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# Hisham SatiNew York University Abu Dhabi · Mathematics

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- Jun 2019

The full 6d 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 Wess-Zumino term with a twisted homotopy Whitehead integral formula, which we establish.

- May 2019

We provide a systematic approach to twisting differential KO-theory leading to a construction of the corresponding twisted differential Atiyah-Hirzebruch spectral sequence (AHSS). We relate and contrast the degree two and the degree one twists, whose description involves appropriate local systems. Along the way, we provide a complete and explicit identification of the differentials at the $E_2$ and $E_3$ pages in the topological case, which has been missing in the literature and which is needed for the general case. The corresponding differentials in the refined theory reveal an intricate interplay between topological and geometric data, the former involving the flat part and the latter requiring the construction of the twisted differential Pontrjagin character. We illustrate with examples and applications from geometry, topology and physics. For instance, quantization conditions show how to lift differential $4k$-forms to twisted differential KO-theory leading to integrality results, while considerations of anomalies in type I string theory allow for characterization of twisted differential Spin structures.

- May 2019

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.

- May 2019

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.

- May 2019

We review how core structures of string/M‐theory emerge as higher structures in super homotopy theory; namely from systematic analysis of the brane bouquet of universal invariant higher central extensions growing out of the superpoint. Since super homotopy theory is immensely rich, to start with we consider this in the rational/infinitesimal approximation which ignores torsion‐subgroups in brane charges and focuses on tangent spaces of super space‐time. Already at this level, super homotopy theory discovers all super p‐brane species, their intersection laws, their M/IIA‐, T‐ and S‐duality relations, their black brane avatars at ADE‐singularities, including their instanton contributions, and, last not least, Dirac charge quantization: for the D‐branes it recovers twisted K‐theory, rationally, but for the M‐branes it gives cohomotopy cohomology theory. We close with an outlook on the lift of these results beyond the rational/infinitesimal approximation to a candidate formalization of microscopic M‐theory in super homotopy theory.

- Apr 2019

We show that all the expected anomaly cancellations in M-theory follow from charge-quantizing the C-field in the non-abelian cohomology theory twisted Cohomotopy. Specifically, we show that such cocycles exhibit all of the following: (1) the half-integral shifted flux quantization condition, (2) the cancellation of the total M5-brane anomaly, (3) the M2-brane tadpole cancellation, (4) the cancellation of the $W_7$ spacetime anomaly, (5) the C-field integral equation of motion, and (6) the C-field background charge. Along the way, we find that the calibrated N=1 exceptional geometries (Spin(7), $G_2$, SU(3), SU(2)) are all induced from the classification of twists in Cohomotopy. Finally we show that the notable factor of 1/24 in the anomaly polynomial reflects the order of the 3rd stable homotopy group of spheres.

We provide a systematic approach to describing the Ramond-Ramond (RR) fields as elements in twisted differential K-theory. This builds on a series of constructions by the authors on geometric and computational aspects of twisted differential K-theory, which to a large extent were originally motivated by this problem. In addition to providing a new conceptual framework and a mathematically solid setting, this allows us to uncover interesting and novel effects. Explicitly, we use our recently constructed Atiyah-Hirzebruch spectral sequence (AHSS) for twisted differential K-theory to characterize the RR fields and their quantization, which involves interesting interplay between geometric and topological data. We illustrate this with the examples of spheres, tori, and Calabi-Yau threefolds.

We review how core structures of string/M-theory emerge as higher structures in super homotopy theory; namely from systematic analysis of the brane bouquet of universal invariant higher central extensions growing out of the superpoint. Since super homotopy theory is immensely rich, to start with we consider this in the rational/infinitesimal approximation which ignores torsion-subgroups in brane charges and focuses on tangent spaces of super space-time. Already at this level, super homotopy theory discovers all super $p$-brane species, their intersection laws, their M/IIA-, T- and S-duality relations, their black brane avatars at ADE-singularities, including their instanton contributions, and, last not least, Dirac charge quantization: for the D-branes it recovers twisted K-theory, rationally, but for the M-branes it gives cohomotopy cohomology theory. We close with an outlook on the lift of these results beyond the rational/infinitesimal approximation to a candidate formalization of microscopic M-theory in super homotopy theory.

We describe an efficient algorithm that computes, for any finite group G, the linear span of its virtual permutation representations inside all its linear representations, hence the image of the canonical morphism $\beta$ from the Burnside ring to the representation ring. We use this to determine the image and cokernel of $\beta$ for binary Platonic groups, hence for finite subgroups of SU(2), over $k \in \{\mathbb{Q}, \mathbb{R}, \mathbb{C}\}$. We find explicitly that for the three exceptional subgroups and for the first seven binary dihedral subgroups, $\beta$ surjects onto the sub-lattice of the real representation ring spanned by the integer-valued characters. We conjecture that, generally, this holds true for all the binary dihedral groups.

- Dec 2018

In this note we provide a new perspective on the topological parts of several action functionals in string and M-theory. We show that rationally these can be viewed as large gauge transformations corresponding to variations of higher structures, such as String, Fivebrane, and Ninebrane structures.

In this note we provide a new perspective on the topological parts of several action functionals in string and M-theory. We show that rationally these can be viewed as large gauge transformations corresponding to variations of higher structures, such as String, Fivebrane, and Ninebrane structures.

We provide a systematic and detailed treatment of differential refinements of KO-theory. We explain how various flavors capture geometric aspects in different but related ways, highlighting the utility of each. While general axiomatics exist, no explicit constructions seem to have appeared before. This fills a gap in the literature in which K-theory is usually worked out leaving KO-theory essentially untouched, with only scattered partial information in print. We compare to the complex case, highlighting which constructions follow analogously and which are much more subtle. We construct a pushforward and differential refinements of genera, leading to a Riemann-Roch theorem for $\widehat{\rm KO}$-theory. We also construct the corresponding Atiyah-Hirzebruch spectral sequence (AHSS) and explicitly identify the differentials, including ones which mix geometric and topological data. This allows us to completely characterize the image of the Pontrjagin character. Then we illustrate with examples and applications, including higher tangential structures, Adams operations, and a differential Wu formula.

A key open problem in M-theory is the mechanism of "gauge enhancement", which supposedly makes M-branes exhibit the nonabelian gauge degrees of freedom that are 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 invariant meaning, the problem is really the lift to M-theory of the twisted K-theory classification of D-brane charges. Here we show how this problem has a solution by universal constructions in super homotopy theory, at least rationally. 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, at the level of rational homotopy theory, that 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.

A key open problem in M-theory is the mechanism of "gauge enhancement", which supposedly makes M-branes exhibit the nonabelian gauge degrees of freedom that are 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 invariant meaning, the problem is really the lift to M-theory of the twisted K-theory classification of D-brane charges. Here we show how this problem has a solution by universal constructions in super homotopy theory, at least rationally. 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, at the level of rational homotopy theory, that 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.

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 4-sphere-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. At each stage the full Green-Schwarz action functional for the given fundamental brane species appears, as the datum associated to the morphisms in the orbit category.

By analyzing super-torsion and brane super-cocycles, we derive a new duality in M-theory, which takes the form of a higher version of T-duality in string theory. This involves a new topology change mechanism abelianizing the 3-sphere associated with the C-field topology to the 517-torus associated with exceptional-generalized super-geometry. Finally we explain parity symmetry in M-theory within exceptional-generalized super-spacetime at the same level of spherical T-duality, namely as an isomorphism on 7-twisted cohomology.

