Post on 03-Jan-2017
Oxford Particle Theory Group
Conformal)field)theory)and)quantum)gravity)
Gauge&string,duality,,holography,,AdS&CFT,correspondence,
String)theory)and)phenomenology)
La8ce)field)theory)
Phenomenology)of)electroweak)and)strong)interac>ons)
Physics)beyond)the)Standard)Model)
Par>cle)astrophysics)and)cosmology)
Mike%Teper%
Andrei%Starinets%John%Wheater%
Subir%Sarkar%Joe%Conlon%Giulia%Zanderighi%
Uli%Haisch% John%March>Russell%
Founded%1963,by,Dick,Dalitz,,Emeri@:,Jack,Paton,,Ian,Aitchison,,John,Taylor,,Chris,Llewellyn&Smith,,
Frank,Close,,Graham,Ross,,
Andre%Lukas%
Supported,by:,STFC,,EU,,Royal,Society,…,,
Juan%Rojo%
+,presently,12%postdocs%&,~25%DPhil%students%%%%
hMp://www2.physics.ox.ac.uk/research/par@cle&theory,
Jorge%Casalderrey>Solana%
Staff
Jorge Casalderrey-Solana Royal Society Fellow jorge.casalderrey@physics.ox.ac.uk
t
z
Energy Density
๏ A new state of matterQuark Gluon Plasma
๏ A strongly coupled fluid
๏ Extremely high temperaturesT > 2×1012 K = 170 MeV
Pb Pb
๏ LHC: Pb-Pb @ √s= 2.76 TeV
๏ What are the properties of QGP?• How is it generated?• What are its transport coefficients?• How does it interact with probes?
๏Techniques that we use• Holography: QFT ⟷ Gravity• pQCD• Effective field theory methods
Heavy Ion Collisions, Holography and Jets
bounds on light spin-0 states in dark
matter context
indirect probes of Higgs trilinear coupling
flavour anomalies seen
at LHCb
stop bounds in presence of flavour
mixing
ULI HAISCH DALITZ 2
Andre LukasMain interest: string theory, with emphasis on
• Calabi-Yau manifolds, vector bundles and heterotic model building
• Flux compactifications and non-CY manifolds
• (Computational) algebraic geometry and string theory
• M-theory compactifications and F-theory
• String cosmology
Current students: Challenger Mishra, Chuang Sun, Stefan Blasneag Capacity for one new students this year.
compactifications, model building and phenomenology. More specifically:
Major theme: “Getting the standard model from string theory”
John March-Russell
• Mystery of the Higgs & the Weak Scale:!
!
• Dark matter:!
!
• Strong CP problem: !
!
• Gauge unification? String Theory pheno? Family replication? CP-violation? Hidden Sectors? Gravity Waves? Inflation? Baryogenesis? Why 4D? Cosmo constant?…!
Approaches to the “Hierarchy Problem” — (non-minimal) supersymmetry? — unusual strong-coupling dynamics? —new symmetries? — extra dimensions? — physical naturalness?
What is it? How does it interact with SM sector? Unusual/new observational signals? New production mechanisms?
Is the Peccei-Quinn axion the solution or something new? Are there other super-light weakly coupled bosons like axion? How to discover them?
Dalitz #6
• Some recent papers (Oxford PhD students):
Twin Higgs Asymmetric Dark Matter IGG, Robert Lasenby, JMR
Twin Higgs WIMP Dark Matter IGG,RL,JMR
Signatures of Large Composite Dark Matter States Edward Hardy, RL, JMR, Stephen West
Auto-Concealment of Supersymmetry in Extra Dimensions Savas Dimopoulos, Kiel Howe, JMR, James Scoville
Big Bang Synthesis of Nuclear Dark Matter Edward Hardy, RL, JMR, Stephen West
Rare Flavor Processes in Maximally Natural Supersymmetry IGG, JMR
Retrofitted Natural Supersymmetry from a U(1) EH, JMR Precision Unification in -SUSY with a 125 GeV Higgs� EH, JMR, James Unwin
Natural Scherk-Schwarz Theories of the Weak Scale Isabel García García, Kiel Howe, JMR
Axion Mediation
Asymmetric Dark Matter via Spontaneous Co-Genesis
Masha Baryakhtar, EH, JMR
JMR, Matthew McCullough
Juan Rojo!!
