Hidden supersymmetry and quadratic deformations of the space-time conformal superalgebra
We analyze the structure of the family of quadratic superalgebras, introduced in J Phys A
44(23):235205 (2011), for the quadratic deformations of N = 1 space-time conformal supersymmetry.
We characterize in particular the ‘zero-step’ modules for this case. In such modules,
the odd generators vanish identically, and the quadratic superalgebra is realized on a single
irreducible representation of the even subalgebra (which is a Lie algebra). In the case under
study, the quadratic deformations of N = 1 space-time conformal supersymmetry, it is shown
that each massless positive energy unitary irreducible representation (in the standard classification
of Mack), forms such a zero-step module, for an appropriate parameter choice amongst the
quadratic family (with vanishing central charge). For these massless particle multiplets therefore,
quadratic supersymmetry is unbroken, in that the supersymmetry generators annihilate
all physical states (including the vacuum state), while at the same time, superpartners do not
exist.

QUANTUM BACKGROUND INDEPENDENCE IN STRING THEORY
Not only in physical string theories, but also in some highly simplified situations,
background independence has been difficult to understand. It is argued that
the “holomorphic anomaly” of Bershadsky, Cecotti, Ooguri, and Vafa gives a fundamental
explanation of some of the problems. Moreover, their anomaly equation
can be interpreted in terms of a rather peculiar quantum version of background
independence: in systems afflicted by the anomaly, background independence does
not hold order by order in perturbation theory, but the exact partition function as
a function of the coupling constants has a background independent interpretation
as a state in an auxiliary quantum Hilbert space. The significance of this auxiliary
space is otherwise unknown.

No, string theory does not need SUSY: Calabi-Yau compactifications of non-supersymmetric heterotic string theory
Phenomenological explorations of heterotic strings have conventionally focused primarily on the E8×E8
theory. We consider smooth compactifications of all three ten-dimensional heterotic theories to exhibit
the many similarities between the non-supersymmetric SO(16)×SO(16) theory and the related
supersymmetric E8×E8 and SO(32) theories. In particular, we exploit these similarities to determine
the bosonic and fermionic spectra of Calabi-Yau compactifications with line bundles of the nonsupersymmetric
string. We use elements of four-dimensional supersymmetric effective field theory to
characterize the non-supersymmetric action at leading order and determine the Green-Schwarz induced
axion-couplings. Using these methods we construct a non-supersymmetric Standard Model(SM)-like
theory. In addition, we show that it is possible to obtain SM-like models from the standard embedding
using at least an order four Wilson line. Finally, we make a proposal of the states that live on five
branes in the SO(16)×SO(16) theory and find under certain assumptions the surprising result that
anomaly factorization only admits at most a single brane solution.

Solving M-theory with the Conformal Bootstrap
We use the conformal bootstrap to perform a precision study of 3d maximally supersymmetric
(N = 8) SCFTs that describe the IR physics on N coincident M2-branes placed either
in flat space or at a C 4/Z2 singularity. First, using the explicit Lagrangians of ABJ(M) [1,2]
and recent supersymmetric localization results, we calculate certain half and quarter-BPS
OPE coefficients, both exactly at small N, and approximately in a large N expansion that we
perform to all orders in 1/N. Comparing these values with the numerical bootstrap bounds
leads us to conjecture that these theories obey an OPE coefficient minimization principle.
We then use this conjecture as well as the extremal functional method to reconstruct the first
few low-lying scaling dimensions and OPE coefficients for both protected and unprotected
multiplets that appear in the OPE of two stress tensor multiplets for all values of N. We
also calculate the half and quarter-BPS operator OPE coefficients in the SU(2)k × SU(2)−k
BLG theory for all values of the Chern-Simons coupling k, and show that generically they
do not obey the same OPE coefficient minimization principle.