By analyzing super-torsion and brane super-cocycles, we derive a new duality in M-theory, which takes the form of a higher version of T-duality in string theory. This involves a new topology change mechanism abelianizing the 3-sphere associated with the C-field topology to the 517-torus associated with exceptional-generalized super-geometry. Finally we explain parity symmetry in M-theory within exceptional-generalized super-spacetime at the same level of spherical T-duality, namely as an isomorphism on 7-twisted cohomology.

We establish a higher generalization of super L-infinity-algebraic T-duality of super WZW-terms for super p-branes. In particular, we demonstrate spherical T-duality of super M5-branes propagating on exceptional-geometric 11d super spacetime.

Degree one twisting of Deligne cohomology, as a differential refinement of integral cohomology, was established in previous work. Here we consider higher degree twists. The Rham complex, hence de Rham cohomology, admits twists of any odd degree. However, in order to consider twists of integral cohomology we need a periodic version. Combining the periodic versions of both ingredients leads us to introduce a periodic form of Deligne cohomology. We demonstrate that this theory indeed admits a twist by a gerbe of any odd degree. We present the main properties of the new theory and illustrate its use with examples and computations, mainly via a corresponding twisted differential Atiyah-Hirzebruch spectral sequence.

We combine Sullivan models from rational homotopy theory with Stasheff's $L_\infty$-algebras to describe a duality in string theory. Namely, what in string theory is known as topological T-duality between $K^0$-cocycles in type IIA string theory and $K^1$-cocycles in type IIB string theory, or as Hori's formula, can be recognized as a Fourier-Mukai transform between twisted cohomologies when looked through the lenses of rational homotopy theory. We show this as an example of topological T-duality in rational homotopy theory, which in turn can be completely formulated in terms of morphisms of $L_\infty$-algebras.

Deligne cohomology can be viewed as a differential refinement of integral cohomology, hence captures both topological and geometric information. On the other hand, it can be viewed as the simplest nontrivial version of a differential cohomology theory. While more involved differential cohomology theories have been explicitly twisted, the same has not been done to Deligne cohomology, although existence is known at a general abstract level. We work out what it means to twist Deligne cohomology, by taking degree one twists of both integral cohomology and de Rham cohomology. We present the main properties of the new theory and illustrate its use with examples and applications. Given how versatile Deligne cohomology has proven to be, we believe that this explicit and utilizable treatment of its twisted version will be useful.

We construct the Atiyah-Hirzebruch spectral sequence (AHSS) for twisted differential generalized cohomology theories. This generalizes to the twisted setting the authors' corresponding earlier construction for differential cohomology theories, as well as to the differential setting the AHSS for twisted generalized cohomology theories, including that of twisted K-theory by Rosenberg and Atiyah-Segal. In describing twisted differential spectra we build on the work of Bunke-Nikolaus, but we find it useful for our purposes to take an approach that highlights direct analogies with classical bundles and that is at the same time amenable for calculations. We will, in particular, establish that twisted differential spectra are bundles of spectra equipped with a flat connection. Our prominent case will be twisted differential K-theory, for which we work out the differentials in detail. This involves differential refinements of primary and secondary cohomology operations the authors developed in earlier papers. We illustrate our constructions and computational tools with examples.

Twisted Morava K-theory, along with computational techniques, including a universal coefficient theorem and an Atiyah-Hirzebruch spectral sequence, was introduced by Craig Westerland and the first author. We employ these techniques to compute twisted Morava K-theory of all connective covers of the stable orthogonal group and stable unitary group, and their classifying spaces, as well as spheres and Eilenberg-MacLane spaces. This extends to the twisted case some of the results of Ravenel and Wilson and of Kitchloo, Laures, and Wilson for Morava K-theory. This also generalizes to all chromatic levels computations by Khorami (and in part those of Douglas) at chromatic level one, i.e. for the case of twisted K-theory. We establish that for natural twists in all cases, there are only two possibilities: either that the twisted Morava homology vanishes, or that it is isomorphic to untwisted homology. We also provide a variant on the twist of Morava K-theory, with mod 2 cohomology in place of integral cohomology.

We develop a theory of parametrized geometric cobordism by introducing smooth Thom stacks. This requires identifying and constructing a smooth representative of the Thom functor acting on vector bundles equipped with extra geometric data, leading to a geometric refinement of the the Pontrjagin-Thom construction in stacks. We demonstrate that the resulting theory generalizes the parametrized cobordism of Galatius-Madsen-Tillman-Weiss. The theory has the feature of being both versatile and general, allowing for the inclusion of families of various geometric data, such as metrics on manifolds and connections on vector bundles, as in recent work of Cohen-Galatius-Kitchloo and Ayala.

Deligne cohomology can be viewed as a differential refinement of integral cohomology, hence captures both topological and geometric information. On the other hand, it can be viewed as the simplest nontrivial version of a differential cohomology theory. While more involved differential cohomology theories have been explicitly twisted, the same has not been done to Deligne cohomology, although existence is known at a general abstract level. We work out what it means to twist Deligne cohomology, by taking degree one twists of both integral cohomology and de Rham cohomology. We present the main properties of the new theory and illustrate its use with examples and applications. Given how versatile Deligne cohomology has proven to be, we believe that this explicit and utilizable treatment of its twisted version will be useful.

The study of higher tangential structures, arising from higher connected covers of Lie groups (String, Fivebrane, Ninebrane structures), require considerable machinery for a full description, especially for connections to geometry and applications. With utility in mind, in this paper we study these structures at the rational level and by considering Lie groups as a starting point for defining each of the higher structures, making close connection to $p_i$-structures. We indicatively call these (rational) Spin-Fivebrane and Spin-Ninebrane structures. We study the space of such structures and characterize their variations, which reveal interesting effects whereby variations of higher structures are arranged to systematically involve lower ones. These patterns turn out to arise in applications, which we demonstrate within M-theory and string theory. We also study the homotopy type of the gauge group corresponding to bundles equipped with the higher rational structures that we define.

We compute the $L_\infty$-theoretic dimensional reduction of the F1/D$p$-brane super $L_\infty$-cocycles with coefficients in rationalized twisted K-theory from the 10d type IIA and type IIB super Lie algebras down to 9d. We show that the two resulting coefficient $L_\infty$-algebras are naturally related by an $L_\infty$-isomorphism which we find to act on the super $p$-brane cocycles by the infinitesimal version of the rules of topological T-duality and inducing an isomorphism between $K^0$ and $K^1$, rationally. Moreover, we show that these $L_\infty$-algebras are the homotopy quotients of the RR-charge coefficients by the "T-duality Lie 2-algebra". We find that the induced $L_\infty$-extension is a gerby extension of a 9+(1+1) dimensional (i.e. "doubled") T-duality correspondence super-spacetime, which serves as a local model for T-folds. We observe that this still extends, via the D0-brane cocycle of its type IIA factor, to a 10+(1+1)-dimensional super Lie algebra. Finally we observe that this satisfies expected properties of a local model space for F-theory elliptic fibrations.