STFC Rutherford Fellow !ERC Starting Grant PI
Neural Network Parton Distribution Functions (PDFs)!Encode non-perturbative dynamics for momentum distributions of quarks and gluons in the proton!Accurate PDFs required for Higgs couplings, high-mass BSM production and precision SM observables!Recently: PDFs with LHC data, PDFs with QED corrections, PDFs with all-order resummations, nuclear PDFs!
! Jet reconstruction and substructure!
Boosted final states crucial in many relevant SM and BSM measurements, like VH(->bb). !Jet substructure improves signal discrimination and bkg rejection!Recently: scale-invariant resonance tagging, new physics in Higgs pair-production, strong diHiggs in vector-boson-fusion!
!NLO event generators!
LHC phenomenology requires N(N)LO Monte Carlo generators, with high automation and possibility of merging!Recently: aMCfast, fast interface to automated NLO and NLO+PS calculations from aMCatNLO, new tunes of Pythia8!
Early&universe:&Reconstruc*on+of+the+primordial+spectrum+of+scalar/tensor+fluctua*ons+�+features+may+indicate+departures+from+‘slow9roll’+during+infla*on+…++also+BICEP2+gravita*onal+wave+spectrum+has+wrong+sign!+
High&energy&neutrinos:&I+par*cipate+in+the+IceCube&expt.+which+has+discovered+cosmic+high+energy+neutrinos+…+now+determining+flavour+composi*on+++es*ma*ng+background+from+atmospheric+charm+++measuring+neutrino+oscilla*ons,
Dark&ma6er:&Par*cular+interest+in+asymmetric+relic+par*cles+and+their+phenomenology+–+and+aKempts+at+detec*on+by+both+terrestrial+and+cosmic+experiments+…++Recent:+DM+content+of+dSphs+++Self9interac*ons+in+A3827?+,
arXiv:1506.01354+
BICEP2+
arXiv:1510.03338+
arXiv:1412.7719+
arXiv:1504.0657+arXiv:1504.02048+
Late&universe:&The+SN+Ia+Hubble+diagram+shows+<3σ+evidence+for+cosmic+accelera*on+(=>+‘dark+energy’)++++In+the+ΛCDM+cosmology,+there+should+be+∼0.1+Bullet+Cluster9like+systems+(whereas+>10+have+been+seen!)+
arXiv:1502.03376+ arXiv:1507.04005+
I+am+at+Oxford+only+during+term+9+rest+of+the+*me+at+NBI+Copenhagen+++Students:&David&Kraljic,&Jim&Talbert&(++Jeppe+Nielsen+&+Amel+Durakovic++@+NBI)+
Closed'strings'picture:'dynamics'of'strings''&'branes'at'low'energy'is'described'by'string'theory'in'curved'space'in'higher'dim.'
''''''conjectured'exact''equivalence'
Open'strings'picture:'dynamics'of'strings''&'branes'at'low'energy'is'desribed'by''a'quantum'field'theory'without'gravity''
AdSBCFT'correspondence''''''''''''''''GaugeBgravity'duality' GaugeBstring'duality' Holography'
STRONG'COUPLING' WEAK'COUPLING'
Allows'study'of'correlaPon'funcPons,'Wilson'loops,'thermodynamics,'transport,''nonBequilibrium'behavior,'turbulence,'quantum'quenches'etc'in'STRONGLY'interacPng'
systems'(of'some'class)'by'using'their'DUAL'weak'gravity'descripPon'''
Oxford'Holography'Group:'J.'Casalderrey'Solana'(RSURF),'T.Andrade,'C.Eling,'A.Ficnar,''V.Keranen,'P.Kleinert,'J.Probst,'N.Gushterov,'A.Starinets'
Mike Teper – Lattice Field Theory
physics of strongly-coupled field theories e.g. gauge theories
recent past:
• SU(N) at large N
=⇒ SU(3) is close to SU(∞) for basic physical quantities
currently:
• calculating the spectrum of confining flux tubes
– effective string description of flux tubes
with Andreas Athenodorou – former student:
e.g. arXiv:1303.5946, 1103.4547, 1007.4720
some recent stringy analyses using our results:
Aharony and Komargodski, arXiv:1302.6257
Dubovsky, Flauger and Gorbenko, arXiv: 1301.2325
• SO(N) gauge theories in D = 2 + 1 and D = 3 + 1
