Preliminary version July 10, 2013 n atcePyis,Jl 03 uhm UK. Durham, 2013, July Physics”, Particle and otns1 Contents Contents ilorpy49 Bibliography ∗ .0Knmtc 25 24 ...... 26 . . . . 17 ...... 22 11 ...... 9 ...... 4 ...... for . . . . algebra . . . . . cluster ...... The . . algebras . . . . . 0.12 . cluster . . . . to . . 6 . . Introduction ...... 0.11 . . . . . 19 . . . . Kinematics . . . . values . 0.10 . . four Zeta . . . . transcendentality . . . . on . . . . details 0.9 . . . Functions . More . . Polylogarithm . . . on . . . Preliminaries . . 0.8 . . Mathematical ...... 0.7 . . Coproduct . . . 2 symbol . . . the 0.6 . . Integrating . . . . . invariance . 0.5 . . Homotopy . . Symbols . 0.4 functions. Transcendental values . zeta and integrals 0.3 polylogarithms Iterated characters: of cast 0.2 The 0.1 .3Pisnbakt 37 43 . . . 39 ...... geometry . projective . . of . Elements notions . mathematical of . Glossary . .2 . . .1 . brackets Poisson 0.13 oe o h umrSho Pllgrtm saBig ewe ubrTheory Number between Bridge a as “Polylogarithms School Summer the for Notes oyoaihsadpyia applications physical and Polylogarithms rsinVergu Cristian uy1,2013 10, July G ( ,n k, 1 31 ...... ) ∗ Preliminary version July 10, 2013 in u eto utsm ftems omntypes. common defini- general most a the give of not some will just We mention polylogarithms. but of tion, types many are There polylogarithms characters: of cast The 0.1 2 • • • • ecall we hs oyoaihscnb rte ntrso ocao polylog- Goncharov of terms in as written arithms be can polylogarithms These polydisc a in convergent are series power These with ah ntecmlxpae hyaemlivle ucin.W call We functions. multi-valued are They n plane. complex the in paths For polylogarithms. Goncharov the the where is notation Another series power as defined polylogarithms Multiple h cascl oyoaihsaedfie y(n tde yEuler, by studied simpler. (and are by Kummer) which defined Abel, cases are particular polylogarithms find “classical” can The we functions these From have we Obviously Li h egt(rtasednaiy ftepolylogarithm. the of transcendentality) (or weight the n n eavalues zeta and 1 G I ··· G ( n a function. k ( 0 k ( a ; x ; a h et and depth the 1 x a 1 x , . . . , = ) a , . . . , G ruet r eesdi the in reversed are arguments Li ( n a R 1 1 G 0 ,...,n x a , . . . , k n ( ( = ) t a ; − dt k a 1 a ( a , . . . , n h nerto otusaetknaogsome along taken are contours integration The . x +1 − 1 n x , . . . , n = ) ; 1) x 1 Li k = ) + n G Z n ; ( · · · x  a k x 0 a Z = ) | 0 = ) n = ) a 0 , . . . , +1 n n x 1 {z k k a , . . . , − t I t 1 X h weight. the p 1 (0; − ≤ − ∞ dt =1 dt 0 } p 1 X a a a x <...

integral in iterated that universal, An show are can above one defined and polylogarithms practice the in frequently appear integrals Iterated vacuum of the [REFS]. computation in amplitudes the also of scattering and in also theory [REFS] are theory appear field values They in zeta diagrams multiple physicists. The to interest theorists. old number One The is for MZV). Euler, teresting relations. (or by value” algebraic zeta found interesting “multiple example, satisfy a values is zeta side multiple hand right the where Li Λ polylogarithm Aomoto The form fw oki rjciesaeaddaietehprlnst points, 2( of to configurations hyperplanes in on the points depends dualize polylogarithm and Aomoto space the projective then in work we If an have we Suppose situation. general more n a considered Aomoto Li polylogarithm The h ucinLi function The have also We Li smlx∆ -simplex Li n n 1 = (1) ( 1 x ( ω x = ) M = ) X ihlgrtmcplso e fhprlns( hyperplanes of set a on poles logarithmic with CP − ζ − a ewitni em fhproaihs hs argu- whose hyperlogarithms, of terms in written be can ( G n L G n  Li − .Mr eeal,Li generally, More ).  endb e fhprlns( hyperplanes of set a by defined | 0 1 Λ n 1 , . . . , . x 1 ( ( n n {z z 1 ; x ( − = ) L sjs h sa logarithm usual the just is ) 1  R 0 0 } L , . . . , n = z , Z ( ω x 1 z 0 − 1 a ewitnas written be can ) ≤ 1 ; ···◦ · ◦· 1 Z −  0 n t 1 1 ; = ≤ ζ n M t (1 t ( z 2 ω − − L ≤ 0 , dt n M , . . . , sepooiin1 nrf 2]fra for [22] ref. in 18 proposition (see 0 ... Z )= 2) x L , . . . , frtoa 1-forms rational of ≤ 0 − n x t 1 n 1 ≤ dt ,...,n = t z ζ G n dt Z n 3.Sc dniisaein- are identities Such (3). k t = )  ; 1 0 (1 1 M x | 0 , . . . , ∧ , . . . , 1 0 n Z {z M , . . . , dt . . . − − ∆ L α 2 L 0 t 0 )= 1) 0 ∧ } L , . . . , ω xaso fstring of expansion = , M dt 1 t t − . 1 ; n n ω n ) i ln(1 .  ζ M narational a on n ( n an and ) = n 0 M , . . . , 1 − n , . . . , Z 0 x x n ) dt . t +1) (10) (9) (8) (7) n k Li n ), ). 