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We study M-theory on a Calabi-Yau fourfold with a smooth surface $S$ of$A_{N-1}$ singularities. The resulting three-dimensional theory has a$\mathcal{N}=2$ $SU(N)$ gauge theory sector, which we obtain from a twisteddimensional reduction of a seven-dimensional $\mathcal{N}=1$ $SU(N)$ gaugetheory on the surface $S$. A variant of the Vafa-Witten equations governs themoduli space of the gauge theory, which, for a trivial $SU(N)$ principal bundleover $S$, admits a Coulomb and a Higgs branch. In M-theory these two gaugetheory branches arise from a resolution and a deformation to smooth Calabi-Yaufourfolds, respectively. We find that the deformed Calabi-Yau fourfoldassociated to the Higgs branch requires for consistency a non-trivial four-formbackground flux in M-theory. The flat directions of the flux-inducedsuperpotential are in agreement with the gauge theory prediction for the modulispace of the Higgs branch. We illustrate our findings with explicit examplesthat realize the Coulomb and Higgs phase transition in Calabi-Yau fourfoldsembedded in weighted projective spaces. We generalize and enlarge this class ofexamples to Calabi-Yau fourfolds embedded in toric varieties with an $A_{N-1}$singularity in codimension two.

We study the four-dimensional low-energy effective N=1 supergravity theory of the dimensional reduction of M-theory on G2-manifolds, which are constructed by Kovalev’s twisted connected sum gluing suitable pairs of asymptotically cylindrical Calabi-Yau threefolds XL/R augmented with a circle S1. In the Kovalev limit the Ricci-flat G2-metrics are approximated by the Ricci-flat metrics on XL/R and we identify the universal modulus—the Kovalevton—that parametrizes this limit. We observe that the low-energy effective theory exhibits in this limit gauge theory sectors with extended supersymmetry. We determine the universal (semi-classical) Kähler potential of the effective N=1 supergravity action as a function of the Kovalevton and the volume modulus of the G2-manifold. This Kähler potential fulfills the no-scale inequality such that no anti-de-Sitter vacua are admitted. We describe geometric degenerations in XL/R, which lead to non-Abelian gauge symmetries enhancements with various matter content. Studying the resulting gauge theory branches, we argue that they lead to transitions compatible with the gluing construction and provide many new explicit examples of G2-manifolds. [ABSTRACT FROM AUTHOR]

We consider deformations of torsion-free G2 structures, defined by the G2-invariant 3-form φ and compute the expansion of $${\ast \varphi }$$ to fourth order in the deformations of φ. By considering M-theory compactified on a G2 manifold, the G2 moduli space is naturally complexified, and we get a Kähler metric on it. Using the expansion of $${\ast \varphi }$$, we work out the full curvature of this metric and relate it to the Yukawa coupling. [ABSTRACT FROM AUTHOR]

We perform a refined count of BPS states in the compactification of M-theory on $${K3 \times T^2}$$ , keeping track of the information provided by both the $${SU(2)_L}$$ and $${SU(2)_R}$$ angular momenta in the SO(4) little group. Mathematically, this four variable counting function may be expressed via the motivic Donaldson-Thomas counts of $${K3 \times T^2}$$ , simultaneously refining Katz, Klemm, and Pandharipande's motivic stable pairs counts on K3 and Oberdieck-Pandharipande's Gromov-Witten counts on $${K3 \times T^2}$$ . This provides the first full answer for motivic curve counts of a compact Calabi-Yau threefold. Along the way, we develop a Hodge-elliptic genus for Calabi-Yau manifolds-a new counting function for BPS states that interpolates between the Hodge polynomial and the elliptic genus of a Calabi-Yau. [ABSTRACT FROM AUTHOR]

We analyze relations between BPS degeneracies related to Labastida-Mariño-Ooguri-Vafa (LMOV) invariants and algebraic curves associated to knots. We introduce a new class of such curves, which we call extremal A-polynomials, discuss their special properties, and determine exact and asymptotic formulas for the corresponding (extremal) BPS degeneracies. These formulas lead to nontrivial integrality statements in number theory, as well as to an improved integrality conjecture, which is stronger than the known M-theory integrality predictions. Furthermore, we determine the BPS degeneracies encoded in augmentation polynomials and show their consistency with known colored HOMFLY polynomials. Finally, we consider refined BPS degeneracies for knots, determine them from the knowledge of super-A-polynomials, and verify their integrality. We illustrate our results with twist knots, torus knots, and various other knots with up to 10 crossings. [ABSTRACT FROM AUTHOR]

