34 research outputs found
Information geometry in quantum field theory: lessons from simple examples
Motivated by the increasing connections between information theory and
high-energy physics, particularly in the context of the AdS/CFT correspondence,
we explore the information geometry associated to a variety of simple systems.
By studying their Fisher metrics, we derive some general lessons that may have
important implications for the application of information geometry in
holography. We begin by demonstrating that the symmetries of the physical
theory under study play a strong role in the resulting geometry, and that the
appearance of an AdS metric is a relatively general feature. We then
investigate what information the Fisher metric retains about the physics of the
underlying theory by studying the geometry for both the classical 2d Ising
model and the corresponding 1d free fermion theory, and find that the curvature
diverges precisely at the phase transition on both sides. We discuss the
differences that result from placing a metric on the space of theories vs.
states, using the example of coherent free fermion states. We compare the
latter to the metric on the space of coherent free boson states and show that
in both cases the metric is determined by the symmetries of the corresponding
density matrix. We also clarify some misconceptions in the literature
pertaining to different notions of flatness associated to metric and non-metric
connections, with implications for how one interprets the curvature of the
geometry. Our results indicate that in general, caution is needed when
connecting the AdS geometry arising from certain models with the AdS/CFT
correspondence, and seek to provide a useful collection of guidelines for
future progress in this exciting area.Comment: 36 pages, 2 figures; added new section and appendix, miscellaneous
improvement
Multicritical Symmetry Breaking and Naturalness of Slow Nambu-Goldstone Bosons
We investigate spontaneous global symmetry breaking in the absence of Lorentz
invariance, and study technical Naturalness of Nambu-Goldstone (NG) modes whose
dispersion relation exhibits a hierarchy of multicritical phenomena with
Lifshitz scaling and dynamical exponents . For example, we find NG modes
with a technically natural quadratic dispersion relation which do not break
time reversal symmetry and are associated with a single broken symmetry
generator, not a pair. The mechanism is protected by an enhanced `polynomial
shift' symmetry in the free-field limit.Comment: 5 pages, 1 figure; v2: minor typos corrected, references adde
New Heat Kernel Method in Lifshitz Theories
We develop a new heat kernel method that is suited for a systematic study of
the renormalization group flow in Horava gravity (and in Lifshitz field
theories in general). This method maintains covariance at all stages of the
calculation, which is achieved by introducing a generalized Fourier transform
covariant with respect to the nonrelativistic background spacetime. As a first
test, we apply this method to compute the anisotropic Weyl anomaly for a
(2+1)-dimensional scalar field theory around a z=2 Lifshitz point and
corroborate the previously found result. We then proceed to general scalar
operators and evaluate their one-loop effective action. The covariant heat
kernel method that we develop also directly applies to operators with spin
structures in arbitrary dimensions.Comment: 47 pages, 1 figure; v2: appendix C updated, minor typos corrected,
references adde
Scalar Field Theories with Polynomial Shift Symmetries
We continue our study of naturalness in nonrelativistic QFTs of the Lifshitz
type, focusing on scalar fields that can play the role of Nambu-Goldstone (NG)
modes associated with spontaneous symmetry breaking. Such systems allow for an
extension of the constant shift symmetry to a shift by a polynomial of degree
in spatial coordinates. These "polynomial shift symmetries" in turn protect
the technical naturalness of modes with a higher-order dispersion relation, and
lead to a refinement of the proposed classification of infrared Gaussian fixed
points available to describe NG modes in nonrelativistic theories. Generic
interactions in such theories break the polynomial shift symmetry explicitly to
the constant shift. It is thus natural to ask: Given a Gaussian fixed point
with polynomial shift symmetry of degree , what are the lowest-dimension
operators that preserve this symmetry, and deform the theory into a
self-interacting scalar field theory with the shift symmetry of degree ? To
answer this (essentially cohomological) question, we develop a new
graph-theoretical technique, and use it to prove several classification
theorems. First, in the special case of (essentially equivalent to
Galileons), we reproduce the known Galileon -point invariants, and find
their novel interpretation in terms of graph theory, as an equal-weight sum