with Richard Lau – recent student:
arXiv:1208.4547 + several in pipeline
• SU(N) gauge theories in D = 2 + 1
1
John Wheater
Discretized models of Quantum Gravity & Quantum Geometry
Matrix Models Rigorous Combinatorics
Manifolds Graphs
Statistical Mechanics
Quantum Field Theory
Numerical Simulation
JW, Ben Niedner (just submitted), collaborators in Copenhagen & Reykjavik
My research focuses on LHC phenomenology
Recent/ongoing work includes
• NLO calculations
• first NNLO+Parton Shower for Higgs and Drell Yan production
• resummation of soft/collinear logs in Higgs production with a jet-veto
• finite mass effects in Higgs production
• electroweak ZZ production & application to anomalous couplings
• fully differential calculation of VBF Higgs production at NNLO
Giulia Zanderighicurrently on leave a 5y leave at CERN
Postdocs
Christopher Eling
Postdoctoral researcher in Andrei Starinets’ group
Past research: “quantum gravity phenomenology", alternative theories ofgravity with local Lorentz violation
Currently: Holography (quantum gravity in d + 1 dimensions equivalentto QFT in d dimensions, e.g. AdS/CFT, but also more general)
Mapping between fluid dynamics and dynamics of black holes- learnsomething new about gravity from hydrodynamics and visa versa, e.g. fluidturbulence in terms of a random measure”Spacetime thermodynamics"- Einstein equations fromthermodynamics/quantum entanglement, emergence of spacetime fromunderlying quantum system
Research interests & activity
• QCD spin structure of the proton and its phenomenologywhat is the nature of the spin of the proton?how do quarks and gluons contribute to the proton spin?how do quarks and gluons hadronize into final-state particles?
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• PDF determination for LHC phenomenology and beyond
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• I work in Juan’s group and I recently joined
Personal path
• born in Aosta, Italy
• Ms.Sc.1, Ph.D.2, PD3
• glad to join Oxford Physics
Emanuele R. Nocera (Dalitz, 9b) A two-minute slide 15 October 2015 1 / 1
Emeriti & Visitors
Graham Ross - Room 4.7
Higgs discovery !
The origin of mass! − c, τ , b,W , Z, t
!!
− MHiggs ???
•• MH ∼ MH"
Scale symmetry – classical & quantum -
SUSY
• Inflation!!
Natural, hybrid Gabriel German
Scale independent Chris Hill
• Scanning
MH = 0
MH ∼ (MH0 + λ φ )
Name % %%%%= %Chris%Allton%“Home”% %%%%= %Swansea%University%Rugby%Team %%%%= %Wallabies%Research%Area%= %QCD%at%high%temperatures%Research%Tool%%= %LaDce%Gauge%Theory%%
+A (x)µA (x)µ
|P >i |P >0
Creates a towerof states x,y,z
t
0 0.2 0.4 0.6 0.8 1 1.2 1.4
r [fm]
2.5
3
3.5
4
VC(r)
[GeV
]
-3 -2 -1 0 1 2 3
ω
0
ρ(ω)
T = 1.90TCT = 1.52TCT = 1.27TCT = 1.09TCT = 0.95TCT = 0.84TCT = 0.76TCT = 0.24TC
024681012
0
1
2
3
4
5
9 10 11 12 13 14 15
⇢(!)/m
2 b
9 10 11 12 13 14 15! (GeV)
9 10 11 12 13 14 15
⌥T/Tc = 0.76
0.84
0.840.95
0.951.09
1.091.27
1.271.52
1.521.90
Nucleons%+%Parity% Charmonium%PotenHal% Upsilon%MelHng%
Name: Gabriel German.Visiting until end of July, 2016.Working with Prof. Graham G. Ross and (possibly) with Prof. Subir Sarkar.
Purpose of the visit: to study details of Hybrid Natural Inflation (HNI) both theoretically and in the light of Planck and other experiments.
HNI was proposed in 2010 by Graham G. Ross and GG in two papers:
HNI from non Abelian discrete symmetry, Hybrid Natural Low Scale Inflation,Phys.Lett. B684 (2010) 199-204. Phys.Lett. B691 (2010) 117-120.