- Jun 2016

We show that supercocycles on super $L_\infty$-algebras capture, at the rational level, the twisted cohomological charge structure of the fields of M-theory and of type IIA string theory. We show that rational 4-sphere-valued supercocycles for M-branes in M-theory descend to supercocycles in type IIA string theory to yield the Ramond-Ramond fields predicted by the rational image of twisted K-theory, with the twist given by the B-field. In particular, we derive the M2/M5 $\leftrightarrow$ F1/Dp/NS5 correspondence via dimensional reduction of sphere-valued $L_\infty$ supercocycles in rational homotopy theory.

We consider spectral sequences in smooth generalized cohomology theories, including differential generalized cohomology theories. The main differential spectral sequences will be of the Atiyah-Hirzebruch (AHSS) type, where we provide a filtration by the Cech resolution of smooth manifolds. This allows for systematic study of torsion in differential cohomology. We apply this in detail to smooth Deligne cohomology, differential topological complex K-theory, and to a smooth extension of integral Morava K-theory that we introduce. In each case we explicitly identify the differentials in the corresponding spectral sequences, which exhibit an interesting and systematic interplay between (refinement of) classical cohomology operations, operations involving differential forms, and operations on cohomology with U(1) coefficients.

We characterize primary operations in differential cohomology via stacks, and illustrate by differentially refining Steenrod squares and Steenrod powers explicitly. This requires a delicate interplay between integral, rational, and mod p cohomology, as well as cohomology with U(1) coefficients and differential forms. Along the way we develop computational techniques in differential cohomology, including a K\"unneth decomposition, that should also be useful in their own right, and point to applications to higher geometry and mathematical physics.

We introduce a periodic form of the iterated algebraic K-theory of ku, the (connective) complex K-theory spectrum, as well as a natural twisting of this cohomology theory by higher gerbes. Furthermore, we prove a form of topological T-duality for sphere bundles oriented with respect to this theory.

We extend Massey products from cohomology to differential cohomology via
stacks, organizing and generalizing existing constructions in Deligne
cohomology. We study the properties and show how they are related to more
classical Massey products in de Rham, singular, and Deligne cohomology. The
setting and the algebraic machinery via stacks allow for computations and make
the construction well-suited for applications. We illustrate with several
examples from differential geometry and mathematical physics.

We uncover higher algebraic structures on Noether currents and BPS charges.
It is known that equivalence classes of conserved currents form a Lie algebra.
We show that at least for target space symmetries of higher parameterized
WZW-type sigma-models this naturally lifts to a Lie (p+1)-algebra structure on
the Noether currents themselves. Applied to the Green-Schwarz-type action
functionals for super p-brane sigma-models this yields super Lie (p+1)-algebra
refinements of the traditional BPS brane charge extensions of supersymmetry
algebras. We discuss this in the generality of higher differential geometry,
where it applies also to branes with (higher) gauge fields on their
worldvolume. Applied to the M5-brane sigma-model we recover and properly
globalize the M-theory super Lie algebra extension of 11-dimensional
superisometries by 2-brane and 5-brane charges. Passing beyond the
infinitesimal Lie theory we find cohomological corrections to these charges in
higher analogy to the familiar corrections for D-brane charges as they are
lifted from ordinary cohomology to twisted K-theory. This supports the proposal
that M-brane charges live in a twisted cohomology theory.

We combine rational homotopy theory and higher Lie theory to describe the Wess-Zumino-Witten (WZW) term in the M5-brane sigma model. We observe that this term admits a natural interpretation as a twisted 7-cocycle on super-Minkowski spacetime with coefficients in the rational 4-sphere. This exhibits the WZW term as an element in twisted cohomology, with the twist given by the cocycle of the M2-brane. We consider integration of this rational situation to differential cohomology and differential cohomotopy.

String structures have played an important role in algebraic topology, via
elliptic genera and elliptic cohomology, in differential geometry, via the
study of higher geometric structures, and in physics, via partition functions.
We extend the description of String structures from connected covers of the
definite-signature orthogonal group ${\rm O}(n)$ to the indefinite-signature
orthogonal group O(p, q), i.e. from the Riemannian to the pseudo-Riemannian
setting. This requires that we work at the unstable level, which makes the
discussion more subtle than the stable case. Similar, but much simpler,
constructions hold for other noncompact Lie groups such as the unitary group
U(p, q) and the symplectic group Sp(p, q). This extension provides a starting
point for an abundance of constructions in (higher) geometry and applications
in physics.

For G = G(ℝ), a split, simply connected, semisimple Lie group of rank n and K the maximal compact subgroup of G, we give a method for computing Iwasawa coordinates of K∖G using the Chevalley generators and the Steinberg presentation. When K∖G is a scalar coset for a supergravity theory in dimensions ≥3, we determine the action of the integral form G(ℤ) on K∖G. We give explicit results for the action of the discrete U-duality groups SL2(ℤ) and E
7(ℤ) on the scalar cosets SO(2)∖SL2(ℝ) and [SU(8)/{ ± Id}]∖E
7(+7)(ℝ) for type IIB supergravity in ten dimensions and 11-dimensional supergravity reduced to D = 4 dimensions, respectively. For the former, we use this to determine the discrete U-duality transformations on the scalar sector in the Borel gauge and we describe the discrete symmetries of the dyonic charge lattice. We determine the spectrum-generating symmetry group for fundamental BPS solitons of type IIB supergravity in D = 10 dimensions at the classical level and we propose an analog of this symmetry at the quantum level. We indicate how our methods can be used to study the orbits of discrete U-duality groups in general.

String structures in degree four are associated with cancellation of
anomalies of string theory in ten dimensions. Fivebrane structures in degree
eight have recently been shown to be associated with cancellation of anomalies
associated to the NS5-brane in string theory as well as the M5-brane in
M-theory. We introduce and describe "Ninebrane structures" in degree twelve and
demonstrate how they capture some anomaly cancellation phenomena in M-theory.
Along the way we also define certain variants, considered as intermediate cases
in degree nine and ten, which we call "2-Orientation" and "2-Spin structures",
respectively. As in the lower degree cases, we also discuss the natural twists
of these structures and characterize the corresponding topological groups
associated to each of the structures, which likewise admit refinements to
differential cohomology.

- Apr 2014

We enhance the action of higher abelian gauge theory associated to a gerbe on
an M5-brane with an action of a torus T^n (n>1), by a noncommutative
T^n-deformation of the M5-brane, showing that the partition function associated
to this enhanced action is a modular form, which is a purely noncommutative
geometry phenomenon since the usual theory only has a Z_2-symmetry. In
particular, S-duality in 6-dimensional higher abelian gauge theory is shown to
be, in this sense, on par with the usual 4-dimensional case. The ingredients of
the noncommutative action and equations of motion include the deformed Hodge
duality, deformed wedge product, and the noncommutative integral over the
noncommutative space obtained by strict deformation quantization.