3 - n − 1 ( t ) . Preliminary version July 10, 2013 nevl[0 interval 4 eeec o hsscini h ae 5 yKTChen. K-T by [5] notation paper the is section this for reference A integrals Iterated 0.2 α P path inverse the define Also, product the define manifold), hsi eusv definition. recursive a is This where path the Z [0 : i a f 1 • • • h ipetieae nerli noedmnin eitouea introduce We dimension. one in is integral iterated simplest The If If b i fyuaentfmla ihtento fplbc,hr stedfiiin If definition. the is here pullback, of notion the with familiar not are you If , dx f 1] function ˜ where r like behaves energy state ground whose CFT the original of the and recover spectrum mass the radius thermodynamic of of the cylinder use a on to theory is r the them formulating by check (TBA), to ansatz way Bethe One made. In in- are with tions operator. theories relevant field two-dimensional a massive tegrable by obtain CFT sup- we two-dimensional cases example, For a some deform models. we integrable they pose in way appear One also Polylogarithms ways. integrals. Feynman several computing in when is appear appear polylogarithms physics, In Motives, w α 1 α i h rudsaeenergy state ground The . ( 1 → → t sa1fr on 1-form a is ∗ w , . . . , , and ) w α 1]. dt X n hsyed ioaih dniis e es 1,15]. [17, refs. See identities. dilogarithm yields this and 0 i by ,te edfietepullback the define we then ), ···◦ · ◦· stepullback the is c β steeetv eta hre nmn ae fitrs the interest of cases many In charge. central effective the is S E K r r ah ie maps (i.e. paths are arx nodrt n this find to order In matrix. ( r -om on 1-forms are ter,etc. -theory, f r Z r a eepesdi em fdlgrtm ntelimit the in dilogarithms of terms in expressed be can ) α ( t w X ) dt 1 with , ◦ · · · ◦ = 1 Z x a fte1-form the of b i γ αβ w  oelclcodnts and coordinates, local some = r Z X E a = α ob h path the be to hnw en h trtditga on integral iterated the define we then , t ( − f r Z 1 1 a ecmue rmteknowledge the from computed be can ) 0 by , ( ,β α, α 1 u S ∗ α ) w du arx nteU limit UV the In matrix. ∗ γ w = w [0 : ( 1 ◦ · · · ◦ i S t P = ) ◦ · · · ◦ ntepath the on arxeatysm assump- some exactly matrix i , f 1] i α α dx dt f → (1 i olwdb h path the by followed r dt α − − ∗ hsi -omo the on 1-form a is This . 1 X w ( α t u r ). , ) samp(norcase our (in map a is where du α .  CONTENTS f r X ( t r ) dt. ssome is → we 0 (11) (12) − w 6 π r β c ˜ = , . Preliminary version July 10, 2013 ..IEAE INTEGRALS ITERATED 0.2. hr h u soe l ( all over is sum the where σ σ { sets the implies (13) eq. in Then, eusv ro sstefloigrcriedfiiino h hffl product shuffle the of definition recursive following The the difficulty. much uses too proof without recursive recursively extended be can argument This Z 1 ( − αβ ( w r , . . . , Z Z of 1 w hs trtditgashv h olwn properties following the have integrals iterated These The ewl utf q 3o w-iesoa case. two-dimensional a on 13 eq. justify will We h u vralte( the all over sum The ihrsett opsto ftepaths the of composition to respect With hneautn h trtditga nteivrept eget we path inverse the on integral iterated the evaluating When Z ( 0 0 1 w 1 r ≤ 1 α ◦ · · · ◦ r ···◦ · ◦· t 1 f + w 2 1 + logarithms and are that integrated way restriction trivially this be the in can impose form we differential later This clear fraction rational become the will which reasons For definitions. form general ferential We the giving integrals. before iterated examples as some functions giving transcendental by some start define now Symbols us Let functions. Transcendental 0.3 integrals. iterated of case the for calculus of theorem fundamental where endpoint that say we Then F deformed α 6 z 0 endpoints. manifold the the of case part some this keep in which transformations fixed, continuous by related paths path of the class of endpoints) the forms differential . < z < . by 3 2 4 3 h eteapei wc trtdintegral iterated twice a is example next The dif- a of integral an is consider will we integral iterated simplest The fw osdrtept needn functional independent path the consider we If ups etk ahin path a take we Suppose hl eomn h path the deforming While o o o a hn bu hsitga sbigtknaogtera xs for axis, real the along taken being as integral this about think can you now For terms, mathematical more In w .O ore hsfnto a eaayial otne o ope ausof values complex for continued analytically be can function this course, Of 1. k 2 α seautda h npito h ah hsi aial the basically is This path. the of endpoint the at evaluated is hl epn t n onsfie,wtotcagn h au of value the changing without fixed, points end its keeping while (1) Li w ∈ w 2 ( i X = z a have. may F = ) hnw have we then , d sidpneto h path the on independent is ln f Z dF ( f 0 x α z ( eartoa rcinwt ainlcoefficients. rational with fraction rational a be ) eaiehmtp ls sdfie steequivalence the as defined is class homotopy relative A . [ F x  α α ,where ), [ = ] Z α eaodcosn n iglrte lk oe)ta the that poles) (like singularities any crossing avoid we 0 = ] t F α 1 w in sidpneto h oooycas(eaieto (relative class homotopy the on independent is du − k Z Z α X u α w x  w n edfieafntoa ftepath the of functional a define we and 1 ∈ 1 dt ◦ · · · ◦ t ◦ · · · ◦ X = and − w Z w k α 0 f . k 4 z − ( ftept a elocally be can path the if dt x F 1 , sartoa fraction. rational a is ) [ ln(1 α safnto fthe of function a as ] t − t ) . CONTENTS (20) (19) (21) Preliminary version July 10, 2013 ihLi with get We integration. and summation of order the exchanging and oeta h ieeta om r ree si h trtditga and path. integral forms integration iterated the the of in endpoint as the ordered at are evaluated forms are differential the that Note called small object on an independent in knowledge is this integral condense will the We symbol if path. the function, of multivalued variations a to 0 define from to path some along taken is integral the where expansion series