arXiv subjects

Yusuke Kimura

Publications and source records attributed to Yusuke Kimura.

At least 55 records · Page 3Linked to original sources

arXiv2018

K3 surfaces without section as double covers of Halphen surfaces, and F-theory compactifications

We construct several examples of genus-one fibered K3 surfaces without a global section with type $I_{n}$ fibers, by considering double covers of a special class of rational elliptic surfaces lacking a global section, known as Halphen surfaces of index 2. The resulting K3 surfaces have bisection geometries. F-theory compactifications on these K3 genus-one fibrations without a section times a K3 yield models that have $SU(n)$ gauge symmetries with a discrete $\mathbb{Z}_2$ symmetry.

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arXiv2017

Structure of stable degeneration of K3 surfaces into pairs of rational elliptic surfaces

F-theory/heterotic duality is formulated in the stable degeneration limit of a K3 fibration on the F-theory side. In this note, we analyze the structure of the stable degeneration limit. We discuss whether stable degeneration exists for pairs of rational elliptic surfaces. We demonstrate that, when two rational elliptic surfaces have an identical complex structure, stable degeneration always exists. We provide an equation that systematically describes the stable degeneration of a K3 surface into a pair of isomorphic rational elliptic surfaces. When two rational elliptic surfaces have different complex structures, whether their sum glued along a smooth fiber admits deformation to a K3 surface can be determined by studying the structure of the K3 lattice. We investigate the lattice theoretic condition to determine whether a deformation to a K3 surface exists for pairs of extremal rational elliptic surfaces. In addition, we discuss the configurations of singular fibers under stable degeneration. The sum of two isomorphic rational elliptic surfaces glued together admits a deformation to a K3 surface, the singular fibers of which are twice that of the rational elliptic surface. For special situations, singular fibers of the resulting K3 surface collide and they are enhanced to a fiber of another type. Some K3 surfaces become attractive in these situations. We determine the complex structures and the Weierstrass forms of these attractive K3 surfaces. We also deduce the gauge groups in F-theory compactifications on these attractive K3 surfaces times a K3. $E_6$, $E_7$, $E_8$, $SU(5)$, and $SO(10)$ gauge groups arise in these compactifications.

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arXiv2017

Gauge symmetries and matter fields in F-theory models without section- compactifications on double cover and Fermat quartic K3 constructions times K3

We investigate gauge theories and matter fields in F-theory compactifications on genus-one fibered Calabi-Yau 4-folds without a global section. In this study, genus-one fibered Calabi-Yau 4-folds are built as direct products of a genus-one fibered K3 surface that lacks a section times a K3 surface. We consider i) double covers of $\mathbb{P}^1\times\mathbb{P}^1$ ramified along a bidegree (4,4) curve, and ii) complete intersections of two bidegree (1,2) hypersurfaces in $\mathbb{P}^1\times\mathbb{P}^3$ to construct genus-one fibered K3 surfaces without a section. $E_7$ gauge group arises in some F-theory compactifications on double covers times K3. We show that the tadpole can be cancelled for an F-theory compactification on complete intersection K3 times K3, when complete intersection K3 is isomorphic to the Fermat quartic, and the complex structure of the other K3 surface in the direct product is appropriately chosen.

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arXiv2017

Discrete Gauge Groups in F-theory Models on Genus-One Fibered Calabi-Yau 4-folds without Section

We determine the discrete gauge symmetries that arise in F-theory compactifications on examples of genus-one fibered Calabi-Yau 4-folds without a section. We construct genus-one fibered Calabi-Yau 4-folds using Fano manifolds, cyclic 3-fold covers of Fano 4-folds, and Segre embeddings of products of projective spaces. Discrete $\mathbb{Z}_5$, $\mathbb{Z}_4$, $\mathbb{Z}_3$ and $\mathbb{Z}_2$ symmetries arise in these constructions. We introduce a general method to obtain multisections for several constructions of genus-one fibered Calabi-Yau manifolds. The pullbacks of hyperplane classes under certain projections represent multisections to these genus-one fibrations. We determine the degrees of these multisections by computing the intersection numbers with fiber classes. As a result, we deduce the discrete gauge symmetries that arise in F-theory compactifications. This method applies to various Calabi-Yau genus-one fibrations.

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arXiv2017

Noncommutative Frobenius algebras and open-closed duality

Some equivalence classes in symmetric group lead to an interesting class of noncommutive and associative algebras. From these algebras we construct noncommutative Frobenius algebras. Based on the correspondence between Frobenius algebras and two-dimensional topological field theories, the noncommutative Frobenius algebras can be interpreted as topological open string theories. It is observed that the centre of the algebras are related to closed string theories via open-closed duality. Area-dependent two-dimensional field theories are also studied.

