- page 1/102

  Today I'll talk about my work as a Ph.D. student at the University of Torino.

  I will deal with aspects related to phenomenology from the point of view of
  strings for which I worked both on theoretical and computational levels.

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  The plan is to first introduce a semi-phenomenological description of physics
  from a theory of strings.
  
  These tools will be used throughout the talk as basis for the rest of the
  topics.

  I will then mainly deal with open strings and analyse correlators in the
  presence of twist and spin fields seen as singular points in the string
  propagation.

  From particle physics, I will then move to a string theory description of
  cosmology in the presence of time dependent singularities in time dependent
  orbifods.

  Finally I will focus on computational aspects of string compactifications.

- page 3/102

  I will therefore start with the most geometrical aspects of the discussion,
  reviewing some basics and introducing a framework to deal with open string
  amplitudes in the presence of twist and spin fields.

- page 4/102

  As usual in string theory the starting point is Polyakov's action.
  We start directly from the superstring extension where gamma is the
  worldsheet metric (for the moment generic) and rhos are the 2-dimensional
  "gamma matrices".

- page 5/102

  This description presents lots of symmetries which, at the end of the day,
  provide the key to its success.
  In fact the invariances of the action are such that the theory is conformal
  with a vanishing traceless stress tensor.

- page 6/102

  Using standard field theory methods the stress energy tensor of the
  superstring in D dimensions generates the known Virasoro algebra, extending
  the classical de Witt's algebra.
  At quantum level it presents a central charge whose value depends on the
  dimension of the target space.

- page 9/102

  In order for the stress energy tensor to be a conformal primary field, we
  need to introduce sets of fields known as conformal ghosts.
  These are conformal fields specified by their weight lambda and they are
  introduced as a first order Lagrangian theory.

- page 8/102

  The central charge arising from the algebra of the modes of the ghost sector
  can then be used to compensate the superstring counterpart.
  Consistency of the theory thus fixes the dimension of the target space to be
  a 10-dimensional spacetime.

- page 9/102

  The fact that "everyday" physics in accelerators is 4-dimensional is
  recovered through compactification.
  In simple words we recover 4-dimensional physics at low energy by considering
  the 10-dimensional Minkowski space as a product of the usual 4-dimensional
  spacetime and a 6-(real)-dimensional space.

  The internal space has to obey stringent restrictions: in particular
  it has to be a compact manifold (to "hide" the extra dimensions), it has to
  break most of the supersymmetry present at high energy, and its arising gauge
  algebra has to contain the standard model algebra.

- page 10/102

  These manifolds have been firstly postulated by Eugenio Calabi and later
  proved to exist by Shing Tung Yau, hence the name Calabi-Yau manifolds.
  In the case at hand, they are a specific class of 3-dimensional complex
  manifolds with SU(3) holonomy.
  They must be Ricci-flat or equivalently have a vanishing first Chern class.

- page 11/102

  In general it is not easy to classify these manifolds (as well as computing
  for instance their metric, which is generally not known).
  However, for instance, we will see that the dimension of the complex
  cohomology groups, known as Hodge numbers, will play a strategic role.

- page 12/102

  We solve the equations of motion and the boundary conditions for the string
  propagation (we focus on the bosonic part for the moment).
  The action in fact introduces naturally the Neumann boundary conditions for
  the strings.
  The solution factorises into its holomorphic components (where z is the usual
  coordinate on the complex plane) due to the equations of motion.

- page 13/102

  Now consider for a second the simplest toroidal compactification of closed
  strings, that is suppose we can "hide" the last direction of the string on a
  circle.

  As in usual quantum mechanics this leads to momentum quantization (defined as
  an integer n in units of the compactification radius).

  A closed string can moreover wind an integer number of times around the cycle
  introducing a "winding number" m

  In turn this is reflected on the spectrum of the theory.
  Differently from field theory, shrinking the radius does not decouple the
  modes, but the compact dimension remains.
  In other words exchanging R and 1/R does not modify the theory.

