Discrete Time Evolution and Baxter's Q-operator Christian.Korff@glasgow.ac.uk ( :( ( : a un nu nu 1) Cylindric Macdonald functions and a deformed Verlinde algebra, CMP 318 (2013) 173-246 2) From quantum Bäcklund transformations to TQFT, JPA 49 (2016) 104001 RAQIS 16
Ablowitt chain quantisation Road Ladik of the map : - ' 76 ] Quantum integrate system integrate system [ Abeowik Classical . Ladik 't } { H ,4j } { Hits 2t4j*= zxj jz(Bkjt÷BB* = , #= ) . znj - - 2lnHY*4D ) tgttsttjnttitjtt µ ZDTQFT - ' [ kueishiqgi - boson algebra Poisson algebra of . ] it { }t0th2 [ ) . = - . . - 4g*4j ) , , a- Rpi ) [ pi ,Pj*I= Sijll { Yi ,Xj* } - of ) Sij C l = quantisation etc < q= ,4j}={Yi*,Xg*3=o 4. { it £ < o 1 → . , 4*→±5 ' Bethe integrals of motion ' algebra = = ( B u%* ) his u&* ) " " Ljcu Ljlui ) = . , - algebra matrix matrix YB monodromy → monodromy no , , spectral Baxter 's commuting transfer invariants matrices
. Ladik Ablowitt separation of chain time flow : motion Equations of - Xj*4j ( 4g ; , + Yj + Yj Xj Yj+ , -24J 2- , ) = { . , . 't4j*4jHjEt YE Yit 2.4¥ tarts YE 's ' ) - - , Decomposition Hamiltonian into left right of and movers - = ,? 4j*Xj+ , Hi ,?4j4j*t { He ,Hr}=o H Hr Hit Ho Hr + = , , , , ' time ' flow Auxiliary - 4g*4j ) ,4j= ,4j { HL } ( l 4g , at = - them commute Since 3 flows all consider separately we can , .
,v ) Lj ( u ) . curvature [ j discrete zero matrices Datboux ( u ) Djcu , ( u Dj+ ,v , = : equation , ( ff ) Djlu ,0 ) det Dj and )=o = C v.v / v ) ( Tlj ,Ij* , Ij=YjCot ) ( Xj ,4j* ) ) transform Bcicklund not : , . } structure Poisson { which the Canonical � 1 � . map preserves ) ' 91 ] Blvd )°B( B ( vz , ) [ Veselov � 2 � BCV Commutativity , ) ° = v : - 4j*4j Yj ¥1210 ) ) Yj ' ' ' discretisalion ( 1 time Cot Cot ) : = . , 1997 ] [ Sun 's ot What the quantum analogue of this evolution ? is eqn
1- qmjtl took boson conditions periodic boundary space of - ,\n)=( • • • • • plo >=o Vacuum • : :^ !^ : : im >=H*zM,o . stale . . boson , mot : . Mn 9- mz m mz , li >=01miH n Im '2m ? . particle stale . .nmn ) ) > partition multi d=( d : , ... , , i= , above Example shown n=io : ... ,mj+ P*jlX>=( lm ) > , .mn , , , ... , ( 10,10 , 817,7 X= , 5,5 ,5 , -2,111,1 ) , 4,4 , 3,313,3 K ... ,mj 2+1+2+3 +2+4+1+3 18 fj It = = >= -1 ) Im ,mn , , , ... ; • • quantisation of the Poisson Canonical algebra An → : • • . pitpi ) [ pi ,pj*]=Sija -9.111 is