The supergravity description of various configurations of supersymmetric M-fivebranes wrapped on calibrated cycles of special holonomy manifolds is studied. The description is provided by solutions of eleven-dimensional supergravity which interpolate smoothly between a special holonomy manifold and an event horizon with Anti-de Sitter geometry. For known examples of Anti-de Sitter solutions, the associated special holonomy metric is derived. One explicit Anti-de Sitter solution of M-theory is so treated for fivebranes wrapping each of the following cycles: Kähler cycles in Calabi-Yau two, three- and four-folds; special lagrangian cycles in three- and four-folds; associative three- and co-associative four-cycles in G 2 manifolds; complex lagrangian four-cycles in Sp(2) manifolds; and Cayley four-cycles in Spin(7) manifolds. In each case, the associated special holonomy metric is singular, and is a hyperbolic analogue of a known metric. The analogous known metrics are respectively: Eguchi-Hanson, the resolved conifold and the four-fold resolved conifold; the deformed conifold, and the Stenzel four-fold metric; the Bryant-Salamon-Gibbons-Page-Pope G 2 metrics on an $${\mathbb{R}^4}$$ bundle over S 3, and an $${\mathbb{R}^3}$$ bundle over S 4 or $${\mathbb{CP}^2}$$ ; the Calabi hyper-Kähler metric on $${T^*\mathbb{CP}^2}$$ ; and the Bryant-Salamon-Gibbons-Page-Pope Spin(7) metric on an $${\mathbb{R}^4}$$ bundle over S 4. By the AdS/CFT correspondence, a conformal field theory is associated to each of the new singular special holonomy metrics, and defines the quantum gravitational physics of the resolution of their singularities. [ABSTRACT FROM AUTHOR]

We obtain exact results in α′ for open and closed A-model topological string amplitudes on a large class of toric Calabi-Yau threefolds by using their correspondence with five dimensional gauge theories. The toric Calabi-Yaus that we analyze are obtained as minimal resolution of cones over Y p, q manifolds and give rise via M-theory compactification to SU( p) gauge theories on $${\mathbb{R}^4\times S^1}$$ . As an application we present a detailed study of the local $${\mathbb{F}_2}$$ case and compute open and closed genus zero Gromov-Witten invariants of the $${\mathbb{C}^3/\mathbb{Z}_4}$$ orbifold. We also display the modular structure of the topological wave function and give predictions for higher genus amplitudes. The mirror curve in this case is the spectral curve of the relativistic A1 Toda chain. Our results also indicate the existence of a wider class of relativistic integrable systems associated to generic Y p, q geometries. [ABSTRACT FROM AUTHOR]

We analyse the most general $${\mathcal{N} = 2}$$ supersymmetric solutions of D = 11 supergravity consisting of a warped product of four-dimensional anti-de-Sitter space with a seven-dimensional Riemannian manifold Y. We show that the necessary and sufficient conditions for supersymmetry can be phrased in terms of a local SU(2)-structure on Y. Solutions with non-zero M2-brane charge also admit a canonical contact structure, in terms of which many physical quantities can be expressed, including the free energy and the scaling dimensions of operators dual to supersymmetric wrapped M5-branes. We show that a special class of solutions is singled out by imposing an additional symmetry, for which the problem reduces to solving a second order non-linear ODE. As well as recovering a known class of solutions, that includes the IR fixed point of a mass deformation of the ABJM theory, we also find new solutions which are dual to cubic deformations. In particular, we find a new supersymmetric warped AdS × S solution with non-trivial four-form flux. [ABSTRACT FROM AUTHOR]

The higher gauge field in 11-dimensional supergravity-the C-field-is constrained by quantum effects to be a cocycle in some twisted version of differential cohomology. We argue that it should indeed be a cocycle in a certain twisted nonabelian differential cohomology. We give a simple and natural characterization of the full smooth moduli 3-stack of configurations of the C-field, the gravitational field/background, and the (auxiliary) E-field. We show that the truncation of this moduli 3-stack to a bare 1-groupoid of field configurations reproduces the differential integral Wu structures that Hopkins-Singer had shown to formalize Witten's argument on the nature of the C-field. We give a similarly simple and natural characterization of the moduli 2-stack of boundary C-field configurations and show that it is equivalent to the moduli 2-stack of anomaly free heterotic supergravity field configurations. Finally, we show how to naturally encode the Hořava-Witten boundary condition on the level of moduli 3-stacks, and refine it from a condition on 3-forms to a condition on the corresponding full differential cocycles. [ABSTRACT FROM AUTHOR]