Beauty is Attractive: String-Theory, D-Branes, and Moduli Trapping at Enhanced Symmetry Points
We study quantum effects on moduli dynamics arising from the production of particles
which are light at points of enhanced symmetry in moduli space. The resulting forces trap
the moduli at these points. Moduli trapping occurs in time-dependent quantum field
theory, as well as in systems of moving D-branes, where it leads the branes to combine into
stacks. Trapping also occurs in the presence of gravity, though the range over which the
moduli can roll is limited by Hubble friction. We observe that a scalar field trapped on a
steep potential can induce a stage of acceleration of the universe, which we call trapped
inflation. Moduli trapping ameliorates the cosmological moduli problem and may affect
vacuum selection. In particular, rolling moduli are most powerfully attracted to the points
of greatest symmetry. Given suitable assumptions about the dynamics of the very early
universe, this effect might help to explain why among the plethora of possible vacuum
states of string theory, we appear to live in one with a large number of (spontaneously
broken) symmetries.

The most comprehensive global fits to date of GUT-scale SUSY models with GAMBIT
We present the most comprehensive global fits to date of three supersymmetric models motivated by grand unification: the Constrained Minimal
Supersymmetric Standard Model (CMSSM), and its Non-Universal Higgs Mass generalisations NUHM1 and NUHM2. We include likelihoods from a number of direct
and indirect dark matter searches, a large collection of electroweak precision and flavour observables, direct searches for supersymmetry at LEP and Runs I and II of
the LHC, and constraints from Higgs observables. Our analysis improves on existing results not only in terms of the number of included observables, but also in the
level of detail with which we treat them, our sampling techniques for scanning the parameter space, and our treatment of nuisance parameters. We show that stau
co-annihilation is now ruled out in the CMSSM at more than 95% confidence. Stop co-annihilation turns out to be one of the most promising mechanisms for achieving
an appropriate relic density of dark matter in all three models, whilst avoiding all other constraints. We find high-likelihood regions of parameter space featuring
light stops and charginos, making them potentially detectable in the near future at the LHC. We also show that tonne-scale direct detection will play a largely complementary
role, probing large parts of the remaining viable parameter space, including essentially all models with multi-TeV neutralinos.

On an integrable deformation of Kapustin-Witten systems
In a celebrated work, Kapustin and Witten [1] described the geometric Langlands
program (GLP) in terms of a compactification on a Riemann surface of a certain twisted
version of the N = 4 superymmetric Yang-Mills theory (SYM) in four dimensions. In
such paper, the authors introduced a set of equations after imposing a BRST-like
preservation conditions on a twisted version of N = 4 SYM theory in four dimensions;
these equations are now known as the Kapustin-Witten (KW) equations and have been
the subject of an intensive work in the last decade in physics as well as in mathematics.
In particular, a relation of KW equations with knot theory is also described by Witten
in [2], where the author describes an approach to Khovanov homology using gauge
theory; in that context, the KW equations appear as a localization condition of the
N = 4 SYM theory in four dimensions (see [3] for a review on this topic). The KW
equations are also closed related to another set of equations, recently introduced by
Ward [4] and usually called the (2k)-Hitchin equations; it is important to mention that
these equations are a natural generalization of another set of equations introduced by
Hitchin [5] in a pionnering work in complex geometry; indeed, the article of Hitchin
is the origin of the notion of Higgs bundle in mathematics, a notion that plays an
important role in the physical interpretation of the GLP developed by Kapustin and
Witten. In this article we study an integrable deformation of the Kapustin-Witten
equations. Using the Weyl-Wigner-Moyal-Groenewold description an integrable
⋆-deformation of a Kapustin-Witten system is obtained. Starting from known
solutions of the original equations, some solutions to these deformed equations
are obtained.

The String Landscape, the Swampland, and the Missing Corner. Based on TASI 2017 Lectures by C. Vafa
Abstract: We give a brief overview of the string landscape and techniques used to construct
string compactifications. We then explain how this motivates the notion of the swampland and
review a number of conjectures that attempt to characterize theories in the swampland. We
also compare holography in the context of superstrings with the similar, but much simpler case
of topological string theory. For topological strings, there is a direct definition of topological
gravity based on a sum over a “quantum gravitational foam.” In this context, holography
is the statement of an identification between a gravity and gauge theory, both of which are
defined independently of one another. This points to a missing corner in string dualities
which suggests the search for a direct definition of quantum theory of gravity rather than
relying on its strongly coupled holographic dual as an adequate substitute (Based on TASI
2017 lectures given by C. Vafa).

