Monstrous entanglement

The Monster CFT plays an important role in moonshine and is also conjectured to be the holographic dual to pure gravity in AdS3. We investigate the entanglement and Renyi entropies of this theory along with other extremal CFTs. The Renyi entropies of a single interval on the torus are evaluated using the short interval expansion. Each order in the expansion contains closed form expressions of the modular parameter. The leading terms in the q-series are shown to precisely agree with the universal corrections to Renyi entropies at low temperatures. Furthermore, these results are shown to match with bulk computations of Renyi entropy using the one-loop partition function on handlebodies. We also explore some features of Renyi entropies of two intervals on the plane.Comment: 41 pages, 4 figures; v2: typos corrected, approximates published versio

Numerical Evaluation of Harmonic Polylogarithms

Harmonic polylogarithms $\H(\vec{a};x)$, a generalization of Nielsen's polylogarithms ${S}_{n,p}(x)$, appear frequently in analytic calculations of radiative corrections in quantum field theory. We present an algorithm for the numerical evaluation of harmonic polylogarithms of arbitrary real argument. This algorithm is implemented into a {\tt FORTRAN} subroutine {\tt hplog} to compute harmonic polylogarithms up to weight 4.Comment: 16 pages, LaTeX, minor changes, to appear in Comp. Phys. Com

Area versus Length Distribution for Closed Random Walks

Using a connection between the $q$-oscillator algebra and the coefficients of the high temperature expansion of the frustrated Gaussian spin model, we derive an exact formula for the number of closed random walks of given length and area, on a hypercubic lattice, in the limit of infinite number of dimensions. The formula is investigated in detail, and asymptotic behaviours are evaluated. The area distribution in the limit of long loops is computed. As a byproduct, we obtain also an infinite set of new, nontrivial identities.Comment: 17 page

Explicit solution of the (quantum) elliptic Calogero-Sutherland model

We derive explicit formulas for the eigenfunctions and eigenvalues of the elliptic Calogero-Sutherland model as infinite series, to all orders and for arbitrary particle numbers and coupling parameters. The eigenfunctions obtained provide an elliptic deformation of the Jack polynomials. We prove in certain special cases that these series have a finite radius of convergence in the nome $q$ of the elliptic functions, including the two particle (= Lam\'e) case for non-integer coupling parameters.Comment: v1: 17 pages. The solution is given as series in q but only to low order. v2: 30 pages. Results significantly extended. v3: 35 pages. Paper completely revised: the results of v1 and v2 are extended to all order

Congruences for Taylor expansions of quantum modular forms

Recently, a beautiful paper of Andrews and Sellers has established linear congruences for the Fishburn numbers modulo an infinite set of primes. Since then, a number of authors have proven refined results, for example, extending all of these congruences to arbitrary powers of the primes involved. Here, we take a different perspective and explain the general theory of such congruences in the context of an important class of quantum modular forms. As one example, we obtain an infinite series of combinatorial sequences connected to the "half-derivatives" of the Andrews-Gordon functions and with Kashaev's invariant on $(2m+1,2)$ torus knots, and we prove conditions under which the sequences satisfy linear congruences modulo at least $50\%$ of primes of primes

Does Anonymity Matter in Electronic Limit Order Markets?

Lecture on the first SFB/TR 15 meeting, Gummersbach, July, 18 - 20, 2004We develop a model of limit order trading in which some traders have better information on future price volatility. As limit orders have option-like features, this information is valuable for limit order traders. We solve for informed and uninformed limit order traders’ bidding strategies in equilibrium when limit order traders’ IDs are concealed and when they are visible. In either design, a large (resp. small) spread signals that informed limit order traders expect volatility to be high (resp. low). However the quality of this signal and market liquidity are different in each market design. We test these predictions using a natural experiment. As of April 23, 2001, the limit order book for stocks listed on Euronext Paris became anonymous. For our sample stocks, we find that following this change, the average quoted and effective spreads declined significantly. Consistent with our model, we also find that the size of the spread is a predictor of future price volatility and that the strength of the association between the spread and volatility is weaker after the switch to anonymity.Market Microstructure; Limit Order Trading; Anonymity; Transparency; Liquidity; Volatility Forecasts