We determine the d + 1 dimensional topological field theory, which encodes the higher-form symmetries and their 't Hooft anomalies for d-dimensional QFTs obtained by compactifying M-theory on a non-compact space X. The resulting theory, which we call the Symmetry TFT, or SymTFT for short, is derived by reducing the topological sector of 11d supergravity on the boundary ∂ X of the space X. Central to this endeavour is a reformulation of supergravity in terms of differential cohomology, which allows the inclusion of torsion in cohomology of the space ∂ X , which in turn gives rise to the background fields for discrete (in particular higher-form) symmetries. We apply this framework to 7d super-Yang Mills, where X = C 2 / Γ ADE , as well as the Sasaki–Einstein links of Calabi–Yau three-fold cones that give rise to 5d superconformal field theories. This M-theory analysis is complemented with a IIB 5-brane web approach, where we derive the SymTFTs from the asymptotics of the 5-brane webs. Our methods apply to both Lagrangian and non-Lagrangian theories, and allow for many generalisations. [ABSTRACT FROM AUTHOR]

Mysterious Duality has been discovered by Iqbal, Neitzke, and Vafa (Adv Theor Math Phys 5:769–808, 2002) as a convincing, yet mysterious correspondence between certain symmetry patterns in toroidal compactifications of M-theory and del Pezzo surfaces, both governed by the root system series E k . It turns out that the sequence of del Pezzo surfaces is not the only sequence of objects in mathematics that gives rise to the same E k symmetry pattern. We present a sequence of topological spaces, starting with the four-sphere S 4 , and then forming its iterated cyclic loop spaces L c k S 4 , within which we discover the E k symmetry pattern via rational homotopy theory. For this sequence of spaces, the correspondence between its E k symmetry pattern and that of toroidal compactifications of M-theory is no longer a mystery, as each space L c k S 4 is naturally related to the compactification of M-theory on the k-torus via identification of the equations of motion of (11 - k) -dimensional supergravity as the defining equations of the Sullivan minimal model of L c k S 4 . This gives an explicit duality between algebraic topology and physics. Thereby, we extend Iqbal-Neitzke-Vafa's Mysterious Duality between algebraic geometry and physics into a triality, also involving algebraic topology. Via this triality, duality between physics and mathematics is demystified, and the mystery is transferred to the mathematical realm as duality between algebraic geometry and algebraic topology. Now the question is: Is there an explicit relation between the del Pezzo surfaces B k and iterated cyclic loop spaces of S 4 which would explain the common E k symmetry pattern? [ABSTRACT FROM AUTHOR]

We present ADHM-Nahm data for instantons on the Taub-NUT space and encode these data in terms of Bow Diagrams. We study the moduli spaces of the instantons and present these spaces as finite hyperkähler quotients. As an example, we find an explicit expression for the metric on the moduli space of one SU(2) instanton. We motivate our construction by identifying a corresponding string theory brane configuration. By following string theory dualities we are led to supersymmetric gauge theories with impurities. [ABSTRACT FROM AUTHOR]

A key open problem in M-theory is to explain the mechanism of "gauge enhancement" through which M-branes exhibit the nonabelian gauge degrees of freedom seen perturbatively in the limit of 10d string theory. In fact, since only the twisted K-theory classes represented by nonabelian Chan–Paton gauge fields on D-branes have an invariant meaning, the problem is really the understanding the M-theory lift of the classification of D-brane charges by twisted K-theory. Here we show that this problem has a solution by universal constructions in rational super homotopy theory. We recall how double dimensional reduction of super M-brane charges is described by the cyclification adjunction applied to the 4-sphere, and how M-theory degrees of freedom hidden at ADE singularities are induced by the suspended Hopf action on the 4-sphere. Combining these, we demonstrate that, in the approximation of rational homotopy theory, gauge enhancement in M-theory is exhibited by lifting against the fiberwise stabilization of the unit of this cyclification adjunction on the A-type orbispace of the 4-sphere. This explains how the fundamental D6 and D8 brane cocycles can be lifted from twisted K-theory to a cohomology theory for M-brane charge, at least rationally. [ABSTRACT FROM AUTHOR]

