment: This paper is a follow up to arXiv:2108.02811 with improved theoretical results and other additional results. This new version presents an improved runtime for the algorithm, and fixes an issue present in the previous version

We review briefly the characteristic topological data of Calabi--Yau threefolds and focus on the question of when two threefolds are equivalent through related topological data. This provides an interesting test case for machine learning methodology in discrete mathematics problems motivated by physics., Comment: 6 pages, 3 figures; Contribution to proceedings of 2021 Nankai symposium on Mathematical Dialogues in celebration of S. S. Chern's 110th anniversary

In this letter we discuss infinite statistics and motivate its role in quantum gravity. Then, we connect infinite statistics to a dynamical form of dark energy, and we obtain an expression for the evolution of the Hubble parameter that we compare to observation. The equation of state parameter $w_{eff} < -1$ in this framework., Comment: 15 pages, 1 figure, LaTeX

We discuss the relevance of quantum gravity to the frontier questions in high energy phenomenology: the problems of dark matter, dark energy, and vacuum selection as well as the problems of emergent spacetime and wormholes. Dark matter and dark energy phenomenology, and the problem of vacuum selection are discussed within the context of string theory as a model of quantum gravity. Emergent spacetime and wormholes are discussed in a more general context of effective theories of quantum gravity., Comment: Contribution to Snowmass 2022 (originally "Snowmass 2021")

The boundary operator is a linear operator that acts on a collection of high-dimensional binary points (simplices) and maps them to their boundaries. This boundary map is one of the key components in numerous applications, including differential equations, machine learning, computational geometry, machine vision and control systems. We consider the problem of representing the full boundary operator on a quantum computer. We first prove that the boundary operator has a special structure in the form of a complete sum of fermionic creation and annihilation operators. We then use the fact that these operators pairwise anticommute to produce an $O(n)$-depth circuit that exactly implements the boundary operator without any Trotterization or Taylor series approximation errors. Having fewer errors reduces the number of shots required to obtain desired accuracies., Comment: 16 pages

We review recent efforts to machine learn relations between knot invariants. Because these knot invariants have meaning in physics, we explore aspects of Chern-Simons theory and higher dimensional gauge theories. The goal of this work is to translate numerical experiments with Big Data to new analytic results., Comment: 10 pages, 2 figures, LaTeX, based on a talk given by VJ at the Nankai Symposium on Mathematical Dialogues, August 2021

Using a fully connected feedforward neural network we study topological invariants of a class of Calabi--Yau manifolds constructed as hypersurfaces in toric varieties associated with reflexive polytopes from the Kreuzer--Skarke database. In particular, we find the existence of a simple expression for the Euler number that can be learned in terms of limited data extracted from the polytope and its dual., Comment: 16 pages, 4 figures

We use deep neural networks to machine learn correlations between knot invariants in various dimensions. The three-dimensional invariant of interest is the Jones polynomial $J(q)$, and the four-dimensional invariants are the Khovanov polynomial $\text{Kh}(q,t)$, smooth slice genus $g$, and Rasmussen's $s$-invariant. We find that a two-layer feed-forward neural network can predict $s$ from $\text{Kh}(q,-q^{-4})$ with greater than $99\%$ accuracy. A theoretical explanation for this performance exists in knot theory via the now disproven knight move conjecture, which is obeyed by all knots in our dataset. More surprisingly, we find similar performance for the prediction of $s$ from $\text{Kh}(q,-q^{-2})$, which suggests a novel relationship between the Khovanov and Lee homology theories of a knot. The network predicts $g$ from $\text{Kh}(q,t)$ with similarly high accuracy, and we discuss the extent to which the machine is learning $s$ as opposed to $g$, since there is a general inequality $|s| \leq 2g$. The Jones polynomial, as a three-dimensional invariant, is not obviously related to $s$ or $g$, but the network achieves greater than $95\%$ accuracy in predicting either from $J(q)$. Moreover, similar accuracy can be achieved by evaluating $J(q)$ at roots of unity. This suggests a relationship with $SU(2)$ Chern--Simons theory, and we review the gauge theory construction of Khovanov homology which may be relevant for explaining the network's performance., Comment: v1: 35 pages, 6 figures; v2: 36 pages, 6 figures, figures updated, typos corrected

