# Volker Braun - UniversityT. Austin

## Contact Details

NameVolker Braun |
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AffiliationUniversityT. Austin |
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CityAustin |
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CountryUnited States |
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## Pubs By Year |
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## External Links |
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## Pub CategoriesHigh Energy Physics - Theory (50) Mathematics - Algebraic Geometry (9) High Energy Physics - Phenomenology (4) Mathematics - K-Theory and Homology (2) Mathematics - Algebraic Topology (1) |

## Publications Authored By Volker Braun

We provide the first explicit example of Type IIB string theory compactification on a globally defined Calabi-Yau threefold with torsion which results in a four-dimensional effective theory with a non-Abelian discrete gauge symmetry. Our example is based on a particular Calabi-Yau manifold, the quotient of a product of three elliptic curves by a fixed point free action of Z_2 x Z_2. Its cohomology contains torsion classes in various degrees. Read More

The special geometries of two recently discovered Calabi-Yau threefolds with $h^{11}=1$ are analyzed in detail. These correspond to the 'minimal three-generation' manifolds with $h^{21}=4$ and the `24-cell' threefolds with $h^{21}=1$. It turns out that the one-dimensional complex structure moduli spaces for these manifolds are both very similar and surprisingly complicated. Read More

Global F-theory compactifications whose fibers are realized as complete intersections form a richer set of models than just hypersurfaces. The detailed study of the physics associated with such geometries depends crucially on being able to put the elliptic fiber into Weierstrass form. While such a transformation is always guaranteed to exist, its explicit form is only known in a few special cases. Read More

We argue that M-theory compactified on an arbitrary genus-one fibration, that is, an elliptic fibration which need not have a section, always has an F-theory limit when the area of the genus-one fiber approaches zero. Such genus-one fibrations can be easily constructed as toric hypersurfaces, and various $SU(5)\times U(1)^n$ and $E_6$ models are presented as examples. To each genus-one fibration one can associate a $\tau$-function on the base as well as an $SL(2,\mathbb{Z})$ representation which together define the IIB axio-dilaton and 7-brane content of the theory. Read More

An algorithm to systematically construct all Calabi-Yau elliptic fibrations realized as hypersurfaces in a toric ambient space for a given base and gauge group is described. This general method is applied to the particular question of constructing SU(5) GUTs with multiple U(1) gauge factors. The basic data consists of a top over each toric divisor in the base together with compactification data giving the embedding into a reflexive polytope. Read More

We construct global F-theory GUTs with SU(5) x U(1) gauge group defined by specifying a fully resolved Calabi-Yau fourfold and consistent four-form G-flux. Its specific U(1) charged matter spectrum allows the desired Yukawa couplings, but forbids dangerous proton decay operators. The model we find: (1) does not follow from an underlying higgsed E8 gauge group (2) leaves the class of theories that can be analyzed with current split-spectral cover techniques. Read More

Within the context of the weakly coupled E8 x E8 heterotic string, we study the hidden sector of heterotic standard model compactifications to four-dimensions. Specifically, we present a class of hidden sector vector bundles - composed of the direct sum of line bundles only - that, together with an effective bulk five-brane, renders the heterotic standard model entirely N=1 supersymmetric. Two explicit hidden sectors are constructed and analyzed in this context; one with the gauge group E7 x U(1) arising from a single line bundle and a second with an SO(12) x U(1) x U(1) gauge group constructed from the direct sum of two line bundles. Read More

The problem of counting points is revisited from the perspective of reflexive 4-dimensional polytopes. As an application, the Hilbert series of the 473,800,776 reflexive polytopes (equivalently, their Calabi-Yau hypersurfaces) are computed. Read More

We describe how local toric singularities, including the Toric Lego construction, can be embedded in compact Calabi-Yau manifolds. We study in detail the addition of D-branes, including non-compact flavor branes as typically used in semi-realistic model building. The global geometry provides constraints on allowable local models. Read More