. operator [ Pasquier 1992 ] Baxter 's Quantum transform Gaudin Backlund Q → - chain ) Toda ( fj ' Qcvi Qcnpj ( fj ,fg* ) = : find ( Bj ,pg*li→ .it#n Q fj* = Qcvspjtaa , o at the solvable vertex transfer matrix Define Qcv ) of exactly model as an : - insertion a gravitational insert particles into b- pile of particles inside a a c : of 5 PE with to potential E particle lift = energy one of , . ••I c • → b • o : ::←s¥tI÷=i 4 a##µ •• ' da ] o [ ' ' ' ' Iq . , insertion 1 b D= at c 'll - 4+3+3+1 ql q = a ,b , C , d 2120 E particle picture Boltzmann weight vertex configuration
, Mzz Lattice mm Mio Mz Mz Whittaker y Mzo Mzo configurations & polynomials Man of - Mhm ,µ,( Me . , ,×n( No ) :< . pipit Match Let set and air , INPUT Mio=o ,o " ) In 'zm . nmn t¥ .cn . Mn µ=( vmmmomn .in?mpnTi0 µ=⇐po*Mioa,Mn mimi Tam ( ) Mom KIZCVIIM KIZCVIIM vM= i. . , . . , .io/ailv;q..o . Define then A M m - . . . ( of )Mio( 9- )Mi , , , , l F )Min it th . ' ✓ " . , . , , } the Matrix elements of . " . )= Em ' vomij IT Zlv min ... , xiijs 1 . ' ' ) . :( mjmh d=( 1mi partition the function are . OUTPUT ( cylindrical ) functions skew Whittaker of - v ;q HIM 13 ] 0 ) [ CK ' =P boundaries C > Open : , boundaries Ed ZDP ) > periodic Mio >0 =
± ~ ' Quantum where Backlund transform PjQ(v5 Qlv pjcv ) ) = . insertion transfer matrix the the Qcv ) of to model Bitfjm is row row of - - . " " time ' 16 ] discrete quantum [ flow htm CK k*MjPI*= Pitt 1) YI pjfgcvhfjncvi , ( =( 1- - Qlu ) ,Q*lv ) ]=O [ Discrete time evolution fjtciotspjcot - pj= Q*cvi§jQ*cv FYP §j § v( ; )Pj+ , 1- functional equation ) = - ; ) pj Qliotl Q*fioH" "Q*fioH Qciot * ucotii 't ( ) ot p ; Ucots = , time evolution operator ' 5 Ulot ) - for time step 't - Fyttiotlp , )p , &l9toEfI= ( l ta ; )pj+ , , . ,( ot -
Multivariate Biicklund transforms & matrices fusion TQFT = ? QCK ! Qx BTCV , ) QCV , ) Pxcv Btcvn ZCV ) 0 ) . , ) = " . m> ; of o o ... , J 4 - Whittaker function Fusion matrix of Naujukcqt ' 13 ] matrices [ Fusion number TIM CK of ZD TQFT for fixed particle k . = ¢n§ v. Qxlm 2- cobordism ' > pair of pants < vl ' R in Recurrence for coefficients relations fusion NYjs*kw Nrfdtjuhtka qmim . qmit qm ' ' 's '±a "sNrPBjtjYL qmic qmt ' " )a " " ' ' " " ' ' . ) NEW a- a- ) = . . '±NrB" nisei think '= ' .FI " New →o : of ' phase such )h WZW fusion ' ring model ] [ Math - CK Stoppel Adv C 2009 . , . . .
