Recent zbMATH articles in MSC 14J32https://www.zbmath.org/atom/cc/14J322021-04-16T16:22:00+00:00WerkzeugSystematics of type IIA moduli stabilisation.https://www.zbmath.org/1456.831052021-04-16T16:22:00+00:00"Marchesano, Fernando"https://www.zbmath.org/authors/?q=ai:marchesano.fernando"Prieto, David"https://www.zbmath.org/authors/?q=ai:prieto.david"Quirant, Joan"https://www.zbmath.org/authors/?q=ai:quirant.joan"Shukla, Pramod"https://www.zbmath.org/authors/?q=ai:shukla.pramod-sSummary: We analyse the flux-induced scalar potential for type IIA orientifolds in the presence of \(p\)-form, geometric and non-geometric fluxes. Just like in the Calabi-Yau case, the potential presents a bilinear structure, with a factorised dependence on axions and saxions. This feature allows one to perform a systematic search for vacua, which we implement for the case of geometric backgrounds. Guided by stability criteria, we consider configurations with a particular on-shell F-term pattern, and show that no de Sitter extrema are allowed for them. We classify branches of supersymmetric and non-supersymmetric vacua, and argue that the latter are perturbatively stable for a large subset of them. Our solutions reproduce and generalise previous results in the literature, obtained either from the 4d or 10d viewpoint.A one parameter family of Calabi-Yau manifolds with attractor points of rank two.https://www.zbmath.org/1456.830892021-04-16T16:22:00+00:00"Candelas, Philip"https://www.zbmath.org/authors/?q=ai:candelas.philip"de la Ossa, Xenia"https://www.zbmath.org/authors/?q=ai:de-la-ossa.xenia-c"Elmi, Mohamed"https://www.zbmath.org/authors/?q=ai:elmi.mohamed"van Straten, Duco"https://www.zbmath.org/authors/?q=ai:van-straten.ducoSummary: In the process of studying the \(\zeta\)-function for one parameter families of Calabi-Yau manifolds we have been led to a manifold, first studied by Verrill, for which the quartic numerator of the \(\zeta\)-function factorises into two quadrics remarkably often. Among these factorisations, we find \textit{persistent factorisations}; these are determined by a parameter that satisfies an algebraic equation with coefficients in \(\mathbb{Q}\), so independent of any particular prime. Such factorisations are expected to be modular with each quadratic factor associated to a modular form. If the parameter is defined over \(\mathbb{Q}\) this modularity is assured by the proof of the Serre Conjecture. We identify three values of the parameter that give rise to persistent factorisations, one of which is defined over \(\mathbb{Q}\), and identify, for all three cases, the associated modular groups. We note that these factorisations are due a splitting of Hodge structure and that these special values of the parameter are rank two attractor points in the sense of IIB supergravity. To our knowledge, these points provide the first explicit examples of non-singular, non-rigid rank two attractor points for Calabi-Yau manifolds of full SU(3) holonomy. The values of the periods and their covariant derivatives, at the attractor points, are identified in terms of critical values of the \(L\)-functions of the modular groups. Thus the critical \(L\)-values enter into the calculation of physical quantities such as the area of the black hole in the 4D spacetime. In our search for additional rank two attractor points, we perform a statistical analysis of the numerator of the \(\zeta\)-function and are led to conjecture that the coefficients in this polynomial are distributed according to the statistics of random USp(4) matrices.Supercongruences arising from hypergeometric series identities.https://www.zbmath.org/1456.110052021-04-16T16:22:00+00:00"Liu, Ji-Cai"https://www.zbmath.org/authors/?q=ai:liu.jicaiThis paper refines a supercongruence of \textit{T. Kilbourn} [Acta Arith. 123, No. 4, 335--348 (2006; Zbl 1170.11008)] about the identity \[ a(p)=p^3-2p^2-7-N(p) \] studied by \textit{S. Ahlgren} and \textit{K. Ono} [J. Reine Angew. Math. 518, 187--212 (2000; Zbl 0940.33002)], by \textit{B. van Geemen} and \textit{N. O. Nygaard} [J. Number Theory 53, No. 1, 45--87 (1995; Zbl 0838.11047)], and by \textit{H. A. Verrill} [CRM Proc. Lecture Notes 19, 333--340. Providence, RI: Amer. Math. Soc. (1999; Zbl 0942.14022] in connection to the modular Calabi-Yau threefold for odd primes \(p\) \[ x+\frac{1}{x}+y+\frac{1}{y}+z+\frac{1}{z}+w+\frac{1}{w}=0 \] associated with truncated hypergeometric series.