regular a has It dilogarithm. the around called is function This SYMBOLS FUNCTIONS. TRANSCENDENTAL 0.3. fw aeafnto rte sa trtditga ihdifferential with integral iterated an as written function a have we If enough is ordering their and forms differential the of knowledge The have we 0.2 sec. of notation the In that recursively show can One recursively define we Then, w hc sdntdby denoted is which , 1 z i ( z ,wihcnb on yusing by found be can which 0, = = = ) d − ln Li ln(1 f n i ( hntesmo a ewitnimdaey u if But immediately. written be can symbol the then , z S = ) (Li − z n .Tedffrnilrlto is relation differential The ). − ( Li z z Z )= )) ∂z − n 0 ∂ ( z Li Li z ln(1 d Li = ) n 2 ln(1 − ( ( n z z (1 ( − = ) = ) z Z Li = ) 0 − − t z = ) dt X X z k k t ∞ ∞ ) =1 =1 ) Li ◦ ⊗ X k n ∞ n z k =1 z | d n − | z n − k n 2 1 ln t . ( 1 . ⊗ · · · ⊗ t n z ( n t t ) n ) ◦ · · · ◦ {z . − , n 1 {z − 1 z } d . z . ln } t , (22) (26) (24) (28) (27) (25) (23) 7 Preliminary version July 10, 2013 d eesl obtain easily we use to is example. symbol an the on works compute this to how way see a us then Let form, recursively. (20) a eq. such know don’t we 8 lo ueynmrcentry numeric purely a Also, from example, For is symbol the So get also We have d d Li Li Li Li S S sn hsw find we this Using trigfo h eisexpansion series the from Starting sn hs rpris efind we properties, these Using properties. some satisfy symbols the that clear is it definition, the From (Li (Li a, 1 1 a, , ,b 1 1 0 ( ( ( ( 1 1 ,y x, ,y x, ,y x, ,y x, , , 1 1 ( ( ,y x, y x, = ) = ) = ) = ) d Li )=(1 = )) (1 = )) ⊗ · · · − d − 1 a,b ln d d − 1 ( ln(1 ln(1 ,y x, x y d ( Li − xy − ln( ( = ) y a xy − − ) − xy Li xy · · · ⊗ 1 x x ) Li , a ) 1 d ⊗ Li ) Li ) = ) ( ( a,b ⊗ ln x ,y x, (1 ) ( ⊗ · · · (1 1 b x ,y x, = − − ( d ( c ) xy xy Li ln − x − Li ⊗ · · · ∈ )+(1 a = ) + ) ) x x − a d c (1 Q − ( 1 (1 + ) + ln(1 · · · ⊗ xy ,b d − d n>m ( ie eobecause zero gives x d − ,y x, X )) ln ln(1 ln xy · · · ⊗ x − , y ≥ ) − 0 = y + ) ) 1 Li ⊗ y − ⊗ x efidthat find we n x Li ) (1 1 n a + ) . ,b d y y Li − ⊗ m y − Li ) ln a − ⊗ · · · 1 m ( 0 b y ( (1 x ,a y ,y x, , (1 )+(1 1 ) ( Li ( − ,y x, x − − a,b y ) ) y , d − xy · · · ⊗ − )+ − = ) ln 1 d d xy ) ( ,y x, 1 ln ln ⊗ 1 ) − y ⊗ . c  (1 − x CONTENTS ) y  . .S we So 0. = 1 x − Li 1 − y Li y − a y y ) ( b . ( xy  y xy  Li (33) (29) (31) (36) (30) (34) (32) (35) (37) (38) ) ) , . . 1 ( xy ) . Preliminary version July 10, 2013 sidpneto ml aitoso h path the of variations small on independent is integral the that theorem Stokes via implies This exact. locally is it hnteintegral the then one-form closed a of integral an with start us Let invariance Homotopy 0.4 INVARIANCE HOMOTOPY 0.4. dx of value the then 1, change. point not the does encountering integral without the deformed be to can 0 path from path a along n define and mle that implies d that obtain We ae eoti h olwn xrsinfrteitgaiiycondition integrability the for expression following the obtain we case, condition integrability the implies argument w similar a case, this In an is This ( j w F  ∧ sa xml,cnie h yblo Li of symbol the consider example, an As . osdrnwtecs hr ehv obeieae integral, iterated double a have we where case the now Consider ecnas osdramr eea case general more a consider also can We dy − 2 xdy

.Using 0. = ) 1 −  − − (1 − nerblt condition integrability xy F xdy ydx − 1 ( z − xy = ) −  xy x I )(1 ∧ ydx sidpneto ml aitoso h path the of variations small on independent is R  − dw z  1 − x z w − I ) ∧ 2 dx 1 ensamliaudfnto.I h integration the If function. multivalued a defines + Z hn h ento fteieae integral iterated the of definition the Then, . = x  0 and 0 = (1 I z  Z dy y I 1 = α − + dt − =  w Z . x  1 α t − Z )(1 1 Z ◦ 1 X α − = i,j α  w − w dx dF − w − 2 − 1 x w 1 = y ◦ xdy ln(1  i 1 ) w = ◦ Z − ∧ − 2 w α , − 1 − w  w F xy j . − 1 ydx 1 α 1 1 z − eobtain we , , (40) ) 1 xy o xml if example For . − 2 dy ( . w ,y x,  y − 1  Because . ∧ (1 ne.(8.I this In (38). eq. in ) +  − 1 − xy − dy y w )(1 y 1  w ∧ − 1 = w sclosed, is P w y 1 ) 2 i,j = ! 0. = α w (39) (41) (42) (43) 1 0 = dt − i ∧ is t (44) 9 , . Preliminary version July 10, 2013 nodrt aeti oecer e swr u neape ehave We example. an out work us let clear, more this make to order In if s h oino yblaan ehv the have We again. symbol of notion the use zero. to up o hudw hn bu nerblt nti eea ae Obviously that happen case? may general It this necessary. in integrability that about conditions the think we should How rva a dtie aclain ewe ieettrsaerequired). are terms different between cancellations (detailed way trivial 10 h nerblt odtosare conditions integrability the h hc fitgaiiyi h rttoetisi eysmlrt h one the where to similar very is entries Li two for first the did in we integrability of check The Li of symbol the and this Using fdffrn lengths. different of dw (1 d ehv hw hti hscs oooyivrac od,bti non- a in but holds, invariance homotopy case this in that shown have We S − Li 12 5 h ih a oepesteitgaiiycniin nti aei to is case this in conditions integrability the express to way right The h otgnrlcs is case general most The ngnrl o nieae integral iterated an for general, In (Li (1 xy nfc hr r oeohrsihl oegnrlcss(e 5) like [5]), (see cases general more slightly other some