Despite the plausible necessity of topological protection for realizing scalable quantum computers, the conceptual underpinnings of topological quantum logic gates had arguably remained shaky, both regarding their physical realization as well as their information-theoretic nature. Building on recent results on defect branes in string/M-theory (Sati and Schreiber in Rev Math Phys, 2023. . []) and on their holographically dual anyonic defects in condensed matter theory (Sati and Schreiber in Rev Math Phys 35(03):2350001, 2023. . []), here we explain [as announced in Sati and Schreiber (PlanQC 2022:33, 2022, [], [ncatlab.org/schreiber/show/Topological+Quantum+Programming+in+TED-K])] how the specification of realistic topological quantum gates, operating by anyon defect braiding in topologically ordered quantum materials, has a surprisingly slick formulation in parameterized point-set topology, which is so fundamental that it lends itself to certification in modern homotopically typed programming languages, such as cubical Agda. We propose that this remarkable confluence of concepts may jointly kickstart the development of topological quantum programming proper as well as of real-world application of homotopy type theory, both of which have arguably been falling behind their high expectations; in any case, it provides a powerful paradigm for simulating and verifying topological quantum computing architectures with high-level certification languages aware of the actual physical principles of realistic topological quantum hardware. In companion articles (Sati and Schreiber in The Quantum Monadology, [], Sati and Schreiber in Entanglement of Sections: The pushout of entangled and parameterized quantum information []) [announced in Schreiber (Quantum types via Linear Homotopy Type Theory, talk at Workshop on Quantum Software @ QTML2022, Naples, 2022, [ncatlab.org/schreiber/files/QuantumDataInLHoTT-221117.pdf])], we explain how further passage to “dependent linear” homotopy types naturally extends this scheme to a full-blown quantum programming/certification language in which our topological quantum gates may be compiled to verified quantum circuits, complete with quantum measurement gates and classical control. [ABSTRACT FROM AUTHOR]

This paper extends the correspondence between discrete Cluster Integrable Systems and BPS spectra of five-dimensional N = 1 QFTs on R 4 × S 1 by proving that algebraic solutions of the integrable systems are exact solutions for the system of TBA equations arising from the BPS spectral problem. This statement is exemplified in the case of M-theory compactifications on local del Pezzo Calabi–Yau threefolds, corresponding to q-Painlevé equations and SU(2) gauge theories with matter. A degeneration scheme is introduced, allowing to obtain closed-form expression for the BPS spectrum also in systems without algebraic solutions. By studying the example of local del Pezzo 3, it is shown that when the region in moduli space associated to an algebraic solution is a “wall of marginal stability”, the BPS spectrum contains states of arbitrarily high spin, and corresponds to a 5d uplift of a four-dimensional nonlagrangian theory. [ABSTRACT FROM AUTHOR]

We study extremal non-BPS black holes and strings arising in M-theory compactifications on Calabi–Yau threefolds, obtained by wrapping M2 branes on non-holomorphic 2-cycles and M5 branes on non-holomorphic 4-cycles. Using the attractor mechanism we compute the black hole mass and black string tension, leading to a conjectural formula for the asymptotic volumes of connected, locally volume-minimizing representatives of non-holomorphic, even-dimensional homology classes in the threefold, without knowledge of an explicit metric. In the case of divisors we find examples where the volume of the representative corresponding to the black string is less than the volume of the minimal piecewise-holomorphic representative, predicting recombination for those homology classes and leading to stable, non-BPS strings. We also compute the central charges of non-BPS strings in F-theory via a near-horizon A d S 3 limit in 6d which, upon compactification on a circle, account for the asymptotic entropy of extremal non-supersymmetric 5d black holes (i.e., the asymptotic count of non-holomorphic minimal 2-cycles). [ABSTRACT FROM AUTHOR]