We construct supersymmetric compactifications of E_8 \times E_8 heterotic string theory which realise exactly the massless spectrum of the Minimal Supersymmetric Standard Model (MSSM) at low energies. The starting point is the standard embedding on a Calabi-Yau threefold which has Hodge numbers (h^11,h^21) = (1,4) and fundamental group Z_12, which gives an E_6 grand unified theory with three net chiral generations. The gauge symmetry is then broken to that of the standard model by a combination of discrete Wilson lines and continuous deformation of the gauge bundle. Read More

The 102581 flat toric elliptic fibrations over P^2 are identified among the Calabi-Yau hypersurfaces that arise from the 473800776 reflexive 4-dimensional polytopes. In order to analyze their elliptic fibration structure, we describe the precise relation between the lattice polytope and the elliptic fibration. The fiber-divisor-graph is introduced as a way to visualize the embedding of the Kodaira fibers in the ambient toric fiber. Read More

We further develop the numerical algorithm for computing the gauge connection of slope-stable holomorphic vector bundles on Calabi-Yau manifolds. In particular, recent work on the generalized Donaldson algorithm is extended to bundles with Kahler cone substructure on manifolds with h^{1,1}>1. Since the computation depends only on a one-dimensional ray in the Kahler moduli space, it can probe slope-stability regardless of the size of h^{1,1}. Read More

Calabi-Yau threefolds with h^11(X)=h^21(X)=1 are constructed as free quotients of a hypersurface in the ambient toric variety defined by the 24-cell. Their fundamental groups are SL(2,3), a semidirect product of Z_3 and Z_8, and Z_3 x Q_8. Read More

F-theory models are constructed where the 7-brane has a non-trivial fundamental group. The base manifolds used are a toric Fano variety and a smooth toric threefold coming from a reflexive polyhedron. The discriminant locus of the elliptically fibered Calabi-Yau fourfold can be chosen such that one irreducible component it is not simply connected (namely, an Enriques surface) and supports a non-Abelian gauge theory. Read More

A numerical algorithm is presented for explicitly computing the gauge connection on slope-stable holomorphic vector bundles on Calabi-Yau manifolds. To illustrate this algorithm, we calculate the connections on stable monad bundles defined on the K3 twofold and Quintic threefold. An error measure is introduced to determine how closely our algorithmic connection approximates a solution to the Hermitian Yang-Mills equations. Read More

In order to find novel examples of non-simply connected Calabi-Yau threefolds, free quotients of complete intersections in products of projective spaces are classified by means of a computer search. More precisely, all automorphisms of the product of projective spaces that descend to a free action on the Calabi-Yau manifold are identified. Read More

We describe a simple and robust mechanism that stabilizes all Kahler moduli in Type IIB orientifold compactifications. This is shown to be possible with just one non-perturbative contribution to the superpotential coming from either a D3-instanton or D7-branes wrapped on an ample divisor. This moduli-stabilization mechanism is similar to and motivated by the one used in the fluxless G_2 compactifications of M-theory. Read More

We present a complete intersection Calabi-Yau manifold Y that has Euler number -72 and which admits free actions by two groups of automorphisms of order 12. These are the cyclic group Z_12 and the non-Abelian dicyclic group Dic_3. The quotient manifolds have chi=-6 and Hodge numbers (h^11,h^21)=(1,4). Read More

A three-generation SU(5) GUT, that is 3x(10+5bar) and a single 5-5bar pair, is constructed by compactification of the E_8 heterotic string. The base manifold is the Z_5 x Z_5-quotient of the quintic, and the vector bundle is the quotient of a positive monad. The group action on the monad and its bundle-valued cohomology is discussed in detail, including topological restrictions on the existence of equivariant structures. Read More

We systematically analyse globally consistent SU(5) GUT models on intersecting D7-branes in genuine Calabi-Yau orientifolds with O3- and O7-planes. Beyond the well-known tadpole and K-theory cancellation conditions there exist a number of additional subtle but quite restrictive constraints. For the realisation of SU(5) GUTs with gauge symmetry breaking via U(1)_Y flux we present two classes of suitable Calabi-Yau manifolds defined via del Pezzo transitions of the elliptically fibred hypersurface P_{1,1,1,6,9}[18] and of the Quintic P_{1,1,1,1,1}[5], respectively. Read More