are there fact In 2 .Hwvr ewl o osdrteecssweew obn trtdintegrals iterated combine we where cases these consider not will we However, 0. = , w 1 − 2 ( ) 1 , ,y x, ⊗ 1 , xy ( w x ,y x, 2 = ) ⊗ ) X and r l ⊗ =1 − y )=(1 = )) 1 1 , − 1 y d ( i w 1 (1 X ,y x, ln ⊗ ,...,i 12 − x x r n-om.I hscs h nerblt odto is condition integrability the case this In one-forms. are r w .Teeoew ilfcso h attoetis We entries. two last the on focus will we Therefore ). xy Li − − w i l i 1 ) xy 1 (1 ∧ , ⊗ 1 I ⊗ · · · ⊗ ( w ) − x ,y x, = i ⊗ ⊗ l I +1 xy 5 Z (1 (1 ) = α for 0 = ) − − i Z w − 1 1 ⊗ X w ,...,i , α d y 1 i dw i ( x (1 )+(1 l ∧ w ln(1 ,y x, ) r ∧ 1 i w w − ⊗ l ◦ · · · ◦ w ∧ l i i led optdw n that find we computed already ) +1 1 − y x − i 1 = l dw ◦ · · · ◦ ) +1 (1 + xy y ,for 0, = ⊗ Li ) i r , . . . , u h remaining the but 0 6= l ) +1 x w ⊗ I − r (1 + 2
y , ( w ⊗ · · · ⊗ sidpneto h path the on independent is x ⊗ x i ) r ) − y . i − − ⊗ − 1 = r ucetbtnot but sufficient are 1 d x (1 (1 ln ) r , . . . , − w − ⊗  i xy r y x 1 0 = ) ) ⊗ − y ⊗ ⊗ CONTENTS R − (1 y y . x w ⊗  1. + 1 − ◦ (1 Li w w w y 1 − 2 )+ 2 i ∧ ( k + y xy (46) (45) (48) (47) (49) w add w ) 2 . 12 ) + . , Preliminary version July 10, 2013 et ftetasedna ucin ertoa ucin ihrational with functions allow rational we be If functions coefficients. transcendental of the notion of a ments define unambiguously to differ- of possible integrals be iterated weight). not as transcendentality would function it same well the is lengths, weight write ent transcendentality could of transcendentality. one notion different (if the defined that of means functions this of particular, In coefficients, rational with tions o ow n h ipetfnto ihta ybl eoew start we Before symbol? that with function simplest the of find combination we simpler do a how be find symbol). can to same result the the possible symbol has that is this the which likely it of functions is that transcendental words, it symbol notice other so the we function (in lucky, compute original simplified are the can we than we if “simpler” Then is and, which functions simplified. 7]) of [6, be combination refs. can in suspect (like functions we transcendental of combination to plicated check check to the useful reduces very is it symbol since The manipulation. function, functions algebraic useful. transcendental transcendental is a this between of why identities symbol said not the have compute we to but how shown symbol have We the Integrating 0.5 a value. is transcendental side a left-hand at the evaluated while function, zero one transcendentality transcendentality has side right-hand the has forms the Li with of symbol the that shown have we So ucin fdffrn rncnetlt,weeteprswt h aetranscendentality same the with themselves. parts among the cancel where transcendentality, different of functions get SYMBOL THE INTEGRATING 0.5. (1 6 o ecnfial xli h esnfrtersrcinta h argu- the that restriction the for reason the explain finally can we Now length of integral iterated an as defined is function multivalued a If hsi h usinw iladesi hsscin ie symbol, a given section: this in address will we question the is This com- a have we but identity, an check to want don’t we suppose But transcendentality sa xml ftiilrltos ehv iercmiain ftranscendental of combinations linear have we relations, trivial of example an As − xy ) ⊗  dy y w i ∧ = dx x k d hr r onontrivial no are There . − ln (1 e f 1 − sa ruette ehv ln have we then argument an as i − − , dy i x y ) 1 = ∧ ⊗ dx  x k , . . . , 1 − + − dy 2 dx x y , 1 hnw a httefunction the that say we then , ( ∧ ,y x, ∧ dx x dy y sintegrable. is ) 6 + − aihn iercombina- linear vanishing dx x dx x ∧ ∧ 1 − 1 − − − dy dy y e y   ,where 1, = 0 = + . (50) 11 k Preliminary version July 10, 2013 oapouto oaihswieteatsmercpr,i tstse the satisfies it if dilogarithms. part, to antisymmetric integrated the be while can constraints, logarithms integrated integrability be of can product part a symmetric The to of parts. antisymmetric symbols and have symmetric integrated. They straightforwardly be can logarithms. which one are length functions one transcendentality discards which symbol 8]). the [4, of refs. version (see modified information a less analytic- using appropriate by imposing or like constraints below), ity examples (see methods other by nteeapeo Li procedure of the illustrate example us the Let on complicated. more is part antisymmetric The tak- containing terms in reproduce consists not 0.3 will of sec. method powers in this general, described In method derivatives. the ing by symbol the only computing typically will answers. we partial powerful transcendentality give some to higher three able is for and two be question but transcendentality this us, of to guide cases answer theorems the general In a that known. mention not should we discussion the 12 a yblzr.Sneteeaen eain ewe ucin fdifferent of functions between relations that no means are this there transcendentality, Since zero. symbol has where manipulations algebraic elementary by rewritten be can RHS The − S   Li + 1 − oeepiil,frtesmercpr ecnuse, can we part symmetric the for explicitly, More in decomposed be can functions two transcendentality of symbols The The transcendentality. to according