- page 14/102

  This so called T-duality can also be applied to open strings with a different
  outcome.
  
  Since open strings cannot wind around the compact dimension, the behaviour of
  the spectrum is as in field theory: the compact dimension decouples and the
  open string is constrained on a lower dimensional surface.

  This is the result of introducing Dirichlet boundary conditions on the T-dual
  coordinate, meaning that the endpoints of the string have to reside on the
  same surface.

  The procedure can be applied to several dimensions thus introducing surfaces
  on which the endpoints of the open string live, called D-branes.

- page 15/102

  D-branes naturally introduce preferred directions of motion thus breaking the
  D-dimensional Poincaré symmetry.

  It is in fact possible to show that in D-dimensions, the open string sector
  at massless level contains an Abelian field.

  D-branes split the components into a lower dimensional U(1) field on the
  D-brane and a vector of spacetime scalars.

- page 16/102

  In general we can have strings whose endpoints are on the same D-brane
  leading to U(1) gauge theories as well as stretched strings across different
  branes.

  The Chan-Paton factors can be used to label the endpoints of the strings by
  the position on the D-branes.

  It is then possible to show that when the D-branes are coincident, states can
  be rearranged into enhanced gauge groups (for instance, unitary), thus
  creating non Abelian gauge theories.
  
- page 17/102

  However, physical constraints on the possible constructions pose serious
  questions on the D-brane dispositions.

  A quark can be modelled with a string stretched across two distant branes,
  but such quark would have a mass proportional to the distance to the branes:
  chirality in particular would not be possible to define, while being one of
  the defining features of the Standard Model.

  This is where the possibility to put D-branes at an angle with respect to
  each other becomes crucial.

  While most of the modes will indeed gain a mass, the massless spectrum can
  support chiral states localised at the intersections.

- page 18/102

  In this framework we consider models built with D6-branes.
  We embed them in 6-dimensional internal space without worrying about
  compactification for now.

  In this scenario we study correlators of fields in the presence of twist
  fields, which are a set of conformal fields arising at the intersections.
  These are particularly interesting to compute for instance Yukawa couplings
  in string theory.

- page 19/102

  Using the path integral approach, correlators involving twist fields are
  dominated by the instanton contribution of the classical Euclidean action.

  We focus specifically on its computation in the case of three D-branes
  necessary for the Yukawa couplings.

- page 20/102

  The literature already takes into consideration D-branes embedded as lines in
  a factorised version of the internal space, where the possible relative
  rotations are Abelian in each plane of the space.

  The contribution of the string in this case is proportional to the area of
  the triangle formed on the plane by the three D-branes since the
  string is completely constrained to move on the plane.

- page 21/102
  
  In what follows we consider a non completely factorised space and we focus on
  its 4-dimensional sector.

  After filling the physical 4-dimensional space, we study the remaining
  directions of the D-brane in 4-dimensional internal space using a
  well-adapted frame of reference which is in general rotated with respect to
  global coordinates.

  The rotation is not directly an SO(4) element, but it is an element of a
  Grassmannian: in fact separately rotating the Dirichlet and Neumann
  components does not change anything (we just need to relabel the
  coordinates), as long as no reflections are involved.

  The rotation involved is therefore a representative of a left equivalence
  class of possible rotations.

- page 22/102

  Introducing the usual conformal transformations mapping the strip to the
  complex plane, the intersecting branes are mapped to the real axis.

  D-branes are therefore real intervals on the space, between their
  intersection points.

- page 23/102

  In this definition, the boundary conditions of the strings become a set of
  discontinuities on the real axis, one for each D-brane, and the embedding
  equation specifying the intersections.

- page 24/102

  Instead of dealing directly with the discontinuities, it is more suitable to
  introduce an auxiliary function defined over the entire complex plane by
  gluing functions defined on the upper plane and on the bottom plane on
  arbitrary interval.