We uncover and highlight relations between the M-branes in M-theory and
various topological invariants: the Hopf invariant over Q, Z and Z_2, the
Kervaire invariant, the f-invariant, and the nu-invariant. This requires either
a framing or a corner structure. The canonical framing provides a minimum for
the classical action and the change of framing encodes the structure of the
action and possible anomalies. We characterize the flux quantization condition
on the C-field and the topological action of the M5-brane via the Hopf
invariant, and the dual of the C-field as (a refinement of) an element of Hopf
invariant two. In the signature formulation, the contribution to the M-brane
effective action is given by the Maslov index of the corner. The Kervaire
invariant implies that the effective action of the M5-brane is quadratic. Our
study leads to viewing the self-dual string, which is the boundary of the
M2-brane on the M5-brane worldvolume, as a string theory in the sense of
cobordism of manifolds with corners. We show that the dynamics of the C-field
and its dual are encoded in unified way in the 4-sphere, which suggests the
corresponding spectrum as the generalized cohomology theory describing the
fields. The effective action of the corner is captured by the f-invariant,
which is an invariant at chromatic level two. Finally, considering M-theory on
manifolds with G_2 holonomy we show that the canonical G_2 structure minimizes
the topological part of the M5-brane action. This is done via the nu-invariant
and a variant that we introduce related to the one-loop polynomial.

We formalize higher dimensional and higher gauge WZW-type sigma-model local
prequantum field theory, and discuss its rationalized/perturbative description
in (super-)Lie n-algebra homotopy theory (the true home of the "FDA"-language
used in the supergravity literature). We show generally how the intersection
laws for such higher WZW-type sigma-model branes (open brane ending on
background brane) are encoded precisely in (super-) L-infinity-extension theory
and how the resulting "extended (super-)spacetimes" formalize spacetimes
containing sigma model brane condensates. As an application we prove in Lie
n-algebra homotopy theory that the complete super p-brane spectrum of
superstring/M-theory is realized this way, including the pure sigma-model
branes (the "old brane scan") but also the branes with tensor multiplet
worldvolume fields, notably the D-branes and the M5-brane. For instance the
degree-0 piece of the higher symmetry algebra of 11-dimensional spacetime with
an M2-brane condensate turns out to be the "M-theory super Lie algebra". We
also observe that in this formulation there is a simple formal proof of the
fact that type IIA spacetime with a D0-brane condensate is the 11-dimensional
sugra/M-theory spacetime, and of (prequantum) S-duality for type IIB string
theory. Finally we give the non-perturbative description of all this by higher
WZW-type sigma-models on higher super-orbispaces with higher WZW terms in
stacky differential cohomology.

The first part of this text is a gentle exposition of some basic
constructions and results in the extended prequantum theory of
Chern-Simons-type gauge field theories. We explain in some detail how the
action functional of ordinary 3d Chern-Simons theory is naturally localized
("extended", "multi-tiered") to a map on the universal moduli stack of
principal connections, a map that itself modulates a circle-principal
3-connection on that moduli stack, and how the iterated transgressions of this
extended Lagrangian unify the action functional with its prequantum bundle and
with the WZW-functional. In the second part we provide a brief review and
outlook of the higher prequantum field theory of which this is a first example.
This includes a higher geometric description of supersymmetric Chern-Simons
theory, Wilson loops and other defects, generalized geometry, higher
Spin-structures, anomaly cancellation, and various other aspects of quantum
field theory.

- Jul 2012

The proper action functional of (4k+3)-dimensional U(1)-Chern-Simons theory
including the instanton sectors has a well known description: it is given on
the moduli space of fields by the fiber integration of the cup product square
of classes in degree-(2k+2) differential cohomology. We first refine this
statement from the moduli space to the full higher smooth moduli stack of
fields, to which the higher order-ghost BRST complex is the infinitesimal
approximation. Then we generalize the refined formulation to cup product
Chern-Simons theories of nonabelian and higher nonabelian gauge fields, such as
the nonabelian String^c-2-connections appearing in quantum-corrected
11-dimensional supergravity and M-branes. We discuss aspects of the off-shell
extended geometric pre-quantization (in the sense of extended or multi-tiered
QFT) of these theories, where there is a prequantum U(1)-k-bundle
(equivalently: a U(1)-(k-1)-bundle gerbe) in each codimension k. Examples we
find include moduli stacks for differential T-duality structures as well as the
anomaly line bundles of higher electric/magnetic charges, such as the 5-brane
charges appearing in heterotic supergravity, appearing as line bundles with
connection on the smooth higher moduli stacks of field configurations.

We study the effects of having multiple Spin structures on the partition
function of the spacetime fields in M-theory. This leads to a potential
anomaly which appears in the eta invariants upon variation of the Spin
structure. The main sources of such spaces are manifolds with nontrivial
fundamental group, which are also important in realistic models. We
extend the discussion to the Spinc case and find the phase of
the partition function, and revisit the quantization condition for the
C-field in this case. In type IIA string theory in 10 dimensions, the
(mod 2) index of the Dirac operator is the obstruction to having a
well-defined partition function. We geometrically characterize manifolds
with and without such an anomaly and extend to the case of nontrivial
fundamental group. The lift to KO-theory gives the α-invariant,
which in general depends on the Spin structure. This reveals many
interesting connections to positive scalar curvature manifolds and
constructions related to the Gromov-Lawson-Rosenberg conjecture. In the
12-dimensional theory bounding M-theory, we study similar geometric
questions, including choices of metrics and obtaining elements of
K-theory in 10 dimensions by pushforward in K-theory on the disk fiber.
We interpret the latter in terms of the families index theorem for Dirac
operators on the M-theory circle and disk. This involves
superconnections, eta forms, and infinite-dimensional bundles, and gives
elements in Deligne cohomology in lower dimensions. We illustrate our
discussion with many examples throughout.

- Mar 2012

The study of the partition function in M-theory involves the use of index
theory on a twelve-dimensional bounding manifold. In eleven dimensions, viewed
as a boundary, this is given by secondary index invariants such as the
Atiyah-Patodi-Singer eta-invariant, the Chern-Simons invariant, or the Adams
e-invariant. If the eleven-dimensional manifold itself has a boundary, the
resulting ten-dimensional manifold can be viewed as a codimension two corner.
The partition function in this context has been studied by the author in
relation to index theory for manifolds with corners, essentially on the product
of two intervals. In this paper, we focus on the case of framed manifolds
(which are automatically Spin) and provide a formulation of the refined
partition function using a tertiary index invariant, namely the f-invariant
introduced by Laures within elliptic cohomology. We describe the context
globally, connecting the various spaces and theories around M-theory, and
providing a physical realization and interpretation of some ingredients
appearing in the constructions due to Bunke-Naumann and Bodecker. The
formulation leads to a natural interpretation of anomalies using corners and
uncovers some resulting constraints in the heterotic corner. The analysis for
type IIA leads to a physical identification of various components of eta-forms
appearing in the formula for the phase of the partition function.

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
8-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.

The worldvolume theory of coincident M5-branes is expected to contain a
nonabelian 2-form/nonabelian gerbe gauge theory that is a higher analog of
self-dual Yang-Mills theory. But the precise details -- in particular the
global moduli / instanton / magnetic charge structure -- have remained elusive.
Here we deduce from anomaly cancellation a natural candidate for the
holographic dual of this nonabelian 2-form field, under AdS7/CFT6-duality. We
find this way a 7-dimensional nonabelian Chern-Simons theory of String
2-connection fields, which, in a certain higher gauge, are given locally by
non-abelian 2-forms with values in an affine Kac-Moody Lie algebra. We
construct the corresponding action functional on the entire smooth moduli
2-stack of field configurations, thereby defining the theory globally, at all
levels and with the full instanton structure, which is nontrivial due to the
twists imposed by the quantum corrections. Along the way we explain some
general phenomena of higher nonabelian gauge theory that we need.