discussion the organize will We oevr yuigti ehdw icr oeifrainbecause information some discard we method this using by Moreover, ow aesonta h difference the that shown have we So 2  Li − ⊗ y 2 S y  (1 1 S (1 − 1 π stesmerctno rdc eoe oeie by sometimes denoted product tensor symmetric the is − ( − x − L y rtrspootoa ozt aus uhtrscnb found be can terms Such values. zeta to proportional terms or − y ⊗ (1 1 1 x y , x 1 ) − y ( − )  ,y x,  + y ⊗ x − ) y )=(1 = )) y  Li 1 ⊗ , (1 1 1 − 2 ( ( x − ,y x, − y Li = ) y x − 2 .Rcl that Recall ). − ( ) S y 2 1 (1 + (ln xy ) ln − ) 2 x (1 ⊗ 2 1 − ln ∗ ln − y y y salna obnto of combination linear a is (1 2 y 1 ) ) (1 )+ln(1 x , ⊗ − − − ∗ y 1 x endbelow defined y − ) y ln(1 + ) (1 + − y ⊗ (1 + x x ln(1 ) − = x − − S ) − 2 1 x x ⊗ ln(1 ) ) y ln (1 ) ⊗ 2 − ( S − x Li (1 CONTENTS ) − y . 1 , ) π − 1 y . ( ) × ,y x,

y  = ) , . nand ln = ) (51) (52) (54) (53) ∗ Preliminary version July 10, 2013 oriaemtto,wihwl edsusdi oedti nsc 0.11. sec. in cluster detail of more example in simple discussed it!). a (try be is five will recursion periodicity which this a mutation, of has coordinate relation version recursion modified this slightly that A show to easy is It antisymmetric is symbol whose if is series power the of coefficients ymtymnfs.Dfieasequence a Define among manifest. symmetry people etc. many Schaeffer, by Kummer, rediscovered Hill, and Abel, discovered Spence, five-term been which the has is them It among important identity. most The identities. nontrivial some logarithms. of terms in uniquely ( around Since expansion series regular a wihaetu pto up true are (which Rogers of terms in written be dilogarithms, of numbers π rational the over ln combinations around series linear power as well-defined pressed a having not of cost the at but symbols. using found we identity nontrivial first the is This π SYMBOL THE INTEGRATING 0.5. yblcnb rte sln as written be can symbol (Rogers around − S 2 2 × ehv hw bv httetrswt niymti yblcan symbol antisymmetric with terms the that above shown have we , Li em,wt ainlcecet.Hwvr ti ayt hwthat show to easy is it However, coefficients. rational with terms,

o opt h olwn combination following the compute Now epeethr hsfietr dniyi a hc ae its makes which way a in identity five-term this here present We satisfy they since complicated more is situation the dilogarithms the For fw lo usle ouetelgrtmiette ln( identities logarithm the use to ourselves allow we If ex- be can functions two transcendentality the all that theorem a is It Li function the that Note ln, 2 X n  =1 ∗ 5 − x π S L contains L y × ,btissmo sntatsmerc ecndfieafunction a define can We antisymmetric. not is symbol its but 0, = L (1 ( 1 function) a nand ln ( − n x − ) = ) ! y x ) =  − π π − X n 2 1 2 h nywyfrteeult ohl ttelvlof level the at hold to equality the for way only the =1 L 5 em.Ngetn o o h em ftype of terms the now for Neglecting terms. (1 Li ( x − 2 − πi Li = ) ( 1 2 y x ,te ymti yblcnb integrated be can symbol symmetric a then ), ) (1 ) − × ⊗ 2 − ( 1 2 L 2 ln. x x a ( ln a eldfie oe eisexpansion series power well-defined a has ) x n ucin,wietetrswt symmetric with terms the while functions, + + ) ) 2 (1 ∧ ∗ 1 2 ,y x, a x − .Therefore, 0. = 2 1 n ⊗ a (0 = ) ln( y = n ln(1 + ) (1 eusvl y1 by recursively x − ln(1 ) − 2 1 , x X n )wt ainlcoefficients. rational with 0) =1 5 ) − − ≡ − ( a x x n ln(1 ) − ) 2 1 , 1 a (1 n +1 − − − xy ) y x a ∧ Li = ) ) ln = ) n ∧ a = n x, x π a 0 = n × 1 0. = x , − 1 nand ln 1 ( , ln + ∗ a ,y x, (56) (57) (55) (58) n has +1 Z 13 ) y 5 . . 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Rogers is ,a x, b m a hntesmo fteproduct the of symbol the then , n X n a 2 ) =1 5 n tt = = ) ρ L (( ( 1 1 b ( a a 1 − a 1 − n ⊗ · · · ⊗ 1 ⊗ · · · ⊗ = ) xy ⊗ x a , ρ X n ( ρ =1 a 5 b b 2 1 m 3 F (Li a ⊗ · · · ⊗ ⊗ = ) n = ) and 2 (( tt ( 1 1 a a a − n 1 − 1 ( ln + ) b G ⊗ · · · ⊗ ⊗ xy a 1 y n (( ⊗ · · · ⊗ ) aesymbols have a , a − 2 a ⊗ · · · ⊗ a n 4 n − a 1 n = ⊗ G F b ) ln m tt ,a y, ρ )=0 = )) a ( a a n sgvnb h shuffle the by given is ( n 1 = ) b ) 2 ⊗ · · · ⊗ tt 5 ⊗ · · · ⊗ a . 1 1 = π 6 ( 2 ⊗ · · · ⊗ b . 1 ⊗ · · · ⊗ − ζ CONTENTS a 2 = (2) xy. b n m − 1 )) ) a . . b n π m 6 2 (63) (62) (61) (59) (60) and ))+ so , Preliminary version July 10, 2013 hr ehv sdtesotadnotation shorthand by denoted the have used we have we where type called n icso eghoea h end. the at groups. one length of pieces and Li the of ρ symbol the with start by and re- this coassociative, algebraic and is the coproduct applied it to This all is applying order yet). it understand available which in to not to that is need function understanding is we transcendental marks coproduct the between quotation a lations the is for this reason that show (the functions on uct” for Li structure of symbol the has this Algebraically the function. coproduct. under single pick a functions a we of of of If out classes sev- map) equivalence produce it. symbol we accurately way of (more this parts In functions functions. only those eral actual therefore integrate to and correspond to holds symbols still is shorter condition integrability idea the One pieces appropriate handle. to plicated ..ITGAIGTESYMBOL THE INTEGRATING 0.5. noeaincnntdtc trilogarithms. detect