We study the BPS particle spectrum of five-dimensional superconformal field theories on R 4 × S 1 with one-dimensional Coulomb branch, by means of their associated BPS quivers. By viewing these theories as arising from the geometric engineering within M-theory, the quivers are naturally associated to the corresponding local Calabi–Yau threefold. We show that the symmetries of the quiver, descending from the symmetries of the Calabi–Yau geometry, together with the affine root lattice structure of the flavor charges, provide equations for the Kontsevich–Soibelman wall-crossing invariant. We solve these equations iteratively: the pattern arising from the solution is naturally extended to an exact conjectural expression, that we provide for the local Hirzebruch F 0 , and local del Pezzo d P 3 and d P 5 geometries. Remarkably, the BPS spectrum consists of two copies of suitable 4d N = 2 spectra, augmented by Kaluza-Klein towers. [ABSTRACT FROM AUTHOR]

The Kauffman bracket skein algebra is a quantization of the algebra of regular functions on the SL 2 character variety of a topological surface. We realize the skein algebra of the 4-punctured sphere as the output of a mirror symmetry construction based on higher genus Gromov–Witten theory and applied to a complex cubic surface. Using this result, we prove the positivity of the structure constants of the bracelets basis for the skein algebras of the 4-punctured sphere and of the 1-punctured torus. This connection between topology of the 4-punctured sphere and enumerative geometry of curves in cubic surfaces is a mathematical manifestation of the existence of dual descriptions in string/M-theory for the N = 2 N f = 4 SU(2) gauge theory. [ABSTRACT FROM AUTHOR]

Using M-theory in physics, Cho et al. (JHEP 2020:115 (2020) recently outlined a program that connects two parallel subjects of three dimensional manifolds, namely, geometric topology and quantum topology. They suggest that classical topological invariants such as Chern-Simons invariants of SL (2 , C) -flat connections and SL (2 , C) -adjoint Reidemeister torsions of a three manifold can be packaged together to produce a (2 + 1) -topological quantum field theory, which is essentially equivalent to a modular tensor category. It is further conjectured that every modular tensor category can be obtained from a three manifold and a semi-simple Lie group. In this paper, we study this program mathematically, and provide strong support for the feasibility of such a program. The program produces an algorithm to generate the potential modular T-matrix and the quantum dimensions of a candidate modular data. The modular S-matrix follows from essentially a trial-and-error procedure. We find premodular tensor categories that realize candidate modular data constructed from Seifert fibered spaces and torus bundles over the circle that reveal many subtleties in the program. We make a number of improvements to the program based on our examples. Our main result is a mathematical construction of the modular data of a premodular category from each Seifert fibered space with three singular fibers and a family of torus bundles over the circle with Thurston SOL geometry. The modular data of premodular categories from Seifert fibered spaces can be realized using Temperley-Lieb-Jones categories and the ones from torus bundles over the circle are related to metaplectic categories. We conjecture that a resulting premodular category is modular if and only if the three manifold is a Z 2 -homology sphere, and condensation of bosons in the resulting properly premodular categories leads to either modular or super-modular tensor categories. [ABSTRACT FROM AUTHOR]

In this paper we propose a definition of torsion refined Gopakumar–Vafa (GV) invariants for Calabi–Yau threefolds with terminal nodal singularities that do not admit Kähler crepant resolutions. Physically, the refinement takes into account the charge of five-dimensional BPS states under a discrete gauge symmetry in M-theory. We propose a mathematical definition of the invariants in terms of the geometry of all non-Kähler crepant resolutions taken together. The invariants are encoded in the A-model topological string partition functions associated to non-commutative (nc) resolutions of the Calabi–Yau. Our main example will be a singular degeneration of the generic Calabi–Yau double cover of P 3 and leads to an enumerative interpretation of the topological string partition function of a hybrid Landau–Ginzburg model. Our results generalize a recent physical proposal made in the context of torus fibered Calabi–Yau manifolds by one of the authors and clarify the associated enumerative geometry. [ABSTRACT FROM AUTHOR]

A key open problem in M-theory is the identification of the degrees of freedom that are expected to be hidden at ADE-singularities in spacetime. Comparison with the classification of D-branes by K-theory suggests that the answer must come from the right choice of generalized cohomology theory for M-branes. Here we show that real equivariant Cohomotopy on superspaces is a consistent such choice, at least rationally. After explaining this new approach, we demonstrate how to use Elmendorf's Theorem in equivariant homotopy theory to reveal ADE-singularities as part of the data of equivariant S 4 -valued super-cocycles on 11d super-spacetime. We classify these super-cocycles and find a detailed black brane scan that enhances the entries of the old brane scan to cascades of fundamental brane super-cocycles on strata of intersecting black M-brane species. We find that on each singular stratum the black brane's instanton contribution, namely its super Nambu–Goto/Green–Schwarz action, appears as the homotopy datum associated to the morphisms in the orbit category. [ABSTRACT FROM AUTHOR]