  This "doubling trick" transforms the discontinuity into a monodromy factor,
  when looping any base point through the glued interval.

- page 25/102

  The fact that rotations are non Abelian leads to two different monodromies
  for a base points starting in the upper plane or in the bottom plane.

  This is a general feature, but the 3 D-branes case simplifies this enough.

- page 26/102

  Dealing with these 4 x 4 matrices is delicate.

  Using a known isomorphism between SO(4) matrices and two copies of SU(2) x
  SU(2), the monodromy matrix can be cast into a tensor product of two 2 x 2
  matrices.
  In this matrix form the solution is therefore given by 2-dimensional basis of
  holomorphic functions with three regular singular points.

- page 27/102

  The usual SL invariance allows us to fix the monodromies in 0, 1 and
  infinity.

  The overall solution will then be a superposition of all possible basis.

  Given the previous properties we therefore look for a basis of Gauss
  hypergeometric functions.

- page 28/102

  Since we deal with rotations, the parameters of the hypergeometric functions
  involved are indeed connected to the rotation vectors.

  However the choice is not unique and labelled by the
  periodicity of the rotations.

- page 29/102

  The reason is a huge redundancy in the description: using the free parameters
  of the rotations we should in fact fix all degrees of freedom in the
  solution, which at the moment is an infinite sum involving an infinite amount
  of free parameters.

  In fact we only showed that the rotation matrix is equivalent to a
  monodromy matrix from which can build an overparametrised solution.

  Using contiguity relations we can then restrict the sum over independent
  functions.

  Finally requiring the Euclidean action to be finite restricts the sum to only
  two terms (the particular terms surviving in the sum depend on the rotation
  vectors but they are never more than two).

  Fixing the intersection points eventually determines the free constants in the solution.

- page 30/102

  The physical interpretation of the solution is finally straightforward in the
  Abelian case, where the action can be reduced to the sum of the areas of the
  internal triangles (this is a general result even for a generic number of
  D-branes).

- page 31/102

  In the non Abelian case we considered there is no simple way to write the
  action using global data.
  However the contribution to the Euclidean action is larger than the Abelian
  case: the strings are in fact no longer constrained on a plane and, in order
  to stretch across the boundaries, they have to form a small bump while
  detaching from the D-brane.

  The Yukawa coupling in this case is therefore suppressed with respect to the
  Abelian case.

- page 32/102

  We then turn the attention to fermions and the computation of correlators
  involving spin fields.
  Though ideally extending some ideas, we abandon the intersecting D-brane
  scenario, and we introduce point-like defects on one boundary of the
  worldsheet in such a way that the superstring undergoes a change of its
  boundary conditions when meeting a defect.

- page 33/102

  It is possible to show that in this case the Hamiltonian of the theory
  is only piecewise conserved.

- page 34/102

  Suppose now that we could expand the field on a basis of solutions to the
  boundary conditions and work, as before, on the entire complex plane.

- page 35/102

  Ideally we would be interested in extracting the modes in order to perform
  any computation of amplitudes.
  The definition of the operation is connected to a dual basis whose form is
  completely fixed by the original field (which we know) and the request of
  time independence.

- page 36/102

  The resulting algebra of the operators is in fact defined through such
  operation and it is therefore time independent.

- page 37/102

  Differently from what done in the bosonic case, we focus on U(1) boundary
  change operators.
  The resulting monodromy on the complex plane is therefore a phase factor.

- page 38/102

  As in the previous case we can write a basis of solutions which incorporates
  the behaviour looping the point-like defects.
  Consequently we can also define a dual basis.

  Both fields are defined up to integer factors, since we are still dealing
  with rotations.

- page 39/102

  In order to compute amplitudes we then need to define the space on which the
  representation of the algebra acts.
  We define an excited vacuum, annihilated by positive frequency modes, and the
  lowest energy vacuum (from the strip definition).

