Exact results for some lattice sums in 2, 4, 6 and 8 dimensions

Zucker, I. J.

doi:10.1088/0305-4470/7/13/011

Cited by 75 publications

(44 citation statements)

References 5 publications

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“…We computed the partition function on T 2 × R in the limit of large fermion flavor number, N f , using strategies similar to those employed for the computation on the three-sphere It is only convergent for Re s > 1/2, but can be defined by analytically continuing outside of this domain. Specifically, it can be expressed in terms of the special functions λ and β [44]:…”

Section: Discussionmentioning

confidence: 99%

Spectrum of conformal gauge theories on a torus

Thomson

Sachdev

2017

Phys. Rev. B

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Many model quantum spin systems have been proposed to realize critical points or phases described by 2+1 dimensional conformal gauge theories. On a torus of size L and modular parameter τ , the energy levels of such gauge theories equal (1/L) times universal functions of τ . We compute the universal spectrum of QED3, a U(1) gauge theory with N f two-component massless Dirac fermions, in the large N f limit. We also allow for a Chern-Simons term at level k, and show how the topological k-fold ground state degeneracy in the absence of fermions transforms into the universal spectrum in the presence of fermions; these computations are performed at fixed N f /k in the large N f limit.2

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Section: Discussionmentioning

confidence: 99%

Spectrum of conformal gauge theories on a torus

Thomson

Sachdev

2017

Phys. Rev. B

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“…. , 1 2 ), then for any p ∈ IN Z Ip ( p+1 p ; 0, − 1 2 ) < 0 [219,220,126]. In this case the potential has a minimum at finite distance…”

Section: The Static Potential On Toroidal Spacesmentioning

confidence: 98%

“…For untwisted fields X ⊥ ( g = 0) and for any p ∈ IN one has Z Ip ( p+1 p ; 0, 0) > 0 (see for example Refs. [219,220,126]). This means that for the untwisted toroidal p-brane α(0) > 0 (the Casimir forces are attractive) and the p-brane tends to collapse [157,189,215].…”

Section: The Static Potential On Toroidal Spacesmentioning

confidence: 99%

Quantum fields and extended objects in space-times with constant curvature spatial section

et al. 1996

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The heat-kernel expansion and ζ-regularization techniques for quantum field theory and extended objects on curved space-times are reviewed. In particular, ultrastatic space-times with spatial section consisting in manifold with constant curvature are discussed in detail. Several mathematical results, relevant to physical applications are presented, including exact solutions of the heat-kernel equation, a simple exposition of hyperbolic geometry and an elementary derivation of the Selberg trace formula. With regards to the physical applications, the vacuum energy for scalar fields, the one-loop renormalization of a self-interacting scalar field theory on a hyperbolic space-time, with a discussion on the topological symmetry breaking, the finite temperature effects and the Bose-Einstein condensation, are considered. Some attempts to generalize the results to extended objects are also presented, including some remarks on path integral quantization, asymptotic properties of extended objects and a novel representation for the one-loop (super)string free energy.

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“…With ζ(s) denoting the Riemann zeta function, results contained in [42] tell us that Equating with (5.12) gives (5.11).…”

Section: The Quaternion Lagrange-gauss Algorithmmentioning

confidence: 99%

Volumes and distributions for random unimodular complex and quaternion lattices

Forrester

Zhang

2018

Journal of Number Theory

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Two themes associated with invariant measures on the matrix groups SL N (F), with F = R, C or H, and their corresponding lattices parametrised by SL N (F)/SL N (O), O being an appropriate Euclidean ring of integers, are considered. The first is the computation of the volume of the subset of SL N (F) with bounded 2-norm or Frobenius norm. Key here is the decomposition of measure in terms of the singular values. The form of the volume, for large values of the bound, is relevant to asymptotic counting problems in SL N (O). The second is the problem of lattice reduction in the case N = 2. A unified proof of the validity of the appropriate analogue of the Lagrange-Gauss algorithm for computing the shortest basis is given. A decomposition of measure corresponding to the QR decomposition is used to specify the invariant measure in the coordinates of the shortest basis vectors. With F = C this allows for the exact computation of the PDF of the first minimum (for O = Z[i] and Z[(1 + √ −3)/2]), and the PDF of the second minimum and that of the angle between the minimal basis vectors (for O = Z[i]). It also encodes the specification of fundamental domains of the corresponding quotient spaces. Integration over the latter gives rise to certain number theoretic constants, which are also present in the asymptotic forms of the PDFs of the lengths of the shortest basis vectors. Siegel's mean value gives an alternative method to compute the arithmetic constants, allowing in particular the computation of the leading form of the PDF of the first minimum for F = H and O the Hurwitz integers, for which direct integration was not possible. arXiv:1709.08960v2 [math.NT]

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Exact results for some lattice sums in 2, 4, 6 and 8 dimensions

Cited by 75 publications

References 5 publications

Spectrum of conformal gauge theories on a torus

Spectrum of conformal gauge theories on a torus

Quantum fields and extended objects in space-times with constant curvature spatial section

Volumes and distributions for random unimodular complex and quaternion lattices

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