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 that 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. [ABSTRACT FROM AUTHOR]

The recently conjectured knots-quivers correspondence (Kucharski et al. in Phys Rev D 96(12):121902, 2017. arXiv:1707.02991, Adv Theor Math Phys 23(7):1849–1902, 2019. arXiv:1707.04017) relates gauge theoretic invariants of a knot K in the 3-sphere to the representation theory of a quiver Q K associated to the knot. In this paper we provide geometric and physical contexts for this conjecture within the framework of Ooguri-Vafa large N duality (Ooguri and Vafa in Nucl Phys B 577:419–438, 2000), that relates knot invariants to counts of holomorphic curves with boundary on L K , the conormal Lagrangian of the knot in the resolved conifold, and corresponding M-theory considerations. From the physics side, we show that the quiver encodes a 3d N = 2 theory T [ Q K ] whose low energy dynamics arises on the worldvolume of an M5 brane wrapping the knot conormal and we match the (K-theoretic) vortex partition function of this theory with the motivic generating series of Q K . From the geometry side, we argue that the spectrum of (generalized) holomorphic curves on L K is generated by a finite set of basic disks. These disks correspond to the nodes of the quiver Q K and the linking of their boundaries to the quiver arrows. We extend this basic dictionary further and propose a detailed map between quiver data and topological and geometric properties of the basic disks that again leads to matching partition functions. We also study generalizations of A-polynomials associated to Q K and (doubly) refined version of LMOV invariants (Ooguri and Vafa 2000; Labastida and Marino in Commun Math Phys 217(2):423–449, 2001. arXiv:hep-th/0004196; Labastida et al. in JHEP 11:007, 2000. arXiv:hep-th/0010102; Aganagic and Vafa in Large N duality, mirror symmetry, and a Q-deformed A-polynomial for knots. arXiv:1204.4709; Fuji et al. in Nucl Phys B 867:506–546, 2013. arXiv:1205.1515). [ABSTRACT FROM AUTHOR]

: We study solutions of the Bogomolny equation on ℝ2×𝕊1 with prescribed singularities. We show that the Nahm transform establishes a one-to-one correspondence between such solutions and solutions of the Hitchin equations on a punctured cylinder with the eigenvalues of the Higgs field growing at infinity in a particular manner. The moduli spaces of solutions have natural hyperkähler metrics of a novel kind. We show that these metrics describe the quantum Coulomb branch of certain 𝒩=2 d=4 supersymmetric gauge theories on ℝ3×𝕊1. The Coulomb branches of the corresponding uncompactified theories have been previously determined by E. Witten using the M-theory fivebrane. We show that the Seiberg-Witten curves of these theories are identical to the spectral curves associated to solutions of the Bogomolny equation on ℝ2×𝕊1. In particular, this allows us to rederive Witten's results without recourse to the M-theory fivebrane. [ABSTRACT FROM AUTHOR]

We introduce a \emph{new} invariant for $\sigma $ models (and foliations more generally) using the \emph{even} pairing between K-homology and cyclic homology. We try to calculate it for the simplest case of foliations, namely principal bundles. We end up by discussing some possible physical applications including quantumgravity and M-Theory. In particular forM-Theory we propose an explicit topological Lagrangian and then using S-duality we conjecture on the existence of certain plane fields on S 11. To my brother Demetrios [ABSTRACT FROM AUTHOR]

When an M-theory fivebrane wraps a holomorphic surface ? in a Calabi–Yau 3-fold X the low energy dynamics is that of a black string in 5 dimensional ? =1 supergravity. The infrared dynamics on the string worldsheet is an ? = (0,4) 2D conformal field theory. Assuming the 2D CFT can be described as a nonlinear sigma model, we describe the target space geometry of this model in terms of the data of X and ?. Variations of weight two Hodge structures enter the construction of the model in an interesting way. [ABSTRACT FROM AUTHOR]