not can operation an ( operations the applying → n eslttersligsmo npee flnt w ttebeginning the at two length of pieces in symbol resulting the split we and Li ∆ Li 7 ntefloigw ilmsl eitrse nytaohrcoproduct, another yet in interested be mostly will we following the In hspoeuecnb undaon.W tr ydfiiga“coprod- a defining by start We around. turned be can procedure This Li of case the In the applying after even Still, k X k ecudas oa(1 a do also could We 3 =0 n 1 ( 0 = x n δ ∗ ⊗ Li ( ) x hc iesfo yngetn h rdcsadas em of terms also and products the neglecting by ∆ from differs which n , . . . , −→ − k S = ) ( and x n n ) ( X ⊗ k ie eoti h symbol. the obtain we times (1 x =0 n ntoprs n flength of one parts, two in ) ⊗ ∗ ln − ( n hmw nert hmt functions: to them integrate we them and n − x n − − ) [(1 1 | ⊗ k n π h rcdr sbs ecie na xml.We example. an on described best is procedure The . 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C 2 ((1 2 1 ⊗ } 3 y ) ∗ 1 ⊗ C 1  + ) 3 ∧ } ) − + )   y ∗ 2 ( − −−−−→ x (Λ  −−−→ B x xy x Λ  3 − ⊗ → →{ 7→ x )+ 3 2 } y 1 + ⊗ − } 2 B 1 xy 2 2 ( ) C − 2 C − 2 B xy B { { −  ∧ = xy 1) ∗ ⊗ ∗ x xy x 2 − 3 1 ), xy 2 } ( y y y } ⊗  −{ − ⊗ { ((1 xy x 2 ) 2 } 1  x 2  C ⊗ 2 − 2 { − { ∧ )+ 1) (Λ  { } 3 x B ⊗ − 2 ∗ y − 2 3 − + ( 1) 2 } + 2 n Λ and ( { ∧ x xy + 1 1 C y 3 − xy → } y { } ∧ {  ∗ − 2 1 { (1  } 2 y ), xy . ) { x y 2 − B } ⊗ 3 y 2 ∧ − } (94) ) 2 + 2 { } } 2 + 2 3 x ((1 B xy x 3 2 xy { − ⊗ { − } } { { − 2 { 3 3 1 ) (Λ − opnnso these of components y −  xy { − ∧ − 1 } xy 2 ⊗ (1 xy 2 x − } C x ) { ∧ y 2 } ( ∗ } − +(1 } ∧ y B xy 3 fLi of ) (97) 0 = 3 3 }  xy y 3 x  { − 3 ) ⊗ − ⊗  ) ⊗ − . } ∧ ⊗ ( C 2 1)+ y y x x . y 2 } ∗ ) , y, + )+ 2 3 pto up , − ∧ (  ,y x, y 1)+ (93) (92) (98) (99) (96) (95) ⊗ ) . 23 y ). Preliminary version July 10, 2013 coefficients by denote We poly- than understand to easier mysterious. are pretty which still values, are zeta but logarithms, the discuss us Let values Zeta 0.9 permutation. x x, 24 hrfr,teee eavle r xrsil spwr of powers as expressible are values zeta even the Therefore, where functions over sum we where offiinsin coefficients etlt orfnto,u opout.I eaegvnol neeetof element an only given are we If products. to { up function, four dentality ihi serf 1,e.27]) eq. [19, ref. 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(1 | 1 τ 3 ⊗ 2 1) | B hc c spruain farguments of permutations as act which + −  = + ) 2 ( n xy − σ  τ x X k 1 ∞ ( ( =0 − ζ ( y x 2(2 2 − xy x 1 ( ) B ) { ) ,m n, B  − − y )+ 1) 2 n k − 1 } n k 3 )! ! 1) t 2 and ⊗ k + ) 1 (2 { ∧ y .  1  1 π { xy ζ ) − | − 3 y σ 2 ( } n + n | } σ τ 3 , 2 stesgaueo the of signature the is ( ( + { { − y x 1 ) ) m B − , xy ) 3 . x ⊗ π } } 3 2 C 3  CONTENTS ihrational with { − ∗ ⊗ compatible ( y x } 3 −  (103) (102) (100) (105) (104) (101) 1)+ ⊗ y Preliminary version July 10, 2013 a’ vnpoethat prove even can’t ζ where to equivalent is conjecture the Then quotient the take we them eliminate To weight. lower Z Conjecture. following the have we algebra Lie graded free a independent. 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Preliminary version July 10, 2013 G superconformal dual the of subgroup conformal group. coordinates this in dual interested be the mostly acts surprising a subgroup has bosonic the also symmetry, theory superconformal Mills this Besides broken. is 26 oec on nda pc ecnascaeatopaein two-plane a associate can we space dual in point each to aho hmapoint in a line them a of to each construction, this intersect projectivize planes in we point corresponding If their line. if a separated in light-like are space dual in eed nmmnu wsosadi aiyivrat hnw uthave must we then invariant, parity is that and twistors momentum on depends twistors momentum Z conjugate construct can variables The momenta the giving that X by denote X will we which C form bilinear non-degenerate a variables the for Unlike unconstrained. [24]. ref. in conservation etmtwistors mentum re omake to order ω lershv enitoue nasre fppr 1,1,2 4 yFomin by 14] 2, 12, [11, papers Cluster of algebras. series cluster Zelevinsky. a about and in facts introduced useful been have some present algebras algebras we section cluster this In to Introduction 0.11 of action the by preserved is which form volume a ( i 4 (2 i i ⊥ v ∈ h opeie n opcie ulsaecnb ersne sthe as represented be can space dual compactified and complexified The aiyat stedsrt transformation discrete the as acts Parity h opeie ulcnomlgopat as acts group conformal dual complexified The The orsodn oapoint a to corresponding i pc snttecmlxfidMnosisae.Gvnatwo-plane a Given space). 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W 1 since ecncntuta rhgnltwo-plane orthogonal an construct can we · n W = ) → dual i W = p f M i C ( i Z or W 4 C − uecnomlsmer,whose symmetry, superconformal ∈ i +1 otiigteoii.Therefore, origin. the containing 1 4 1 x W W , . . . , X orsodn to corresponding i · h oetmtitr are twistors momentum the , i ⊥ where , W − x W 1 ntefloigw will we following the In . Z i ∩ i 0. = i Then, . x n ↔ SL X ) i p − . i ⊥ SL i 2 1 (4 W M x , n momentum and 0 = N ow a conclude can we so n , i (4 faquantity a If . i C n soit to associate and ) C points · super-Yang- 4 = ∈ , ). C 4 Wrig this (Warning! C Z 4 orsod a corresponds SL ntemo- the on ) i CONTENTS w points Two . Z ∈ i (4 Z and h X , i ijkl C ∈ i Z .In ). (108) since CP i ω i we = 3 is f . 