discrete time evolution and baxter's q-operator · functor z: zcob → vec # if-vector spaces...
TRANSCRIPT
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
Discrete Time Evolution and Baxter's Q-operator
RAQIS 16
a ( nu ( nu ::( un
Road map: quantisation of the Ablowitt - Ladik chain
Classical integrate system [Abeowik . Ladik ' 76 ] Quantum integrate system
zxj = { H ,4j }, 2t4j*= { Hits
't }#= .jz(Bkjt÷BB*- znj )
µ -
tgttsttjnttitjtt'
- 2lnHY*4D )[ kueishiqgi
Poisson algebra of- boson algebra
[ .
,. ] = - it { .
,. }t0th2 )
{ Yi ,Xj* } = Sij C l - 4g*4j ) [ pi ,Pj*I= Sijll -
of ) a- Rpi )
quantisation
{ 4..,4j}={Yi*,Xg*3=o £ → it,
o < q=etc < 1
4*→±5integrals of motion ' Bethe algebra
'
=
ZDTQFTLjcu ) = his.
"u%* ) Ljlui = ( B,
"u&* )
→ monodromy matrix,
no monodromy matrix,
YB - algebraspectral invariants Baxter 's commuting transfer matrices
Ablowitt . Ladik chain : separation of time flow
Equations of motion
2- Xj = Yj+ , -24J + Yj . ,
- Xj*4j ( 4g ; ,+ Yj . ,
)
{2.4¥- YE, tarts
's- YE't4j*4jHjEtYit '
)
Decomposition of Hamiltonian into left - and right movers
H = Hr + Hit Ho,
Hr =
,? 4j*Xj+ , , Hi ,?4j4j*t, ,{ He ,Hr}=o
'
Auxiliary time flow'
at ,4j=- { HL ,4j} = 4g ,( l - 4g*4j )
Since all 3 flows commute,
we can consider them separately .
Datboux matrices : Dj+ ,( u ,v ) Lj ( u ) = [ j
( u ) Djcu ,v ,,
discrete zero . curvature
equation
det Dj C v.v )=o and Djlu ,0 ) = ( ff )
v )Bcicklund transform : ( Xj ,4j* ) /( Tlj ,Ij* )
,not
, Ij=YjCot)
�1� Canonical map which preserves the Poisson structure { .
). }
�2� Commutativity : BCV ,) ° Blvd = B ( vz )°B( v
,) [ Veselov
'
91 ]
'
time'
discretisalion : Yj ¥1210 '= ( 1 - 4j*4j Cot ) ) Yj . ,
Cot )
[ Sun 's 1997 ] ot
What is the quantum analogue of this evolution eqn ?
of - boson took space periodic boundary conditions
• •
• ••
Vacuum : plo>=o •
mot . boson stale : im >=H*zM,o,:^
.!^
..
: :
9- mz mz m,
Mn
n
multi . particle stale : li >=01miH ) > partition d=( d, ,
...
,,\n)=(Im'2m?. .nmn )
i= ,
P*jlX>=( 1-qmjtl)lm, ,
... ,mj+ ,,
... .mn >,
Example shown above : n=io
X= ( 10,10 , 817,7, 5,5 ,5
, 4,4 , 3,313,3 , -2,111,1 )
fj It >= Im, ,
... ,mj -1,... ,mn ) K = 2+1+2+3 +2+4+1+3 = 18
•
•
→ Canonical quantisation of the Poisson algebra : An •
•
[ pi ,pj*]=Sija-9.111 . pitpi ) ;is
Quantum Backlund transform → Baxter 's Q . operator [ Pasquier - Gaudin 1992 ]( Todachain )
( Bj ,pg*li→ ( fj ,fg* )fj = Qcnpj Qcvi
'
fj*= Qcvspjtaa ,.it#n : find Q
Define Qcv ) as the transfer matrix of an exactly solvable vertex model :
of- insertion : insert a particles into a pile of b- c particles inside a gravitational
potential with of= 5 PE
,E energy to lift one particle .