The entropy of $1/16$-th BPS AdS$_5$ black holes can be microscopically accounted for by the superconformal index of the $\mathcal{N}=4$ super-Yang-Mills theory. One way to compute this is through a Cardy-like limit of a formula for the index obtained in [1] using the "$S$-transformation" of the elliptic $\Gamma$ function. In this paper, we derive more general $SL(3,\mathbb{Z})$ modular properties of the elliptic $\Gamma$ function. We then use these properties to obtain a three integer parameter family of generalized Cardy-like limits of the $\mathcal{N}=4$ superconformal index. From these limits, we obtain entropy formulae that have a similar form as that of the original AdS$_5$ black hole, up to an overall rescaling of the entropy. We interpret this both on the field theory and the gravitational side. Finally, we comment on how our work suggests a generalization of the Farey tail to four dimensions., Comment: 63 pages + appendices. v2: refs added, various corrections and improvements, discussion section rewritten. v3: corrected defn normalized partition function, added tau=sigma limit modular property in section 3.5 + its physical interpretation in section 5.2. (published version)

Ricci flat metrics for Calabi-Yau threefolds are not known analytically. In this work, we employ techniques from machine learning to deduce numerical flat metrics for the Fermat quintic, for the Dwork quintic, and for the Tian-Yau manifold. This investigation employs a single neural network architecture that is capable of approximating Ricci flat Kaehler metrics for several Calabi-Yau manifolds of dimensions two and three. We show that measures that assess the Ricci flatness of the geometry decrease after training by three orders of magnitude. This is corroborated on the validation set, where the improvement is more modest. Finally, we demonstrate that discrete symmetries of manifolds can be learned in the process of learning the metric., Comment: v2: 42 pages, figures improved, discrete symmetries section added, discussions enhanced, references added

We present a simple phenomenological formula which approximates the hyperbolic volume of a knot using only a single evaluation of its Jones polynomial at a root of unity. The average error is just $2.86$% on the first $1.7$ million knots, which represents a large improvement over previous formulas of this kind. To find the approximation formula, we use layer-wise relevance propagation to reverse engineer a black box neural network which achieves a similar average error for the same approximation task when trained on $10$% of the total dataset. The particular roots of unity which appear in our analysis cannot be written as $e^{2\pi i / (k+2)}$ with integer $k$; therefore, the relevant Jones polynomial evaluations are not given by unknot-normalized expectation values of Wilson loop operators in conventional $SU(2)$ Chern$\unicode{x2013}$Simons theory with level $k$. Instead, they correspond to an analytic continuation of such expectation values to fractional level. We briefly review the continuation procedure and comment on the presence of certain Lefschetz thimbles, to which our approximation formula is sensitive, in the analytically continued Chern$\unicode{x2013}$Simons integration cycle., Comment: v1: 26 + 19 pages, 15 figures; v2: 27 + 18 pages, figures updated, references added, journal version

In the $\Lambda$CDM model, dark energy is viewed as a constant vacuum energy density, the cosmological constant in the Einstein--Hilbert action. This assumption can be relaxed in various models that introduce a dynamical dark energy. In this letter, we argue that the mixing between infrared and ultraviolet degrees of freedom in quantum gravity lead to infinite statistics, the unique statistics consistent with Lorentz invariance in the presence of non-locality, and yield a fine structure for dark energy. Introducing IR and UV cutoffs into the quantum gravity action, we deduce the form of $\Lambda$ as a function of redshift and translate this to the behavior of the Hubble parameter., Comment: 14 pages, 2 figures

We compute the superconformal index of the $\mathcal{N}=4$ $SU(N)$ Yang-Mills theory through a residue calculation. The method is similar in spirit to the Bethe Ansatz formalism, except that all poles are explicitly known, and we do not require specialization of any of the chemical potentials. Our expression for the index allows us to revisit the Cardy limit using modular properties of four-dimensional supersymmetric partition functions. We find that all residues contribute at leading order in the Cardy limit. In a specific region of flavour chemical potential space, close to the two unrefined points, in fact all residues contribute universally. These universal residues precisely agree with the entropy functions of the asymptotically AdS$_5$ black hole and its "twin saddle" respectively. Finally, we discuss how our formula is suited to study the implications of four-dimensional modularity for the index beyond the Cardy limit., Comment: 36+11 pages, 4 figures, v2: added refs + minor clarifications and corrections, v3: added ref, corrected comment about [6] in introduction, v4: published version