To study topological aspects of the partition function of the NS5-brane in
type IIA string theory, we define a cohomology class whose vanishing is a
necessary condition for this function to be well-defined. This leads to various
topological conditions, including a twisted Fivebrane structure as well as
secondary cohomology operations arising from a K-theoretic description. We
explain how these operations also generate the topological part of the action
as well as the phase of the partition function. Part of the discussion also
applies to the M5-brane.

We study global gravitational anomalies in type IIB string theory with
nontrivial middle cohomology. This requires the study of the action of
diffeomorphisms on this group. Several results and constructions, including
some recent vanishing results via elliptic genera, make it possible to consider
this problem. Along the way, we describe in detail the intersection pairing and
the action of diffeomorphisms, and highlight the appearance of various
structures, including the Rochlin invariant and its variants on the mapping
torus.

Studying the topological aspects of M-branes in M-theory leads to various
structures related to Wu classes. First we interpret Wu classes themselves as
twisted classes and then define twisted notions of Wu structures. These
generalize many known structures, including Pin^- structures, twisted Spin
structures in the sense of Distler-Freed-Moore, Wu-twisted differential
cocycles appearing in the work of Belov-Moore, as well as ones introduced by
the author, such as twisted Membrane and twisted String^c structures. In
addition, we introduce Wu^c structures, which generalize Pin^c structures, as
well as their twisted versions. We show how these structures generalize and
encode the usual structures defined via Stiefel-Whitney classes.

For an integral cohomology class H of degree n+2 on a space X, we define
twisted Morava K-theory K(n)(X; H) at the prime 2, as well as an integral
analogue. We explore properties of this twisted cohomology theory, study a
twisted Atiyah-Hirzebruch spectral sequence, and give a universal coefficient
theorem (in the spirit of Khorami). We extend the construction to define
twisted Morava E-theory, and provide applications to string theory and
M-theory.

Recent years have seen noteworthy progress in the mathematical formulation of
quantum field theory and perturbative string theory. We give a brief survey of
these developments. It serves as an introduction to the more detailed
collection "Mathematical Foundations of Quantum Field Theory and Perturbative
String Theory".

- Feb 2011

We interpret heterotic M-theory in terms of h-cobordism, that is the
eleven-manifold is a product of the ten-manifold times an interval is
translated into a statement that the former is a cobordism of the latter which
is a homtopy equivalence. In the non-simply connected case, which is important
for model building, the interpretation is then in terms of s-cobordism, so that
the cobordism is a simple-homotopy equivalence. This gives constraints on the
possible cobordisms depending on the fundamental groups and hence provides a
characterization of possible compactification manifolds using the Whitehead
group-- a quotient of algebraic K-theory of the integral group ring of the
fundamental group-- and a distinguished element, the Whitehead torsion. We also
consider the effect on the dynamics via diffeomorphisms and general dimensional
reduction, and comment on the effect on F-theory compactifications.

The actions, anomalies, and quantization conditions allow the M2-brane and
the M5-brane to support, in a natural way, structures beyond Spin on their
worldvolumes. The main examples are twisted String structures. This also
extends to twisted String^c structures, which we introduce and relate to
twisted String structures. The relation of the C-field to Chern-Simons theory
suggests the use of the String cobordism category to describe the M2-brane.

M-theory can be defined on closed manifolds as well as on manifolds with
boundary. As an extension, we show that manifolds with corners appear naturally
in M-theory. We illustrate this with four situations: The lift to bounding
twelve dimensions of M-theory on Anti de Sitter spaces, ten-dimensional
heterotic string theory in relation to twelve dimensions, and the two M-branes
within M-theory in the presence of a boundary. The M2-brane is taken with (or
as) a boundary and the worldvolume of the M5-brane is viewed as a tubular
neighborhood. We then concentrate on (variant) of the heterotic theory as a
corner and explore analytical and geometric consequences. In particular, we
formulate and study the phase of the partition function in this setting and
identify the corrections due to the corner(s). The analysis involves
considering M-theory on disconnected manifolds, and makes use of the extension
of the Atiyah-Patodi-Singer index theorem to manifolds with corners and the
b-calculus of Melrose.

- Dec 2010

We consider geometric and analytical aspects of M-theory on a manifold with
boundary Y. The partition function of the C-field requires summing over
harmonic forms. When Y is closed Hodge theory gives a unique harmonic form in
each de Rham cohomology class, while in the presence of a boundary the
Hodge-Morrey-Friedrichs decomposition should be used. This leads us to study
the boundary conditions for the C-field. The dynamics and the presence of the
dual to the C-field gives rise to a mixing of boundary conditions with one
being Dirichlet and the other being Neumann. We describe the mixing between the
corresponding absolute and relative cohomology classes via Poincar\'e duality
angles, which we also illustrate for the M5-brane as a tubular neighborhood.
Several global aspects are then considered. We provide a systematic study of
the extension of the E8 bundle and characterize obstructions. Considering Y as
a fiber bundle, we describe how the phase looks like on the base, hence
providing dimensional reduction in the boundary case via the adiabatic limit of
the eta invariant. The general use of the index theorem leads to a new effect
given by a gravitational Chern-Simons term CS on Y whose restriction to the
boundary would be a generalized WZW model. This suggests that holographic
models of M-theory can be viewed as a sector within this index-theoretic
approach.

The equations of motion and the Bianchi identity of the C-field in M-theory
are encoded in terms of the signature operator. We then reformulate the
topological part of the action in M-theory using the signature, which leads to
connections to the geometry of the underlying manifold, including positive
scalar curvature. This results in a variation on the miraculous cancellation
formula of Alvarez-Gaum\'e and Witten in twelve dimensions and leads naturally
to the Kreck-Stolz s-invariant in eleven dimensions. Hence M-theory detects
diffeomorphism type of eleven-dimensional (and seven-dimensional) manifolds,
and in the restriction to parallelizable manifolds classifies topological
eleven-spheres. Furthermore, requiring the phase of the partition function to
be anomaly-free imposes restrictions on allowed values of the s-invariant.
Relating to string theory in ten dimensions amounts to viewing the bounding
theory as a disk bundle, for which we study the corresponding phase in this
formulation.

Studying the M-branes leads us naturally to new structures that we call
Membrane-, Membrane^c-, String^K(Z,3)- and Fivebrane^K(Z,4)-structures, which
we show can also have twisted counterparts. We study some of their basic
properties, highlight analogies with structures associated with lower levels of
the Whitehead tower of the orthogonal group, and demonstrate the relations to
M-branes.

