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We demonstrate how non-simply-laced gauge and flavor symmetries arise in F-theory on spaces with non-isolated singularities. The breaking from a simply-laced symmetry to one that is non-simply-laced is induced by Calabi-Yau complex structure deformation. In all examples the deformation maintains non-isolated singularities but is accompanied by a splitting of an I 1 seven-brane that opens new loops in the geometry near a non-abelian seven-brane. The splitting also arises in the moduli space of a probe D3-brane, which upon traversing the new loop experiences a monodromy that acts on 3-7 string junctions on the singular space. The monodromy reduces the symmetry algebra, which is the flavor symmetry of the D3-brane and the gauge symmetry of the seven-brane, to one that is non-simply-laced. A collision of the D3-brane with the seven-brane gives rise to a 4d $$ \mathcal{N}=1 $$ N = 1 SCFT with a non-simply-laced flavor symmetry.

Singular fiber resolution does not describe the spontaneous breaking of gauge symmetry in F-theory, as the corresponding branch of the moduli space does not exist in the theory. Accordingly, even non-abelian gauge theories have not been fully understood in global F-theory compactifications. We present a systematic discussion of using singularity deformation, which does describe the spontaneous breaking of gauge symmetry in F-theory, to study non-abelian gauge symmetry. Since this branch of the moduli space also exists in the defining M-theory compactification, it provides the only known description of gauge theory states which exists in both pictures; they are string junctions in F-theory. We discuss how global deformations give rise to local deformations, and also give examples where local deformation can be utilized even in models where a global deformation does not exist. Utilizing deformations, we study a number of new examples, including non-perturbative descriptions of $SU(3)$ and $SU(2)$ gauge theories on seven-branes which do not admit a weakly coupled type IIb description. It may be of phenomenological interest that these non-perturbative descriptions do not exist for higher rank $SU(N)$ theories., 30 pages. v2: Updated codes, added references, and discussed how local deformation can be utilized even when a global deformation does not exist (the case of non-Higgsable clusters). v3: final version, published in Communications in Mathematical Physics

We study D3-brane theories that are dually described as deformations of two different $$ \mathcal{N}=2 $$ superconformal theories with massless monopoles and dyons. These arise at the self-intersection of a seven-brane in F-theory, which cuts out a link on a small three-sphere surrounding the self-intersection. The spectrum is studied by taking small loops in the three-sphere, yielding a link-induced monodromy action on string junction D3-brane states, and subsequently quotienting by the monodromy. This reduces the differing flavor algebras of the $$ \mathcal{N}=2 $$ theories to the same flavor algebra, as required by duality, and projects out charged states, yielding an $$ \mathcal{N}=1 $$ superconformal theory on the D3-brane. In one, a deformation of a rank one Argyres-Douglas theory retains its SU(2) flavor symmetry and exhibits a charge neutral flavor triplet that is comprised of electron, dyon, and monopole string junctions. From duality we argue that the monodromy projection should also be imposed away from the conformal point, in which case the D3-brane field theory appears to exhibit confinement of electrons, dyons, and monopoles. We will address the mathematical counterparts in a companion paper.

We prove generating series formulae for suitable twisted characteristic classes of symmetric products of a singular complex quasi-projective variety. More concretely, we study homology Hirzebruch classes for motivic coefficients, as well as for complexes of mixed Hodge modules. As a special case, we obtain a generating series formula for the (intersection) homology Hirzebruch classes of symmetric products. In some cases, the latter yields a similar formula for twisted homology L-classes generalizing results of Hirzebruch-Zagier and Moonen. Our methods also apply to the study of Todd classes of (complexes of) coherent sheaves, as well as Chern classes of (complexes of) constructible sheaves, generalizing to arbitrary coefficients results of Moonen and resp. Ohmoto., Paper rewritten to also include generating series formulae for motivic homology Hirzebruch classes, Todd classes of coherent sheaves, as well as Chern classes of constructible sheaves. Some special cases of L-class formulae are also obtained. Moreover, we give a mixed-Hodge-module-free proof for the case of motivic homology Hirzebruch classes

