Virasoro Symmetries of the Extended Toda Hierarchy

We prove that the extended Toda hierarchy of \cite{CDZ} admits nonabelian Lie algebra of infinitesimal symmetries isomorphic to the half of the Virasoro algebra. The generators $L_m$, $m\geq -1$ of the Lie algebra act by linear differential operators onto the tau function of the hierarchy. We also prove that the tau function of a generic solution to the extended Toda hierarchy is annihilated by a combination of the Virasoro operators and the flows of the hierarchy. As an application we show that the validity of the Virasoro constraints for the $CP^1$ Gromov-Witten invariants and their descendents implies that their generating function is the logarithm of a particular tau function of the extended Toda hierarchy.Comment: A remark at the end of Section 5 is added; more detailed explanations in Appendix; references adde

The generalized no-ghost theorem for N=2 SUSY critical strings

We prove the no-ghost theorem for the N=2 SUSY strings in (2,2) dimensional flat Minkowski space. We propose a generalization of this theorem for an arbitrary geometry of the N=2 SUSY string theory taking advantage of the N=4 SCA generators present in this model. Physical states are found to be the highest weight states of the N=4 SCA.Comment: 13

A New Description of the E_6 Singularity

We discuss a new type of Landau-Ginzburg potential for the E_6 singularity of the form $W=const+(Q_1(x)+P_1(x)\sqrt{P_2(x)})/x^3$ which featured in a recent study of heterotic/typeII string duality. Here $Q_1,P_1$ and $P_2$ are polynomials of degree 15,10 and 10, respectively. We study the properties of the potential in detail and show that it gives a new and consistent description of the E_6 singularity.Comment: LaTeX, 13 pages, no figure

Complexity Analysis of Precedence Terminating Infinite Graph Rewrite Systems

The general form of safe recursion (or ramified recurrence) can be expressed by an infinite graph rewrite system including unfolding graph rewrite rules introduced by Dal Lago, Martini and Zorzi, in which the size of every normal form by innermost rewriting is polynomially bounded. Every unfolding graph rewrite rule is precedence terminating in the sense of Middeldorp, Ohsaki and Zantema. Although precedence terminating infinite rewrite systems cover all the primitive recursive functions, in this paper we consider graph rewrite systems precedence terminating with argument separation, which form a subclass of precedence terminating graph rewrite systems. We show that for any precedence terminating infinite graph rewrite system G with a specific argument separation, both the runtime complexity of G and the size of every normal form in G can be polynomially bounded. As a corollary, we obtain an alternative proof of the original result by Dal Lago et al.Comment: In Proceedings TERMGRAPH 2014, arXiv:1505.06818. arXiv admin note: text overlap with arXiv:1404.619

Formalizing Termination Proofs under Polynomial Quasi-interpretations

Usual termination proofs for a functional program require to check all the possible reduction paths. Due to an exponential gap between the height and size of such the reduction tree, no naive formalization of termination proofs yields a connection to the polynomial complexity of the given program. We solve this problem employing the notion of minimal function graph, a set of pairs of a term and its normal form, which is defined as the least fixed point of a monotone operator. We show that termination proofs for programs reducing under lexicographic path orders (LPOs for short) and polynomially quasi-interpretable can be optimally performed in a weak fragment of Peano arithmetic. This yields an alternative proof of the fact that every function computed by an LPO-terminating, polynomially quasi-interpretable program is computable in polynomial space. The formalization is indeed optimal since every polynomial-space computable function can be computed by such a program. The crucial observation is that inductive definitions of minimal function graphs under LPO-terminating programs can be approximated with transfinite induction along LPOs.Comment: In Proceedings FICS 2015, arXiv:1509.0282