Vertex Expansion Research Articles

Unimodular loop quantum cosmology

Unimodular gravity is based on a modification of the usual Einstein-Hilbert action that allows one to recover general relativity with a dynamical cosmological constant. It also has the interesting property of providing, as the momentum conjugate to the cosmological constant, an emergent clock variable. In this paper we investigate the cosmological reduction of unimodular gravity, and its quantization within the framework of flat homogeneous and isotropic loop quantum cosmology. It is shown that the unimodular clock can be used to construct the physical state space, and that the fundamental features of the previous models featuring scalar field clocks are reproduced. In particular, the classical singularity is replaced by a quantum bounce, which takes place in the same condition as obtained previously. We also find that requirement of semiclassicality demands the expectation value of the cosmological constant to be small (in Planck units). The relation to spin foam models is also studied, and we show that the use of the unimodular time variable leads to a unique vertex expansion.

Loop vertex expansion for Φ2k theory in zero dimension

In this paper we extend the method of loop vertex expansion to interactions with degree higher than 4. As an example we provide through this expansion an explicit proof that the free energy of ϕ2k scalar theory in zero dimension is Borel–Le Roy summable of order k−1. We detail the computations in the case of a ϕ6 interaction.

Asymptotic estimation of the fraction of errors correctable by q-ary LDPC codes

We consider an ensemble of random q-ary LDPC codes. As constituent codes, we use q-ary single-parity-check codes with d = 2 and Reed-Solomon codes with d = 3. We propose a hard-decision iterative decoding algorithm with the number of iterations of the order of the logarithm of the code length. We show that under this decoding algorithm there are codes in the ensemble with the number of correctable errors linearly growing with the code length. We weaken a condition on the vertex expansion of the Tanner graph corresponding to the code.

Loop quantum cosmology and spin foams

Loop quantum cosmology (LQC) is used to provide concrete evidence in support of the general paradigm underlying spin foam models (SFMs). Specifically, it is shown that: (i) the physical inner product in the timeless framework equals the transition amplitude in the deparameterized theory; (ii) this quantity admits a vertex expansion a la SFMs in which the M-th term refers just to M volume transitions, without any reference to the time at which the transition takes place; (iii) the exact physical inner product is obtained by summing over just the discrete geometries; no ‘continuum limit’ is involved; and, (iv) the vertex expansion can be interpreted as a perturbative expansion in the spirit of group field theory. This sum over histories reformulation of LQC also addresses certain other issues which are briefly summarized.

Renormalization of the BCS-BEC crossover by order-parameter fluctuations

We use the functional renormalization group approach with partial bosonization in the particle-particle channel to study the effect of order parameter fluctuations on the BCS-Bose-Einstein condensate (BEC) crossover of superfluid fermions in three dimensions. Our approach is based on a new truncation of the vertex expansion where the renormalization group flow of bosonic two-point functions is closed by means of Dyson-Schwinger equations and the superfluid order parameter is related to the single-particle gap via a Ward identity. We explicitly calculate the chemical potential, the single-particle gap, and the superfluid order parameter at the unitary point and compare our results with experiments and previous calculations.

LQG propagator from the new spin foams

We compute metric correlations in loop quantum gravity with the dynamics defined by the new spin foam models. The analysis is done at the lowest order in a vertex expansion and at the leading order in a large spin expansion. The result is compared to the graviton propagator of perturbative quantum gravity.

Proof of the MHV vertex expansion for all tree amplitudes in 𝒩 = 4 SYM theory

We prove the MHV vertex expansion for all tree amplitudes of = 4 SYM theory. The proof uses a shift acting on all external momenta, and we show that every NkMHV tree amplitude falls off as 1/zk, or faster, for large z under this shift. The MHV vertex expansion allows us to derive compact and efficient generating functions for all NkMHV tree amplitudes of the theory. We also derive an improved form of the anti-NMHV generating function. The proof leads to a curious set of sum rules for the diagrams of the MHV vertex expansion.

A super MHV vertex expansion for 𝒩 = 4 SYM theory

We present a supersymmetric generalization of the MHV vertex expansion for all tree amplitudes in = 4 SYM theory. In addition to the choice of a reference spinor, this super MHV vertex expansion also depends on four reference Grassmann parameters. We demonstrate that a significant fraction of diagrams in the expansion vanishes for a judicious choice of these Grassmann parameters, which simplifies the computation of amplitudes. Even pure-gluon amplitudes require fewer diagrams than in the ordinary MHV vertex expansion. We show that the super MHV vertex expansion arises from the recursion relation associated with a holomorphic all-line supershift. This is a supersymmetric generalization of the holomorphic all-line shift recently introduced in arXiv:0811.3624. We study the large-z behavior of generating functions under these all-line supershifts, and find that they generically provide 1/zk falloff at (Next-to)kMHV level. In the case of anti-MHV generating functions, we find that a careful choice of shift parameters guarantees a stronger 1/zk+4 falloff. These particular all-line supershifts may therefore play an important role in extending the super MHV vertex expansion to = 8 supergravity.