Namely, the author establishes that \[a(p) \equiv p \cdot {{_{4}F_3} \left[ \begin{matrix} \frac{1}{2}, & \frac{1}{2}, & \frac{1}{2}, & \frac{1}{2}\;\\ & 1, & \frac{3}{4}, & \frac{5}{4} \end{matrix} \Big| \; 1 \right]}_ \frac{ p-1}{2} \pmod {p^3}\] for any prime \(p \geq 5 \).
In addition, the paper gives a ``human proof'' of a supercongruence already found by the author [J. Math. Anal. Appl. 471, 613--622 (2019; Zbl 1423.11015)], via the Mathematica package \(Sigma\) supplied by \textit{C. Schneider} [Sémin. Lothar. Comb. 56, B56b, 36 p. (2006; Zbl 1188.05001)], as extension of the \(p\)-adic analogue of a Ramanujan's identity conjectured by \textit{L. van Hamme} [Lect. Notes Pure Appl. Math. 192, 223--236 (1997; Zbl 0895.11051)].
Beyond basic properties of the gamma function (both classical and \(p\)-adic), the Taylor's expansion, and the Wolstenholme's congruence, the theorem-proving recalls some results from \textit{W. N. Bailey} [Generalized hypergeometric series. London: Cambridge University Press (1935; Zbl 0011.02303)], from \textit{L. Long} and \textit{R. Ramakrishna} [Adv. Math. 290, 773--808 (2016; Zbl 1336.33018)], and from \textit{F. J. W. Whipple} [Proc. London Math Soc. 24, 247--263 (1926; JFM 51.0283.03)].
Reviewer: Enzo Bonacci (Latina)The pluricanonical systems of a product-quotient variety.https://www.zbmath.org/1456.140582021-04-16T16:22:00+00:00"Favale, Filippo F."https://www.zbmath.org/authors/?q=ai:favale.filippo-francesco"Gleissner, Christian"https://www.zbmath.org/authors/?q=ai:gleissner.christian"Pignatelli, Roberto"https://www.zbmath.org/authors/?q=ai:pignatelli.robertoSummary: We give a method for the computation of the plurigenera of a product-quotient manifold, and two different types of applications of it: to the construction of Calabi-Yau threefolds and to the determination of the minimal model of a product-quotient surface of general type.
For the entire collection see [Zbl 07237934].Correspondence scrolls.https://www.zbmath.org/1456.140482021-04-16T16:22:00+00:00"Eisenbud, David"https://www.zbmath.org/authors/?q=ai:eisenbud.david"Sammartano, Alessio"https://www.zbmath.org/authors/?q=ai:sammartano.alessioThis paper introduces schemes called ``correspondence scrolls'' and gives a general study of them. For a closed subscheme \(Z\) in \(\Pi _{i=1}^n {\mathbb A}^{a_i+1}\), defined by a multigraded ideal \(I\subset A:={\Bbbk}[x_{i,j} : 1\le i \le n, 0 \le j \le a_i]\), and for \(\mathbf{b}=(b_1, \ldots , b_n) \in \mathbb{N}_+^n\), consider the homomorphism \({\Bbbk}[z_{i,\alpha }] \rightarrow A/I\) which sends a variable \(z_{i, \alpha }\) to the monomial \(x_i^\alpha\), of degree \(b_i\). Here \(x_i^ {\alpha }\) denotes \(x_{i,0}^{\alpha _0} \cdots x_{i, a_i}^{\alpha _{a_i}}\). The kernel of the above map defines a closed projective subscheme \(C(Z, {\mathbf b}) \subset { \mathbb P}^N\) (\(N= \sum \binom{a_i+b_i}{a_i}-1)\), called \textit{correspondence scroll}. This definition includes classical correspondences as well as interesting non-classical ones: rational normal scrolls, double structures which are degenerate \(K3\) surfaces, degenerate Calabi-Yau threefolds, etc. Many invariants or properties of correspondence scrolls are studied: dimension, degree, nonsingularity, Cohen-Macaulay and Gorenstein property and others. The paper is very well written and invites to further research.