over all labeled trees with vertices. Then we extend the classification to
and find a whole host of new invariants, including those that represent
the most relevant (or least irrelevant) deformations of the corresponding
Gaussian fixed points, and we study their uniqueness.Comment: 70 pages. v2: minor clarifications, typos corrected, a reference
adde
The edge of chaos: quantum field theory and deep neural networks
We explicitly construct the quantum field theory corresponding to a general class of deep neural networks encompassing both recurrent and feedforward architectures. We first consider the mean-field theory (MFT) obtained as the leading saddlepoint in the action, and derive the condition for criticality via the largest Lyapunov exponent. We then compute the loop corrections to the correlation function in a perturbative expansion in the ratio of depth T to width N, and find a precise analogy with the well-studied O(N) vector model, in which the variance of the weight initializations plays the role of the 't Hooft coupling. In particular, we compute both the O(1) corrections quantifying fluctuations from typicality in the ensemble of networks, and the subleading O(T/N) corrections due to finite-width effects. These provide corrections to the correlation length that controls the depth to which information can propagate through the network, and thereby sets the scale at which such networks are trainable by gradient descent. Our analysis provides a first-principles approach to the rapidly emerging NN-QFT correspondence, and opens several interesting avenues to the study of criticality in deep neural networks
Non-Lorentzian IIB Supergravity from a Polynomial Realization of SL(2,R)
We derive the action and symmetries of the bosonic sector of non-Lorentzian
IIB supergravity by taking the non-relativistic string limit. We find that the
bosonic field content is extended by a Lagrange multiplier that implements a
restriction on the Ramond-Ramond fluxes. We show that the SL(2,R)
transformation rules of non-Lorentzian IIB supergravity form a novel, nonlinear
polynomial realization. Using classical invariant theory of polynomial
equations and binary forms, we will develop a general formalism describing the
polynomial realization of SL(2,R) and apply it to the special case of
non-Lorentzian IIB supergravity. Using the same formalism, we classify all the
relevant SL(2,R) invariants. Invoking other bosonic symmetries, such as the
local boost and dilatation symmetry, we show how the bosonic part of the
non-Lorentzian IIB supergravity action is formed uniquely from these SL(2,R)
invariants. This work also points towards the concept of a non-Lorentzian
bootstrap, where bosonic symmetries in non-Lorentzian supergravity are used to
bootstrap the bosonic dynamics in Lorentzian supergravity, without considering
the fermions.Comment: 43 page
Branched SL(2,ℤ) duality
We investigate how SL(2,ℤ) duality is realized in nonrelativistic type IIB superstring theory, which is a self-contained corner of relativistic string theory. Within this corner, we realize manifestly SL(2,ℤ)-invariant (p, q)-string actions. The construction of these actions imposes a branching between strings of opposite charges associated with the two-form fields. The branch point is determined by these charges and the axion background field. Both branches must be incorporated in order to realize the full SL(2,ℤ) group. Besides these string actions, we also construct D-instanton and D3-brane actions that manifestly realize the branched SL(2,ℤ) symmetry
Towards quantifying information flows: relative entropy in deep neural networks and the renormalization group
We investigate the analogy between the renormalization group (RG) and deep neural networks, wherein subsequent layers of neurons are analogous to successive steps along the RG. In particular, we quantify the flow of information by explicitly computing the relative entropy or Kullback-Leibler divergence in both the one- and two-dimensional Ising models under decimation RG, as well as in a feedforward neural network as a function of depth. We observe qualitatively identical behavior characterized by the monotonic increase to a parameter-dependent asymptotic value. On the quantum field theory side, the monotonic increase confirms the connection between the relative entropy and the c-theorem. For the neural networks, the asymptotic behavior may have implications for various information maximization methods in machine learning, as well as for disentangling compactness and generalizability. Furthermore, while both the two-dimensional Ising model and the random neural networks we consider exhibit non-trivial critical points, the relative entropy appears insensitive to the phase structure of either system. In this sense, more refined probes are required in order to fully elucidate the flow of information in these models