The adjoint representations of the Lie algebras of the classical groups SU(n), SO(n), and Sp(n) are, respectively, tensor, antisymmetric, and symmetric products of two vector spaces, and hence are matrix representations. We consider the analogous products of three vector spaces and study when they appear as summands in Lie algebra decompositions. The Z3-grading of the exceptional Lie algebras provide such summands and provides representations of classical groups on hypermatrices. The main natural application is a formal study of three-junctions of strings and membranes. Generalizations are also considered. Comment: 24 pages, 4 figures

We consider the topological and geometric structures associated with cohomological and homological objects in M-theory. For the latter, we have M2-branes and M5-branes, the analysis of which requires the underlying spacetime to admit a String structure and a Fivebrane structure, respectively. For the former, we study how the fields in M-theory are associated with the above structures, with homotopy algebras, with twisted cohomology, and with generalized cohomology. We also explain how the corresponding charges should take values in Topological Modular forms. We survey background material and related results in the process. Comment: 56 pages, typos and minor corrections, published version

The reduction of the $E_8$ gauge theory to ten dimensions leads to a loop
group, which in relation to twisted $K$-theory has a Dixmier–Douady class
identified with the Neveu–Schwarz $H$-field. We give an interpretation of
the degree two part of the eta form by comparing the adiabatic limit of the
eta invariant with the one loop term in type IIA. More generally, starting
with a $G$-bundle, the comparison for manifolds with String structure
identifies $G$ with $E_8$ and the representation as the adjoint, due to an
interesting appearance of the dual Coxeter number. This makes possible
a description in terms of a generalized Wess-Zumino-Witten model at
the critical level. We also discuss the relation to the index gerbe, the
possibility of obtaining such bundles from loop space, and the symmetry
breaking to finite-dimensional bundles. We discuss the implications of
this and we give several proposals.

In the background effective field theory of heterotic string theory, the Green-Schwarz anomaly cancellation mechanism plays a key role. Here we reinterpret it and its magnetic dual version in terms of, differential twisted String- and differential twisted Fivebrane-structures that generalize the notion of Spin-structures and Spin-lifting gerbes and their differential refinement to smooth Spin-connections. We show that all these structures can be encoded in terms of nonabelian cohomology, twisted nonabelian cohomology, and differential twisted nonabelian cohomology, extending the differential generalized abelian cohomology as developed by Hopkins and Singer and shown by Freed to formalize the global description of anomaly cancellation problems in higher gauge theories arising in string theory. We demonstrate that the Green-Schwarz mechanism for the H
3-field, as well as its magnetic dual version for the H
7-field define cocycles in differential twisted nonabelian cohomology that may be called, respectively, differential twisted Spin(n)-, String(n)- and Fivebrane(n)- structures on target space, where the twist in each case is provided by the obstruction to lifting the classifying map of the gauge bundle through a higher connected cover of U(n) or O(n). We show that the twisted Bianchi identities in string theory can be captured by the (nonabelian) L
∞-algebra valued differential form data provided by the differential refinements of these twisted cocycles.

The massless supermultiplet of eleven-dimensional supergravity can be
generated from the decomposition of certain representation of the exceptional
Lie group F4 into those of its maximal compact subgroup Spin(9). In an earlier
paper, a dynamical Kaluza-Klein origin of this observation is proposed with
internal space the Cayley plane, OP2, and topological aspects are explored. In
this paper we consider the geometric aspects and characterize the corresponding
forms which contribute to the action as well as cohomology classes, including
torsion, which contribute to the partition function. This involves
constructions with bilinear forms. The compatibility with various string
theories are discussed, including reduction to loop bundles in ten dimensions.

- Mar 2009

Considering the gauge field and its dual in heterotic string theory as a unified field, we show that the equations of motion at the rational level contain a twisted differential with a novel degree seven twist. This generalizes the usual degree three twist that lifts to twisted K-theory and raises the natural question of whether at the integral level the abelianized gauge fields belong to a generalized cohomology theory. Some remarks on possible such extension are given.

Ramond has observed that the massless multiplet of eleven-dimensional supergravity can be generated from the decomposition of certain representation of the exceptional Lie group F4 into those of its maximal compact subgroup Spin(9). The possibility of a topological origin for this observation is investigated by studying Cayley plane, OP2, bundles over eleven-manifolds Y. The lift of the topological terms gives constraints on the cohomology of Y which are derived. Topological structures and genera on Y are related to the corresponding ones on the total space M. The latter, being 27-dimensional, might provide a candidate for `bosonic M-theory'. The discussion leads to a connection with an octonionic version of Kreck-Stolz elliptic homology theory. Comment: Previous version split into two papers, based on a suggestion by the referee. This first part of 25 pages to appear in CNTP

In this note we revisit the subject of anomaly cancelation in string theory
and M-theory on manifolds with String structure and give three observations.
First, that on String manifolds there is no E8 x E8 global anomaly in heterotic
string theory. Second, that the description of the anomaly in the phase of the
M-theory partition function of Diaconescu-Moore-Witten extends from the Spin
case to the String case. Third, that the cubic refinement law of
Diaconescu-Freed-Moore for the phase of the M-theory partition function extends
to String manifolds. The analysis relies on extending from invariants which
depend on the Spin structure to invariants which instead depend on the String
structure. Along the way, the one-loop term is refined via the Witten genus.

We study the cohomological physics of fivebranes in type II and heterotic string theory. We give an interpretation of the one-loop term in type IIA, which involves the first and second Pontrjagin classes of spacetime, in terms of obstructions to having bundles with certain structure groups. Using a generalization of the Green-Schwarz anomaly cancelation in heterotic string theory which demands the target space to have a String structure, we observe that the "magnetic dual" version of the anomaly cancelation condition can be read as a higher analog of String structure, which we call Fivebrane structure. This involves lifts of orthogonal and unitary structures through higher connected covers which are not just 3- but even 7-connected. We discuss the topological obstructions to the existence of Fivebrane structures. The dual version of the anomaly cancelation points to a relation of String and Fivebrane structures under electric-magnetic duality. Comment: 34 pages, presentation improved, minor changes

- Mar 2008

The M-theory field strength and its dual, given by the integral lift of the left-hand side of the equation of motion, both satisfy certain cohomological properties. We study the combined fields and observe that the multiplicative structure on the product of the corresponding degree four and degree eight cohomology fits into that given by Spin K-theory. This explains some earlier results and leads naturally to the use of Spin characteristic classes. We reinterpret the one-loop term in terms of such classes and we show that it is a homotopy invariant. We argue that the various anomalies have natural interpretations within Spin K-theory. In the process, mod 3 reductions play a special role.

We give a generalization of the notion of a Cartan-Ehresmann connection from Lie algebras to L∞algebras and use it to study the obstruction theory of lifts through higher String-like extensions of Lie algebras. It is known that over a D-brane the Kalb-Ramond background field of the string restricts to a 2-bundle with connection (a gerbe) which can be seen as the obstruction to lifting the P U(H)-bundle on the D-brane to a U(H)-bundle. We discuss how this phenomenon generalizes from the ordinary central extension U(1) → U(H) → P U(H) to higher categorical central extensions, like the String-extension BU(1) → String(G) → G. Here the obstruction to the lift is a 3-bundle with connection (a 2-gerbe): the Chern-Simons 3-bundle classified by the first Pontrjagin class. For G = Spin(n) this obstructs the existence of a String-structure. We discuss how to describe this obstruction problem in terms of Lie n-algebras and their corresponding categorified Cartan-Ehresmann connections. Generalizations even beyond String-extensions are then straightforward. For G = Spin(n) the next step is “Fivebrane structures ” whose existence is obstructed by certain generalized Chern-Simons 7-bundles classified by the second Pontrjagin class.

