T.YUKAWA and S.HORATA Soknda/KEK D SIMPLICIAL QUATUM GRAITY Plan of th talk Rvw of th D slcal quantu gravty Rvw of nurcal thods urcal rsult and dscusson
Whr dos th slcal quantu gravty stand? In short dstanc xstnc of th contnuu lt In long dstanc Enstn gravty as th classcal lt
n-slx 3 D slcal anfold Condton : two -slcs attach through on 3-slxfac and on 3-slx s shard by two - slcs. Condton : -slcs sharng on - slx consttut -ball. x. sallst -ball
anfold condtons gv 5 3 5 3 + 3 Eulr rlatonsh 3 + + χ χ : total nubr of -slcs : Eulr charactrstc Choos and 3 χ 5 5 + + 3 as ndndnt aratrs
Slcal gravty Rgg96 Enstn-Hlbrt acton Eucldan S G { d x π g R n + n! + Λ d x cos g } + Λ{ n n a n volu of n-slx + π 5 cos π.99 }
Flds on slcal anfold Lattc -for flds L -for fld -for fld -for fld φ φ j φ jk on a st φ x on a lnk j x x j dx µ φµ x µ ν on a trangl jk dx dx φ x Dual flds on th dual lattc d-cll : d-olygon dual to a st Ld t jk µν d--cll j: fac shard by d-cll and d-cll j j
Dual flds ψ ψ j -for dual fld: on th d-cll -for dual fld: on th d--cll j -for dual fld: ψ jk on th d--cll jk Scalar roducts φψ Dualty rlatons φ l jk... L ψ jk... l j L φ jψ j d ψ jk... l... φ jk... jk σ d ψ a φ < φ ψ > σ jk... φ jk ψ jk t jk : volu of th slx jk : volu of th cll jk
Mass trs M φ < φ ψ Drvatvs d d : L L + rark: d Kntc trs K φ M dφ < dφ dψ > > M M σ φ l j σ j φ j φ φ j dφ j dφ jk φj + φ jk + φ k K σ j φ φ j l j K σ jk φj + φ jk + φk t jk
Mattr flds S M rarks: n X n A n K K n K X + K n A X ο lj X X j l j X ο tjk Aj + Ajk + Ak t jk d a σ a ο s d d Thn for ass trs σ a ο s a { s } n { s } and for kntc trs σ { s } a d
urcal thods Partton functon Z DX DA T S G M S S + S G X S A S Rark: s th su ovr all th ossbl gotrcal confguratons T Mont Carlo ntgraton Gnraton of a Markov chan α α... α α +... M S α TXA wth th wght Dtald balanc Sα Pα α n n α S α+ + + α + P α α α S S G + SM α n α : # of ossbl ovs fro α
Gotrcal ovs Elntary q-ovs q 3 5 5 5 33 3 - - -5-5 - -5 - - - Rark: Ergodcty of lntary ovs
5 5 Mov Mov 33 Mov
Choc of th gotrcal ov Startng a nu confguraton wth a fxd toology Choos th -ov wth th robablty P S G js j G Hat-bath thod 3 q 5 33 5 S G + +
Gotrcal constrants Aftr { }-ovs 5 5 3 3 3 3 3 3 3 + + + + Bsds thr nvarant condtons for bcaus w hav startd a th nu confguraton 5 3
Postvty of th slcal acton Th two nqualty rlatons gv 5 5 Postvty of S G s satsfd Whn or + Λ cos 5 π π whch s always tru for. Λ -.98
Gotrcal ov contnud 3Scfy th ov by choosng a slx for th -ov randoly. Chck th anfold condtons. T T Acct th ov wth th robablty Mtrools thod } n{ T T n n T T P whr n T s # of ossbl ovs fro confguraton T: 5 T s T n T 5 5 3 3 5 s Thn T T n n T T
Mattr ovs 5 Put flds randoly whn a nw vrtx and/or lnks ar addd. 6Mattr ovs ar ad for all X and A flds by th hat-bath thod. P S x X x x + S x S X dx X x ο lj x X j j 7Instad of x w ay choos Th ovr rlaxaton thod X x + X X j ο l j j ο l X j j
Rsults of th nurcal sulatons Calculaton of artton functons + A X T S M Z α S M α S M z α c ζ Wth a fxd toology S and n X n A.
c S W calculat z G S G + by tunng as. c wth a odfd acton
Scalar curvatur < >/ < d d > ln z c For larg t s boundd.. < >.5 nar.5: 5 ovs donat branchd olyr has nar.: ovs donat cruld surfac has
3 Curvatur susctablty χ ln χ d d z c > > < <
Short dstanc corrlaton n t C t { ο t ο n t t s a trangl havng a coon lnk wth th trangl t t }
Dscussons Is th analogy to th D gravty OK? In D th scalng rlaton of DDK holds xactly: z DDK αω zddk αω + Q In th DT calculaton w xct Z DT z DT T ζ T ωχ Thn w hav a corrsondnc: Z DT a ζ a a a λ dz a DT a c c c λ
In D th conforal gravty assus th scalng ωχ αω βω αω b C C z z Whl n DT w calculat S DT M Z α ζ α z S DT c M Th contnuous lt s takn as a DT c a a a a Z ζ DT z d λ ωχ αω βω αω ζ ζ b Thus
Th artton functon xctd fro th conforal gravty Z DT ϕ µ ν c whr α β ar solutons of α α β b β b wth b n X + 6nA + 538 36 Ths fors do not allow us to ak us of th old MIBU algorth and w hav to chck all th old rsults.