The model generalise Natural Inflation in the sense that a sub-Planckian scale of (pseudo) Nambu-Goldstone symmetry breaking is generated.
The inflationary sector is simply given by the potential
The model is of the hybrid type
Before July 2016 we should be able to know where to put the model
or
STEPHEN WEST
DARK MATTER AND HIDDEN SECTORS
BEYOND WIMP DM - FREEZE-IN, NUCLEAR DARK MATTER…
LINK TO MATTER-ANTIMATTER ASYMMETRY - ASYMMETRIC DARK MATTER
ASTRO-PARTICLE PHYSICS
BLACK HOLES
BSM PHYSICS AT THE LHC
EARLY UNIVERSE COSMOLOGY
ELECTROWEAK SYMMETRY BREAKING
DARK MATTER CONSEQUENCES FOR STARS
LONG LIVED STATES
ANYTHING BSM INCLUDING DARK MATTER!
COLLIDING PARTICLES IN THE FIELDS OF…
EXTENDED HIGGS SECTORS
HOW DO WE PROBE MORE COMPLICATED HIDDEN SECTORS?
DARK MATTER DIRECT AND INDIRECT DETECTION
Neighbours
1
Higgs boson and EWSB � mH natural or fine-tuned ? Æ if natural: what new physics/symmetry? � does it regularize the divergent VLVL cross-section at high M(VLVL) ? Or is there a new dynamics ? � elementary or composite Higgs ? � is it alone or are there other Higgs bosons ? � origin of couplings to fermions � coupling to dark matter ? � does it violate CP ? � cosmological EW phase transition
Neutrinos: � ν masses and and their origin � what is the role of H(125) ? � Majorana or Dirac ? � CP violation � additional species Æ sterile ν ?
Dark matter: � composition: WIMP, sterile neutrinos, axions, other hidden sector particles, .. � one type or more ? � only gravitational or other interactions ?
The two epochs of Universe’s accelerated expansion: � primordial: is inflation correct ? which (scalar) fields? role of quantum gravity? � today: dark energy (why is Λ so small?) or gravity modification ?
Quarks and leptons: � why 3 families ? � masses and mixing � CP violation in the lepton sector � matter and antimatter asymmetry � baryon and charged lepton number violation
Physics at the highest E-scales: � how is gravity connected with the other forces ? � do forces unify at high energy ?
Outstanding Questions in Particle Physics circa 2014 … there has never been a better time to be a particle physicist!
PP Strategic Plan
• Accelerator based: Searches for BSM physics & SM benchmarks
• Neutrinos
• Dark Universe
• Silicon and related work near the IP
• Accelerator Physics (JAI)
• Advanced Data Acquisition
• Advanced particle ID
The themes are: intertwined and overlapping, have many links to Theory and Astro
We place an extremely high priority on interactions with theory and seek to
systematically enhance those interactions going forward
Each theme self-organized to develop a strategy within the context of Oxford, the UK, the
European & emerging global strategy for particle physics, broadly interpreted.
Two scenarios: blue sky 10 - 20 years
tempered by reality over next few years
3
Higgs boson and EWSB � mH natural or fine-tuned ? Æ if natural: what new physics/symmetry? � does it regularize the divergent VLVL cross-section at high M(VLVL) ? Or is there a new dynamics ? � elementary or composite Higgs ? � is it alone or are there other Higgs bosons ? � origin of couplings to fermions � coupling to dark matter ? � does it violate CP ? � cosmological EW phase transition
Neutrinos: � ν masses and and their origin � what is the role of H(125) ? � Majorana or Dirac ? � CP violation � additional species Æ sterile ν ?
Dark matter: � composition: WIMP, sterile neutrinos, axions, other hidden sector particles, .. � one type or more ? � only gravitational or other interactions ?
The two epochs of Universe’s accelerated expansion: � primordial: is inflation correct ? which (scalar) fields? role of quantum gravity? � today: dark energy (why is Λ so small?) or gravity modification ?
Quarks and leptons: � why 3 families ? � masses and mixing � CP violation in the lepton sector � matter and antimatter asymmetry � baryon and charged lepton number violation
Physics at the highest E-scales: � how is gravity connected with the other forces ?
� do forces unify at high energy ?