We present a vacuum of heterotic M-theory whose observable sector has the MSSM spectrum with the addition of one extra pair of Higgs-Higgs conjugate superfields. The quarks/leptons have a realistic mass hierarchy with a naturally light first family. The double elliptic structure of the Calabi-Yau compactification threefold leads to two ``stringy'' selection rules. Read More

A numerical algorithm for explicitly computing the spectrum of the Laplace-Beltrami operator on Calabi-Yau threefolds is presented. The requisite Ricci-flat metrics are calculated using a method introduced in previous papers. To illustrate our algorithm, the eigenvalues and eigenfunctions of the Laplacian are computed numerically on two different quintic hypersurfaces, some Z_5 x Z_5 quotients of quintics, and the Calabi-Yau threefold with Z_3 x Z_3 fundamental group of the heterotic standard model. Read More

We study aspects of worldsheet instantons relevant to a heterotic standard model. The non-simply connected Calabi-Yau threefold used admits Z_3 x Z_3 Wilson lines, and a more detailed investigation shows that the homology classes of curves are H_2(X,Z)=Z^3+Z_3+Z_3. We compute the genus-0 prepotential, this is the first explicit calculation of the Gromov-Witten invariants of homology classes with torsion (finite subgroups). Read More

We extend previous computations of Calabi-Yau metrics on projective hypersurfaces to free quotients, complete intersections, and free quotients of complete intersections. In particular, we construct these metrics on generic quintics, four-generation quotients of the quintic, Schoen Calabi-Yau complete intersections and the quotient of a Schoen manifold with Z_3 x Z_3 fundamental group that was previously used to construct a heterotic standard model. Various numerical investigations into the dependence of Donaldson's algorithm on the integration scheme, as well as on the Kahler and complex structure moduli, are also performed. Read More

We apply mirror symmetry to the problem of counting holomorphic rational curves in a Calabi-Yau threefold X with Z_3 x Z_3 Wilson lines. As we found in Part A [hep-th/0703182], the integral homology group H_2(X,Z)=Z^3 + Z_3 + Z_3 contains torsion curves. Using the B-model on the mirror of X as well as its covering spaces, we compute the instanton numbers. Read More

As a first step towards studying vector bundle moduli in realistic heterotic compactifications, we identify all holomorphic rational curves in a Calabi-Yau threefold X with Z_3 x Z_3 Wilson lines. Computing the homology, we find that H_2(X,Z)=Z^3+Z_3+Z_3. The torsion curves complicate our analysis, and we develop techniques to distinguish the torsion part of curve classes and to deal with the non-toric threefold X. Read More

As a first step towards computing instanton-generated superpotentials in heterotic standard model vacua, we determine the Gromov-Witten invariants for a Calabi-Yau threefold with fundamental group pi_1(X)=Z_3 x Z_3. We find that the curves fall into homology classes in H_2(X,Z)=Z^3+(Z_3+Z_3). The unexpected appearance of the finite torsion subgroup in the homology group complicates our analysis. Read More

We study a new mechanism to dynamically break supersymmetry in the E8xE8 heterotic string. As discussed recently in the literature, a long-lived, meta-stable non-supersymmetric vacuum can be achieved in an N=1 SQCD whose spectrum contains a sufficient number of light fundamental flavors. In this paper, we present, within the context of the hidden sector of the weakly and strongly coupled heterotic string, a slope-stable, holomorphic vector bundle on a Calabi-Yau threefold for which all matter fields are in the fundamental representation and are massive at generic points in moduli space. Read More

It is shown that four-dimensional N=1 supersymmetric QCD with massive flavors in the fundamental representation of the gauge group can be realized in the hidden sector of E8xE8 heterotic string vacua. The number of flavors can be chosen to lie in the range of validity of the free-magnetic dual, using which one can demonstrate the existence of long-lived meta-stable non-supersymmetric vacua. This is shown explicitly for the gauge group Spin(10), but the methods are applicable to Spin(Nc), SU(Nc) and Sp(Nc) for a wide range of color index Nc. Read More