We give a generalization of the notion of a Cartan-Ehresmann connection from Lie algebras to L∞-algebras and use it to study the obstruction theory of lifts through higher String-like extensions of Lie algebras. It is known that over a D-brane the Kalb-Ramond background field of the string restricts to a 2-bundle with connection (a gerbe) which can be seen as the obstruction to lifting the P U (H)-bundle on the D-brane to a U (H)-bundle. We discuss how this phenomenon generalizes from the ordinary central extension U (1) → U (H) → P U (H) to higher categorical central extensions, like the String-extension BU (1) → String(G) → G. Here the obstruction to the lift is a 3-bundle with connection (a 2-gerbe): the Chern-Simons 3-bundle classified by the first Pontrjagin class. For G = Spin(n) this obstructs the existence of a String-structure. We discuss how to describe this obstruction problem in terms of Lie n-algebras and their corresponding categorified Cartan-Ehresmann connections. Generalizations even beyond String-extensions are then straightforward. For G = Spin(n) the next step is "Fivebrane structures" whose existence is obstructed by certain generalized Chern-Simons 7-bundles classified by the second Pontrjagin class.

We give a generalization of the notion of a Cartan-Ehresmann connection from Lie algebras to L
∞-algebras and use it to study the obstruction theory of lifts through higher String-like extensions of Lie algebras. We find (generalized) Chern-Simons and BF-theory functionals this way and describe aspects of their parallel transport and quantization.
It is known that over a D-brane the Kalb-Ramond background field of the string restricts to a 2-bundle with connection (a gerbe) which can be seen as the obstruction to lifting the PU(H)-bundle on the D-brane to a U(H)-bundle. We discuss how this phenomenon generalizes from the ordinary central extension U(1) → U(H) → PU(H) to higher categorical central extensions, like the String-extension BU(1) → String(G) → G. Here the obstruction to the lift is a 3-bundle with connection (a 2-gerbe): the Chern-Simons 3-bundle classified by the first Pontrjagin class. For G = Spin(n) this obstructs the existence of a String-structure. We discuss how to describe this obstruction problem in terms of Lie n-algebras and their corresponding categorified Cartan-Ehresmann connections. Generalizations even beyond String-extensions are then straightforward. For G = Spin(n) the next step is “Fivebrane structures” whose existence is obstructed by certain generalized Chern-Simons 7-bundles classified by the second Pontrjagin class.

The dimensional reduction of the E8 gauge theory in eleven dimensions leads to a loop bundle in ten dimensional type IA string theory. We show that the restriction to the Neveu-Schwarz sector leads naturally to a sigma model with target space E8 with the ten-dimensional spacetime as the source. The corresponding bundle has a structure group the group of based loops, whose classifying space we study. We explore some consequences of this proposal such as possible Lagrangians and existence of flat connections.

The reduction of the E8 gauge theory to ten dimensions leads to a loop group, which in relation to twisted K-theory has a Dixmier-Douady class identified with the Neveu-Schwarz H-field. We give an interpretation of the degree two part of the eta-form by comparing the adiabatic limit of the eta invariant with the one loop term in type IIA. More generally, starting with a G-bundle, the comparison for manifolds with String structure identifies G with E8 and the representation as the adjoint, due to an interesting appearance of the dual Coxeter number. This makes possible a description in terms of a generalized WZW model at the critical level. We also discuss the relation to the index gerbe, the possibility of obtaining such bundles from loop space, and the symmetry breaking to finite-dimensional bundles. We discuss the implications of this and we give several proposals. Comment: 35 pages, minor corrections, reference added

We derive a formula for D3-brane charge on a compact spacetime, which includes torsion corrections to the tadpole cancellation condition. We use this to classify D-branes and RR fluxes in type II string theory on RP 3 × RP 2k+1 × S 6−2k with torsion H-flux and to demonstrate the conjectured T-duality to S 3 × S 2k+1 × S 6−2k with no flux. When k = 1, H = 0 and so the K-theory that classifies fluxes is twisted. When k = 2 the square of the H-flux yields an S-dual Freed-Witten anomaly which is canceled by a D3-brane insertion that ruins the dual K-theory flux classification. When k = 3 the cube of H is nontrivial and so the D3 insertion may itself be inconsistent and the compactification unphysical. Along the way we provide a physical interpretation for the AHSS in terms of the boundaries of branes within branes.

Sometimes a homology cycle of a nonsingular compactification manifold
cannot be represented by a nonsingular submanifold. We want to know
whether such nonrepresentable cycles can be wrapped by D-branes. A
brane wrapping a representable cycle carries a K-theory charge if and
only if its Freed-Witten anomaly vanishes. However some K-theory
charges are only carried by branes that wrap nonrepresentable cycles. We provide two examples of Freed-Witten anomaly-free D6-branes wrapping nonrepresentable cycles in the presence of a trivial NS 3-form flux. The first occurs in type IIA string theory compactified on the
Sp(2) group manifold and the second in IIA on a product of lens spaces.
We find that the first D6-brane carries a K-theory charge while the second does not.

Elliptic curves play a natural and important role in elliptic cohomology. In earlier work with I. Kriz, thes elliptic curves were interpreted physically in two ways: as corresponding to the intersection of M2 and M5 in the context of (the reduction of M-theory to) type IIA and as the elliptic fiber leading to F-theory for type IIB. In this paper we elaborate on the physical setting for various generalized cohomology theories, including elliptic cohomology, and we note that the above two seemingly unrelated descriptions can be unified using Sen's picture of the orientifold limit of F-theory compactification on K3, which unifies the Seiberg-Witten curve with the F-theory curve, and through which we naturally explain the constancy of the modulus that emerges from elliptic cohomology. This also clarifies the orbifolding performed in the previous work and justifies the appearance of the w_4 condition in the elliptic refinement of the mod 2 part of the partition function. We comment on the cohomology theory needed for the case when the modular parameter varies in the base of the elliptic fibration. Comment: 23 pages, typos corrected, minor clarifications

- Nov 2005

In a previous work [H. Sati, M-theory and characteristic classes, JHEP 0508 (2005) 020, hep-th/0501245] we introduced characters and classes built out of the M-theory four-form and the Pontrjagin classes, which we used to express the Chern–Simons and the one-loop terms in a way that makes the topological structures behind them more transparent. In this paper we further investigate such classes and the corresponding candidate generalized cohomology theories. In particular, we study the flux quantization conditions that arise in this context.

In previous work we derived the topological terms in the
M-theory action in terms of certain characters that we defined. In
this note, we propose the extention of these characters to include the
dual fields. The unified treatment of the M-theory four-form field
strength and its dual leads to several observations. In particular we
elaborate on the possibility of a twisted cohomology theory with a
twist given by degrees higher than three.

- May 2005

In the presence of background Neveu–Schwarz flux, the description of the Ramond–Ramond fields of type IIB string theory using twisted K-theory is not compatible with S-duality. We argue that other possible variants of twisted K-theory would still not resolve this issue. We propose instead a connection of S-duality with elliptic cohomology, and a possible T-duality relation of this to a previous proposal for IIA theory, and higher-dimensional limits. In the process, we obtain some other results which may be interesting on their own. In particular, we prove a conjecture of Witten that the 11-dimensional spin cobordism group vanishes on K(Z,6), which eliminates a potential new θ-angle in type IIB string theory.