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( b,c = ) x x ler,which algebra, 1 1 x , x , 7 2 ,w en- we ), = cluster x (110) (109) 2 . 27 Preliminary version July 10, 2013 w-yls(ar farw on nopst ietosbtentover- two between and directions target) opposite and in connected, origin going to same tices). arrows restrict the of will with we (pairs (arrows following two-cycles loops the without In quivers graph. finite oriented an is of quiver matrices Cartan 1). to tab. correspond (see algebras algebras cluster Lie finite simple the that notice we and if variables cluster of number finite a has it that if shown only be can It ables. Lie simple and algebras cluster two. finite rank between at algebras Correspondence 1: Table 28 prtoso h nta quiver: initial the on operations vertex at mutating pnec ihse-ymti arcs new xa reigo the of ordering as an defined is fix matrix we skew-symmetric once The matrices, vertices. skew-symmetric with spondence cluster. initial the obtain at mutation The • • • o uvrwt ie vertex given a with quiver a For A algebras. cluster and quivers between link the describe now us Let vari- cluster of number infinite an generically has algebra cluster This ahqie ftersrce yedfie bv si n-ooecorre- one-to-one in is above defined type restricted the of quiver Each o ahpath each for eoealtetocce htmyhv formed. have may that two-cycles the all remove with incident edges the on arrows the all reverse (1 (1 (1 ( ,c b, bc , , , 3) 2) 1) ≤ ) .I enwfr h matrices the form now we If 3. atnmatrix Cartan b ij k − − − 2 2 2 k 2 1 2 3 2 2 (#arrows = i sa nouin hnapidtiei ucsinwe succession in twice applied when involution; an is h e uvri bandb pligtefollowing the applying by obtained is quiver new The . − − − → 1 1 1


k → j i algebra Lie eada arrow an add we  i − → 2 G A C c 2 2 2 j k ) − 2 − edfieanwqie bandby obtained quiver new a define we b  (#arrows , ykndiagram Dynkin i      _jt ks → j j → k i ) . CONTENTS period 8 7 5 (111) (112) Preliminary version July 10, 2013 bantequiver the obtain of components the arrow functions each two for with together quiver a is a erpeetdb edquiver seed a by represented be can a by and from vertices obtained of set be the can they but above, nonvanishing, is above terms vertex the at of mutation one only Since ALGEBRAS CLUSTER TO INTRODUCTION 0.11. lse lerso emti type geometric of algebras cluster algebra. D is it but at mutation uainat mutation a ihteudrtnigta nepypouti e ooe h mutation The one. unchanged. to variables set is product empty at an that understanding the with variable a eaina h vertex the at relation quiver diag( = k oeta nti aetematrix the case this in that Note The ewl emsl neetdi pca ls fcutragba,named algebras, cluster of class special a in interested mostly be will We The with quiver a with start we If changes x A 1 A skew-symmetrizable ( ←−− 2 d b,c x ( c,b (1) i lse ler a eepesdb quiver a by expressed be can algebra cluster b ) ecnuetese-ymti matrix skew-symmetric the use can we , x ij ) lse lerscnntb bandfo uvra described as quiver a from obtained be not can algebras cluster 2 x . . . , x e 1+ = k elcsi by it replaces ij 1 x 2 x , to      elcsi by it replaces 1 b 1+ uhthat such ) b i v hsrpoue h xhnerl fthe of rule exchange the reproduces This . x ij 0 k − v 0 → 1 ( x x , ( e 2 c = v h matrix the e x k 0 k ij ( ) ←−− k j e 1 .Tetematrix the The ). (          endb q 14 n evsteohrcluster other the leaves and (114) eq. by defined by x c,b ) ehave we E 2 k 0 b b b − ) ij ij ij = h e fegso uvr hnavle quiver valued a then quiver, a of edges of set the b x , hsmasta hr sadaoa matrix diagonal a is there that means This . ij + − 2 i Db x | , hl fe uainat mutation a after while , b Y e e between arrow no 2 0 b b ik x ik ik = > sa ro between arrow an is between arrow an is b hyaeas ecie yqies but quivers, by described also are They . sse-ymti.Nwtealgebra the Now skew-symmetric. is 1 0 d b b 0 n kj kj rnfrsto transforms ( = x 1+ i etcsadascaet ahvertex each to associate and vertices x , ) , x i b ik 2 v x 1 1+ 1 ( v x + −−→ e b 1 ( x ≡ ij b,c : 2 audquiver valued sntse-ymti anymore, skew-symmetric not is i if if if if ) → V | b ) x b ≡ Y 1 ik b b b k 5 sdfie as defined is = x < . ik ik ik x { ∈ 2 0 b , b , b 3 fe uainat mutation a After . d N x kj j n eesstearw A arrow. the reverses and kj kj i − v 2 b ,j i, i ≤ b 0 ( and ik > < e and ie by given } ij , 0 b , 0 0 , ) b j i 2 , odfieamutation a define to fw eoeby denote we If . ij d and where , and j, . : = x → E x 2 1 j, − i, eoti the obtain we → b . A ji N v ne a Under . ( ( x b,c e uhthat such 2 ij ) Then, . ) cluster 1 x (114) (113) (115) , A 2 1 ( are b,c we 29 V i ) Preliminary version July 10, 2013 hycnb aee by labeled be can They where Pl¨ucker relations satisfy They coordinates. ( by minors these initial the of columns the Grassmannian the of in points in related points therefore have as them the consider all properly of rescale action the the can Using by we matrix. Moreover, the transformation. of linear columns the by SL given need coordinates we having why points is This Grassmannian. the in the identify