••I c
• →
• ob
4 •• 'o o at:::←s¥tI÷=i Iq
' ' ' '. [ da ]
, a##µ1 insertion
q4+3+3+1
= ql'll D= at b - c
a ,b ,C
,d E 2120
particle picture Boltzmann weight vertex configuration
Lattice configurations & of- Whittaker polynomials
Let
MeMatch. , ,×n( No ) and set air pipit,
INPUT
µ=(In'zm . .cn .
nmn " ) t¥
µ=⇐po*Mioa,Mn. ..in?mpnTi0
Mom,
i. .
Mn. , ( )MioM
, ,Mhm, ,. . . m,o Define A
-then
✓,
it th . ' ( of )Mio(9- )Mi,
" . l F )Min. ,
yMzoMz,MzzMz}Mzovmmmomn. "Manmm.
the Matrix elements ofmimi... min
Zlv )= EmTam'
,
vM=IT vomij
. 1 xiijs
d=( 1mi. :( mjmh ' ' ) are the partition function .
OUTPUT
( cylindrical ) skew of - Whittaker functions
HIM[ CK'
13 ] Open boundaries
Mio=o: C
KIZCVIIM> =P
.,µ,(v ;q ,
0 )
periodic boundariesMio >0
:<KIZCVIIM> = Ed ZDP
.io/ailv;q..o)
~
Quantum Backlund transform pjcv ) = Qlv ) PjQ(v5'
where
Qcv ) is the row - to - row transfer matrix of the of. insertion model .
htm [ CK ' 16 ] discrete quantum"
time flow"
±YI =(1-
pjfgcvhfjncvik*MjPI*=Pitt, (Bitfjm-1)
Discrete time evolution [ Qlu),Q*lv ) ]=O
§; =Q*cvi§jQ*cv ) §j- pj= - v( 1-FYP ; )Pj+ , functional equation
p ;( ot ) =
Q*fioH"Qciot) pj Qliotl"Q*fioH,
ucotii Ucots't
*- 5
'
Ulot ) time evolution operator
&l9toEfI= (l - Fyttiotlp; )pj+, ta-
fjtciotspjcot, )p, . ,( ot ,
for time step 't
Multivariate Biicklund transforms & TQFT fusion matrices
BTCV,) o ... o
Btcvn. ,
) m> ZCV ) = QCV ,) " .
QCK!= ? Qx Pxcv ; of ,
0 )
J 4Fusion matrix
of- Whittaker function
TIM [ CK'
13 ] Fusion matrices of ZD TQFT for fixed particle number k.
v.Naujukcqt< vl Qxlm > = ¢n§ '
pair of pants'
2- cobordism
R in
Recurrence relations for fusion coefficients
a- qmim"
) NYjs*kw'
= a- qm. " ") NEW '±a.qmit'
' '''sNrfdtjuhtka.qmic " )a .qmt '
'"sNrPBjtjYL"
of →o : nisei'=New'±NrB"think .FI'"
such)h - WZW fusion ring'phase model
'
[ CK. ,
C. Stoppel Adv
.
Math. 2009 ]
ZD TQFT ± Symmetric Frobenius algebras [ Aliyah
19881Functor Z : Zcob → Vec
#
IF - vector
spaceswith dimV< co Z C On )=V ZCO )=V*
s'
multiplication m : ZCSYOZCS 's → ZCS 'sZ(&÷j§)E Horn (
ZCSYOZCS's
,ZCS
'
) )( commutative )
aoxbt ab
invariant
[email protected]. ) )
bilinear form IF
cab ,c > = < a ,bc >
unit element e :
F→z(s
') ZCO ) E Horn ( IF
,ZCS
'
) )1 H I
TQFT partition function Z(€YFfE¥c::# )
genus g Surface
ZD TQFT operator version of- bosons
ZCQ) E Vec
. ,
F=Z[ 9- it' '
] Bethe algebra Bmkc End ( Fr ,)
si p÷j§)eHom(Zcssozcs
's ,Zcs 's )
Q×Qµ=FNdi%9*÷ of ]
IF 1-Z ( @ )eHom( ZCSYOZCS 's ,ZC . ) ) < Qx,Qµ>=£xµ*Fh[mica ) ]q !