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arXiv2016

Flavour singlets in gauge theory as Permutations

Gauge-invariant operators can be specified by equivalence classes of permutations. We develop this idea concretely for the singlets of the flavour group $SO(N_f)$ in $U(N_c)$ gauge theory by using Gelfand pairs and Schur-Weyl duality. The singlet operators, when specialised at $N_f =6$, belong to the scalar sector of ${\cal N}=4$ SYM. A simple formula is given for the two-point functions in the free field limit of $g_{YM}^2 =0$. The free two-point functions are shown to be equal to the partition function on a 2-complex with boundaries and a defect, in a topological field theory of permutations. The permutation equivalence classes are Fourier transformed to a representation basis which is orthogonal for the two-point functions at finite $N_c , N_f$. Counting formulae for the gauge-invariant operators are described. The one-loop mixing matrix is derived as a linear operator on the permutation equivalence classes.

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arXiv2016

Gauge Groups and Matter Fields on Some Models of F-theory without Section

We investigate F-theory on an elliptic Calabi-Yau 4-fold without a section to the fibration. To construct an elliptic Calabi-Yau 4-fold without a section, we introduce families of elliptic K3 surfaces which do not admit a section. A product K3 $\times$ K3, with one of the K3's chosen from these families of elliptic K3 surfaces without a section, realises an elliptic Calabi-Yau 4-fold without a section. We then compactify F-theory on such K3 $\times$ K3's. We determine the gauge groups and matter fields which arise on 7-branes for these models of F-theory compactifications without a section. Since each K3 $\times$ K3 constructed does not have a section, gauge groups arising on 7-branes for F-theory models on constructed K3 $\times$ K3's do not have $U(1)$-part. Interestingly, exceptional gauge group $E_6$ appears for some cases.

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arXiv2015

Negative anomalous dimensions in N=4 SYM

We elucidate aspects of the one-loop anomalous dimension of $so(6)$-singlet multi-trace operators in $\mathcal{N}=4\ SU(N_c)$ SYM at finite $N_c$. First, we study how $1/N_c$ corrections lift the large $N_c$ degeneracy of the spectrum, which we call the operator submixing problem. We observe that all large $N_c$ zero modes acquire non-positive anomalous dimension starting at order $1/N_c^2$, and they mix only among the operators with the same number of traces at leading order. Second, we study the lowest one-loop dimension of operators of length equal to $2N_c$. The dimension of such operators becomes more negative as $N_c$ increases, which will eventually diverge in a double scaling limit. Third, we examine the structure of level-crossing at finite $N_c$ in view of unitarity. Finally we find out a correspondence between the large $N_c$ zero modes and completely symmetric polynomials of Mandelstam variables.

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arXiv2014

Multi-matrix models and Noncommutative Frobenius algebras obtained from symmetric groups and Brauer algebras

It has been understood that correlation functions of multi-trace operators in ${\cal N}=4$ SYM can be neatly computed using the group algebra of symmetric groups or walled Brauer algebras. On the other hand such algebras have been known to construct 2D topological field theories (TFTs). After reviewing the construction of 2D TFTs based on symmetric groups, we construct 2D TFTs based on walled Brauer algebras. In the construction, the introduction of a dual basis manifests a similarity between the two theories. We next construct a class of 2D field theories whose physical operators have the same symmetry as multi-trace operators constructed from some matrices. Such field theories correspond to non-commutative Frobenius algebras. A matrix structure arises as a consequence of the noncommutativity. Correlation functions of the Gaussian complex multi-matrix models can be translated into correlation functions of the two-dimensional field theories.

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arXiv2014

The Noether-Lefschetz Problem and Gauge-Group-Resolved Landscapes: F-Theory on K3 x K3 as a Test Case

Four-form flux in F-theory compactifications not only stabilizes moduli, but gives rise to ensembles of string vacua, providing a scientific basis for a stringy notion of naturalness. Of particular interest in this context is the ability to keep track of algebraic information (such as the gauge group) associated with individual vacua while dealing with statistics. In the present work, we aim to clarify conceptual issues and sharpen methods for this purpose, using compactification on ${\rm K3} \times {\rm K3}$ as a test case. Our first approach exploits the connection between the stabilization of complex structure moduli and the Noether-Lefschetz problem. Compactification data for F-theory, however, involve not only a four-fold (with a given complex structure) $Y_4$ and a flux on it, but also an elliptic fibration morphism $Y_4 \longrightarrow B_3$, which makes this problem complicated. The heterotic-F-theory duality indicates that elliptic fibration morphisms should be identified modulo isomorphism. Based on this principle, we explain how to count F-theory vacua on ${\rm K3} \times {\rm K3}$ while keeping the gauge group information. Mathematical results reviewed/developed in our companion paper are exploited heavily. With applications to more general four-folds in mind, we also clarify how to use Ashok-Denef-Douglas' theory of the distribution of flux vacua in order to deal with statistics of sub-ensembles tagged by a given set of algebraic/topological information. As a side remark, we extend the heterotic/F-theory duality dictionary on flux quanta and elaborate on its connection to the semistable degeneration of a K3 surface.

