7,520 research outputs found
Sum of Squares Lower Bounds from Symmetry and a Good Story
In this paper, we develop machinery which makes it much easier to prove sum of squares lower bounds when the problem is symmetric under permutations of [1,n] and the unsatisfiability of our problem comes from integrality arguments, i.e. arguments that an expression must be an integer. Roughly speaking, to prove SOS lower bounds with our machinery it is sufficient to verify that the answer to the following three questions is yes:
1) Are there natural pseudo-expectation values for the problem?
2) Are these pseudo-expectation values rational functions of the problem parameters?
3) Are there sufficiently many values of the parameters for which these pseudo-expectation values correspond to the actual expected values over a distribution of solutions which is the uniform distribution over permutations of a single solution?
We demonstrate our machinery on three problems, the knapsack problem analyzed by Grigoriev, the MOD 2 principle (which says that the complete graph K_n has no perfect matching when n is odd), and the following Turan type problem: Minimize the number of triangles in a graph G with a given edge density. For knapsack, we recover Grigoriev\u27s lower bound exactly. For the MOD 2 principle, we tighten Grigoriev\u27s linear degree sum of squares lower bound, making it exact. Finally, for the triangle problem, we prove a sum of squares lower bound for finding the minimum triangle density. This lower bound is completely new and gives a simple example where constant degree sum of squares methods have a constant factor error in estimating graph densities
The effect of convolving families of L-functions on the underlying group symmetries
L-functions for GL_n(A_Q) and GL_m(A_Q), respectively, such that, as N,M -->
oo, the statistical behavior (1-level density) of the low-lying zeros of
L-functions in F_N (resp., G_M) agrees with that of the eigenvalues near 1 of
matrices in G_1 (resp., G_2) as the size of the matrices tend to infinity,
where each G_i is one of the classical compact groups (unitary, symplectic or
orthogonal). Assuming that the convolved families of L-functions F_N x G_M are
automorphic, we study their 1-level density. (We also study convolved families
of the form f x G_M for a fixed f.) Under natural assumptions on the families
(which hold in many cases) we can associate to each family L of L-functions a
symmetry constant c_L equal to 0 (resp., 1 or -1) if the corresponding
low-lying zero statistics agree with those of the unitary (resp., symplectic or
orthogonal) group. Our main result is that c_{F x G} = c_G * c_G: the symmetry
type of the convolved family is the product of the symmetry types of the two
families. A similar statement holds for the convolved families f x G_M. We
provide examples built from Dirichlet L-functions and holomorphic modular forms
and their symmetric powers. An interesting special case is to convolve two
families of elliptic curves with rank. In this case the symmetry group of the
convolution is independent of the ranks, in accordance with the general
principle of multiplicativity of the symmetry constants (but the ranks persist,
before taking the limit N,M --> oo, as lower-order terms).Comment: 41 pages, version 2.1, cleaned up some of the text and weakened
slightly some of the conditions in the main theorem, fixed a typ
JuliBootS: a hands-on guide to the conformal bootstrap
We introduce {\tt JuliBootS}, a package for numerical conformal bootstrap
computations coded in {\tt Julia}. The centre-piece of {\tt JuliBootS} is an
implementation of Dantzig's simplex method capable of handling arbitrary
precision linear programming problems with continuous search spaces. Current
supported features include conformal dimension bounds, OPE bounds, and
bootstrap with or without global symmetries. The code is trivially
parallelizable on one or multiple machines. We exemplify usage extensively with
several real-world applications. In passing we give a pedagogical introduction
to the numerical bootstrap methods.Comment: 29 page
Higgs and SUSY Searches at LHC
I start with a brief introduction to Higgs mechanism and supersymmetry. Then
I discuss the theoretical expectations, current limits and search strategies
for Higgs boson(s) at LHC --- first in the SM and then in the MSSM. Finally I
discuss the signatures and search strategies for the superparticles.Comment: Typos and figure styles corrected; LaTeX (28 pages) including 13 ps