Quantum chromodynamics (QCD) is the theory of the strong interaction. The fundamental particles of QCD, quarks and gluons, carry colour charge and form colourless bound states at low energies. The hadronic bound states of primary interest to us are the mesons and the baryons. From knowledge of the meson spectrum, we use neural networks and Gaussian processes to predict the masses of baryons with 90.3% and 96.6% accuracy, respectively. These results compare favourably to the constituent quark model. We as well predict the masses of pentaquarks and other exotic hadrons., Comment: 25 pages, 3 figures, 1 table

Extremal black holes in general dimensions are well known to contain an AdS$_2$ factor in their near-horizon geometries. If the extremal limit is taken in conjunction with a specific vanishing horizon limit, the so-called Extremal Vanishing Horizon (EVH) limit, the AdS$_2$ factor lifts to a locally AdS$_3$ factor with a pinching angular direction. In this paper, we study the EVH limit of asymptotically AdS black holes which preserve some supersymmetry. The primary example we consider is the 1/16$^{\rm th}$ BPS asymptotically AdS$_5$ black hole, whose EVH limit has an AdS$_3$ factor in its near-horizon geometry. We also consider the near-EVH limit of this black hole, in which the near-horizon geometry instead contains an extremal BTZ factor. We employ recent results on the large-$N$ limit of the superconformal index of the dual CFT$_4$ to understand the emergence of a CFT$_2$ in the IR of the CFT$_4$, which is the field theory dual to the emergence of the locally AdS$_3$ factor in the near-horizon geometry. In particular, we show that the inverse Laplace transform of the superconformal index, yielding the black hole entropy, becomes equivalent to the derivation of a Cardy formula for the dual CFT$_2$. Finally, we examine the EVH limit of supersymmetric black holes in other dimensions., Comment: 46 pages; v2 reference added, typos corrected

We study the dynamics of genetic code evolution. The model of Vetsigian et al. [1] and Vetsigian [2] uses the mechanism of horizontal gene transfer to demonstrate convergence of the genetic code to a near universal solution. We reproduce and analyze the algorithm as a dynamical system. All the parameters used in the model are varied to assess their impact on convergence and optimality score. We show that by allowing specific parameters to vary with time, the solution exhibits attractor dynamics. Finally, we study automorphisms of the genetic code arising due to this model. We use this to examine the scaling of the solutions in order to re-examine universality and find that there is a direct link to mutation rate., Comment: 38 pages

Supervised machine learning can be used to predict properties of string geometries with previously unknown features. Using the complete intersection Calabi-Yau (CICY) threefold dataset as a theoretical laboratory for this investigation, we use low $h^{1,1}$ geometries for training and validate on geometries with large $h^{1,1}$. Neural networks and Support Vector Machines successfully predict trends in the number of K\"ahler parameters of CICY threefolds. The numerical accuracy of machine learning improves upon seeding the training set with a small number of samples at higher $h^{1,1}$., Comment: 18 pages, 3 figures

An important conjecture in knot theory relates the large-$N$, double scaling limit of the colored Jones polynomial $J_{K,N}(q)$ of a knot $K$ to the hyperbolic volume of the knot complement, $\text{Vol}(K)$. A less studied question is whether $\text{Vol}(K)$ can be recovered directly from the original Jones polynomial ($N = 2$). In this report we use a deep neural network to approximate $\text{Vol}(K)$ from the Jones polynomial. Our network is robust and correctly predicts the volume with $97.6\%$ accuracy when training on $10\%$ of the data. This points to the existence of a more direct connection between the hyperbolic volume and the Jones polynomial., Comment: 18 pages, 9 figures, updated figures

Restriction enzymes recognize and bind to specific sequences on invading bacteriophage DNA. Like a key in a lock, these proteins require many contacts to specify the correct DNA sequence. Using information theory we develop an equation that defines the number of independent contacts, which is the dimensionality of the binding. We show that EcoRI, which binds to the sequence GAATTC, functions in 24 dimensions. Information theory represents messages as spheres in high dimensional spaces. Better sphere packing leads to better communications systems. The densest known packing of hyperspheres occurs on the Leech lattice in 24 dimensions. We suggest that the single protein EcoRI molecule employs a Leech lattice in its operation. Optimizing density of sphere packing explains why 6 base restriction enzymes are so common., Comment: Version 1: 31 pages, 3 figures, 1 table; Version 2: 33 pages, 3 figures, 1 table, responses to reviewers, new refs