Homology Hirzebruch characteristic classes for singular varieties have been recently defined by Brasselet-Schuermann-Yokura as an attempt to unify previously known characteristic class theories for singular spaces (e.g., MacPherson-Chern classes, Baum-Fulton-MacPherson Todd classes, and Goresky-MacPherson L-classes, respectively). In this note we define equivariant analogues of these classes for singular quasi-projective varieties acted upon by a finite group of algebraic automorphisms, and show how these can be used to calculate the homology Hirzebruch classes of global quotient varieties. We also compute the new classes in the context of monodromy problems, e.g., for varieties that fiber equivariantly (in the complex topology) over a connected algebraic manifold. As another application, we discuss Atiyah-Meyer type formulae for twisted Hirzebruch classes of global orbifolds., Comment: v2: updates include a motivic approach, as well as an Atiyah-Meyer formula for global orbifolds, including a defect formula

We study the behavior of Hodge-theoretic genera under morphisms of complex algebraic varieties. We prove that the additive $\chi_y$-genus which arises in the motivic context satisfies the so-called ``stratified multiplicative property", which shows how to compute the invariant of the source of a proper surjective morphism from its values on various varieties that arise from the singularities of the map. By considering morphisms to a curve, we obtain a Hodge-theoretic analogue of the Riemann-Hurwitz formula. We also study the contribution of monodromy to the $\chi_y$-genus of a smooth projective family, and prove an Atiyah-Meyer type formula for twisted $\chi_y$-genera. This formula measures the deviation from multiplicativity of the $\chi_y$-genus, and expresses the correction terms as higher-genera associated to cohomology classes of the quotient of the total period domain by the action of the monodromy group. By making use of Saito's theory of mixed Hodge modules, we also obtain formulae of Atiyah-Meyer type for the corresponding Hirzebruch characteristic classes., Comment: V2: No major changes in content, but some results are stated in greater generality (original proofs not altered); examples added; corrected a calculation in Example 2.9(2) following a comment of J. Steenbrink; updated references, removed typos, improved exposition; the section on intersection homology genera removed (it will be incorporated in a future paper)

This note studies the behavior of Euler characteristics and of intersection homology Euler characteristics under proper morphisms of algebraic (respectively, analytic) varieties. The methods also yield, for algebraic (respectively, analytic) varieties, formulae comparing these two kinds of Euler characteristics. The main results are direct consequences of the calculus of constructible functions and Grothendieck groups of constructible sheaves. Similar formulae for Hodgetheoretic invariants of algebraic varieties under morphisms were announced by the first and third authors in [5, 14]. c � 2007 Wiley Periodicals, Inc.

The aim of this paper is to study the behavior of Hodge-theoretic (intersection homology) genera and their associated characteristic classes under proper morphisms of complex algebraic varieties. We obtain formulae that relate (parametrized families of) global invariants of a complex algebraic variety X to such invariants of singularities of proper algebraic maps defined on X. Such formulae severely constrain, both topologically and analytically, the singularities of complex maps, even between smooth varieties. Similar results were announced by the first and third author in [13, 32]. © 2007 Wiley Periodicals, Inc.

Many four-dimensional supersymmetric compactifications of F-theory contain gauge groups that cannot be spontaneously broken through geometric deformations. These “non-Higgsable clusters” include realizations of SU(3), SU(2), and SU(3) × SU(2), but no SU(n) gauge groups or factors with n > 3. We study possible realizations of the standard model in F-theory that utilize non-Higgsable clusters containing SU(3) factors and show that there are three distinct possibilities. In one, fields with the non-abelian gauge charges of the standard model matter fields are localized at a single locus where non-perturbative SU(3) and SU(2) seven-branes intersect; cancellation of gauge anomalies implies that the simplest four-dimensional chiral SU(3) × SU(2) × U(1) model that may arise in this context exhibits standard model families. We identify specific geometries that realize non-Higgsable SU(3) and SU(3) × SU(2) sectors. This kind of scenario provides a natural mechanism that could explain the existence of an unbroken QCD sector, or more generally the appearance of light particles and symmetries at low energy scales., United States. Dept. of Energy (Contract DE-SC00012567), National Science Foundation (U.S.) (Grant PHYS-1066293)

This paper establishes an Atiyah-type characteristic class formula for both the twisted signature and the twisted L-classes of a singular space. The result, in its present form, applies to coefficient systems that satisfy a transversality condition near the singularities. This condition is frequently automatically satisfied, e.g. on supernormal spaces or for local systems arising in certain geometric mapping situations. We obtain applications to stratified maps by combining our formula with the Cappell-Shaneson signature formula.