Reviewer: Nicolae Manolache (Bucureşti)A hypergeometric version of the modularity of rigid Calabi-Yau manifolds.https://www.zbmath.org/1456.110732021-04-16T16:22:00+00:00"Zudilin, Wadim"https://www.zbmath.org/authors/?q=ai:zudilin.wadimThis paper considers the fourteen one-parameter families of Calabi-Yau
threefolds whose periods are expressed in terms of hypergeometric functions.
For these fourteen families, periods are solutions of hypergeometric equations
with parameter \((r, 1-r, t, 1-t)\), where
\begin{multline*}
(r,t)=\Big(\frac{1}{2},\frac{1}{2}\Big),\Big(\frac{1}{2},\frac{1}{3}\Big),\Big(\frac{1}{2},\frac{1}{4}\Big),
\Big(\frac{1}{2},\frac{1}{6}\Big),\Big(\frac{1}{3}\Big),\Big(\frac{1}{3},\frac{1}{4}\Big),\Big(\frac{1}{3},\frac{1}{6}\Big),\\
\Big(\frac{1}{4},\frac{1}{4}\Big),\Big(\frac{1}{4},\frac{1}{6}\Big),\Big(\frac{1}{6},\frac{1}{6}\Big),\Big(\frac{1}{5},\frac{2}{5}\Big),
\Big(\frac{1}{8},\frac{3}{8}\Big),\Big(\frac{1}{10},\frac{3}{10}\Big),\Big(\frac{1}{12},\frac{5}{12}\Big).
\end{multline*}
At a conifold point, any of these Calabi-Yau threefolds becomes rigid,
and the \(p\)-th coefficient \(a(p)\) of the corresponding modular form of weight \(4\)
can be recovered from the truncated partial sums of the corresponding
hypergeometric series modulo a higher power of \(p\), where \(p\) is any good prime \(>5\).
This paper discusses relationships between the critical values of the \(L\)-series of the modular form
and the values of a related basis of solutions to the hypergeometric differential equation.
It is numerically observed that the critical \(L\)-values are \(\mathbb{Q}\)-proportional to the
hypergeometric values \(F_1(1), F_2(1), F_3(1)\), where \(F_j(z)\) are solutions of the hypergeometric
equation for the hypergeometric function \(F_0(z)=_4F_3(z)\) with parameters \((r, 1-r, t, 1-t)\).
This confirms the prediction of Golyshev concerning gamma structures [\textit{V. Golyshev} and \textit{A. Mellit}, J. Geom. Phys. 78, 12--18 (2014; Zbl 1284.33001)].