± Symmetric Frobenius algebras [ Aliyah ZD TQFT Z Zcob Vec Functor : → # ZCSYOZCS 19881 vector with dimV< C On IF )=V* )=V ZCO Z - co ' s spaces : F→z( Z(&÷j§)E Horn ( ) ) multiplication 's ' ZCS ZCSYOZCS 's ZCS 's → m : c.isiZCsyoxzC5s-EZl@1eHomCZCsY0ZCss.ZC , commutative ( ) ab aoxbt ) ) invariant . bilinear form IF a ,bc cab > > < ,c = ' ) ) ' ) ZCO ( unit Horn IF element ) E ZCS s e , I 1 H Z(€YFfE¥c :# partition function ) TQFT : Surface genus g
bosons ZCQ ZD si operator TQFT of - version ) ' ( Fr , ) Vec F=Z[ ' Bmkc End E 9- it ] Bethe Zcssozcs algebra . , p÷j§)eHom( 's ,Zcs 's ) ,Qµ>=£xµ*Fh[ Q×Qµ=FNdi%9* ÷ of ] IF 1- < Qx ) ) )eHom( mica ) ]q ! @ 's ,ZC Z ( ZCSYOZCS . bilinear invariant form nmn imizmz ×= ... ' ) ) O ( Horn ... ,n ) Qx Qx Z ) IF Qcn ( ZCS E = , , unit element EXETER ,§Qx§*)9 ZC :# " Tr ( partition function TQFT ) : surface genus g
bosons ZCQ ZD operator : TQFT of - version 'µY9* ) ' ( Fr , ) Vec E=Z[ ' Bmkc End E 9- it ] Bethe Zcssozcs algebra . , &÷j§)eHom( Q×Qµ= ? Nd ( 's ,Zcs 's ) ,Qµ>=£xµ*Fh[ ÷ of ] IF 1- < Qx ) ) )eHom( mica ) ]q ! @ Z ( ZCSYOZCS 's ZC . , bilinear invariant form , zmz nmn |m ×= ... ' ) ) O ( Horn ... ,n ) Qx Qx Z ) IF Qcn ( ZCS E = , , unit element EXETER ,§Qx§*)9 ZC :# " Tr ( partition function TQFT ) : surface genus g describe terms AIM the time discrete the dynamics of TQFT in :
. operators Q±cv)=§,ovrQ±r Two Q Epn as .PE#EHR*Fi*an cnzz.pitanHRI.ch#Mlpnanggaea* Qtr . . ( Fla. ( of Ian , ( of la ( of Ian Qtr . " . . , identities GZNU " T±M0 Functional N=#ofq bosons 0(u1= - ' 13,16 CK )Q+Cul= Qtcuqijtocu )QHuq ' Tcu ) TQ+ equation , QTuq3+o(uq2)QTuq2 ) Qtutlu QT equation )= ) ' quantum ]=[Q±r,Q±s]=O Qtcu ) Qtuqtj - unq2N Qtcuq 's QTU ) Wronskian ' I = also imply the The functional )°Blk)=BWoBw relations quantum analogue of BH , ) . [ Qtr ,TsI=[ Qtr ,Q 's Cor . ' Bethe ' algebra boson C - of algebra commutative commutative non .
variables with the the . commutative Because quantum are dealing in case we non , the - Baxter eqn equation defining Dfzluiv Dfduiv Esjlvl Ltsjlv Darboux transformation replaced with the Yang is now : ) Ly Lytu lu ) ) ) ) = . Tr Lncul . :L Q± to Tcu , lu ) This define allows similar ) one in a way as = . operators Q± for the 'z± boson [ CK 2016 ] model 2013 of , ' ' fight ) " = ( trim MIMI ija 1 = ( vmqm M * M ) Eons ' , mm . , mm . , . ( formal - boson algebra ) operators Current with coefficients in in power series Trench Qttv v of = Four E QE " explicitly ) Cr ) known = . .
Omitted the from discussion Hall polynomial Ny Combinatorial ( q ) N ,Iu ( OIEZ to compute z [ of 1 approach D e f. ,u = ? ftp.vlq.tl Pvlx ;qH Px ,µ( functions Recall skew ) Macdonald x ; of it : = ? Nstuvlq ) Pv ( x ; q - Whittaker functions x ;q , , o ) Px ,d,µ( 0 ) cylindnc → of - fusion such )r - bosons the WZW D coefficients k=# off are - ,o such ) tilting lens or category of Ko ( ) I E E TQFT Ue modules g=o , = el "k+h with ( ) QFT Chern . Simons e : interpretation Geometric at ? → to of
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