We study elliptic fibrations by analyzing suitable deformations of the fibrations and vanishing cycles. We introduce geometric string junctions and describe some of their properties. We show how the structure of the geometric string junctions is naturally related to the Lie algebra structures of the associated singularities. One application in physics is in F-theory, where our novel approach connecting deformations and Lie algebras describes the structure of generalized type IIB seven-branes and string junction states which end on them., 16 pages. v2: additional comments and references

We utilize the deformation theory of algebraic singularities to study charged matter in compactifications of M-theory, F-theory, and type IIa string theory on elliptically fibered Calabi-Yau manifolds. In F-theory, this description is more physical than that of resolution. We describe how two-cycles can be identified and systematically studied after deformation. For ADE singularities, we realize non-trivial ADE representations as sublattices of Z^N, where N is the multiplicity of the codimension one singularity before deformation. We give a method for the determination of Picard-Lefschetz vanishing cycles in this context and utilize this method for one-parameter smooth deformations of ADE singularities. We give a general map from junctions to weights and demonstrate that Freudenthal's recursion formula applied to junctions correctly reproduces the structure of high-dimensional ADE representations, including the 126 of SO(10) and the 43,758 of E_6. We identify the Weyl group action in some examples, and verify its order in others. We describe the codimension two localization of matter in F-theory in the case of heterotic duality or simple normal crossing and demonstrate the branching of adjoint representations. Finally, we demonstrate geometrically that deformations correctly reproduce the appearance of non-simply-laced algebras induced by monodromy around codimension two singularities, showing the reduction of D_4 to G_2 in an example. A companion mathematical paper will follow., Comment: 30 pages + references, appendices. v2: references and two figures added, typos corrected

The Goresky-MacPherson L-class of a PL pseudomanifold piecewise-linearly embedded in a PL manifold in a possibly nonlocally flat way, can be computed in terms of the Hirzebruch-Thom L-class of the manifold and twisted L-classes associated to the singularities of the embedding, as was shown by Cappell and Shaneson. These formulae are refined here by analyzing the twisted classes. We treat the case of Blanchfield local systems that extend into the singularities as well as cases where they do not extend. In the latter situation, we consider fibered embeddings of strata and 4-dimensional singular sets, using work of Banagl. Rho-invariants enter the picture.

The Milnor-Hirzebruch class of a locally complete intersection X in an algebraic manifold M measures the difference between the (Poincare dual of the) Hirzebruch class of the virtual tangent bundle of X and, respectively, the Brasselet-Schuermann-Yokura (homology) Hirzebruch class of X. In this note, we calculate the Milnor-Hirzebruch class of a globally defined algebraic hypersurface X in terms of the corresponding Hirzebruch invariants of singular strata in a Whitney stratification of X. Our approach is based on Schuermann's specialization property for the motivic Hirzebruch class transformation of Brasselet-Schuermann-Yokura. The present results also yield calculations of Todd, Chern and L-type characteristic classes of hypersurfaces., v2: title changed, exposition improved and expanded, references added

For smooth manifolds, Atiyah and Meyer studied contributions of monodromy to usual signatures. In this note we obtain Atiyah-Meyer type formulae for equivariant Hodge-theoretic genera of complex algebraic varieties. Equivariant Hirzebruch genera y(X;g) of a quasi-projective variety X acted upon by a nite group of algebraic automorphisms are dened by combining the group action with the information encoded by the Hodge ltration of the mixed Hodge structure in cohomology. While for a projective algebraic manifold y(X;g) can by computed by the Atiyah- Singer holomorphic Lefschetz theorem, we derive a Atiyah-Meyer type formula for y(X;g) in the case when X is not necessarily smooth or compact, but just bers equivariantly (in the complex topology) over a compact algebraic manifold. These results apply to computing Hodge-theoretic invariants of orbit spaces. We also obtain some results comparing equivariant Hodge-theoretic genera of the range and domain of an equivariant algebraic map in terms of its singularities.