In this note we show that the Chern-Simons and the
one-loop terms in the M-theory action can be written in terms of new
characters involving the M-theory four-form and the string classes.
This sheds a new light on the topological structure behind M-theory
and suggests the construction of a theory of `higher'
characteristic classes.

This paper, in a sense, completes a series of three papers. In the previous two hep-th/0404013, hep-th/0410293, we have explored the possibility of refining the K-theory partition function in type II string theories using elliptic cohomology. In the present paper, we make that more concrete by defining a fully quantized free field theory based on elliptic cohomology of 10-dimensional spacetime. Moreover, we describe a concrete scenario how this is related to compactification of F-theory on an elliptic curve leading to IIA and IIB theories. We propose an interpretation of the elliptic curve in the context of elliptic cohomology. We discuss the possibility of orbifolding of the elliptic curves and derive certain properties of F-theory. We propose a link of this to type IIB modularity, the structure of the topological Lagrangian of M-theory, and Witten's index of loop space Dirac operators. Comment: 27 pages; clarifications added and typos corrected; to appear in JHEP

The topological part of the M-theory partition function was shown by Witten to be encoded in the index of an E8 bundle in eleven dimensions. This partition function is, however, not automatically anomaly-free. We observe here that the vanishing W_7=0 of the Diaconescu-Moore-Witten anomaly in IIA and compactified M-theory partition function is equivalent to orientability of spacetime with respect to (complex-oriented) elliptic cohomology. Motivated by this, we define an elliptic cohomology correction to the IIA partition function, and propose its relationship to interaction between 2-branes and 5-branes in the M-theory limit.

- Mar 2004

We show that the recently demonstrated absence of the van Dam–Veltman–Zakharov discontinuity for massive spin-3/2 with a Λ term is an artifact of the tree approximation, and that the discontinuity reappears at one loop. As a numerical check on the calculation, we rederive the vanishing of the one-loop beta function for D=11 supergravity on AdS4×S7 level-by-level in the Kaluza–Klein tower.

Recently, Diaconescu, Moore and Witten provided a nontrivial link between K-theory and M-theory, by deriving the partition function of the Ramond-Ramond fields of Type IIA string theory from an E8 gauge theory in eleven dimensions. We give some relations between twisted K-theory and M-theory by adapting the method of Diaconescu-Moore-Witten and Moore-Saulina. In particular, we construct the twisted K-theory torus which defines the partition function, and also discuss the problem from the E8 loop group picture, in which the Dixmier-Douady class is the Neveu-Schwarz field. In the process of doing this, we encounter some mathematics that is new to the physics literature. In particular, the eta differential form, which is the generalization of the eta invariant, arises naturally in this context. We conclude with several open problems in mathematics and string theory.

- Jan 2004
- The Launching of La Belle Epoque of High Energy Physics and Cosmology, A Festschrift for Paul Frampton in His 60th Year and Memorial Tributes to Behram Kursunoglu (1922-2003) - 32nd Coral Gables Conference

- Jan 2003

This dissertation is devoted to studying classical and quantum massive
fields in Anti-de Sitter space. Those can arise in various ways. First,
classical massive fields arising from Kaluza-Klein compactifications on
spheres are studied. The supersymmetry of the reduction to massive
supergravities in 4, 5 and 7 dimensions is studies and the effective
supersymmetry transformations of the fermionic superpartners of those
massive breathing and squashing modes are derived. Then, the one-loop
quantum properties of the graviton and the gravitino are studied and a
quantum discontinuity between massive and massless propagators is shown
to persist. Then, a process of dynamical generation of mass is
considered in which conformal field composites provide that mass. The
relevance to brane-world scenarios, in particular Randall-Sundrum and
Karch-Randall models is stressed.

Diaconescu, Moore and Witten have shown that the topological part of the M-theory partition function is an invariant of an E8 gauge bundle over the 11-dimensional bulk. This presents a puzzle as an 11d gauge theory cannot exhibit linearly realized supersymmetry. One possibility is that the gauge theory is nonsupersymmetric and flows to 11d SUGRA only in the infrared, with SUSY arising as a low energy accidental degeneracy. Although no such gauge theory has been constructed, any such construction must satisfy a number of constraints in order to correctly reproduce the known 10-dimensional physics on each boundary component. We analyze these constraints and in particular use them to attempt an approximate construction of the 11d gravitino as a condensate of the gauge theory fields. Comment: 14 pages, no figures

We report on the recent work on the van Dam-Veltman-Zakharov discontinuity for massive and partially massless gravitons in A(dS) space at one-loop.

We perform a one‐loop test of the holographic interpretation of the Karch‐Randall model, whereby a massive graviton appears on an AdS4
brane in an AdS5 bulk. Within the AdS/CFT framework, we examine the quantum corrections to the graviton propagator on the brane, and demonstrate that they induce a graviton mass in exact agreement with the Karch‐Randall result. Interestingly enough, at one loop order, the spin 0, spin 1/2 and spin 1 loops contribute to the dynamically generated (mass)2 in the same 1 : 3 : 12 ratio as enters the Weyl anomaly and the 1/r
3 corrections to the Newtonian gravitational potential.

We show that the recently demonstrated absence of the usual discontinuity for massive spin 2 with a Lambda term is an artifact of the tree approximation, and that the discontinuity reappears at one loop.

- Jul 2001

It has recently been shown that the Randall–Sundrum brane-world may be obtained from an appropriate doubled D3-brane configuration in type IIB theory. This corresponds, in five dimensions, to a sphere compactification of the original IIB theory with a non-trivial breathing mode supporting the brane. In this paper, we shall study the supersymmetry of this reduction to massive five-dimensional supergravity, and derive the effective supersymmetry transformations for the fermionic superpartners to the breathing mode. We also consider the sphere compactifications of eleven-dimensional supergravity to both four and seven dimensions. For the compactifications on S5 and S7, we include a squashing mode scalar and discuss the truncation from N=8 to N=2 supersymmetry.

In a previous paper we showed that the absence of the van Dam–Veltman–Zakharov discontinuity as M2→0 for massive spin 2 with a Λ term is an artifact of the tree approximation, and that the discontinuity reappears at one loop, as a result of going from five degrees of freedom to two. In this Letter we show that a similar classical continuity but quantum discontinuity arises in the “partially massless” limit M2→2Λ/3, as a result of going from five degrees of freedom to four.

- Mar 1999

We construct the non-linear Kaluza-Klein ans\"atze describing the embeddings of the U(1)^3, U(1)^4 and U(1)^2 truncations of D=5, D=4 and D=7 gauged supergravities into the type IIB string and M-theory. These enable one to oxidise any associated lower dimensional solutions to D=10 or D=11. In particular, we use these general ans\"atze to embed the charged AdS_5, AdS_4 and AdS_7 black hole solutions in ten and eleven dimensions. The charges for the black holes with toroidal horizons may be interpreted as the angular momenta of D3-branes, M2-branes and M5-branes spinning in the tranverse dimensions, in their near-horizon decoupling limits. The horizons of the black holes coincide with the worldvolumes of the branes. The Kaluza-Klein ans\"atze also allow the black holes with spherical or hyperbolic horizons to be reinterpreted in D=10 or D=11. Comment: Latex, 35 pages; minor typos corrected