Iterated residue, toric forms and Witten genus

We introduce the notion of {\em iterated residue} to study generalized Bott manifolds. When applying the iterated residues to compute the Borisov-Gunnells toric form and the Witten genus of certain toric varieties as well as complete intersections, we obtain interesting vanishing results and some theta function identities, one of which is a twisted version of a classical Rogers-Ramanujan type formula.Comment: 19 page

Some Identities of Ramanujan's q-Continued Fractions of Order Fourteen and Twenty-Eight, and Vanishing Coefficients

We deduce $q$-continued fractions $S_{1}(q)$, $S_{2}(q)$ and $S_{3}(q)$ of order fourteen, and continued fractions $V_{1}(q)$, $V_{2}(q)$ and $V_{3}(q)$ of order twenty-eight from a general continued fraction identity of Ramanujan. We establish some theta-function identities for the continued fractions and derive some colour partition identities as applications. Some vanishing coefficients results arising from the continued fractions are also offered.Comment: 8 Page

CFT Description of String Theory Compactified on Non-compact Manifolds with G_2 Holonomy

We construct modular invariant partition functions for strings propagating on non-compact manifolds of G_2 holonomy. Our amplitudes involve a pair of N=1 minimal models M_m, M_{m+2} (m=3,4,...) and are identified as describing strings on manifolds of G_2 holonomy associated with A_{m-2} type singularity. It turns out that due to theta function identities our amplitudes may be cast into a form which contain tricritical Ising model for any m. This is in accord with the results of Shatashvili and Vafa. We also construct a candidate partition function for string compactified on a non-compact Spin(7) manifold.Comment: It is found that tricritical Ising model is contained in our amplitues in agreement with the results of Shatashvili and Vafa. Manuscript is revised accordingly. A new reference is also adde

String Theory on G_2 Manifolds Based on Gepner Construction

We study the type II string theories compactified on manifolds of $G_2$ holonomy of the type ({Calabi-Yau 3-fold} \times S^1)/\bz_2 where $CY_3$ sectors realized by the Gepner models. We construct modular invariant partition functions for $G_2$ manifold for arbitrary Gepner models of the Calabi-Yau sector. We note that the conformal blocks contain the tricritical Ising model and find extra massless states in the twisted sectors of the theory when all the levels $k_i$ of minimal models in Gepner constructions are even.Comment: 20 pages, no figure, improvement on some technical points in the discussions of twisted sector

Modular Invariance in Superstring on Calabi-Yau n-fold with A-D-E Singularity

We study the type II superstring theory on the background \br^{d-1,1}\times X_n, where $X_n$ is a Calabi-Yau $n$-fold ($2n+d=10$) with an isolated singularity, by making use of the holographically dual description proposed by Giveon-Kutasov-Pelc (hep-th/9907178). We compute the toroidal partition functions for each of the cases $d=6,4,2$, and obtain manifestly modular invariant solutions classified by the standard $A-D-E$ series corresponding to the type of singularities on $X_n$. Partition functions of these modular invariants all vanish due to theta function identities and are consistent with the presence of space-time supersymmetry.Comment: typos corrected, to appear in Nucl. Phys.

Integer Matrix Exact Covering Systems and Product Identities for Theta Functions

In this paper, we prove that there is a natural correspondence between product identities for theta functions and integer matrix exact covering systems. We show that since $\mathbb{Z}^n$ can be taken as the disjoint union of a lattice generated by $n$ linearly independent vectors in $\mathbb{Z}^n$ and a finite number of its translates, certain products of theta functions can be written as linear combinations of other products of theta functions. We firstly give a general theorem to write a product of $n$ theta functions as a linear combination of other products of theta functions. Many known identities for products of theta functions are shown to be special cases of our main theorem. Several entries in Ramanujan's notebooks as well as new identities are proved as applications, including theorems for products of three and four theta functions that have not been obtained by other methods

Orbifold boundary states from Cardy's condition

Boundary states for D-branes at orbifold fixed points are constructed in close analogy with Cardy's derivation of consistent boundary states in RCFT. Comments are made on the interpretation of the various coefficients in the explicit expressions, and the relation between fractional branes and wrapped branes is investigated for $\mathbb{C}^2/\Gamma$ orbifolds. The boundary states are generalised to theories with discrete torsion and a new check is performed on the relation between discrete torsion phases and projective representations.Comment: LaTeX2e, 50 pages, 5 figures. V3: final version to appear on JHEP (part of a section moved to an appendix, titles of some references added, one sentence in the introduction expanded