Let X and Y be spaces and f: X -Y be a map between them. Then a large family of problems in topology (and geometry) involve the attempt to relate the invariants of X and Y via f. This attempt has been successful often for the case X and Y manifolds and f the projection of a fibre bundle, for many types of important invariants. For example, the result of ChernHirzebruch-Serre [CHS] asserts that for Y simply connected, the signature of X is the product of the signatures of Y and of the fibre f 1 (y) . In this paper we undertake the study of maps in which the fiber may vary from point to point. The spaces X and Y will be any Whitney stratified spaces with even codimension strata (e.g., algebraic varieties) and f will be a stratified map. This means essentially that for V a component of a stratum of Y (thus V is an open manifold), f 1V is a union of strata of X and the restriction of f to f71V is a locally trivial fibre bundle. For example, subanalytic maps of subanalytic sets can be stratified, with respect to Whitney

In this paper I would like to survey and strengthen some of the connections between topology and other areas of mathematics, pure and applied, and areas beyond mathematics as well. One type of problem that appears in many areas of mathematics, applied mathematics, physics, economics, and probably other sciences concerns the summing of the values of a function over a discrete set of points in a prescribed region of space. Slightly more precisely, let L be a lattice in Euclidean space ℝ n , such as the set of points with integral co-ordinates. Let S ⊂ ℝ n be a region, and let f be a (reasonable) function. Then how can one write (or approximate) the sum ∑n∊L∩S f(x) in terms of quantities that are continuous or continuously computed from f and the (static) geometry of the region S? For example, if f ≡ 1, one is asking for a computation of #(L ∩ S), the number of lattice points in S, in terms of its geometry.

This paper announces results on the behavior of some important algebraic and topological invariants --- Euler characteristic, arithmetic genus, and their intersection homology analogues; the signature, etc. --- and their associated characteristic classes, under morphisms of projective algebraic varieties. The formulas obtained relate global invariants to singularities of general complex algebraic (or analytic) maps. These results, new even for complex manifolds, are applied to obtain a version of Grothendieck-Riemann-Roch, a calculation of Todd classes of toric varieties, and an explicit formula for the number of integral points in a polytope in Euclidean space with integral vertices., 8 pages

global topological invariant to be studied will be the L-classes Li(X) E Hi(X; Q), X a stratified pseudomanifold with even-codimension strata. For manifolds, these characteristic classes are the Poincare duals of the Hirzebruch polynomials in the Pontrjagin classes; Goresky-MacPherson extended the definition to stratified spaces with even-codimension strata using intersection homology [GM1]. Cheeger [Ch] defined L-classes analytically using his development of L2cohomology theory of singular Riemannian spaces. See also [S], [CSW]. The importance of these classes is illustrated by the famous theorem of Browder and Novikov: The homeomorphism type of a compact simply connected manifold of dimension at least five is determined by its homotopy type and its L-classes, up to a finite number of possibilities. (For 4-manifolds, the homotopy type alone suffices [F].) By recent work of Cappell-Weinberger (see [We]), similar results hold for stratified spaces with simply connected even-codimension strata and simply connected links, with respect to isovariant homotopy type and L-classes of the space and its strata. The study of L-classes will take place in the context of a stratified pseudomanifold X7 of real dimension n, embedded, say piecewise linearly, in a manifold Mm' of dimension m = n + 2. For example X might be a hypersurface in a projective complex algebraic variety. For X a smoothly or PL locally flatly embedded submanifold, the L-classes are given by the classical formula L(X) = [X] n i*/(P(M) u (1 +X f the total Hirzebruch L-polynomial, P(M) E H*(M) the total Pontrjagin class of M, i the inclusion, and X the Poincare dual of i [X].

This note announces some results on the relationship between global invariants and local topological structure. The first section gives a local-global formula for Pontrjagin classes or L -classes. The second section describes a corresponding decomposition theorem on the level of complexes of sheaves. A final section mentions some related aspects of “singular knot theory” and the study of nonisolated singularities. Analogous equivariant analogues, with local-global formulas for Atiyah-Singer classes and their relations to G -signatures, will be presented in a future paper.

Let f be a periodic differentiable map from a sphere to itself. A well-known conjecture of Smith asserts that in many cases (e.g., when the fixed points are isolated) the derivatives of f at its fixed points, regarded as Jacobian matrices, are linearly similar. Here we give counterexamples to this conjecture. The results show that, in many cases, these Jacobian matrices are only nonlinearly similar. This uses our recent discovery of orthogonal matrices which are nonlinearly similar without being linearly similar. Some results on general smooth actions of finite groups on differentiable manifolds are presented; the topological equivalence of their tangential representations at the fixed points is studied.