- page 40/102

  Vacua need to be consistent, leading to conditions labelled by an integer
  factor relating the basis of solutions with its dual (and ultimately the
  algebra of operators).
  The vacuum must always be correctly normalised and the description of physics
  using any two of the vacuum definitions should be consistently equivalent.

- page 41/102

  To avoid having overlapping in- and out-annihilators, the label L must
  vanish.

- page 42/102

  In this framework, the stress energy tensor displays as expected a time
  dependence due to the presence of the point-like defects.
  Specifically it shows that in each defect we have a primary boundary changing
  operators (whose weight depends on the monodromy) and which creates the
  excited vacuum from the invariant vacuum.
  This is by all means an excited spin field.

  Finally (and definitely fascinating), the stress energy tensor obeys the
  canonical OPE, that is the theory is still conformal (even though there is a
  time dependence). 

- page 43/102

  In formulae, the excited vacuum used in computations is thus created by a
  radially ordered product of excited spin fields hidden in the defects.

- page 44/102

  We are therefore in a position to compute the correlators involving such spin
  fields (however since we cannot compute the normalisation, we can compute
  only quantities not involving it).
  For instance we reproduce the known result of bosonization.

  Moreover, since we have complete control over the algebra of the fermionic
  fields, we can also compute any correlator involving both spin and matter
  fields.

- page 45/102

  We therefore showed that semi-phenomenological models need the ability to
  compute correlators involving twist and spin fields.

  We then introduced a framework to compute the instanton contribution to the
  correlators using intersecting D-branes and we showed how to compute
  correlators in the fermionic case involving spin fields as point-like defects
  on the string worldsheet.

  The question would now be how to extend this to non Abelian spin fields and,
  most importantly, to twist fields, where there is no framework such as
  bosonization.

- page 46/102

  After considering defects and singular points in particle physics, I will
  deal with time dependent singularities in cosmology.

- page 47/102

  The reason is that as string theory is considered a theory of everything, its
  phenomenological description should in fact include both strong and
  electroweak forces as well as gravity.

- page 48/102

  In particular from the gravity side, we would like to have a better view of
  the cosmological implications in string theory.

- page 49/102

  For instance we could try to study Big Bang models to gain some better
  insight with respect to field theory.

- page 50/102

  For this, one way would be to build toy models of singularities in time, in
  which the singular point exists in one specific moment, rather than place.

- page 51/102

  A simple way to make it so is to build toy models from time-dependent
  orbifolds which can model singularities as their fixed points.

- page 52/102

  In the past people already dealt with such problem finding divergences in the
  computation of amplitudes.
  The presence of such divergences in N-point correlators is however usually
  associated to a gravitational backreaction due to exchange of gravitons.

- page 53/102

  However the 4-tachyon amplitude in string theory is divergent already in the
  open string sector at tree level: in other words genuine gauge theories.

  The effective field theory interpretation would be a 4-point interaction of
  scalar fields (higher spins would only spoil the behaviour).

- page 54/102

  To investigate further, we consider the so called Null Boost Orbifold.

  The construction starts from D-dimensional Minkowski spacetime through a
  change of coordinates.

- page 55/102
  
  The orbifold is then built through the periodic identification of one
  coordinate along the direction of its Killing vector, which leads to momentum
  quantization.

- page 56/102

  From these identifications, we can build scalar wave functions obeying the
  standard equations of motion.

  Notice the behaviour in the time direction u which already takes a peculiar
  form and the presence of the quantized momentum in a strategic place.

- page 57/102

  In order to introduce the divergence problem we first consider a theory of
  scalar QED.

- page 58/102

  When computing the interactions between the fields, the terms involved are
  entirely defined by two main integrals.

  It might not be immediately visible, but given the behaviour of the scalar
  functions, any vertex interaction with more than 3 fields diverges.

- page 59/102

  The reason for the divergence is connected to the "strategically placed"
  quantized momentum.

