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Michaelmas 2019 Michaelmas 2019 http://www2.physics.ox.ac.uk/research/particle-theory Applicants resident in the UK will usually be called for interview in March; overseas applicants may be interviewed by phone/skype. Preliminary offers (subject to satisfactory examination results) are made at this time, both for places and for STFC studentships. Oxford Particle Theory Group Founded in 1963 by Richard (Dick) Dalitz Former members: Jack Paton, Ian Aitchison, John Taylor, Chris Llewellyn- John Wheater 1986- Smith, Frank Close, Graham Ross*, Mike Andrei Starinets 2008- Conformal field theory Teper*, Giulia Zanderighi, Uli Haisch Gauge-string duality, & quantum gravity holography, AdS-CFT + presently 6 postdocs & ~15 DPhil students + Visitors: Christopher Herzog (KCL) Stephen West (RHUL) Lorenzo Tancredi RSURF 2019- Andre Lukas 2004- + Many associates in Astro, PP & Maths Inst String theory and phenomenology Supported by: UKRI, EU, Royal Society … Lucian Harland-Lang ERF 2017- Subir Sarkar 1990/98- Joe Conlon 2008/14- John March-Russell 2002- Gavin Salam 2018- Fabrizio Caola 2019- Particle astrophysics Physics Beyond the Phenomenology of EW & strong interactions and cosmology Standard Model Oxford alumnus shares Nobel Prize in Physics 2016 Staff JOSEPH CONLON String theory and compactifications String phenomenology Also interested in astroparticle physics, cosmology and BSM Current interest: the swampland – how does ultraviolet consistency constrainJoe low-energy effective Conlon Lagrangians? Students: Filippo Revello (2nd year), Sirui Ning (1st year) Andre Lukas Main interest: string theory, with emphasis on compactifications, model building and phenomenology. More specifically: • Calabi-Yau manifolds, vector bundles and heterotic model building • Flux compactifications and non-CY manifolds • Machine learning andAll this string suggests a di↵erent theory approach to machine learning of line bundle cohomology which incor- porates information about the mathematical structure of the problem. This is what we turn to • M-theory compactificationsnow. and F-theory • String cosmology 4 Learning cohomology formulae As discussed earlier, there are strong indications that cohomology dimensions can be described in terms of a relatively simple mathematical structure. The Picard lattice is divided into regions - fre- quently, but not always cones - in each of which the cohomology dimensions are given by a polynomial Major theme: “Gettingwith the degree equal standard to the complex dimension model of the manifold. Thisfrom behaviour hasstring been empirically theory” observed for a number of examples [11,15], including surfaces and three-folds, and it has been proven for classes of surfaces [14] as well as for some three-folds, such as the bi-cubic [11]. Our hypothesis For example arXiv:1906.08730: here is simply that this structure is fairly general and in particular applies to all examples we will study. Where currently no mathematical proof is available, further evidence for the hypothesis will in fact be provided by the approach to machine learning discussed in this section. region 1 Machine Learning Line Bundle Cohomology region 2 region 1 4.1 Network structure region 3 region 2 10 10 10 region 4 region 3 k2 k 10 The structure of the network we will be using0 is indicated in Fig. 4. 2 0 1 2 1 1 k 10 region 4 10 2 0 region 5 k2 Callum R. Brodie , Andrei Constantin , Rehan Deen , Andre Lukas 10 0 -10 region-10 5 10 -10 region 6 -10 0 region 6 k 0 -10 1 not0identified -10 k1 0 1 k1 not identified k Rudolf Peierls Centre for Theoretical Physics, University of Oxford, -10 -10 1 0 0 Parks Road, Oxford OX1 3PU, UK 0 h n1 n1 n2 -10 0 -10 k0 -10 k0 -10 Z R R R k0 n2 k0 k (W1, b1) (W , b ) 10 10 σ 2 2 σ R 10 10 2Pembroke College, University of Oxford, OX1 1DW, UK R Mansfield College, University of Oxford, OX1 3TF, UK Figure 6: RegionsFigure in the 6: PicardRegions lattice in the for Picard the lattice zeroth for cohomology the zeroth cohomology of dP2, determined of dP , determined from the from trained the network trained network in in n2 2 Fig. 4 for (n1,n2Fig.)=(84 for, 8)(nand1,n2 a)=(8 trainingR , 8) and box a of training size kmax box of= size15.k Themax figure= 15. The on the figure left on shows the left the shows regions the regionsobtained obtained N after step (2) ofafter theZ algorithm. step (2) of theThe algorithm. figure on· The the figure right onshows the rightthe “cleaned-up” shows the “cleaned-up” regions obtained regions obtained after step after (4) step of (4) of x =(ki,kikj,...) the algorithm.