Non-extremal black holes are endowed with geometric invariants related to their horizon areas. We extend earlier work on hot attractor black holes to higher dimensions and add a scalar potential. In addition to the event and Cauchy horizons, when we complexify the radial coordinate, non-extremal black holes will generically have other horizons as well. We prove that the product of all of the horizon areas is independent of variations of the asymptotic moduli further generalizing the attractor mechanism for extremal black holes. In the presence of a scalar potential, as typically appears in gauged supergravity, we find that the product of horizon areas is not necessarily the geometric mean of the extremal area, however. We outline the derivation of horizon invariants for stationary backgrounds., Comment: 39 pages, 3 figures, v2 references and clarifications added

We apply techniques in natural language processing, computational linguistics, and machine-learning to investigate papers in hep-th and four related sections of the arXiv: hep-ph, hep-lat, gr-qc, and math-ph. All of the titles of papers in each of these sections, from the inception of the arXiv until the end of 2017, are extracted and treated as a corpus which we use to train the neural network Word2Vec. A comparative study of common n-grams, linear syntactical identities, word cloud and word similarities is carried out. We find notable scientific and sociological differences between the fields. In conjunction with support vector machines, we also show that the syntactic structure of the titles in different sub-fields of high energy and mathematical physics are sufficiently different that a neural network can perform a binary classification of formal versus phenomenological sections with 87.1% accuracy, and can perform a finer five-fold classification across all sections with 65.1% accuracy., Comment: 50 pages, 6 figures

The latest techniques from Neural Networks and Support Vector Machines (SVM) are used to investigate geometric properties of Complete Intersection Calabi-Yau (CICY) threefolds, a class of manifolds that facilitate string model building. An advanced neural network classifier and SVM are employed to (1) learn Hodge numbers and report a remarkable improvement over previous efforts, (2) query for favourability, and (3) predict discrete symmetries, a highly imbalanced problem to which both Synthetic Minority Oversampling Technique (SMOTE) and permutations of the CICY matrix are used to decrease the class imbalance and improve performance. In each case study, we employ a genetic algorithm to optimise the hyperparameters of the neural network. We demonstrate that our approach provides quick diagnostic tools capable of shortlisting quasi-realistic string models based on compactification over smooth CICYs and further supports the paradigm that classes of problems in algebraic geometry can be machine learned., Comment: 22 pages, 10 figures. V2: Added new results of using permutations of the CICY matrix to reduce the class imbalance when predicting discrete symmetries. V3: Corrected typing errors in table 5

In previous work, we have commenced the task of unpacking the $473,800,776$ reflexive polyhedra by Kreuzer and Skarke into a database of Calabi-Yau threefolds (see http://www.rossealtman.com). In this paper, following a pedagogical introduction, we present a new algorithm to isolate Swiss cheese solutions characterized by "holes," or small 4-cycles, descending from the toric divisors inherent to the original four dimensional reflexive polyhedra. Implementing these methods, we find $2,268$ explicit Swiss cheese manifolds, over half of which have $h^{1,1}=6$. Many of our solutions have multiple large cycles. Such Swiss cheese geometries facilitate moduli stabilization in string compactifications and provide flat directions for cosmological inflation., Comment: 36 pages, 2 figures, and 2 tables

We study the likelihood which relative minima of random polynomial potentials support the slow-roll conditions for inflation. Consistent with renormalizability and boundedness, the coefficients that appear in the potential are chosen to be order one with respect to the energy scale at which inflation transpires. Investigation of the single field case illustrates a window in which the potentials satisfy the slow-roll conditions. When there are two scalar fields, we find that the probability depends on the choice of distribution for the coefficients. A uniform distribution yields a $0.05\%$ probability of finding a suitable minimum in the random potential whereas a maximum entropy distribution yields a $0.1\%$ probability., Comment: 33 pages, 12 figures, 1 table, LaTeX, v3: matching version appearing in publication