ATLAS
LHC-b
DUNE HyperK MINOS T2K SNO+0vvββ
LZ
LSST
HL-LHC/HE-LHC/ ILC/CLIC/FCC
Oxford is one of the few universities on the planet that has an experimental programme addressing most of these questions in the same department as an active theory groupp!
Underpinned by strong theory group - YOU!
1
DARK UNIVERSE
DRAFT
A green paper produced for the particle physics strategy exercise by members of the
sub-departments of astrophysics, particle physics, and theory
Editors: Joe Conlon, Jo Dunkley, Pedro Ferreira, Hans Kraus, Subir Sarkar,
Ian Shipsey
Authors: James Binney, Joe Conlon, Roger Davies, Julien Devriendt Jo Dunkley,
Pedro Ferreira, Uli Haisch, Hans Kraus, John March-Russell, Lance Miller,
John Maggorian, Subir Sarkar, Ian Shipsey, Adrianne Slyz
! !
In addition to PP many authors from Theory and Astro
1""
Oxford'LHCb'group:'New'Physics'searches'with'quark'flavour'measurements'"
1. Introduction'
Flavour"physics"boasts"a"long"history"of"discovery"physics"by"making"precision"measurements"of"decays"sensitive"to"high"mass9scale"currents"running"in"virtual"loop"processes."LHCb"is"currently"the"luminosity9frontier"experiment"and"has"had"a"hugely"successful"period"of"operation"since"machine"startup"in"2010."The"experiment"has"collected"a"total"of"3"fb91"of"data,"an"achievement"made"possible"by"the"experiment’s"ability"to"run"at"over"twice"its"design"luminosity."The"Oxford"group"has"firmly"established"itself"as"one"of"the"most"productive"groups"in"the"LHCb"collaboration"and"we"are"proud"that"we"have"played"a"central"role"in"the"acquisition"and"analysis"of"these"world9leading"results."LHCb,"together"with"its"upgrade,"will"provide"a"rich"source"of"physics"measurements"for"at"least"the"next"fifteen"years.""
The"Oxford"group’s"long9term"strategy"for"physics"exploitation"is"dictated"by"our"core"specialities"in"the"design"and"operation"of"the"Ring"Imaging"Cherenkov"and"the"silicon"microvertex"detectors,"as"well"as"our"responsibility"for"the"calibration"of"the"experiment’s"particle"identification."Our"LHCb"physics"planning"was"key"to"us"joining"the"CLEO9c"experiment"in"2006,"where"we"made"several"important"measurements"with"quantum9correlated"D9decays,"results"which"provide"essential"input"to"the"CKM"and"charm"measurements"we"make"on"LHCb"today."Our"strategic"goal"for"physics"analysis"broadly"spans"the"following"three"areas:"""
Precision"CKM"metrology"where"we"have"led"several"flagship"analyses,"most"notably"the"first"LHCb"measurement"of"the"unitarity9triangle"angle" ."Here"and"in"the"future"our"CLEO9c"quantum9correlated"results will"play"an"important"role."
Charm"physics"where"we"have"been"instrumental"in"making"the"first"measurements"of"CP"violation,"with"the"future"aim"to"pursue"the"accuracy"down"to"the"Standard"Model"level;""
Exclusive"diffractive"production"in"pp"collisions"to"rigorously"test"QCD,"where"we"have"pioneered"the"physics"and"also"the"instrumentation"for"the"new"Herschel"forward"detector."""
In"addition,"group"members"make"important"contributions"in"the"following"areas:"
Excited"heavy9flavour"spectroscopy:"exploring"the"limits"of"the"quark"model;" Electroweak"loop"decays:"exploiting"sensitivity"to"particles"at"a"high9mass"scale.""
The"Oxford"LHCb"group"has"three"permanent"academic"staff"members,"Neville"Harnew,"Malcolm"John"and"Guy"Wilkinson."We"also"have"two"STFC9supported"postdocs,"Sneha"Malde"and"Christopher"Thomas"with"a"third"soon"to"start,"Paolo"Gandini,"an"EU"Fellow,"Denis"Derkach,"and"an"Oxford9supported"postdoc"working"on"the"upgrade,"Pavel"Jalocha.""We"have"around"8"graduate"students"over"3.5"years"of"their"term."In"addition"we"have"an"electronic"(Rui"Gao)"and"mechanical"engineer"(Matthew"Brock),"as"well"as"a"physicist"programmer"(Stig"Topp9Jorgensen)."We"also"make"regular"use"of"the"Oxford"Electronic"and"Mechanical"Workshops."