It is shown that strongly coupled heterotic M-theory with anti-five-branes in the S^1/Z_2 bulk space can have meta-stable vacua which break N=1 supersymmetry and have a small, positive cosmological constant. This is demonstrated for the "minimal" heterotic standard model. This vacuum has the exact MSSM matter spectrum in the observable sector, a trivial hidden sector vector bundle and both five-branes and anti-five-branes in the bulk space. Read More

The observable sector of the "minimal heterotic standard model" has precisely the matter spectrum of the MSSM: three families of quarks and leptons, each with a right-handed neutrino, and one Higgs-Higgs conjugate pair. In this paper, it is explicitly proven that the SU(4) holomorphic vector bundle leading to the MSSM spectrum in the observable sector is slope-stable. Read More

In this paper, we present a formalism for computing the Yukawa couplings in heterotic standard models. This is accomplished by calculating the relevant triple products of cohomology groups, leading to terms proportional to Q*H*u, Q*Hbar*d, L*H*nu and L*Hbar*e in the low energy superpotential. These interactions are subject to two very restrictive selection rules arising from the geometry of the Calabi-Yau manifold. Read More

We show the existence of realistic vacua in string theory whose observable sector has exactly the matter content of the MSSM. This is achieved by compactifying the E_8 x E_8 heterotic superstring on a smooth Calabi-Yau threefold with an SU(4) gauge instanton and a Z_3 x Z_3 Wilson line. Specifically, the observable sector is N=1 supersymmetric with gauge group SU(3)_C x SU(2)_L x U(1)_Y x U(1)_{B-L}, three families of quarks and leptons, each family with a right-handed neutrino, and one Higgs-Higgs conjugate pair. Read More

We construct Gepner models in terms of coset conformal field theories and compute their twisted equivariant K-theories. These classify the D-brane charges on the associated geometric backgrounds and therefore agree with the topological K-theories. We show this agreement for various cases, in particular the Fermat quintic. Read More

In this paper, we present a formalism for computing the non-vanishing Higgs
mu-terms in a heterotic standard model. This is accomplished by calculating the
cubic product of the cohomology groups associated with the vector bundle moduli
(phi), Higgs (H) and Higgs conjugate (Hbar) superfields. This leads to terms
proportional to phi H Hbar in the low energy superpotential which, for non-zero
moduli expectation values, generate moduli dependent mu-terms of the form

In previous papers, we introduced a heterotic standard model and discussed its basic properties. The Calabi-Yau threefold has, generically, three Kahler and three complex structure moduli. The observable sector of this vacuum has the spectrum of the MSSM with one additional pair of Higgs-Higgs conjugate fields. Read More

Stable, holomorphic vector bundles are constructed on an torus fibered, non-simply connected Calabi-Yau threefold using the method of bundle extensions. Since the manifold is multiply connected, we work with equivariant bundles on the elliptically fibered covering space. The cohomology groups of the vector bundle, which yield the low energy spectrum, are computed using the Leray spectral sequence and fit the requirements of particle phenomenology. Read More

In a previous paper, we introduced a heterotic standard model and discussed its basic properties. This vacuum has the spectrum of the MSSM with one additional pair of Higgs-Higgs conjugate fields and a small number of uncharged moduli. In this paper, the requisite vector bundles are formulated; specifically, stable, holomorphic bundles with structure group SU(N) on smooth Calabi-Yau threefolds with Z_3 x Z_3 fundamental group. Read More

Within the context of the E_8 x E_8 heterotic superstring compactified on a smooth Calabi-Yau threefold with an SU(4) gauge instanton, we show the existence of simple, realistic N=1 supersymmetric vacua that are compatible with low energy particle physics. The observable sector of these vacua has gauge group SU(3)_C x SU(2)_L x U(1)_Y x U(1)_{B-L}, three families of quarks and leptons, each with an additional right-handed neutrino, two Higgs-Higgs conjugate pairs, a small number of uncharged moduli and no exotic matter. The hidden sector contains non-Abelian gauge fields and moduli. Read More