Recursion relations, generating functions, and unitarity sums in 𝒩 = 4 SYM theory

We prove that the MHV vertex expansion is valid for any NMHV tree amplitude of N=4 SYM. The proof uses induction to show that there always exists a complex deformation of three external momenta such that the amplitude falls off at least as fast as 1/z for large z. This validates the generating function for n-point NMHV tree amplitudes. We also develop generating functions for anti-MHV and anti-NMHV amplitudes. As an application, we use these generating functions to evaluate several examples of intermediate state sums on unitarity cuts of 1-, 2-, 3- and 4-loop amplitudes. In a separate analysis, we extend the recent results of arXiv:0808.0504 to prove that there exists a valid 2-line shift for any n-point tree amplitude of N=4 SYM. This implies that there is a BCFW recursion relation for any tree amplitude of the theory.

Towards far-from-equilibrium quantum field dynamics: A functional renormalisation-group approach

Dynamic equations for quantum fields far from equilibrium are derived by use of functional renormalisation group techniques. The obtained equations are non-perturbative and lead substantially beyond mean-field and quantum Boltzmann type approximations. The approach is based on a regularised version of the generating functional for correlation functions where times greater than a chosen cutoff time are suppressed. As a central result, a time evolution equation for the non-equilibrium effective action is derived, and the time evolution of the Green functions is computed within a vertex expansion. It is shown that this agrees with the dynamics derived from the 1 / N -expansion of the two-particle irreducible effective action.

Three-body scattering from nonperturbative flow equations

We consider fermion-dimer scattering in the presence of a large positive scattering length in the frame of functional renormalization group equations. A flow equation for the momentum dependent fermion-dimer scattering amplitude is derived from first principles in a systematic vertex expansion of the exact flow equation for the effective action. The resummation obtained from the nonperturbative flow is shown to be equivalent to the one performed by the integral equation by Skorniakov and Ter-Martirosian (STM). The flow equation approach allows to integrate out fermions and bosons simultaneously, in line with the fact that the bosons are not fundamental but build up gradually as fluctuation induced bound states of fermions. In particular, the STM result for atom-dimer scattering is obtained by choosing the relative cutoff scales of fermions and bosons such that the fermion fluctuations are integrated out already at the initial stage of the RG evolution.

On tree width, bramble size, and expansion

A bramble in a graph G is a family of connected subgraphs of G such that any two of these subgraphs have a nonempty intersection or are joined by an edge. The order of a bramble is the least number of vertices required to cover every subgraph in the bramble. Seymour and Thomas [P.D. Seymour, R. Thomas, Graph searching and a min-max theorem for tree-width, J. Combin. Theory Ser. B 58 (1993) 22–33] proved that the maximum order of a bramble in a graph is precisely the tree width of the graph plus one. We prove that every graph of tree width at least k has a bramble of order Ω ( k 1 / 2 / log 2 k ) and size polynomial in n and k, and that for every k there is a graph G of tree width Ω ( k ) such that every bramble of G of order k 1 / 2 + ϵ has size exponential in n. To prove the lower bound, we establish a close connection between linear tree width and vertex expansion. For the upper bound, we use the connections between tree width, separators, and concurrent flows.

On graphs for which every planar immersion lifts to a knotted spatial embedding

We call a graph G intrinsically linkable if there is a way to assign over/under information to any planar immersion of G such that the associated spatial embedding contains a pair of nonsplittably linked cycles. We define intrinsically knottable graphs analogously. We show there exist intrinsically linkable graphs that are not intrinsically linked. (Recall a graph is intrinsically linked if it contains a pair of nonsplittably linked cycles in every spatial embedding.) We also show there are intrinsically knottable graphs that are not intrinsically knotted. In addition, we demonstrate that the property of being intrinsically linkable (knottable) is not preserved by vertex expansion.

Constructive ϕ 4 Field Theory without Tears

We propose to treat the � 4 Euclidean theory constructively in a simpler way. Our method, based on a new kind of ”loop vertex expansion”, no longer requires the painful intermediate tool of cluster and Mayer expansions.

