On algebraic equations satisfied by hypergeometric correlators in WZW models. II

We give an explicit description of "bundles of conformal blocks" in Wess-Zumino-Witten models of Conformal field theory and prove that integral representations of Knizhnik-Zamolodchikov equations constructed earlier by the second and third authors are in fact sections of these bundles.Comment: 32 pp., amslate

The Extreme Claim, Psychological Continuity and the Person Life View

Marya Schechtman has raised a series of worries for the Psychological Continuity Theory of personal identity (PCT) stemming out of what Derek Parfit called the ‘Extreme Claim’. This is roughly the claim that theories like it are unable to explain the importance we attach to personal identity. In her recent Staying Alive (2014), she presents further arguments related to this and sets out a new narrative theory, the Person Life View (PLV), which she sees as solving the problems as well as bringing other advantages over the PCT. I look over some of her earlier arguments and responses to them as a way in to the new issues and theory. I argue that the problems for the PCT and advantages that the PLV brings are all merely apparent, and present no reason for giving up the former for the latter

Am I my brother’s keeper? on personal identity and responsibility

The psychological continuity theory of personal identity has recently been accused of not meeting what is claimed to be a fundamental requirement on theories of identity - to explain personal moral responsibility. Although they often have much to say about responsibility, the charge is that they cannot say enough. I set out the background to the charge with a short discussion of Locke and the requirement to explain responsibility, then illustrate the accusation facing the theory with details from Marya Schechtman. I aim some questions at the challengers' reading of Locke, leading to an argument that the psychological continuity theory can say all that it needs to say about responsibility, and so is not in any grave predicament, at least not with regard to this particular charge.Web of Scienc

Technological fictions and personal identity: on Ricoeur, Schechtman and analytic thought experiments

It is notable when philosophers in one tradition take seriously the work in another and engage with it. This is certainly the case when Paul Ricoeur engages with the thought of Derek Parfit on personal identity. He sees it as worth engaging with, but as emblematic of errors in the analytic approach to the topic, especially when it comes to methodology. But he is, in a fairly clear way, taking the analytic debate on its own terms. Marya Schechtman’s work is also noteworthy in this regard. Although she writes in the analytic tradition, in many ways she has represented thinking like Ricoeur’s in the tradition – pressing concerns that echo his, and demanding that the debate needs to take notice. I will focus on complaints that both of them present, which I think are closely related, about the thought experiments that feature large in analytic discussions of personal identity, especially in the seminal work of Parfit. The complaints relate both to those devices and to the theory they have produced. I want to offer something of a defence of both.IS

Representations of affine Lie algebras, elliptic r-matrix systems, and special functions

There were some errors in paper hep-th/9303018 in formulas 6.1, 6.6, 6.8, 6.11. These errors have been corrected in the present version of this paper. There are also some minor changes in the introduction.Comment: 33 pages, no figure

Unitarity of the Knizhnik-Zamolodchikov-Bernard connection and the Bethe Ansatz for the elliptic Hitchin systems

We work out finite-dimensional integral formulae for the scalar product of genus one states of the group $G$ Chern-Simons theory with insertions of Wilson lines. Assuming convergence of the integrals, we show that unitarity of the elliptic Knizhnik-Zamolodchikov-Bernard connection with respect to the scalar product of CS states is closely related to the Bethe Ansatz for the commuting Hamiltonians building up the connection and quantizing the quadratic Hamiltonians of the elliptic Hitchin system.Comment: 24 pages, latex fil

An Integrable Model of Quantum Gravity

We present a new quantization scheme for $2D$ gravity coupled to an $SU(2)$ principal chiral field and a dilaton; this model represents a slightly simplified version of stationary axisymmetric quantum gravity. The analysis makes use of the separation of variables found in our previous work [1] and is based on a two-time hamiltonian approach. The quantum constraints are shown to reduce to a pair of compatible first order equations, with the dilaton playing the role of a ``clock field''. Exact solutions of the Wheeler-DeWitt equation are constructed via the integral formula for solutions of the Knizhnik-Zamolodchiokov equations.Comment: 12 page

Geometry of q-Hypergeometric Functions as a Bridge between Yangians and Quantum Affine Algebras

The rational quantized Knizhnik-Zamolodchikov equation (qKZ equation) associated with the Lie algebra $sl_2$ is a system of linear difference equations with values in a tensor product of $sl_2$ Verma modules. We solve the equation in terms of multidimensional $q$-hypergeometric functions and define a natural isomorphism between the space of solutions and the tensor product of the corresponding quantum group $U_q(sl_2)$ Verma modules, where the parameter $q$ is related to the step $p$ of the qKZ equation via $q=e^{pi i/p}$. We construct asymptotic solutions associated with suitable asymptotic zones and compute the transition functions between the asymptotic solutions in terms of the trigonometric $R$-matrices. This description of the transition functions gives a new connection between representation theories of Yangians and quantum loop algebras and is analogous to the Kohno-Drinfeld theorem on the monodromy group of the differential Knizhnik-Zamolodchikov equation. In order to establish these results we construct a discrete Gauss-Manin connection, in particular, a suitable discrete local system, discrete homology and cohomology groups with coefficients in this local system, and identify an associated difference equation with the qKZ equation.Comment: 66 pages, amstex.tex (ver. 2.1) and amssym.tex are required; misprints are correcte

Interplay between quasi-periodicity and disorder in quantum spin chains in a magnetic field

We study the interplay between disorder and a quasi periodic coupling array in an external magnetic field in a spin-1/2 XXZ chain. A simple real space decimation argument is used to estimate the magnetization values where plateaux show up. The latter are in good agreement with exact diagonalization results on fairly long XX chains. Spontaneous susceptibility properties are also studied, finding a logarithmic behaviour similar to the homogeneously disordered case.Comment: 5 RevTeX pages, 5 Postscript figures include