We explore the question of which shape a manifold is compelled to take when immersed in another one, provided it must be the extremum of some functional. We consider a family of functionals which depend quadratically on the extrinsic curvatures and on projections of the ambient curvatures. These functionals capture a number of physical setups ranging from holography to the study of membranes and elastica. We present a detailed derivation of the equations of motion, known as the shape equations, placing particular emphasis on the issue of gauge freedom in the choice of normal frame. We apply these equations to the particular case of holographic entanglement entropy for higher curvature three dimensional gravity and find new classes of entangling curves. In particular, we discuss the case of New Massive Gravity where we show that non-geodesic entangling curves have always a smaller on-shell value of the entropy functional. Then we apply this formalism to the computation of the entanglement entropy for dual logarithmic CFTs. Nevertheless, the correct value for the entanglement entropy is provided by geodesics. Then, we discuss the importance of these equations in the context of classical elastica and comment on terms that break gauge invariance., Comment: 54 pages, 8 figures. Significantly improved version, accepted for publication in Annals of Physics. New section on logarithmic CFTs. Detailed derivation of the shape equations added in appendix B. Typos corrected, clarifications added

Brane tilings describe Lagrangians (vector multiplets, chiral multiplets, and the superpotential) of four dimensional $\mathcal{N}=1$ supersymmetric gauge theories. These theories, written in terms of a bipartite graph on a torus, correspond to worldvolume theories on $N$ D$3$-branes probing a toric Calabi-Yau threefold singularity. A pair of permutations compactly encapsulates the data necessary to specify a brane tiling. We show that geometric consistency for brane tilings, which ensures that the corresponding quantum field theories are well behaved, imposes constraints on the pair of permutations, restricting certain products constructed from the pair to have no one-cycles. Permutations without one-cycles are known as derangements. We illustrate this formulation of consistency with known brane tilings. Counting formulas for consistent brane tilings with an arbitrary number of chiral bifundamental fields are written down in terms of delta functions over symmetric groups., Comment: 20 pages, 5 figures, LaTeX

Non-extremal black holes in $\mathcal{N}=2$ supergravity have two horizons, the geometric mean of whose areas recovers the horizon area of the extremal black hole obtained from taking a smooth zero temperature limit. In prior work (arxiv:1410.3478), using the attractor mechanism, we deduced the existence of several moduli independent invariant quantities obtained from averaging over a decoupled inter-horizon region. We establish that non-extremal geometries at the Reissner--Nordstr\"om point, where the scalar moduli are held fixed, can be lifted to solutions in supergravity with a near-horizon AdS$_3\times$S$^2$. These solutions have the same entropy and temperature as the original black hole and therefore allow an interpretation of the underlying gravitational degrees of freedom in terms of CFT$_2$. Symmetries of the moduli space enable us to explicate the origin of entropy in the extremal limit., Comment: References added

We present a complete classification of the vacuum geometries of all renormalizable superpotentials built from the fields of the electroweak sector of the MSSM. In addition to the Severi and affine Calabi-Yau varieties previously found, new vacuum manifolds are identified; we thereby investigate the geometrical implication of theories which display a manifest matter parity (or R-parity) via the distinction between leptonic and Higgs doublets, and of the lepton number assignment of the right-handed neutrino fields. We find that the traditional R-parity assignments of the MSSM more readily accommodate the neutrino see-saw mechanism with non-trivial geometry than those superpotentials that violate R-parity. However there appears to be no geometrical preference for a fundamental Higgs bilinear in the superpotential, with operators that violate lepton number, such as \nu H\bar{H}, generating vacuum moduli spaces equivalent to those with a fundamental bilinear., Comment: 29 pages, LaTeX

We explore the distribution of topological numbers in Calabi-Yau manifolds, using the Kreuzer-Skarke dataset of hypersurfaces in toric varieties as a testing ground. While the Hodge numbers are well-known to exhibit mirror symmetry, patterns in frequencies of combination thereof exhibit striking new patterns. We find pseudo-Voigt and Planckian distributions with high confidence and exact fit for many substructures. The patterns indicate typicality within the landscape of Calabi-Yau manifolds of various dimension., Comment: 62 pages, 22 figures in main text, LaTeX, v.3: section 2.5 added; minor edits; matches version in CMP

We discuss a precise relation between the Veneziano amplitude of string theory, rewritten in terms of ratios of the Riemann zeta function, and two elementary criteria for the Riemann hypothesis formulated in terms of integrals of the logarithm and the argument of the zeta function. We also discuss how the integral criterion based on the argument of the Riemann zeta function relates to the Li criterion for the Riemann hypothesis. We provide a new generalization of this integral criterion. Finally, we comment on the physical interpretation of our recasting of the Riemann hypothesis in terms of the Veneziano amplitude., Comment: 33 pages, LaTeX; v.2 minor edits