(W3, b3) The legend labels the regions as in Eqs. (4.3), (4.4) Abstract gθ the algorithm. The legend labels the regions as in Eqs. (4.3), (4.4) We investigate di↵erent approaches to machine learning of line bundle cohomology on complex polynomial fit topolynomial each of thefit to six each regions of the (step six regions (3) of (step the algorithm)(3) of the algorithm) leads to leads to surfaces as well as on Calabi-Yau three-folds. Standard function learning based on simple fully Figure 4: Structure of the network to learn piecewise polynomial cohomology formulae. The dot in the element connected networks with logistic sigmoids is reviewed and its main features and shortcomings are 1+ 3 k + 1 k2 + 1 k 1 k2 + 1 k 1 k2 in region 1 , on the right indicates a dot product, taken between the two input vectors. 3 1 2 2 10 2 10 2 2 11− 2 11 22 2 − 2 2 discussed. It has been observed recently that line bundle cohomology can be described by dividing 1+ 2 k0 + 2 k0 + 2 k1 2 2 k1 + 2 k2 2 k2 in region 1 , 81+2k0 +−k + k1 + k0k1−+ k2 + k0k2 + k1k2 in region 2 , the Picard lattice into certain regions in each of which the cohomology dimension is described 2 0 Let us discuss how this network relates to the underlying mathematical81+2 structurek0 + k>0 of+ k line31 + bundlek01k12 + k1 2 + k10k22 + k1k2 in region 2 , >1+ k0 + k + k2 k in region 3 , by a polynomial formula. Based on this structure, we set up a network capablecohomology. of identifying The training data used for this network is of the formh (2.160>( ), that(3k)) is = we>1 are2 adding2 1 2 the10 2 2 − 2 2 (4.3) >1+dP2 k0 + >k +3 k2 1 2k 1 1 2 in region 3 , the regions and their associated polynomials, thereby e↵ectively generating a conjecture for the 0 >O 2 >2 1+0 k20 + k20 +2 k1 k1 in region 4 , monomials kikj and, for three-folds, also the monomialsh ( dPkik2j(kkl ))to = the> input vector>ki, so2 that−2 the 2 2 (4.3) > 3 <1 2 3 1 1 1 2 − correct cohomology formula. For complex surfaces, we also set up a network which learns certain O >1+ k0 + k1+0 + kk+1 k2k1 in regionin 4 region, 5 . problem becomes e↵ectively piecewise linear. > 2 2 2 20 −2 20 rigid divisors which appear in a recently discovered master formula for cohomology dimensions. < 3 >1 2 The upper part of the network in Fig. 4, denoted by g✓,where✓ =(W1+1, b1,Wk02+, b>2),k00 is a two-layer in region in 5 region. 6 . 2 >2 network with n1 and n2 neurons in each layer, whose input is simply> the vector k.> We can think of >0> in region 6 . the output, g✓(k) as a “binary code” which indicates theUsing region these within equations> which to the determine: vector k theresides. exact regions (step (4)) leads to the plot on the right-hand- In other words, this upper branch of the network detects the various> regions in the Picard lattice. Using these equationsside of Fig. to6 determine.:> In step (5), the we exact thenregions determine (step the inequalities(4)) leadsto which the describe plot on those the right-hand- regions. They are Current students: Stefan Blasneag,The width Callum (n1,n2) of this Brodie, network should be Pandora of the ordergiven of by the expectedDominiak number of regions. The underlying assumption is that the regionsside are of Fig.given6 as. In intersections step (5), we of half-spaces, then determine which theis the inequalities case which describe those regions. They are for most examples studied to date. given by Region 1: k1 0 k2 0 k0 + k1 + k2 0 − ≥ − ≥ ≥ The lower part of the network is a simple linear layer withRegionn neurons 2: and weights/biases (Wk0, b+)k1 + k2 < 0 k0 + k1 0 k0 + k2 0 2 3 3 ≥ ≥ which represent the coefficients of the variousRegion polynomial 1: kRegion equations.1 0 3: Byk2k multiplying<00 k0 +k thesek1 +0 weightsk2 0 k + k 0 − ≥ −− ≥1 − 2 ≥ ≥ 0 2 ≥ (4.4) into the monomial input vector x =(k ,kRegionk ,...) this 2: layerRegion outputs 4: a vectork which0 k0 +k consistsk<1 +0 k of2 < the0 k0 + k1 0 k0 + k2 k 0+ k 0 i i j − 1 ≥ − 2 ≥ ≥0 2 ≥ values of n2 polynomials in k. Dotting thisRegion vector into 3: thekRegion “binary< 0 5: vector”k k1 from<00 thek upper2 < 0 network k + k 0 k0 0 − 1 −−2 ≥ − 0 2 ≥ ≥(4.4) selects the specific linear combination of polynomialsRegion 4: associatedkRegion to0 6: the regionotherwisek < 0 in which k resides. k + k 0 − 1 ≥ − 2 0 2 ≥ Region 5: k < 0 k < 0 k 0 In− summary,1 the− network2 has learned the formula for the dimensions of the zeroth0 ≥ line bundle coho- Region12 6: otherwise mology on dP2. By applying the master formula (2.4)todP2, it can be shown that the above result arXiv:1906.08730v1 [hep-th] 20 Jun 2019 is indeed correct on the entire Picard lattice.
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