BKM Lie superalgebra for the Z_5 orbifolded CHL string

We study the Z_5-orbifolding of the CHL string theory by explicitly constructing the modular form tilde{Phi}_2 generating the degeneracies of the 1/4-BPS states in the theory. Since the additive seed for the sum form is a weak Jacobi form in this case, a mismatch is found between the modular forms generated from the additive lift and the product form derived from threshold corrections. We also construct the BKM Lie superalgebra, tilde{G}_5, corresponding to the modular form tilde{Delta}_1 (Z) = tilde{Phi}_2 (Z)^{1/2} which happens to be a hyperbolic algebra. This is the first occurrence of a hyperbolic BKM Lie superalgebra. We also study the walls of marginal stability of this theory in detail, and extend the arithmetic structure found by Cheng and Dabholkar for the N=1,2,3 orbifoldings to the N=4,5 and 6 models, all of which have an infinite number of walls in the fundamental domain. We find that analogous to the Stern-Brocot tree, which generated the intercepts of the walls on the real line, the intercepts for the N >3 cases are generated by linear recurrence relations. Using the correspondence between the walls of marginal stability and the walls of the Weyl chamber of the corresponding BKM Lie superalgebra, we propose the Cartan matrices for the BKM Lie superalgebras corresponding to the N=5 and 6 models.Comment: 30 pages, 2 figure

BKM Lie superalgebras from counting twisted CHL dyons

Following Sen[arXiv:0911.1563], we study the counting of (`twisted') BPS states that contribute to twisted helicity trace indices in four-dimensional CHL models with N=4 supersymmetry. The generating functions of half-BPS states, twisted as well as untwisted, are given in terms of multiplicative eta products with the Mathieu group, M_{24}, playing an important role. These multiplicative eta products enable us to construct Siegel modular forms that count twisted quarter-BPS states. The square-roots of these Siegel modular forms turn out be precisely a special class of Siegel modular forms, the dd-modular forms, that have been classified by Clery and Gritsenko[arXiv:0812.3962]. We show that each one of these dd-modular forms arise as the Weyl-Kac-Borcherds denominator formula of a rank-three Borcherds-Kac-Moody Lie superalgebra. The walls of the Weyl chamber are in one-to-one correspondence with the walls of marginal stability in the corresponding CHL model for twisted dyons as well as untwisted ones. This leads to a periodic table of BKM Lie superalgebras with properties that are consistent with physical expectations.Comment: LaTeX, 32 pages; (v2) matches published versio

A pp-adic Approach to the Weil Representation of Discriminant Forms Arising from Even Lattices

Suppose that $M$ is an even lattice with dual $M^{*}$ and level $N$. Then the group $Mp_{2}(\mathbb{Z})$, which is the unique non-trivial double cover of $SL_{2}(\mathbb{Z})$, admits a representation $\rho_{M}$, called the Weil representation, on the space $\mathbb{C}[M^{*}/M]$. The main aim of this paper is to show how the formulae for the $\rho_{M}$-action of a general element of $Mp_{2}(\mathbb{Z})$ can be obtained by a direct evaluation which does not depend on ``external objects'' such as theta functions. We decompose the Weil representation $\rho_{M}$ into $p$-parts, in which each $p$-part can be seen as subspace of the Schwartz functions on the $p$-adic vector space $M_{\mathbb{Q}_{p}}$. Then we consider the Weil representation of $Mp_{2}(\mathbb{Q}_{p})$ on the space of Schwartz functions on $M_{\mathbb{Q}_{p}}$, and see that restricting to $Mp_{2}(\mathbb{Z})$ just gives the $p$-part of $\rho_{M}$ again. The operators attained by the Weil representation are not always those appearing in the formulae from 1964, but are rather their multiples by certain roots of unity. For this, one has to find which pair of elements, lying over a matrix in $SL_{2}(\mathbb{Q}_{p})$, belong to the metaplectic double cover. Some other properties are also investigated.Comment: 29 pages, shortened a lo

Estimating prognosis and palliation based on tumour marker CA 19-9 and quality of life indicators in patients with advanced pancreatic cancer receiving chemotherapy

To investigate the prognostic value of quality of life (QOL) relative to tumour marker carbohydrate antigen (CA) 19-9, and the role of CA 19-9 in estimating palliation in patients with advanced pancreatic cancer receiving chemotherapy

Irinotecan plus folinic acid/continuous 5-fluorouracil as simplified bimonthly FOLFIRI regimen for first-line therapy of metastatic colorectal cancer

BACKGROUND: Combination therapy of irinotecan, folinic acid (FA) and 5-fluorouracil (5-FU) has been proven to be highly effective for the treatment of metastatic colorectal cancer. However, in light of safety and efficacy concerns, the best combination regimen for first-line therapy still needs to be defined. The current study reports on the bimonthly FOLFIRI protocol consisting of irinotecan with continuous FA/5-FU in five German outpatient clinics, with emphasis on the safety and efficiency, quality of life, management of delayed diarrhea, and secondary resection of regressive liver metastases. METHODS: A total of 35 patients were treated for metastatic colorectal cancer. All patients received first-line treatment according to the FOLFIRI regimen, consisting of irinotecan (180 mg/m(2)), L-FA (200 mg/m(2)) and 5-FU bolus (400 mg/m(2)) on day 1, followed by a 46-h continuous infusion 5-FU (2400 mg/m(2)). One cycle contained three fortnightly administrations. Staging was performed after 2 cycles. Dosage was reduced at any time if toxicity NCI CTC grade III/IV was observed. Chemotherapy was administered only to diarrhea-free patients. RESULTS: The FOLFIRI regimen was generally well tolerated. It was postponed for one-week in 51 of 415 applications (12.3%). Dose reduction was necessary in ten patients. Grade III/IV toxicity was rare, with diarrhea (14%), nausea/vomiting (12%), leucopenia (3%), neutropenia (9%) and mucositis (3%). The overall response rate was 31% (4 CR and 7 PR), with disease control in 74%. After primary chemotherapy, resection of liver metastases was achieved in three patients. In one patient, the CR was confirmed pathologically. Median progression-free and overall survival were seven and 17 months, respectively. CONCLUSIONS: The FOLFIRI regimen proved to be safe and efficient. Outpatient treatment was well tolerated. Since downstaging was possible, combinations of irinotecan and continuous FA/5-FU should further be investigated in neoadjuvant protocols