3,786 research outputs found
Kronecker's Double Series and Exact Asymptotic Expansion for Free Models of Statistical Mechanics on Torus
For the free models of statistical mechanics on torus, exact asymptotic
expansions of the free energy, the internal energy and the specific heat in the
vicinity of the critical point are found. It is shown that there is direct
relation between the terms of the expansion and the Kronecker's double series.
The latter can be expressed in terms of the elliptic theta-functions in all
orders of the asymptotic expansion.Comment: REVTeX, 22 pages, this is expanded version which includes exact
asymptotic expansions of the free energy, the internal energy and the
specific hea
K-Rational D-Brane Crystals
In this paper the problem of constructing spacetime from string theory is
addressed in the context of D-brane physics. It is suggested that the knowledge
of discrete configurations of D-branes is sufficient to reconstruct the motivic
building blocks of certain Calabi-Yau varieties. The collections of D-branes
involved have algebraic base points, leading to the notion of K-arithmetic
D-crystals for algebraic number fields K. This idea can be tested for D0-branes
in the framework of toroidal compactifications via the conjectures of Birch and
Swinnerton-Dyer. For the special class of D0-crystals of Heegner type these
conjectures can be interpreted as formulae that relate the canonical Neron-Tate
height of the base points of the D-crystals to special values of the motivic
L-function at the central point. In simple cases the knowledge of the
D-crystals of Heegner type suffices to uniquely determine the geometry.Comment: 36 page
Determination of modular elliptic curves by Heegner points
For every integer N ≥ 1, consider the set K(N) of imaginary quadratic fields such that, for each K in K(N), its discriminant D is an odd, square-free integer congruent to 1 modulo 4, which is prime to N and a square modulo 4N. For each K, let c = ([x]−[∞]) be the divisor class of a Heegner point x of discriminant D on the modular curve X = X(0)(N) as in [GZ]. (Concretely, such an x is the image of a point z in the upper half plane H such that both z and Nz are roots of integral, definite, binary quadratic forms of the same discriminant D ([B]).) Then c defines a point rational over the Hilbert class field H of K on the Jacobian J = J(0)(N) of X. Denote by cK the trace of c to K
Modular and duality properties of surface operators in N=2* gauge theories
We calculate the instanton partition function of the four-dimensional N=2*
SU(N) gauge theory in the presence of a generic surface operator, using
equivariant localization. By analyzing the constraints that arise from
S-duality, we show that the effective twisted superpotential, which governs the
infrared dynamics of the two-dimensional theory on the surface operator,
satisfies a modular anomaly equation. Exploiting the localization results, we
solve this equation in terms of elliptic and quasi-modular forms which resum
all non-perturbative corrections. We also show that our results, derived for
monodromy defects in the four-dimensional theory, match the effective twisted
superpotential describing the infrared properties of certain two-dimensional
sigma models coupled either to pure N=2 or to N=2* gauge theories.Comment: 51 pages, v3: references added, typos fixed, footnote added, some
small changes in the text, appendix B streamlined. Matches the published
versio
Lam\'e polynomials, hyperelliptic reductions and Lam\'e band structure
The band structure of the Lam\'e equation, viewed as a one-dimensional
Schr\"odinger equation with a periodic potential, is studied. At integer values
of the degree parameter l, the dispersion relation is reduced to the l=1
dispersion relation, and a previously published l=2 dispersion relation is
shown to be partially incorrect. The Hermite-Krichever Ansatz, which expresses
Lam\'e equation solutions in terms of l=1 solutions, is the chief tool. It is
based on a projection from a genus-l hyperelliptic curve, which parametrizes
solutions, to an elliptic curve. A general formula for this covering is
derived, and is used to reduce certain hyperelliptic integrals to elliptic
ones. Degeneracies between band edges, which can occur if the Lam\'e equation
parameters take complex values, are investigated. If the Lam\'e equation is
viewed as a differential equation on an elliptic curve, a formula is
conjectured for the number of points in elliptic moduli space (elliptic curve
parameter space) at which degeneracies occur. Tables of spectral polynomials
and Lam\'e polynomials, i.e., band edge solutions, are given. A table in the
older literature is corrected.Comment: 38 pages, 1 figure; final revision
Classical torus conformal block, N=2* twisted superpotential and the accessory parameter of Lame equation
In this work the correspondence between the semiclassical limit of the DOZZ
quantum Liouville theory on the torus and the Nekrasov-Shatashvili limit of the
N=2* (Omega-deformed) U(2) super-Yang-Mills theory is used to propose new
formulae for the accessory parameter of the Lame equation. This quantity is in
particular crucial for solving the problem of uniformization of the
one-punctured torus. The computation of the accessory parameters for torus and
sphere is an open longstanding problem which can however be solved if one
succeeds to derive an expression for the so-called classical Liouville action.
The method of calculation of the latter has been proposed some time ago by
Zamolodchikov brothers. Studying the semiclassical limit of the four-point
function of the quantum Liouville theory on the sphere they have derived the
classical action for the Riemann sphere with four punctures. In the present
work Zamolodchikovs idea is exploited in the case of the Liouville field theory
on the torus. It is found that the Lame accessory parameter is determined by
the classical Liouville action on the one-punctured torus or more concretely by
the torus classical block evaluated on the saddle point intermediate classical
weight. Secondly, as an implication of the aforementioned correspondence it is
obtained that the torus accessory parameter is related to the sum of all
rescaled column lengths of the so-called "critical" Young diagrams extremizing
the instanton "free energy" for the N=2* gauge theory. Finally, it is pointed
out that thanks to the known relation the sum over the "critical" column
lengths can be expressed in terms of a contour integral in which the integrand
is built out of certain special functions.Comment: 41 pages, published versio
Dimensionally Reduced SYM_4 as Solvable Matrix Quantum Mechanics
We study the quantum mechanical model obtained as a dimensional reduction of
N=1 super Yang-Mills theory to a periodic light-cone "time". After mapping the
theory to a cohomological field theory, the partition function (with periodic
boundary conditions) regularized by a massive term appears to be equal to the
partition function of the twisted matrix oscillator. We show that this
partition function perturbed by the operator of the holonomy around the time
circle is a tau function of Toda hierarchy. We solve the model in the large N
limit and study the universal properties of the solution in the scaling limit
of vanishing perturbation. We find in this limit a phase transition of
Gross-Witten type.Comment: 29 pages, harvmac, 1 figure, formulas in appendices B and C correcte
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