files containing 11 figures; Invited talk at the 5th Workshop on High Energy
Physics Phenomenology (WHEPP-5), Pune, India, 12 - 25 January 199
Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents
We use the conformal bootstrap to perform a precision study of the operator
spectrum of the critical 3d Ising model. We conjecture that the 3d Ising
spectrum minimizes the central charge c in the space of unitary solutions to
crossing symmetry. Because extremal solutions to crossing symmetry are uniquely
determined, we are able to precisely reconstruct the first several Z2-even
operator dimensions and their OPE coefficients. We observe that a sharp
transition in the operator spectrum occurs at the 3d Ising dimension
Delta_sigma=0.518154(15), and find strong numerical evidence that operators
decouple from the spectrum as one approaches the 3d Ising point. We compare
this behavior to the analogous situation in 2d, where the disappearance of
operators can be understood in terms of degenerate Virasoro representations.Comment: 55 pages, many figures; v2 - refs and comments added, to appear in a
special issue of J.Stat.Phys. in memory of Kenneth Wilso
Limitations of semidefinite programs for separable states and entangled games
Semidefinite programs (SDPs) are a framework for exact or approximate
optimization that have widespread application in quantum information theory. We
introduce a new method for using reductions to construct integrality gaps for
SDPs. These are based on new limitations on the sum-of-squares (SoS) hierarchy
in approximating two particularly important sets in quantum information theory,
where previously no -round integrality gaps were known: the set of
separable (i.e. unentangled) states, or equivalently, the
norm of a matrix, and the set of quantum correlations; i.e. conditional
probability distributions achievable with local measurements on a shared
entangled state. In both cases no-go theorems were previously known based on
computational assumptions such as the Exponential Time Hypothesis (ETH) which
asserts that 3-SAT requires exponential time to solve. Our unconditional
results achieve the same parameters as all of these previous results (for
separable states) or as some of the previous results (for quantum
correlations). In some cases we can make use of the framework of
Lee-Raghavendra-Steurer (LRS) to establish integrality gaps for any SDP, not
only the SoS hierarchy. Our hardness result on separable states also yields a
dimension lower bound of approximate disentanglers, answering a question of
Watrous and Aaronson et al. These results can be viewed as limitations on the
monogamy principle, the PPT test, the ability of Tsirelson-type bounds to
restrict quantum correlations, as well as the SDP hierarchies of
Doherty-Parrilo-Spedalieri, Navascues-Pironio-Acin and Berta-Fawzi-Scholz.Comment: 47 pages. v2. small changes, fixes and clarifications. published
versio
Solving the 3D Ising Model with the Conformal Bootstrap
We study the constraints of crossing symmetry and unitarity in general 3D
Conformal Field Theories. In doing so we derive new results for conformal
blocks appearing in four-point functions of scalars and present an efficient
method for their computation in arbitrary space-time dimension. Comparing the
resulting bounds on operator dimensions and OPE coefficients in 3D to known
results, we find that the 3D Ising model lies at a corner point on the boundary
of the allowed parameter space. We also derive general upper bounds on the
dimensions of higher spin operators, relevant in the context of theories with
weakly broken higher spin symmetries.Comment: 32 pages, 11 figures; v2: refs added, small changes in Section 5.3,
Fig. 7 replaced; v3: ref added, fits redone in Section 5.
Finite-size scaling tests for SU(3) lattice gauge theory with color sextet fermions
The observed slow running of the gauge coupling in SU(3) lattice gauge theory
with two flavors of color sextet fermions naturally suggests it is a theory
with one relevant coupling, the fermion mass, and that at zero mass correlation
functions decay algebraically. I perform a finite-size scaling study on
simulation data at two values of the bare gauge coupling with this assumption
and observe a common exponent for the scaling of the correlation length with
the fermion mass, y_m ~ 1.5. An analysis of the scaling of valence Dirac
eigenvalues at one of these bare couplings produces a similar number.Comment: 23 pages, revtex, 13 figure
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