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Nicol\\`o Piazzalunga
Publication date
2018
fields
Physics
and research's language is
English
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We study the rank N magnificent four theory, which is the supersymmetric localization of U(N) super-Yang-Mills theory with matter (a super-group U(N|N) gauge theory) on a Calabi-Yau fourfold. Our theory contains the higher rank Donaldson-Thomas theory of threefolds. We conjecture an explicit formula for the partition function Z, and report on the performed checks. The partition function Z has a free field representation. Surprisingly, it depends on the Coulomb and mass parameters in a simple way. We also clarify the definition of the instanton measure.
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We compute characteristic numbers of elliptically fibered fourfolds with multisections or non-trivial Mordell-Weil groups. We first consider the models of type E$_{9-d}$ with $d=1,2,3,4$ whose generic fibers are normal elliptic curves of degree $d$. We then analyze the characteristic numbers of the $Q_7$-model, which provides a smooth model for elliptic fibrations of rank one and generalizes the E$_5$, E$_6$, and E$_7$-models. Finally, we examine the characteristic numbers of $G$-models with $G=text{SO}(n)$ with $n=3,4,5,6$ and $G=text{PSU}(3)$ whose Mordell-Weil groups are respectively $mathbb{Z}/2mathbb{Z}$ and $mathbb{Z}/3 mathbb{Z}$. In each case, we compute the Chern and Pontryagin numbers, the Euler characteristic, the holomorphic genera, the Todd-genus, the L-genus, the A-genus, and the eight-form curvature invariant from M-theory.
This work addresses nonperturbative effects in both matrix models and topological strings, and their relation with the large-order behavior of the 1/N expansion. We study instanton configurations in generic one-cut matrix models, obtaining explicit results for the one-instanton amplitude at both one and two loops. The holographic description of topological strings in terms of matrix models implies that our nonperturbative results also apply to topological strings on toric Calabi-Yau manifolds. This yields very precise predictions for the large-order behavior of the perturbative genus expansion, both in conventional matrix models and in topological string theory. We test these predictions in detail in various examples, including the quartic matrix model, topological strings on the local curve, and Hurwitz theory. In all these cases we provide extensive numerical checks which heavily support our nonperturbative analytical results. Moreover, since all these models have a critical point describing two-dimensional gravity, we also obtain in this way the large-order asymptotics of the relevant solution to the Painleve I equation, including corrections in inverse genus. From a mathematical point of view, our results predict the large-genus asymptotics of simple Hurwitz numbers and of local Gromov-Witten invariants.
Whenever available, refined BPS indices provide considerably more information on the spectrum of BPS states than their unrefined version. Extending earlier work on the modularity of generalized Donaldson-Thomas invariants counting D4-D2-D0 brane bound states in type IIA strings on a Calabi-Yau threefold $mathfrak{Y}$, we construct the modular completion of generating functions of refined BPS indices supported on a divisor class. Although for compact $mathfrak{Y}$ the refined indices are not protected, switching on the refinement considerably simplifies the construction of the modular completion. Furthermore, it leads to a non-commutative analogue of the TBA equations, which suggests a quantization of the moduli space consistent with S-duality. In contrast, for a local CY threefold given by the total space of the canonical bundle over a complex surface $S$, refined BPS indices are well-defined, and equal to Vafa-Witten invariants of $S$. Our construction provides a modular completion of the generating function of these refined invariants for arbitrary rank. In cases where all reducible components of the divisor class are collinear (which occurs e.g. when $b_2(mathfrak{Y})=1$, or in the local case), we show that the holomorphic anomaly equation satisfied by the completed generating function truncates at quadratic order. In the local case, it agrees with an earlier proposal by Minahan et al for unrefined invariants, and extends it to the refined level using the afore-mentioned non-commutative structure. Finally, we show that these general predictions reproduce known results for $U(2)$ and $U(3)$ Vafa-Witten theory on $mathbb{P}^2$, and make them explicit for $U(4)$.
We study crepant resolutions of Weierstrass models of $text{SU}(2)!times!text{SU}(3)$-models, whose gauge group describes the non-abelian sector of the Standard Model. The $text{SU}(2)!times!text{SU}(3)$-models are elliptic fibrations characterized by the collision of two Kodaira fibers with dual graphs that are affine Dynkin diagrams of type $widetilde{text{A}}_1$ and $widetilde{text{A}}_2$. Once we eliminate those collisions that do not have crepant resolutions, we are left with six distinct collisions that are related to each other by deformations. Each of these six collisions has eight distinct crepant resolutions whose flop diagram is a hexagon with two legs attached to two adjacent nodes. Hence, we consider 48 distinct resolutions that are connected to each other by deformations and flops. We determine topological invariants---such as Euler characteristics, Hodge numbers, and triple intersections of fibral divisors---for each of the crepant resolutions. We analyze the physics of these fibrations when used as compactifications of M-theory and F-theory on Calabi--Yau threefolds yielding 5d ${mathcal N}=1$ and 6d ${mathcal N}=(1,0)$ supergravity theories respectively. We study the 5d prepotential in the Coulomb branch of the theory and check that the six-dimensional theory is anomaly-free and compatible with a 6d uplift from a 5d theory.
Alday, Gaiotto, and Tachikawa conjectured relations between certain 4d N=2 supersymmetric field theories and 2d Liouville conformal field theory. We study generalizations of these relations to 4d theories with surface operators. For one type of surface operators the corresponding 2d theory is the WZW model, and for another type - the Liouville theory with insertions of extra degenerate fields. We show that these two 4d theories with surface operators exhibit an IR duality, which reflects the known relation (the so-called separation of variables) between the conformal blocks of the WZW model and the Liouville theory. Furthermore, we trace this IR duality to a brane creation construction relating systems of M5 and M2 branes in M-theory. Finally, we show that this duality may be expressed as an explicit relation between the generating functions for the changes of variables between natural sets of Darboux coordinates on the Hitchin moduli space.
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