  When when all quantized momenta vanish, in the limit of small u (that is near
  the singularity) the integrands develop isolated zeros preventing the
  convergence.
  In fact, in this case, even a distributional interpretation (not unlike the
  derivative of a delta function) fails.

- page 60/102

  So far the situation is therefore somewhat troublesome.

  Moreover, obvious ways to regularise the theory do not work: for instance
  adding a Wilson line does not cure the problem as divergences also involve
  neutral strings which would not feel the regularisation.

  In fact the problems seem to arise from the vanishing volume in phase space
  along the compact direction: the issue looks like geometrical, rather than
  strictly gravitational.

- page 61/102

  Since the field theory fails to give a reasonable value for amplitudes
  involving time-dependent singularities, we could therefore ask whether string
  theory can shed some light.

- page 62/102

  The relevant divergent integrals are in fact present also in string theory.
  They arise from interactions of massive vertices like what is shown here.

  These vertices are usually overlooked as they do not play in general a
  relevant role at low energy.
  However it is possible that near the singularity they might actually give
  their contribution.
  These vertices are involved at low energy in the definition of contact terms
  in the effective field theory, which therefore does not account for them.

- page 63/102

  In this sense even string theory cannot give a solution to the problem.
  In other words since the effective theory does not even exist, its high
  energy completion is not capable of providing a better description.

- page 64/102

  There is however one geometric way to escape this.
  Since the issues are related to a vanishing phase space volume, analytically
  speaking it is sufficient to add a non compact direction to the orbifold in
  which the particle is "free to escape".

- page 65/102

  While the Generalised Null Boost Orbifold has basically the same definition
  through one of its Killing vector, the presence of the additional direction
  acts in a different way on the definition of the scalar functions.

  As you can see the new time behaviour ensures better convergence properties,
  and the presence of the continuous momentum ensures that no isolated zeros
  are present at any time.
  Even in the worst case scenario, the arising amplitudes would still have a
  distributional interpretation.

- page 66/102

  We therefore showed that divergences in the simplest theories are present
  both in field theory and string theory and that in the presence of
  singularities, the string massive states start to play a role.

  The nature of the divergences is however due to vanishing volumes in phase
  space and cannot be classified as simply a gravitational backreaction.
  In fact the introduction of "escape routes" for fields grants a
  distributional interpretation of the amplitudes.

  It is also possible to show that this is not restricted to "null boost" types
  of orbifolds, but even other kinds of orbifolds present the same issues.

- page 67/102

  In summary we showed that the divergences cannot be regarded as simply
  gravitational, but even gauge theories (that is the open sector of the string
  theory) present issues.

  Their nature is however subtle and connected to the interaction of string
  massive modes (or contact terms in the low energy formulation) which are not
  usually taken into account.

- page 68/102 

  We finally move to the last section.

  After the analysis of semi-phenomenological analytical models, we now
  consider a computational task related to compactifications of
  extra-dimensions using machine learning.

- page 69/102

  We focus on Calabi-Yau manifolds in three complex dimensions.

  Due to their properties and their symmetries, the relevant topological
  invariants are two Hodge numbers.

  As the number of possible compact Calabi-Yau 3-folds is a huge, we focus on a
  subset.

- page 70/102

  Specifically we focus on manifolds built as intersections of hypersurfaces in
  projective spaces, that is intersections of several homogeneous equations in
  the complex coordinates of the manifold.

  As we are interested in studying these manifolds as topological spaces, for
  each equation and projective space we do not care about the coefficients, but
  only the exponents, or better the degree of the equation in a given
  coordinate.

- page 71/102

  The intersections can be generalised to multiple projective spaces and
  equations and the manifold can be characterised by a matrix containing the
  powers of the coordinates in each equation.

  The problem in which we are interested is therefore to be able to take the so
  called "configuration matrix" of the manifolds and predict the value of the
  Hodge numbers.

- page 72/102

  The real issue is now how to treat the configuration matrix and how to build
  such map.