R_D(∗): A possible hint for natural supersymmetry with R-parity violation
Recently, several B-physics experiments have reported an appreciable deviation from the Standard
Model (SM) in the tree-level observables RD(∗) ; the combined weighted average now stands at
≈ 4σ. We first show the anomaly necessarily implies model-independent collider signals of the form
pp → bτ ν that should be expediously searched for at ATLAS/CMS as a complementary test of the
anomaly. Next we suggest a possible interconnection of the anomaly with the radiative stability
of the Standard Model Higgs boson and point to a minimal effective supersymmetric scenario with
R-parity violation as the underlying cause. We also comment on the possibility of simultaneously
explaining the recently reported RK(∗) anomaly in this setup.

Edward Witten: Symmetry and Emergence
I discuss gauge and global symmetries in particle physics, condensed matter physics, and
quantum gravity. In a modern understanding of particle physics, global symmetries are approximate
and gauge symmetries may be emergent. (Based on a lecture at the April, 2016 meeting
of the American Physical Society in Salt Lake City, Utah.)

On conservation laws for the supersymmetric sigma model
Abstract. We derive conservation laws for Dirac-harmonic maps and their extensions to manifolds
that have isometries, where we mostly focus on the spherical case. In addition, we discuss
several geometric and analytic applications of the latter.

"Even the Harmonic Oscillator — the Most Basic of All Quantum Systems — Exhibits Supersymmetry!"
Solvability of the ubiquitous quantum harmonic oscillator relies on a spectrum generating osp(1|2)
superconformal symmetry. We study the problem of constructing all quantum mechanical models
with a hidden osp(1|2) symmetry on a given space of states. This problem stems from interacting
higher spin models coupled to gravity. We find interesting new realizations of supersymmetry
(SUSY) in quantum mechanics where

**Grassmann parity equals wavefunction parity**.Phenomenology with F-theory SU(5)
We explore the low energy phenomenology of an F-theory based SU(5) model which, in
addition to the known quarks and leptons, contains Standard Model (SM) singlets, and
vector-like color triplets and SU(2) doublets. Depending on their masses and couplings,
some of these new particles may be observed at the LHC and future colliders. We discuss
the restrictions by CKM constraints on their mixing with the ordinary down quarks of the
three chiral familes. The model is consistent with gauge coupling unification at the usual
supersymmetric GUT scale, dimension five proton decay is adequately suppressed, while
dimension-six decay mediated by the superheavy gauge bosons is enhanced by a factor of
5-7. The third generation charged fermion Yukawa couplings yield the corresponding lowenergy
masses in reasonable agreement with observations. The hierarchical nature of the
masses of lighter generations is accounted for via non-renormalisable interactions, with the
perturbative vacuum expectation values (vevs) of the SM singlet fields playing an essential
role.

Metastring Theory and Modular Space-time
String theory is canonically accompanied with a space-time interpretation which determines
S-matrix-like observables, and connects to the standard physics at low energies in the guise of
local effective field theory. Recently, we have introduced a reformulation of string theory which
does not rely on an a priori space-time interpretation or a pre-assumption of locality. This
metastring theory is formulated in such a way that stringy symmetries (such as T-duality) are
realized linearly. In this paper, we study metastring theory on a flat background and develop
a variety of technical and interpretational ideas. These include a formulation of the moduli
space of Lorentzian worldsheets, a careful study of the symplectic structure and consequently
consistent closed and open boundary conditions, and the string spectrum and operator algebra.
What emerges from these studies is a new quantum notion of space-time that we refer to as a
quantum Lagrangian or equivalently a modular space-time. This concept embodies the standard
tenets of quantum theory and implements in a precise way a notion of relative locality. The
usual string backgrounds (non-compact space-time along with some toroidally compactified
spatial directions) are obtained from modular space-time by a limiting procedure that can be
thought of as a correspondence limit.

Quantum geometry of elliptic Calabi-Yau manifolds
We study the quantum geometry of the class of Calabi-Yau threefolds, which
are elliptic fibrations over a two-dimensional toric base. A holomorphic anomaly
equation for the topological string free energy is proposed, which is iterative in
the genus expansion as well as in the curve classes in the base. T-duality on
the fibre implies that the topological string free energy also captures the BPSinvariants
of D4-branes wrapping the elliptic fibre and a class in the base. We
verify this proposal by explicit computation of the BPS invariants of 3 D4-branes
on the rational elliptic surface.