Harnew"and"Wilkinson"were"heavily"involved"already"at"the"inception"of"LHCb,"and"group"members"have"since"taken"leadership"roles"in"the"collaboration."Wilkinson,"was"the"first"elected"Physics"Coordinator"of"the"experiment"and"has"since"been"elected"Spokesperson,"a"position"which"he"will"occupy"until"June"2017."Harnew"has"had"a"49year"term"as"Collaboration"Board"Chairman"of"LHCb"and"
Accelerator Based Searches for Physics Beyond the Standard Model
and the Measurement of Standard Model Benchmark Processes
Oxford ATLAS Group Vision and Strategy
DRAFT 23.11.2014
A green paper produced for the particle physics strategy exercise
Alan Barr, Daniela Bortoletto, Amanda CooperSarkar, Elizabeth Gallas, Claire Gwenlan, Chris
Hays, Todd Huffman, Cigdem Issever, Juan Rojo, Ian Shipsey, Jeff Tseng, Tony Weidberg
1
Recent Theory/Experimental Particle Physics Collaborations Joint Supervision: • Uli Haisch/Guy Wilkinson: supervised Rhorry Gauld who studied the potential of top
measurement at LHCb • Juan/Rojo/ AMC-S : supervise Franceso Giuli who is involved with High-Mass Drell-Yan
measurements in ATLAS, both for BSM physics and as an SM measurement to be input to PDF fits
• John March-Russell/Alan Barr : supervised James Scovell who studied Supersymmetric Searches in ATLAS– and Jesse Liu has just joined them
Faculty Interaction: • Giulia Zanderighi/Chris Hays: estimation of uncertainties on Higgs boson cross-sections
involving measurements with jet binning • Juan Rojo/AMC-S/Claire Gwenlan: inclusion of ATLAS data into PDF fits including MCFast
techniques– most recently W+c with Valerio Bertone • Guido Bell/AMC-S/CG: calculations of the transverse momentum of the Z boson for inclusion
in PDF fits • Guido Bell/Chris Hays: accurate modelling of the transverse momentum of the Higgs boson • Juan Rojo/Cigdem Issever: feasibility study for the measurement of Higgs self-coupling in the
4b state with boosted techniques PDRAs • Re was involved with the development of POWHEG and MiNLOand Monni with theory
uncertainties in jet vetoes
Astrophysics
• Circa 29 faculty
• Head of sub-dept: Prof Steve Balbus
• Areas
• Instrumentation (ELT,SKA,CBASS, WEAVE, ...)
• Observations (CMB, Optical, Radio, IR. Xray)
• Theory (compact objects, galaxies, large scale structure, early universe)
Monday, 12 October 15
Cosmology• Jo Dunkley (CMB)- Planck, ACTs
• Julien Devriendt (Galaxy evolution and formation)- computational and analytical techniques
• Pedro G Ferreira (Cosmology theory)- Early universe, general relativity, large scale structure
• Lance Miller (Weak Lensing)- CFHTLenS, Euclid, KIDS
• Joe Silk (Astroparticle and Galaxies)- dark matter, large scale structure, CMB
• Adrianne Slyz (Galaxy evolution and formation)- computational and analytical techniques
Monday, 12 October 15
Beecroft Institute• David Alonso (LSS)
• Tessa Baker (GR, LSS)
• Rafael Alves Batista (Astroparticle)
• Erminia Calabrese (CMB)
• Elisa Chisari (WL)
• Giuseppe Congedo (WL, GWaves)
• Julian Mertens (WL)
• Sigurd Naess (CMB)
• Johannes Noller (GR, Early Universe)
• Mark Richardson (LSS, Galaxies)
• David Sloan (Inflation, Early Universe)
• Niraj Welikala (WL, Galaxies)
• Hans Winther (GR, Early Universe)
Monday, 12 October 15
String Theory Group (Maths Branch)• James Sparks ; AdS/CFT
• Fernando Alday ; integrability, strong coupling limit of N=4 SYM
• Lionel Mason and Andrew Hodges ; twistor string theory, amplitudes of N=4 SYM amplitudes.
• Xenia de la Ossa and PC ; CY manifolds and heterotic string theory, non-Kähler manifolds.