- page 73/102

  We use a machine learning approach.

  In very simple words it means that we want to find a new representation of
  the input (possibly parametrized by some weights which we can tune and
  control) such that the predicted Hodge numbers are as close as possible to
  the correct result.

  In this sense the machine has to learn some way to transform the input to get
  a result close to what in the computer science literature is called the
  "ground truth".

  The measure of proximity or distance from the true value is called "loss
  function" or "Lagrangian function" (with a slight abuse of naming
  conventions).
  The machine then learns some way to minimise this function (for instance
  using gradient descent methods and updating the parameters).

- page 74/102

  We thus exchange the difficult problem of finding an analytical solution with
  an optimisation problem (it does not imply "easy", but it is at least
  doable).

- page 75/102

  In order to learn the best way of doing this, we can rely on a vast computer
  science literature and use large physics datasets containing lots of samples
  from which to infer a structure.

- page 76/102

  In this sense the approach can merge techniques from physics, mathematics and
  computer science benefiting from advancements in all fields.

- page 77/102

  The approach can furthermore provide a good way to analyse data and infer
  structure and advance hypothesis, which could end up overlooked using
  traditional brute force algorithms.

  In this case we focus on the prediction of two Hodge numbers with very
  different distributions and ranges.
  The data we consider were computed using top of the class computing power at
  CERN in the 80s, with a huge effort by the string theory community.

  In this sense Complete Intersection Calabi-Yau manifolds are a good starting
  point to investigate the application of machine learning techniques because
  they are well studied and characterised.

- page 78/102

  The dataset we use contains less than 10000 manifolds (in machine learning
  terms it is still small).

  From these we remove product spaces (recognisable by their block diagonal
  form of the configuration matrix) and we remove very high values of the Hodge
  numbers to avoid learning "extremal configurations".
  Mind that we only remove them from the training data which the machine
  actually uses to learn.

  In this sense we are simply not giving the machine "extremal" configurations
  in an attempt to push as far as possible the application: should the machine
  learn a good representation, it will automatically be capable of learning
  also those configurations without a human manually feeding them.
  
  We then define three separate folds: the largest contains training data used
  by the machine to adjust the parametrisation, 10% of the data is then used
  for intermediate evaluation of the process, while the last subset is used to
  give the final predictions.

  Differently from the validation set, the test set has not been seen by the
  machine and therefore can reliably test the generalisation ability of the
  algorithm.

  Differently from some previous approaches we consider this as a regression
  task in the attempt to let the machine learn a true map between the
  configuration matrix and the Hodge numbers (in case we can also discuss the
  classification approach as it has some interesting applications itself).

- page 79/102

  The pruned distribution of the Hodge numbers therefore presents less outliers
  than the initial dataset, but as you can see we expect the result to be
  similar even without the procedure, since the number of outliers removed is
  small.

  This first analysis however proved to be a good success to get higher
  results.

- page 80/102

  The pipeline we adopt is the same used at industrial level by companies and
  data scientists.
  We in fact heavily rely on data analysis to improve as much as possible the
  output.

- page 81/102

  This for instance can be done by including additional information with
  respect to the configuration matrix, that is by feeding the machine variables
  which can be manually derived: by definition they are redundant but can be
  used to easily learn a pattern.

  In fact as we can see most of the features such as the number of projective
  spaces or the number of equations in the matrix are heavily correlated with
  the Hodge numbers.
  
  Moreover using algorithms to produce a ranking of the variables such as
  decision trees shows that such "engineered features" are much more important
  than the configuration matrix itself.

- page 82/102

  Using the "engineered data", we now get to the choice of the algorithm.
  There is no general rule for this, even though there might be good guidelines
  to follow.

- page 83/102

  Though the approach is clearly "supervised" in the sense that the machine
  learns by approximating a known result, we also tried other approaches in an
  attempt to generate additional information which the machine could use.