Reviewer: Noriko Yui (Kingston)On \(\alpha '\)-effects from \(D\)-branes in \(4d\) \( \mathcal{N} = 1\).https://www.zbmath.org/1456.813402021-04-16T16:22:00+00:00"Weissenbacher, Matthias"https://www.zbmath.org/authors/?q=ai:weissenbacher.matthiasSummary: In this work we study type IIB Calabi-Yau orientifold compactifications in the presence of space-time filling D7-branes and O7-planes. In particular, we conclude that \(\alpha'^2g_s\)-corrections to their DBI actions lead to a modification of the four-dimensional \(\mathcal{N} = 1\) Kähler potential and coordinates. We focus on the one-modulus case of the geometric background i.e. \(h^{1,1} = 1\) where we find that the \(\alpha'^2g_s \)-correction is of topological nature. It depends on the first Chern form of the four-cycle of the Calabi-Yau orientifold which is wrapped by the D7-branes and O7-plane. This is in agreement with our previous F-theory analysis and provides further evidence for a potential breaking of the no-scale structure at order \(\alpha'^2g_s\). Corrected background solutions for the dilaton, the warp-factor as well as the internal space metric are derived. Additionally, we briefly discuss \(\alpha '\)-corrections from other \(Dp\)-branes.Quiver Yangian from crystal melting.https://www.zbmath.org/1456.812162021-04-16T16:22:00+00:00"Li, Wei"https://www.zbmath.org/authors/?q=ai:li.wei.7|li.wei-wayne|li.wei.10|li.wei.11|li.wei.9|li.wei.4|li.wei.8|li.wei|li.wei.3|li.wei.2"Yamazaki, Masahito"https://www.zbmath.org/authors/?q=ai:yamazaki.masahitoSummary: We find a new infinite class of infinite-dimensional algebras acting on BPS states for non-compact toric Calabi-Yau threefolds. In Type IIA superstring compactification on a toric Calabi-Yau threefold, the D-branes wrapping holomorphic cycles represent the BPS states, and the fixed points of the moduli spaces of BPS states are described by statistical configurations of crystal melting. Our algebras are ``bootstrapped'' from the molten crystal configurations, hence they act on the BPS states. We discuss the truncation of the algebra and its relation with D4-branes. We illustrate our results in many examples, with and without compact 4-cycles.Non-simply-connected symmetries in 6D SCFTs.https://www.zbmath.org/1456.814272021-04-16T16:22:00+00:00"Dierigl, Markus"https://www.zbmath.org/authors/?q=ai:dierigl.markus"Oehlmann, Paul-Konstantin"https://www.zbmath.org/authors/?q=ai:oehlmann.paul-konstantin"Ruehle, Fabian"https://www.zbmath.org/authors/?q=ai:ruehle.fabianSummary: Six-dimensional \(\mathcal{N} = (1, 0)\) superconformal field theories can be engineered geometrically via F-theory on elliptically-fibered Calabi-Yau 3-folds. We include torsional sections in the geometry, which lead to a finite Mordell-Weil group. This allows us to identify the full non-abelian group structure rather than just the algebra. The presence of torsion also modifies the center of the symmetry groups and the matter representations that can appear. This in turn affects the tensor branch of these theories. We analyze this change for a large class of superconformal theories with torsion and explicitly construct their tensor branches. Finally, we elaborate on the connection to the dual heterotic and M-theory description, in which our configurations are interpreted as generalizations of discrete holonomy instantons.Distributions of extremal black holes in Calabi-Yau compactifications.https://www.zbmath.org/1456.830422021-04-16T16:22:00+00:00"Hulsey, George"https://www.zbmath.org/authors/?q=ai:hulsey.george"Kachru, Shamit"https://www.zbmath.org/authors/?q=ai:kachru.shamit"Yang, Sungyeon"https://www.zbmath.org/authors/?q=ai:yang.sungyeon"Zimet, Max"https://www.zbmath.org/authors/?q=ai:zimet.maxSummary: We study non-supersymmetric extremal black hole excitations of 4d \(\mathcal{N} = 2\) supersymmetric string vacua arising from compactification on Calabi-Yau threefolds. The values of the (vector