In this paper, we show the equivalence between two seemingly distinct 2d TQFTs: one comes from the "Coulomb branch index" of the class S theory T [Σ, G] on L(k, 1) × S¹, the other is the LG "equivariant Verlinde formula", or equivalently partition function of LGC complex Chern-Simons theory on Σ × S¹. We first derive this equivalence using the M-theory geometry and show that the gauge groups appearing on the two sides are naturally G and its Langlands dual LG. When G is not simply-connected, we provide a recipe of computing the index of T [Σ, G] as summation over the indices of T [Σ, Ğ] with non-trivial background 't Hooft fluxes, where Ğ is the universal cover of G. Then we check explicitly this relation between the Coulomb index and the equivariant Verlinde formula for G = SU(2) or SO(3). In the end, as an application of this newly found relation, we consider the more general case where G is SU(N) or PSU(N) and showthat equivariantVerlinde algebra can be derived using field theory via (generalized) Argyres-Seiberg duality. We also attach a Mathematica notebook that can be used to compute the SU(3) equivariant Verlinde coefficients. [ABSTRACT FROM AUTHOR]

The full 6d Hopf–Wess–Zumino term in the action functional for the M5-brane is anomalous as traditionally defined. What has been missing is a condition implying the higher analogue of level quantization familiar from the 2d Wess–Zumino term. We prove that the anomaly cancellation condition is implied by the hypothesis that the C-field is charge-quantized in twisted Cohomotopy theory. The proof follows by a twisted/parametrized generalization of the Hopf invariant, after identifying the full 6d Hopf–Wess–Zumino term with a twisted homotopy Whitehead integral formula, which we establish. [ABSTRACT FROM AUTHOR]

We formulate a refinement of SU( N) Chern-Simons theory on a three-manifold M via an index in the (2, 0) theory on N M5 branes. The refined Chern-Simons theory is defined on any M with a semi-free circle action. We give an explicit solution of the theory, in terms of a one-parameter refinement of the S and T matrices of Chern-Simons theory, related to the theory of Macdonald polynomials. The ordinary and refined Chern-Simons theory are similar in many ways; for example, the Verlinde formula holds in both. Refined Chern-Simons theory gives rise to new topological invariants of Seifert three-manifolds and torus knots inside them. We conjecture that the invariants are certain indices on knot homology groups. For torus knots in S colored by fundamental representation, the index equals the Poincaré polynomials of the knot homology theory categorifying the HOMFLY polynomial. As a byproduct, we show that our theory on S has a large- N dual which is the refined topological string on $${X=\mathcal{O}(-1) \oplus \mathcal{O}(-1) \rightarrow {\rm I\!P}^1}$$ ; this supports the conjecture by Gukov, Schwarz and Vafa relating the spectrum of BPS states on X to SL knot homology. We also provide a matrix model description of some amplitudes of the refined Chern-Simons theory on S. [ABSTRACT FROM AUTHOR]

A refinement of the stable pair invariants of Pandharipande and Thomas for non-compact Calabi-Yau spaces is introduced based on a virtual Bialynicki-Birula decomposition with respect to a $${\mathbb{C}^{*}}$$ action on the stable pair moduli space, or alternatively the equivariant index of Nekrasov and Okounkov. This effectively calculates the refined index for M-theory reduced on these Calabi-Yau geometries. Based on physical expectations we propose a product formula for the refined invariants extending the motivic product formula of Morrison, Mozgovoy, Nagao, and Szendroi for local $${\mathbb{P}^1}$$ . We explicitly compute refined invariants in low degree for local $${\mathbb{P}^2}$$ and local $${\mathbb{P}^1\,\times\,\mathbb{P}^1}$$ and check that they agree with the predictions of the direct integration of the generalized holomorphic anomaly and with the product formula. The modularity of the expressions obtained in the direct integration approach allows us to relate the generating function of refined PT invariants on appropriate geometries to Nekrasov's partition function and a refinement of Chern-Simons theory on a lens space. We also relate our product formula to wall crossing. [ABSTRACT FROM AUTHOR]

Phenomenological implications of the volume of the Calabi-Yau threefolds on the hidden and observable M-theory boundaries, together with slope stability of their corresponding vector bundles, constrain the set of Kähler moduli which give rise to realistic compactifications of the strongly coupled heterotic string. When vector bundles are constructed using extensions, we provide simple rules to determine lower and upper bounds to the region of the Kähler moduli space where such compactifications can exist. We show how small these regions can be, working out in full detail the case of the recently proposed Heterotic Standard Model. More explicitly, we exhibit Kähler classes in these regions for which the visible vector bundle is stable. On the other hand, there is no polarization for which the hidden bundle is stable. [ABSTRACT FROM AUTHOR]