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arXiv2013

On the Classification of Elliptic Fibrations modulo Isomorphism on K3 Surfaces with large Picard Number

Motivated by a problem originating in string theory, we study elliptic fibrations on K3 surfaces with large Picard number modulo isomorphism. We give methods to determine upper bounds for the number of inequivalent K3 surfaces sharing the same frame lattice. For any given Neron--Severi lattice $S_X$, such a bound on the `multiplicity' can be derived by investigating the quotient of the isometry group of $S_X$ by the automorphism group. The resulting bounds are strongest for large Picard numbers and multiplicities of unity do indeed occur for a number of K3 surfaces with Picard number 20. Under a few extra conditions, a more refined analysis is also possible by explicitly studying the embedding of $S_X$ into the even unimodular lattice ${\rm II}_{1,25}$ and exploiting the detailed structure of the isometry groups of $S_X$ and ${\rm II}_{1,25}$. We illustrate these methods in examples and derive bounds for the number of elliptic fibrations on Kummer surfaces of Picard numbers 17 and 20. As an intermediate step, we also discuss coarser classification schemes and review known results.

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arXiv2013

Non-planar operator mixing by Brauer representations

We study the action of the dilatation operator on the basis of local operators constructed from the elements of the walled Brauer algebra, with non-planar corrections fully taken into account. We will see that the operator mixing can be neatly expressed in terms of the irreducible representations of the algebra. In particular we focus on a role of the integer that determines the number of boxes in the representations.

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arXiv2012

Correlation functions and representation bases in free N=4 Super Yang-Mills

We study exact correlation functions of N=4 SYM at zero coupling. It has been known that it is convenient to label local gauge invariant operators by irreducible representations of symmetric groups/Brauer algebras. We first review the construction of representation bases from the viewpoint of the enhanced symmetry structure of the free theory. We present a basis of multi-matrix models using elements of Brauer algebras, generalising our previous construction for two matrices. We will compute multi-point functions of the basis with the exact N-dependence. In particular we study three-point functions of a class of BPS operators, and we find that they are given by a branching rule of the Brauer algebra. The three-point functions take a factorised form if representations on the operators satisfy a relation.

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arXiv2011

Young diagrams, Brauer algebras, and bubbling geometries

We study the 1/4 BPS geometries corresponding to the 1/4 BPS operators of the dual gauge theory side, in ${\cal N}=4$ SYM. By analyzing asymptotic structure and flux integration of the geometries, we present a mapping between droplet configurations arising from the geometries and Young diagrams of the Brauer algebra. In particular, the integer $k$ classifying the operators in the Brauer basis is mapped to the mixing between the two angular directions.

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arXiv2010

Free particles from Brauer algebras in complex matrix models

The gauge invariant degrees of freedom of matrix models based on an N x N complex matrix, with U(N) gauge symmetry, contain hidden free particle structures. These are exhibited using triangular matrix variables via the Schur decomposition. The Brauer algebra basis for complex matrix models developed earlier is useful in projecting to a sector which matches the state counting of N free fermions on a circle. The Brauer algebra projection is characterized by the vanishing of a scale invariant laplacian constructed from the complex matrix. The special case of N=2 is studied in detail: the ring of gauge invariant functions as well as a ring of scale and gauge invariant differential operators are characterized completely. The orthonormal basis of wavefunctions in this special case is completely characterized by a set of five commuting Hamiltonians, which display free particle structures. Applications to the reduced matrix quantum mechanics coming from radial quantization in N=4 SYM are described. We propose that the string dual of the complex matrix harmonic oscillator quantum mechanics has an interpretation in terms of strings and branes in 2+1 dimensions.

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arXiv2010

Quarter BPS classified by Brauer algebra

We analyse the one-loop dilatation operator with the help of the Brauer algebra. We find some BPS operators in N=4 SYM, which are labelled by irreducible representations of the Brauer algebra. Some of them are quarter BPS operators. The result includes full non-planar corrections. Our construction and proof are based on simple algebraic arguments and are carried out for any number of fields.

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arXiv2009

Non-holomorphic multi-matrix gauge invariant operators based on Brauer algebra

We present an orthogonal basis of gauge invariant operators constructed from some complex matrices for the free matrix field, where operators are expressed with the help of Brauer algebra. This is a generalisation of our previous work for a signle complex matrix. We also discuss the matrix quantum mechanics relevant to N=4 SYM on S^{3} times R. A commuting set of conserved operators whose eigenstates are given by the orthogonal basis is shown by using enhanced symmetries at zero coupling.

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arXiv2008

Enhanced symmetries of gauge theory and resolving the spectrum of local operators

Enhanced global non-abelian symmetries at zero coupling in Yang Mills theory play an important role in diagonalising the two-point functions of multi-matrix operators. Generalised Casimirs constructed from the iterated commutator action of these enhanced symmetries resolve all the multiplicity labels of the bases of matrix operators which diagonalise the two-point function. For the case of U (N) gauge theory with a single complex matrix in the adjoint of the gauge group we have a U(N)^{\times 4} global symmetry of the scaling operator at zero coupling. Different choices of commuting sets of Casimirs, for the case of a complex matrix, lead to the restricted Schur basis previously studied in connection with string excitations of giant gravitons and the Brauer basis studied in connection with brane-anti-brane systems. More generally these remarks can be extended to the diagonalisation for any global symmetry group G. Schur-Weyl duality plays a central role in connecting the enhanced symmetries and the diagonal bases.

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