multiplet) moduli at the black hole horizon are governed by the attractor mechanism. This raises natural questions, such as ``what is the distribution of attractor points on moduli space?'' and ``how many attractor black holes are there with horizon area up to a certain size?'' We employ tools developed by \textit{F. Denef} and \textit{M. R. Douglas} [``Distributions of flux vacua'', J. High Energy Phys. 2004, No. 5, Paper No. 072, 46 p. (2004; \url{doi:10.1088/1126-6708/2004/05/072})] to answer these questions.A construction of infinitely many solutions to the Strominger system.https://www.zbmath.org/1456.813342021-04-16T16:22:00+00:00"Fei, Teng"https://www.zbmath.org/authors/?q=ai:fei.teng.1|fei.teng"Huang, Zhijie"https://www.zbmath.org/authors/?q=ai:huang.zhijie"Picard, Sebastien"https://www.zbmath.org/authors/?q=ai:picard.sebastienFrom the introduction:: As for compact Kähler Calabi-Yau manifolds (treated as solutions to
the Strominger system), it is widely speculated that in each dimension
there are only finitely many deformation types and hence finitely many
sets of Hodge numbers. Moreover, there are no explicit expressions for
Calabi-Yau metrics except for the flat case.\par\vspace{1mm}
In this paper, we demonstrate that the non-Kähler world of solu-
tions to the Strominger system is considerably different. More pre-
cisely, we construct explicit smooth solutions to the Strominger system
on compact non-Kähler Calabi-Yau 3-folds with infinitely many topo-
logical types and sets of Hodge numbersPicard-Vessiot groups of Lauricella's hypergeometric systems \(E_C\) and Calabi-Yau varieties arising integral representations.https://www.zbmath.org/1456.140142021-04-16T16:22:00+00:00"Goto, Yoshiaki"https://www.zbmath.org/authors/?q=ai:goto.yoshiaki"Koike, Kenji"https://www.zbmath.org/authors/?q=ai:koike.kenjiThe authors study the Zariski closure of the monodromy group \(Mon\) of Lauricella's hypergeometric function \(F_C(a,b,c;x)=\sum_{m_1,\ldots ,m_n=0}^{\infty}\frac{(a)_{m_1+\cdots +m_n}(b)_{m_1+\cdots +m_n}}{(c_1)_{m_1}\cdots (c_n)_{m_n}m_1!\cdots m_n!}x_1^{m_1}\cdots x_n^{m_n}\), where \(a,b\in \mathbb{C}\), \(c_i\in \mathbb{C}\setminus \{ 0,-1,-2,\ldots \}\), \((c_i)_{m_i}=\Gamma (c_i+m_i)/\Gamma (c_i)\), and Calabi-Yau varieties arising from its integral representation. When the identity component of \(Mon\) acts irreducibly, then \(\overline{Mon}\cap SL_{2^n}(\mathbb{C})\) is one of the classical groups \(SL_{2^n}(\mathbb{C})\), \(SO_{2^n}(\mathbb{C})\) or \(Sp_{2^n}(\mathbb{C})\).
Reviewer: Vladimir P. Kostov (Nice)M-theory and orientifolds.https://www.zbmath.org/1456.140462021-04-16T16:22:00+00:00"Braun, Andreas P."https://www.zbmath.org/authors/?q=ai:braun.andreas-pSummary: We construct the M-Theory lifts of type IIA orientifolds based on \(K3\)-fibred Calabi-Yau threefolds with compatible involutions. Such orientifolds are shown to lift to M-Theory on twisted connected sum \(G_2\) manifolds. Beautifully, the two building blocks forming the \(G_2\) manifold correspond to the open and closed string sectors. As an application, we show how to use such lifts to explicitly study open string moduli. Finally, we use our analysis to construct examples of \(G_2\) manifolds with different inequivalent TCS realizations.GLSMs for exotic Grassmannians.https://www.zbmath.org/1456.814342021-04-16T16:22:00+00:00"Gu, Wei"https://www.zbmath.org/authors/?q=ai:gu.wei"Sharpe, Eric"https://www.zbmath.org/authors/?q=ai:sharpe.eric-r"Zou, Hao"https://www.zbmath.org/authors/?q=ai:zou.haoSummary: In this paper we explore nonabelian gauged linear sigma models (GLSMs) for symplectic and orthogonal Grassmannians and flag manifolds, checking e.g. global symmetries, Witten indices, and Calabi-Yau conditions, following up a proposal in the math community. For symplectic Grassmannians, we check that Coulomb branch vacua of the GLSM are consistent with ordinary and equivariant quantum cohomology of the space.