invariant bilinear form ×= imizmz ... nmn
Z ( O ) E Horn ( IF,
ZCS'
) ) Qcn,
... ,n ) Qx = Qxunit element
ZCEXETER: :#) TQFT partition function Tr (,§Qx§*)9"
genus g surface
ZD TQFT operator version of- bosons
ZCQ) E Vec
. ,
E=Z[ 9- it' '
] Bethe algebra Bmkc End ( Fr ,)
:(&÷j§)eHom(Zcssozcs
's ,Zcs 's ) Q×Qµ= ? Nd'µY9*÷ of ]
IF 1-Z ( @ )eHom( ZCSYOZCS 's
,ZC . ) ) < Qx,Qµ>=£xµ*Fh[mica ) ]q !
invariant bilinear form ×= |m , zmz ... nmn
Z ( O ) E Horn ( IF,
ZCS'
) ) Qcn,
... ,n ) Qx = Qxunit element
ZCEXETER: :#) TQFT partition function Tr (,§Qx§*)9"
genus g surface
AIM : describe the discrete time dynamics in terms of the TQFT
Two Q . operators Q±cv)=§,ovrQ±r
Qtr . Epn.PE#EHR*Fi*an, as.cnzz.pitanHRI.ch#Mlpnanggaea*
Qtr ( Fla. . " ( of Ian ( of la,
. . ( of Ian
T±M0 Functional identities 0(u1= GZNU "
N=#ofq - bosons
CK'
13,16
Tcu )Q+Cul= Qtcuqijtocu )QHuq'
),
TQ+ equation
Qtutlu )= QTuq3+o(uq2)QTuq2) QT equation)
Qtcu ) Qtuqtj - unq2N Qtcuq 's QTU ) = I'
quantum Wronskian '
The functional relations also imply the quantum analogue of BH . )°Blk)=BWoBw,)
Cor [ Qtr ,TsI=[ Qtr ,Q's
]=[Q±r,Q±s]=O.
' Bethe algebra'
C of- boson
algebracommutative
non .commutative
Because we are dealing with non . commutative variables in the quantum case,
the
equation defining the Darboux transformation is now replaced with the Yang - Baxter eqn :
Dfduiv) Lytu)Esjlvl=Ltsjlv) Ly. lu )Dfzluiv )
This allows one to define Q± in a similar way as Tcu ) = Tr Lncul . :L,
lu )
Q± . operators for the of boson model [ CK 2013,
2016 ]
M 1* M
Eons = ( trim MIMI"
)mm
,
, . ija' = ( vmqm'z± ' ' fight)mm
.
, .
Current operators ( formal power series in v with coefficients in of- boson algebra)
Qttv) = Trench. .E Cr ) = Four QE "
explicitly known
Omitted from the discussion
D Combinatorial approach to compute Ny,u ( q ) e z [ of 1 f.Hall polynomial
Recall skew Macdonald functions : Px ,µ( x ; of it ) = ? ftp.vlq.tl Pvlx ;qH
→ cylindnc of- Whittaker functions Px ,d,µ( x ;q ,
0 ) = ? Nstuvlq ) Pv ( x ; q ,o )
D N ,Iu (OIEZ,o are the such )r - WZW - fusion coefficients k=# off - bosons
TQFTg=o
I Ko ( E ),
E lens or category of Uesuch ) tilting modules
with e = el "k+h ( QFT : Chern . Simons )
→ Geometric interpretation at of to ?