A fundamental problem in contemporary string/M theory is to count the number of inequivalent vacua satisfying constraints in a string theory model. This article contains the first rigorous results on the number and distribution of supersymmetric vacua of type IIb string theories compactified on a Calabi-Yau 3-fold X with flux. In particular, complete proofs of the counting formulas in Ashok-Douglas [AD] and Denef-Douglas [DD1] are given, together with van der Corput style remainder estimates. Supersymmetric vacua are critical points of certain holomorphic sections (flux superpotentials) of a line bundle $$\mathcal{L} \to \mathcal{C}$$ over the moduli space of complex structures on X × T 2 with respect to the Weil-Petersson connection. Flux superpotentials form a lattice of full rank in a 2 b 3( X)-dimensional real subspace $$\mathcal{S} \subset H^0(\mathcal{C}, \mathcal{L})$$ . We show that the density of critical points in $$\mathcal{C}$$ for this lattice of sections is well approximated by Gaussian measures of the kind studied in [DSZ1,DSZ2,AD,DD1]. [ABSTRACT FROM AUTHOR]

Supersymmetric vacua (‘universes’) of string/M theory may be identified with certain critical points of a holomorphic section (the ‘superpotential’) of a Hermitian holomorphic line bundle over a complex manifold. An important physical problem is to determine how many vacua there are and how they are distributed, as the superpotential varies over physically relevant ensembles. In several papers over the last few years, M. R. Douglas and co-workers have studied such vacuum statistics problems for a variety of physical models at the physics level of rigor [Do,AD,DD]. The present paper is the first of a series by the present authors giving a rigorous mathematical foundation for the vacuum statistics problem. It sets down basic results on the statistics of critical points ?s=0 of random holomorphic sections of Hermitian holomorphic line bundles with respect to a metric connection ?, when the sections are endowed with a Gaussian measure. The principal results give formulas for the expected density and number of critical points of fixed Morse index of Gaussian random sections relative to ?. They are particularly concrete for Riemann surfaces. In our subsequent work, the results will be applied to the vacuum statistics problem and to the purely geometric problem of studying metrics which minimize the expected number of critical points. [ABSTRACT FROM AUTHOR]

T-duality acts on circle bundles by exchanging the first Chern class with the fiberwise integral of the H-flux, as we motivate using E8 and also using S-duality. We present known and new examples including NS5-branes, nilmanifolds, lens spaces, both circle bundles over Pn, and the AdS5×S5 to AdS5×P2×S1 with background H-flux of Duff, Lü and Pope. When T-duality leads to M-theory on a non-spin manifold the gravitino partition function continues to exist due to the background flux, however the known quantization condition for G4 receives a correction. In a more general context, we use correspondence spaces to implement isomorphisms on the twisted K-theories and twisted cohomology theories and to study the corresponding Grothendieck-Riemann-Roch theorem. Interestingly, in the case of decomposable twists, both twisted theories admit fusion products and so are naturally rings. [ABSTRACT FROM AUTHOR]

: Enhanced gauge symmetry appears in Type II string theory (as well as F- and M-theory) compactified on Calabi–Yau manifolds containing exceptional divisors meeting in Dynkin configurations. It is shown that in many such cases, at enhanced symmetry points in moduli a braid group acts on the derived category of sheaves of the variety. This braid group covers the Weyl group of the enhanced symmetry algebra, which itself acts on the deformation space of the variety in a compatible way. Extensions of this result are given for nontrivial B-fields on K3 surfaces, explaining physical restrictions on the B-field, as well as for elliptic fibrations. The present point of view also gives new evidence for the enhanced gauge symmetry content in the case of a local A2n-configuration in a threefold having global ℤ/2 monodromy. [ABSTRACT FROM AUTHOR]

We present a nonlinear realization of E 8(8) on a space of 57 dimensions, which is quasiconformal in the sense that it leaves invariant a suitably defined “light cone” in ℝ57. This realization, which is related to the Freudenthal triple system associated with the unique exceptional Jordan algebra over the split octonions, contains previous conformal realizations of the lower rank exceptional Lie groups on generalized space times, and in particular a conformal realization of E 7(7) on ℝ27 which we exhibit explicitly. Possible applications of our results to supergravity and M-Theory are briefly mentioned. [ABSTRACT FROM AUTHOR]