 Award ID(s):
 1911298
 Publication Date:
 NSFPAR ID:
 10233771
 Journal Name:
 Journal of High Energy Physics
 Volume:
 2021
 Issue:
 2
 ISSN:
 10298479
 Sponsoring Org:
 National Science Foundation
More Like this

Abstract We consider holomorphic twists of arbitrary supersymmetric theories in four dimensions. Working in the BV formalism, we rederive classical results characterizing the holomorphic twist of chiral and vector supermultiplets, computing the twist explicitly as a family over the space of nilpotent supercharges in minimal supersymmetry. The BV formalism allows one to work with or without auxiliary fields, according to preference; for chiral superfields, we show that the result of the twist is an identical BV theory, the holomorphic $$\beta \gamma $$ β γ system with superpotential, independent of whether or not auxiliary fields are included. We compute the character of local operators in this holomorphic theory, demonstrating agreement of the free local operators with the usual index of free fields. The local operators with superpotential are computed via a spectral sequence and are shown to agree with functions on a formal mapping space into the derived critical locus of the superpotential. We consider the holomorphic theory on various geometries, including Hopf manifolds and products of arbitrary pairs of Riemann surfaces, and offer some general remarks on dimensional reductions of holomorphic theories along the $$(n1)$$ ( n  1 ) sphere to topological quantum mechanics. We also study an infinitedimensionalmore »

A bstract We initiate the geometric engineering of 2d $$ \mathcal{N} $$ N = (0 , 1) gauge theories on D1branes probing singularities. To do so, we introduce a new class of backgrounds obtained as quotients of CalabiYau 4folds by a combination of an antiholomorphic involution leading to a Spin(7) cone and worldsheet parity. We refer to such constructions as Spin(7) orientifolds . Spin(7) orientifolds explicitly realize the perspective on 2d $$ \mathcal{N} $$ N = (0 , 1) theories as real slices of $$ \mathcal{N} $$ N = (0 , 2) ones. Remarkably, this projection is geometrically realized as Joyce’s construction of Spin(7) manifolds via quotients of CalabiYau 4folds by antiholomorphic involutions. We illustrate this construction in numerous examples with both orbifold and nonorbifold parent singularities, discuss the role of the choice of vector structure in the orientifold quotient, and study partial resolutions.

A bstract We introduce a systematic approach to constructing $$ \mathcal{N} $$ N = 1 Lagrangians for a class of interacting $$ \mathcal{N} $$ N = 2 SCFTs. We analyse in detail the simplest case of the construction, arising from placing branes at an orientifolded ℂ 2 / ℤ 2 singularity. In this way we obtain Lagrangian descriptions for all the R 2 ,k theories. The rank one theories in this class are the E 6 MinahanNemeschansky theory and the C 2 × U(1) ArgyresWittig theory. The Lagrangians that arise from our brane construction manifestly exhibit either the entire expected flavour symmetry group of the SCFT (for even k ) or a fullrank subgroup thereof (for odd k ), so we can compute the full superconformal index of the $$ \mathcal{N} $$ N = 2 SCFTs, and also systematically identify the Higgsings associated to partial closing of punctures.

Abstract We continue the program of proving circuit lower bounds via circuit satisfiability algorithms. So far, this program has yielded several concrete results, proving that functions in
and other complexity classes do not have small circuits (in the worst case and/or on average) from various circuit classes$\mathsf {Quasi}\text {}\mathsf {NP} = \mathsf {NTIME}[n^{(\log n)^{O(1)}}]$ $\mathrm{Quasi}\mathrm{NP}=\mathrm{NTIME}\left[{n}^{{\left(\mathrm{log}n\right)}^{O\left(1\right)}}\right]$ , by showing that$\mathcal { C}$ $C$ admits nontrivial satisfiability and/or$\mathcal { C}$ $C$# SAT algorithms which beat exhaustive search by a minor amount. In this paper, we present a new strong lower bound consequence of having a nontrivial# SAT algorithm for a circuit class . Say that a symmetric Boolean function${\mathcal C}$ $C$f (x _{1},…,x _{n}) issparse if it outputs 1 onO (1) values of . We show that for every sparse${\sum }_{i} x_{i}$ ${\sum}_{i}{x}_{i}$f , and for all “typical” , faster$\mathcal { C}$ $C$# SAT algorithms for circuits imply lower bounds against the circuit class$\mathcal { C}$ $C$ , which may be$f \circ \mathcal { C}$ $f\circ C$stronger than itself. In particular:$\mathcal { C}$ $C$# SAT algorithms forn ^{k}size circuits running in 2^{n}/$\mathcal { C}$ $C$n ^{k}time (for allk ) implyN E X P does not have circuits of polynomial size.$(f \circ \mathcal { C})$ $(f\circ C)$# SAT algorithms for size$2^{n^{{\varepsilon }}}$ ${2}^{{n}^{\epsilon}}$ circuits running in$\mathcal { C}$ $C$ time (for some$2^{nn^{{\varepsilon }}}$ ${2}^{n{n}^{\epsilon}}$ε > 0) implyQ u a s i N P does not have circuits of polynomial size.$(f \circ \mathcal { C})$ $(f\circ C)$Applying
# SAT algorithms from the literature, one immediate corollary of our results is thatQ u a s i N P does not haveE M A J ∘A C C ^{0}∘T H R circuits of polynomialmore » 
In this paper, we make partial progress on a function field version of the dynamical uniform boundedness conjecture for certain onedimensional families ${\mathcal{F}}$ of polynomial maps, such as the family $f_{c}(x)=x^{m}+c$ , where $m\geq 2$ . We do this by making use of the dynatomic modular curves $Y_{1}(n)$ (respectively $Y_{0}(n)$ ) which parametrize maps $f$ in ${\mathcal{F}}$ together with a point (respectively orbit) of period $n$ for $f$ . The key point in our strategy is to study the set of primes $p$ for which the reduction of $Y_{1}(n)$ modulo $p$ fails to be smooth or irreducible. Morton gave an algorithm to construct, for each $n$ , a discriminant $D_{n}$ whose list of prime factors contains all the primes of bad reduction for $Y_{1}(n)$ . In this paper, we refine and strengthen Morton’s results. Specifically, we exhibit two criteria on a prime $p$ dividing $D_{n}$ : one guarantees that $p$ is in fact a prime of bad reduction for $Y_{1}(n)$ , yet this same criterion implies that $Y_{0}(n)$ is geometrically irreducible. The other guarantees that the reduction of $Y_{1}(n)$ modulo $p$ is actually smooth. As an application of the second criterion, we extend results of Morton, Flynn, Poonen, Schaefer, andmore »