Topological Strings on Elliptic Fibrations
We study topological string theory on elliptically fibered Calabi-Yau manifolds using mirror
symmetry. We compute higher genus topological string amplitudes and express these in terms
of polynomials of functions constructed from the special geometry of the deformation spaces.
The polynomials are fixed by the holomorphic anomaly equations supplemented by the expected
behavior at special loci in moduli space. We further expand the amplitudes in the base moduli of
the elliptic fibration and find that the fiber moduli dependence is captured by a finer polynomial
structure in terms of the modular forms of the modular group of the elliptic curve. We further
find a recursive equation which captures this finer structure and which can be related to the
anomaly equations for correlation functions.

Seiberg-Witten-Nekrasov Theory and Modular Properties of 6d Double Elliptic Systems
If super-Yang-Mills theory possesses the exact conformal invariance, there is an additional modular
invariance under the change of the complex bare charge τ = θ2π +4πı g2 −→ − 1τ. The low-energy SeibergWitten
prepotential F(a), however, is not explicitly invariant, because the flat moduli also change a −→ aD =
∂F/∂a. In result, the prepotential is not a modular form and depends also on the anomalous Eisenstein
series E2. This dependence is usually described by the universal MNW modular anomaly equation. We
demonstrate that, in the 6d SU(N) theory with two independent modular parameters τ and τˆ, the modular
anomaly equation changes, because the modular transform of τ is accompanied by an (N-dependent!) shift
of τˆ and vice versa. This is a new peculiarity of double-elliptic systems, which deserves further investigation.

Current Algebra Formulation of M-theory based on E11 Kac-Moody Algebra
Quantum M-theory is formulated using the current algebra technique. The current algebra is based on
a Kac-Moody algebra rather than usual finite dimensional Lie algebra. Specifically, I study the E11 KacMoody
algebra that was shown recently[1] to contain all the ingredients of M-theory. Both the internal
symmetry and the external Lorentz symmetry can be realized inside E11, so that, by constructing the
current algebra of E11, I obtain both internal gauge theory and gravity theory.
The energy-momentum tensor is constructed as the bilinear form of the currents, yielding a system
of quantum equations of motion of the currents/fields. Supersymmetry is incorporated in a natural way.
The so-called “field-current identity” is built in and, for example, the gravitino field is itself a conserved
super-current. One unanticipated outcome is that the quantum gravity equation is not identical to the
one obtained from the Einstein-Hilbert action.

Instantons on Calabi-Yau and hyper-Kähler cones
The instanton equations on vector bundles over Calabi-Yau and hyper-K¨ahler cones can be
reduced to matrix equations resembling Nahm’s equations. We complement the discussion of
Hermitian Yang-Mills (HYM) equations on Calabi-Yau cones, based on regular semi-simple elements,
by a new set of (singular) boundary conditions which have a known instanton solution
in one direction. This approach extends the classic results of Kronheimer by probing a relation
between generalised Nahm’s equations and nilpotent pairs/tuples. Moreover, we consider
quaternionic instantons on hyper-K¨ahler cones over generic 3-Sasakian manifolds and study the
HYM moduli spaces arising in this set-up, using the fact that their analysis can be traced back
to the intersection of three Hermitian Yang-Mills conditions.

Quaternion-Kähler N = 4 Supersymmetric Mechanics
Using the N = 4, 1D harmonic superspace approach, we construct a new type of
N = 4 supersymmetric mechanics involving 4n-dimensional Quaternion-Kähler
(QK) 1D sigma models as the bosonic core. The basic ingredients of our construction
are local N = 4, 1D supersymmetry realized by the appropriate transformations
in 1D harmonic superspace, the general N = 4, 1D superfield vielbein
and a set of 2(n + 1) analytic “matter” superfields representing (n + 1) off-shell
supermultiplets (4, 4, 0). Both superfield and component actions are given for
the simplest QK models with the manifolds HHn = Sp(1, n)/[Sp(1)×Sp(n)] and
HP n = Sp(1+n)/[Sp(1)×Sp(n)] as the bosonic targets. For the general case the
relevant superfield action and constraints on the (4, 4, 0) “matter” superfields are
presented. Further generalizations are briefly discussed.

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