  The first approach is a clustering algorithm, intuitively used to look for a
  notion of "proximity" between the configuration matrices.
  This however did not play a role in the analysis.

  The other is definitely more interesting and it consists in finding a better
  representation of the configuration matrix using less components.
  The idea is therefore to "squeeze" or "concentrate" the information in a
  lower dimensional space (matrices in our case have 180 components, so we are
  trying to aim for something less than that).

- page 84/102

  For the predictions we first relied on traditional regression algorithms,
  such as linear models, support vector machines and boosted decision trees.
  I will not enter into the details and differences between the algorithms, but
  we can indeed discuss them.

- page 85/102

  Let me however say a few words a dimensionality reduction procedure known as
  "principal components analysis" (or PCA for short), since this proved to be
  an important step in the analysis.

  Suppose that we have a rectangular matrix (which could be number of samples
  in the dataset times the number of components of the matrix once it has been
  flattened).

  The idea of PCA is to project the data onto a lower dimensional surface where
  the variance if maximised in order to retain as much information as possible.

  This is usually used to isolate a signal from a noisy background.
  Thus by isolating only the meaningful components of the matrix we can hope to
  help the algorithm.

- page 86/102

  Visually PCA is used to isolate the eigenvalues and eigenvectors of the
  covariance matrix (or the singular values of the matrix) which do not belong
  to the background.

  From random matrix theory we know that the eigenvalues of an independently
  and identically distributed matrix (a Wishart matrix) follow a
  Marchenko-Pastur distribution.

  Such matrix containing a signal would therefore be recognised by the presence
  of eigenvalues outside this probability distribution.
  We could therefore simply keep the corresponding eigenvectors.

  In our case this resulted in an improvement of the accuracy, obtained by
  retaining less than half of the components of the matrix (corresponding to
  99% of the variance of the initial set).

- page 87/102

  As we can see we used several algorithms to evaluate the procedure.

  Previous approaches in the literature mainly relied on the direct application
  of algorithms to the configuration matrix.
  We extended this beyond the previously considered algorithms (mainly support
  vectors) to decision trees and linear models for comparison.

- page 88/102

  Techniques such as feature engineering and PCA provide a huge improvement
  (even with less training data).
  Let me for instance point out the fact the even a simple linear regression
  reaches the same level of accuracy previously reached by more complex
  algorithms, even with much less training data.

  This ultimately can cut computational costs and complexity.

- page 89/102

  However this does not conclude the landscape of algorithms used in machine
  learning.
  In fact we also used neural networks architectures.

  They are a class of function approximators which use (some variants of)
  gradient descent to optimise the weights.
  Their layered structure is key to learn highly non linear and complicated
  functions.

  We focused on two distinct architectures.

  The older fully connected networks were employed in previous attempts at
  predicting the Hodge numbers.
  They rely on a series of matrix operations to create new outputs from
  previous layers.
  In this sense the matrix W and the bias term b are the weights which need to
  be updated.
  Each node is connected to all the outputs, hence the name fully connected or
  densely connected.
  Equivalently this means that the matrix W does not have vanishing
  entries.

  To learn non linear functions this is however not sufficient: an iterated
  application of these linear maps would simply result in a linear function to
  be learned.
  We "break" linearity by introducing an "activation function" at each layer.

  The second architecture is called convolutional from its iterated application
  of "sliding window function" (that is convolutions) applied on the layers.

- page 90/102

  Convolutional networks have several advantages over a fully connected
  approach.

  Since the input in this case does not need to flattened, convolutions retain
  the notion of vicinity between cells in a grid (here we have an example of a
  configuration matrix as seen by a convolutional neural network).

  Since they do not have one weight for each connection, they have a smaller
  number of parameters to update (proportional to the size of the window).
  In our specific case we cut by more than one order of magnitude the number
  of parameters used with respect to fully connected networks.