A torus fibered Calabi-Yau threefold with first homotopy group Z_3 x Z_3 is constructed as a free quotient of a fiber product of two dP_9 surfaces. Calabi-Yau threefolds of this type admit Z_3 x Z_3 Wilson lines. In conjunction with SU(4) holomorphic vector bundles, such vacua lead to anomaly free, three generation models of particle physics with a right handed neutrino and a U(1)_{B-L} gauge factor, in addition to the SU(3)_C x SU(2)_L x U(1)_Y standard model gauge group. Read More

We present an encompassing treatment of D-brane charges in supersymmetric SO(3) WZW models. There are two distinct supersymmetric CFTs at each even level: the standard bosonic SO(3) modular invariant tensored with free fermions, as well as a novel twisted model. We calculate the relevant twisted K-theories and find complete agreement with the CFT analysis of D-brane charges. Read More

I determine the twisted K-theory of all compact simply connected simple Lie groups. The computation reduces via the Freed-Hopkins-Teleman theorem to the CFT prescription, and thus explains why it gives the correct result. Finally I analyze the exceptions noted by Bouwknegt et al. Read More

I review the proposal of Berenstein-Douglas for a completely general definition of Seiberg duality. To give evidence for their conjecture I present the first example of a physical dual pair and explicitly check that it satisfies the requirements. Then I explicitly show that a pair of toric dual quivers is also dual according to their proposal. Read More

We describe the general geometrical framework of brane world constructions in orientifolds of type IIA string theory with D6-branes wrapping 3-cycles in a Calabi-Yau 3-fold, and point out their immediate phenomenological relevance. These branes generically intersect in points, and the patterns of intersections govern the chiral fermion spectra and issues of gauge and supersymmetry breaking in the low energy effective gauge theory on their world volume. In particular, we provide an example of an intersecting brane world scenario on the quintic Calabi-Yau with the gauge group and the chiral spectrum of the Standard Model and discuss its properties in some detail. Read More

We investigate orientifolds of type II string theory on K3 and Calabi-Yau 3-folds with intersecting D-branes wrapping special Lagrangian cycles. We determine quite generically the chiral massless spectrum in terms of topological invariants and discuss both orbifold examples and algebraic realizations in detail. Intriguingly, the developed techniques provide an elegant way to figure out the chiral sector of orientifold models without computing any explicit string partition function. Read More

We present a construction of superconformal field theories for manifolds with Spin(7) holonomy. Geometrically these models correspond to the realization of Spin(7) manifolds as anti-holomorphic quotients of Calabi-Yau fourfolds. Describing the fourfolds as Gepner models and requiring anomaly cancellation we determine the resulting Betti numbers of the Spin(7) superconformal field theory. Read More

We present the construction of exactly solvable superconformal field theories describing Type II string models compactified on compact G_2 manifolds. These models are defined by anti-holomorphic quotients of the form (CY*S^1)/Z_2, where we realize the Calabi-Yau as a Gepner model. In the superconformal field theory the Z_2 acts as charge conjugation implying that the representation theory of a W(2,4,6,8,10) algebra plays an important role in the construction of these models. Read More

We discuss some issues related to D(2p)-D0 branes with background magnetic fluxes respectively, in a T-dual picture, Dp-Dp branes at angles. In particular, we describe the nature of the supersymmetric bound states appearing after tachyon condensation. We present a very elementary derivation of the conditions to be satisfied by such general supersymmetric gauge configurations, which are simply related by T-duality to the conditions for supersymmetric p-cycles in C^p. Read More

The Chern isomorphism determines the free part of the K-groups from ordinary cohomology. Thus to really understand the implications of K-theory for physics one must look at manifolds with K-torsion. Unfortunately there are not many explicit examples, and usually for very symmetric spaces. Read More

**Affiliations:**

^{1}U.T. Austin,

^{2}U.T. Austin

M-theory compactification leads one to consider 7-manifolds obtained by rolling Calabi-Yau threefolds in the web of Calabi-Yau moduli spaces. The resulting 7-space in general has singularities governed by the extremal transition undergone. After providing some background in Sec. Read More