Continuum Annulus Amplitude from the Two-Matrix Model

An explicit expression for continuum annulus amplitudes having boundary lengths $\ell_{1}$ and $\ell_{2}$ is obtained from the two-matrix model for the case of the unitary series; $(p,q) = (m + 1, m)$. In the limit of vanishing cosmological constant, we find an integral representation of these amplitudes which is reproduced, for the cases of the $m = 2~(c=0)$ and the $m \rightarrow \infty~(c=1)$, by a continuum approach consisting of quantum mechanics of loops and a matter system integrated over the modular parameter of the annulus. We comment on a possible relation to the unconventional branch of the Liouville gravity.Comment: 9 pages, OU-HET 190, revised version. A part of the conclusions has been corrected. A new result on integral representation of the annulus amplitudes has been adde

Vacuum Energy Cancellation in a Non-supersymmetric String

We present a nonsupersymmetric orbifold of type II string theory and show that it has vanishing cosmological constant at the one and two loop level. We argue heuristically that the cancellation persists at higher loops.Comment: 31 pages harvmac big, 6 figures. New version includes the 2-loop analysis of hep-th/9810129 and elimination of one of the two heuristic arguments for higher loop cancellatio

M-branes and N=2 Strings

The string field theory of N=(2,1) heterotic strings describes a set of self-dual Yang-Mills fields coupled to self-dual gravity in 2+2 dimensions. We show that the exact classical action for this field theory is a certain complexification of the Green-Schwarz/Dirac-Born-Infeld string action, closely related to the four dimensional Wess-Zumino action describing self-dual gauge fields. This action describes the world-volume of a 2+2d ``M-brane'', which gives rise upon different null reductions to critical strings and membranes. We discuss a number of further properties of N=2 heterotic strings, such as the geometry of null reduction, general features of a covariant formulation, and possible relations to BPS and GKM algebras.Comment: 49 pages, harvmac; 1 figure (uses epsf.tex). References adde

Spherical Casimir energies and Dedekind sums

Casimir energies on space-times having general lens spaces as their spatial sections are shown to be given in terms of generalised Dedekind sums related to Zagier's. These are evaluated explicitly in certain cases as functions of the order of the lens space. An easily implemented recursion approach is used.Comment: 18 pages, 2 figures, v2:typos corrected, inessential equation in Discussion altered. v3:typos corrected, 1 reference and comments added. v4:typos corrected. Ancillary results added in an appendi

Localized Tachyons and the Quantum McKay Correspondence

The condensation of closed string tachyons localized at the fixed point of a C^d/\Gamma orbifold can be studied in the framework of renormalization group flow in a gauged linear sigma model. The evolution of the Higgs branch along the flow describes a resolution of singularities via the process of tachyon condensation. The study of the fate of D-branes in this process has lead to a notion of a ``quantum McKay correspondence.'' This is a hypothetical correspondence between fractional branes in an orbifold singularity in the ultraviolet with the Coulomb and Higgs branch branes in the infrared. In this paper we present some nontrivial evidence for this correspondence in the case C^2/Z_n by relating the intersection form of fractional branes to that of ``Higgs branch branes,'' the latter being branes which wrap nontrivial cycles in the resolved space.Comment: 25 pages; harvma

Closed string tachyons, flips and conifolds

Following the analysis of tachyons and orbifold flips described in hep-th/0412337, we study nonsupersymmetric analogs of the supersymmetric conifold singularity and show using their toric geometry description that they are nonsupersymmetric orbifolds of the latter. Using linear sigma models, we see that these are unstable to localized closed string tachyon condensation and exhibit flip transitions between their two small resolutions (involving 2-cycles), in the process mediating mild dynamical topology change. Our analysis shows that the structure of these nonsupersymmetric conifolds as quotients of the supersymmetric conifold obstructs the 3-cycle deformation of such singularities, suggesting that these nonsupersymmetric conifolds decay by evolving towards their stable small resolutions.Comment: Latex, 22 pgs, 2 figs. v4: matches JHEP version, 29 pgs, 3 figures, more elaborate Introduction, various clarifications adde

Theta Vectors and Quantum Theta Functions

In this paper, we clarify the relation between Manin's quantum theta function and Schwarz's theta vector in comparison with the kq representation, which is equivalent to the classical theta function, and the corresponding coordinate space wavefunction. We first explain the equivalence relation between the classical theta function and the kq representation in which the translation operators of the phase space are commuting. When the translation operators of the phase space are not commuting, then the kq representation is no more meaningful. We explain why Manin's quantum theta function obtained via algebra (quantum tori) valued inner product of the theta vector is a natural choice for quantum version of the classical theta function (kq representation). We then show that this approach holds for a more general theta vector with constant obtained from a holomorphic connection of constant curvature than the simple Gaussian one used in the Manin's construction. We further discuss the properties of the theta vector and of the quantum theta function, both of which have similar symmetry properties under translation.Comment: LaTeX 21 pages, give more explicit explanations for notions given in the tex

A Note on Background (In)dependence

In general quantum systems there are two kinds of spacetime modes, those that fluctuate and those that do not. Fluctuating modes have normalizable wavefunctions. In the context of 2D gravity and ``non-critical'' string theory these are called macroscopic states. The theory is independent of the initial Euclidean background values of these modes. Non-fluctuating modes have non-normalizable wavefunctions and correspond to microscopic states. The theory depends on the background value of these non-fluctuating modes, at least to all orders in perturbation theory. They are superselection parameters and should not be minimized over. Such superselection parameters are well known in field theory. Examples in string theory include the couplings $t_k$ (including the cosmological constant) in the matrix models and the mass of the two-dimensional Euclidean black hole. We use our analysis to argue for the finiteness of the string perturbation expansion around these backgrounds.Comment: 16 page

The Solution Space of the Unitary Matrix Model String Equation and the Sato Grassmannian

The space of all solutions to the string equation of the symmetric unitary one-matrix model is determined. It is shown that the string equation is equivalent to simple conditions on points $V_1$ and $V_2$ in the big cell \Gr of the Sato Grassmannian $Gr$. This is a consequence of a well-defined continuum limit in which the string equation has the simple form \lb \cp ,\cq_- \rb =\hbox{\rm 1}, with \cp and \cq_- $2\times 2$ matrices of differential operators. These conditions on $V_1$ and $V_2$ yield a simple system of first order differential equations whose analysis determines the space of all solutions to the string equation. This geometric formulation leads directly to the Virasoro constraints \L_n\,(n\geq 0), where \L_n annihilate the two modified-KdV \t-functions whose product gives the partition function of the Unitary Matrix Model.Comment: 21 page

Notes on the algebraic curves in (p,q) minimal string theory

Loop amplitudes in (p,q) minimal string theory are studied in terms of the continuum string field theory based on the free fermion realization of the KP hierarchy. We derive the Schwinger-Dyson equations for FZZT disk amplitudes directly from the W_{1+\infty} constraints in the string field formulation and give explicitly the algebraic curves of disk amplitudes for general backgrounds. We further give annulus amplitudes of FZZT-FZZT, FZZT-ZZ and ZZ-ZZ branes, generalizing our previous D-instanton calculus from the minimal unitary series (p,p+1) to general (p,q) series. We also give a detailed explanation on the equivalence between the Douglas equation and the string field theory based on the KP hierarchy under the W_{1+\infty} constraints.Comment: 61 pages, 1 figure, section 2.5 and Appendix B added, references added, final version to appear in JHE