Robust and efficient CAD topology generation using adaptive tolerances

AbstractA topology generation algorithm, commonly known as geometry repairing/healing, is presented to detect commonly found geometrical and topological issues like cracks, gaps, overlaps, intersections, T‐connections and no/invalid topology in the model, process them and build correct topological information. The present algorithm is based on the iterative vertex pair contraction and expansion operations called stitching and filling, respectively. The algorithm closes small gaps/overlaps via the stitching operation and fills larger gaps by adding faces through the filling operation to process the model accurately. Moreover, the topology generation algorithm can process manifold as well as non‐manifold models, which makes the procedure more general and flexible. Most of the existing techniques use a constant tolerance for topology generation and suffer from the reliability issues and cannot preserve small details of the same size of gaps/overlaps. This algorithm uses an automatic and adaptive tolerance that enhances reliability of the process and preserves small features in the model. In this way, the combination of generality, accuracy, reliability and efficiency of this algorithm seems to be a significant improvement over existing techniques. Copyright © 2007 John Wiley & Sons, Ltd.

Universality from very general nonperturbative flow equations in QCD

In the context of very general exact renormalization groups, it will be shown that, given a vertex expansion of the Wilsonian effective action, remarkable progress can be made without making any approximations. Working in QCD we will derive, in a manifestly gauge invariant way, an exact diagrammatic expression for the expectation value of an arbitrary gauge invariant operator, in which many of the non-universal details of the setup do not explicitly appear. This provides a new starting point for attacking nonperturbative problems.

Independent sets in tensor graph powers

AbstractThe tensor product of two graphs, G and H, has a vertex set V(G) × V(H) and an edge between (u,v) and (u′,v′) iff both u u′ ∈ E(G) and v v′ ∈ E(H). Let A(G) denote the limit of the independence ratios of tensor powers of G, lim, α(Gn)/|V(Gn)|. This parameter was introduced in [Brown, Nowakowski, Rall, SIAM J Discrete Math 9 (1996), 290–300], where it was shown that A(G) is lower bounded by the vertex expansion ratio of independent sets of G. In this article we study the relation between these parameters further, and ask whether they are in fact equal. We present several families of graphs where equality holds, and discuss the effect the above question has on various open problems related to tensor graph products. © 2006 Wiley Periodicals, Inc. J Graph Theory

Automatic CAD model topology generation

AbstractComputer aided design (CAD) models often need to be processed due to the data translation issues and requirements of the downstream applications like computational field simulation, rapid prototyping, computer graphics, computational manufacturing, and real‐time rendering before they can be used. Automatic CAD model processing tools can significantly reduce the amount of time and cost associated with the manual processing. The topology generation algorithm, commonly known as CAD repairing/healing, is presented to detect commonly found geometrical and topological issues like cracks, gaps, overlaps, intersections, T‐connections, and no/invalid topology in the model, process them and build correct topological information. The present algorithm is based on the iterative vertex pair contraction and expansion operations called stitching and filling, respectively, to process the model accurately. Moreover, the topology generation algorithm can process manifold as well as non‐manifold models, which makes the procedure more general and flexible. In addition, a spatial data structure is used for searching and neighbour finding to process large models efficiently. In this way, the combination of generality, accuracy, and efficiency of this algorithm seems to be a significant improvement over existing techniques. Results are presented showing the effectiveness of the algorithm to process two‐ and three‐dimensional configurations. Copyright © 2006 John Wiley & Sons, Ltd.

Couplings of vector-spinor representation for SO(10) model building

Higgs multiplet in the vector-spinor representations of SO(10), i.e., the $144+\bar{144}$ multiplet can break the SO(10) gauge symmetry spontaneously in one step down to the Standard Model gauge group symmetry $SU(3)_C\times SU(2)_L\times U(1)_Y$ and a recent analysis has used such vector-spinors for building a new class of SO(10) grand unification models (hep-ph/0506312) . Here we discuss the techniques for the computation of several classes of vector-spinor couplings using the Basic Theorem on the SO(2N) vertex expansion developed by the authors. The computations include the cubic couplings of the vector-spinors with SO(10) tensors, quartic self-couplings of the vector-spinors, and couplings of the vector-spinors with spinor representations of SO(10). The last set include couplings of vector-spinors with the 16-plets of quarks and lepton and with the 16 and $\bar{16}$ of Higgs. These couplings provide a crucial tool for further development of the SO(10) grand unification using vector-spinor representations. These include study of quark-lepton masses, analysis of dimension five operators including baryon and lepton number violating operators, and study of neutrino masses and mixings. Illustrative examples are given for their computation using a sample of vector-spinor couplings.

Global Connectivity And Expansion: Long Cycles and Factors In f-Connected Graphs

Given a function f : ℕ→ℝ, call an n-vertex graph f-connected if separating off k vertices requires the deletion of at least f(k) vertices whenever k≤(n−f(k))/2. This is a common generalization of vertex connectivity (when f is constant) and expansion (when f is linear). We show that an f-connected graph contains a cycle of length linear in n if f is any linear function, contains a 1-factor and a 2-factor if f(k)≥2k+1, and contains a Hamilton cycle if f(k)≥2(k+1)2. We conjecture that linear growth of f suffices to imply hamiltonicity.