Kreuzer and Skarke famously produced the largest known database of Calabi-Yau threefolds by providing a complete construction of all 473,800,776 reflexive polyhedra that exist in four dimensions. These polyhedra describe the singular limits of ambient toric varieties in which Calabi-Yau threefolds can exist as hypersurfaces. In this paper, we review how to extract topological and geometric information about Calabi-Yau threefolds using the toric construction, and we provide, in a companion online database (see http://nuweb1.neu.edu/cydatabase), a detailed inventory of these quantities which are of interest to physicists. Many of the singular ambient spaces described by the Kreuzer-Skarke list can be smoothed out into multiple distinct toric ambient spaces describing different Calabi-Yau threefolds. We provide a list of the different Calabi-Yau threefolds which can be obtained from each polytope, up to current computational limits. We then give the details of a variety of quantities associated to each of these Calabi-Yau such as Chern classes, intersection numbers, and the K\"ahler and Mori cones, in addition to the Hodge data. This data forms a useful starting point for a number of physical applications of the Kreuzer-Skarke list.

We propose a model for large field inflation in heterotic string theory. The construction applies the near alignment mechanism of Kim, Nilles, and Peloso. By including gaugino condensates and world-sheet instanton non-perturbative effects, we obtain a large effective axion decay constant., Comment: 1+16 pages, 1 figure; Numerical example updated to published version

The product of the areas of the event horizon and the Cauchy horizon of a non-extremal black hole equals the square of the area of the horizon of the black hole obtained from taking the smooth extremal limit. We establish this result for a large class of black holes using the second order equations of motion, black hole thermodynamics, and the attractor mechanism for extremal black holes. This happens even though the area of each horizon generically depends on the moduli, which are asymptotic values of scalar fields. The conformal field theory dual to the BTZ black hole facilitates a microscopic interpretation of the result. In addition, we demonstrate that certain quantities which vanish in the extremal case are zero when integrated over the region between the two horizons. We corroborate these conclusions through an analysis of known solutions., Comment: 28 pages, 4 figures, LaTeX; v.2: references added, coffee stains removed; v.3: footnote added, minor typos corrected

We present an intriguing and precise interplay between algebraic geometry and the phenomenology of generations of particles. Using the electroweak sector of the MSSM as a testing ground, we compute the moduli space of vacua as an algebraic variety for multiple generations of Standard Model matter and Higgs doublets. The space is shown to have Calabi-Yau, Grassmannian, and toric signatures which sensitively depend on the number of generations of leptons, as well as inclusion of Majorana mass terms for right-handed neutrinos. We speculate as to why three generations is special., Comment: 51 pages, 7 tables

Using the attractor mechanism for extremal solutions in ${\cal N}=2$ gauged supergravity, we construct a $c$-function that interpolates between the central charges of theories at ultraviolet and infrared conformal fixed points corresponding to anti-de Sitter geometries. The $c$-function we obtain is couched purely in terms of bulk quantities and connects two different dimensional CFTs at the stable conformal fixed points under the RG flow., Comment: 13 pages, revised references and typographic modifications

We explain the origin of the Veronese surface in the vacuum moduli space geometry of the MSSM electroweak sector. While this result appeared many years ago using techniques of computational algebraic geometry, it has never been demonstrated analytically. Here, we present an analytical derivation of the vacuum geometry of the electroweak theory by understanding how the F- and D-term relations lead to the Veronese surface. We moreover give a detailed description of this geometry, realising an extra branch as a zero-dimensional point when quadratic Higgs lifting deformations are incorporated into the superpotential., Comment: 13 pages, 3 tables

In this note, we use entanglement entropy as a tool to explore the universal properties of CFTs dual to extremal BTZ black holes. We demonstrate that the entanglement entropies computed in the CFTs at the boundary of the extremal BTZ and the boundary of the near-horizon limit of the extremal BTZ are in perfect agreement and have the form appropriate to a two-dimensional CFT with only the chiral part excited and the anti-chiral in the ground state. Furthermore, we analyze the universal limits of the entanglement entropy and recover the correct value of the thermal entropy for large entangling intervals and the first-law like relation for the small interval., Comment: 19 pages, v2 references, several clarifications and comments added