  Moreover weights are shared by adjacent cells, meaning that if there is a
  structure to be inferred, this is the way to go to exploit the "artificial
  intelligence" underlying the operations involved.

- page 91/102

  In this sense a convolutional architecture can isolate defining features of
  the output and pass them to the following layer as in the animation.

  For instance, using a computer science analogy, this can be used to classify
  objects given a picture: a convolutional neural network is literally capable
  of isolating what makes a dog a dog and what distinguishes it from a cat
  (even more specific it can separate a Labrador from a Golden Retriever).

- page 92/102

  This has in fact been used in computer vision tasks in recent year for
  pattern recognitions, object detections and spatial awareness tasks (for
  instance to isolate the foreground from the background).
  In this sense this is the closest approximation of artificial intelligence in
  supervised tasks.

- page 93/102

  My contribution in this sense is inspired by deep learning research at
  Google.
  In recent years they were able to devise new architectures using so called
  "inception modules" in which different convolution operations are used
  concurrently.
  The architecture has better generalisation properties since more features can
  be detected and processed at the same time.

- page 94/102

  In our case we decided to go for two concurrent convolutions one scanning
  each equation (the vertical kernel) of the configuration matrix, while a
  second convolutions scans the projective spaces (in horizontal).

  The layer structure is then concatenated until a single output is produced
  (the Hodge number that is).

  The idea is that this way the network can learn a relation between projective
  spaces and equations and recombine them to find a new representation.

- page 95/102

  As we can see even the simple introduction of a traditional convolutional
  kernel (the result shown was reached with a 5x5 kernel function) is
  sufficient to boost the accuracy of the predictions (previous best results in
  2018 reached only 77% of accuracy on h^{1,1}).

- page 96/102

  The introduction of the Inception architecture has major advantages: it uses
  even less parameters than "traditional" convolutional networks, it boosts the
  performance reaching near perfect accuracy, it needs a lot less data (even
  with just 30% of the data for training, the accuracy is already near
  perfect).

  Moreover with this architecture we were able to predict also h^{2,1} with 50%
  accuracy: even if does not look a reliable method to predict it (I agree, for
  now), mind that previous attempts have usually avoided computing it, or they
  reached accuracies as high as 8-9% (even feature engineering could boost it
  only around 35%).

  The network is also solid enough to predict both Hodge numbers at the same
  time: trading a bit of the accuracy for a simpler model, it is in fact
  possible to let the machine learn the existing relation between the Hodge
  numbers without specifically inputing anything (for instance by inserting the
  fact that the difference of the Hodge numbers is the Euler characteristic).

  For more specific info I invite you to take a look at Harold's talk on the
  subject at the recent "string_data" workshop.

- page 97/102

  Deep learning can therefore be used as a predictive method, provided that one
  is able to analyse the data (no black boxes should ever be admitted).

- page 98/102

  It can also be used a source of inspiration for inquiries and investigations
  always provided a good analysis is done beforehand.

- page 99/102

  Deep learning can also be used for generalisation of patterns and relations.
  As always only after careful consideration.

- page 100/102

  Moreover convolutional networks look promising and with a lot of unexplored
  potential.
  This is in fact the first time in which they have been successfully used in
  theoretical physics.

  Finally, this is an interdisciplinary approach in which a lot is yet to be
  learned from different perspective.

- page 101/102

  More directions to investigate now remain.

  In fact one could in principle exploit freedom in representing the
  configuration matrices to learn the best possible representation.

  Otherwise one could start to think about this in a mathematical embedding and
  study what happens for CICY 4-folds (almost one million complete
  intersections).

  Moreover, as I was saying, this could be used as an attempt to study formal
  aspects of deep learning, or even more to directly dive into the "real
  artificial intelligence" and start to study the problem in a reinforcement
  learning environment where the machine automatically learns a task without
  knowing the final result.

- page 102/102

  I will therefore leave the open question as to whether this is actually going
  to be the end or just the start of something else.

  In the meantime I thank you for your attention.