point to same the describing are arx oee,i ecoeaohrsto etr hc r bandfrom obtained are a which by vectors of ones set initial another the chose we if However, matrix. in this describe to order by of action of class of origin the nians a with work matrix to ( this economical full vertices more frozen is it the matrix between links an no have will are there since However, and them. the with of incident edges the and vertices frozen the variables erasing Also, associated The vertices. the called vertices. frozen define are frozen the vertices the between frozen in the arrows mutations to allow allow not not called do do and we we special that are in vertices special the of part 30 C n k ( ie a Given a Dually, ntefloigw ilso o lse aibe rs rmGrassman- from arise variables cluster how show will we following the In analog an define can we type geometric of algebras cluster of case the In k n m × ree onsin points ordered n ugopof subgroup ) hc pnthe span which G I + n b rznvrie,w a avl en ( define naively can we vertices, frozen k ( samliidxwith multi-index a is ,n k, arcsaie sflos on in point a follows: as arises matrices arxa el fteagbahsrank has algebra the If well. as matrix m × GL ) rnia part principal n k teGrassmannian (the ) × m C ( k ( arcso aia rank maximal of matrices × k ,j I j, i, n × ( × k r ntesm qiaec ls) hsparametrization This class). equivalence same the in are ) .W a soit oec on in point each to associate can We ). n n × m n i + arxwith matrix 1 i , . . . , arxcnb huh fas of thought be can matrix n )( lc le ihzrs nta fwrigwt hsfull this with working of Instead zeros. with filled block m GL GL ,l I l, k, arcswihdffrb a by differ which matrices k k matrix. ) CP pae sn these Using -plane. paew a rirrl pick arbitrarily can we -plane k ( ( k k vcosb h aeaon ow hudmore should we so amount same the by -vectors k k ecntasomthese transform can we ) k k .Teedtriat r lokona Pl¨ucker as known also are determinants These ). rnfrainwihpeevsthe preserves which transformation ) ( = ) fsc uvrt eteqie bandby obtained quiver the be to quiver a such of integers − × 1 . n k k ,k I k, i, − arx ewl eoetedtriat of determinants the denote will We matrix. coefficients ≤ nre.TeP¨ce eain en an define Pl¨ucker relations The entries. 2 G n i 1 ( )( i , . . . , CP ecnform can we ,n k, ,l I l, j, k k stesaeof space the is ) − almtie hc ie yaleft a by differ which matrices (all 1 k n k ( + ) nta of instead ahrta etr in vectors than rather rznvertices frozen { ∈ n G vcosw a ul a build can we -vectors n × ( n 1 ,l I l, i, GL ,n k, + n , . . . , ( ree onsin points ordered G n m n k n GL (
sa is ) ( + k G k )( ) ,n k, k ( vcosb h same the by -vectors action. ) ioso type of minors lse variables cluster ( × n ,k I k, j, m 1 = (1) needn vectors independent ,n k, } umti fthe of submatrix ) orsodn to corresponding , oconfigurations to ) ( nrznvertices) unfrozen k n k h uvris quiver The . nequivalence an ) paein -plane pae thought -planes + ) , GL CONTENTS m k matrix ) ( pae we -plane, k ) C /SL C C k k k (116) k We . n We . the , × In . × ( k k b n ) . . Preliminary version July 10, 2013 oriae ntecl fteGasana where Grassmannian the of cell the on coordinates representative l the operation this ( After trix. appropriate an with a of action left the by differ they if equivalent k of classes equivalence as mannian for algebra Grassmannian The cluster The 0.12 by coordinates. and cluster Pl¨ucker coordinates complicated are Indeed, more variables generate example). whose will for cluster we (114, mutation with eq. start (see algebra will cluster we a in relations exchange ( inside fits which matrix square a of point of [25]). ref. in reviewed G also is construction (this [16] ref. in scribed projec- a dimension into of Grassmannian space tive the of Pl¨ucker embedding, called embedding, eie bv od vnwhen even holds above derived of definition the In F FOR ALGEBRA CLUSTER THE 0.12. i 1 ,j i, ≤ ij = ( × k k C ,n k, Y , o edfieamatrix a define we Now Grass- the of description the consider to sufficient is it purposes our For the to similar very look (116) eq. in Pl¨ucker relations the that Notice h osrcino e.[6 sstedfiiino h Grassmannian the of definition the uses [16] ref. of construction The e sexpress us Let omk ta it make to × inside ) l k n n − ( ,where ), hr h first the where io snnsnua,i.e. non-singular, is minor sacstof coset a as ) − n j G − and 1 + k ( h entries The . ,n k, k Y eoboki h pe-ih block. upper-right the in block zero ) f hnw define we Then . ij ,tesubgroup the ), 1 = k k l > i f sthe is ( × ij f G h h GL ij k ntrsof terms in 1 i +1 k ( ,...,i SL − ,n k, ehv iie by divided have we arx efind we matrix, f ,...,k,k ai etr i nthe in lie vectors basis ( ij k + k j n k y ( × ( = j arx ecntasomi oteiett ma- identity the to it transform can we matrix, )
n, ij a lse ler tutr hc a de- was which structure algebra cluster a has ) .B digrw n oun otematrix the to columns and rows adding By 1. + − . F h 1 , l k 1 + − C ij ,...,k − j,...,i h 1 dniymti and matrix identity 1 yaprblcsubgroup parabolic a by ) P ,i o 1 for 1) ,...,k Y l h ≤ +1 k ( i 1 h ,j i, bakt.Teeaetocsst consider: to cases two are There -brackets. otisaltemtie of matrices the all contains + ( k , . . . , n hs oe-etcre sa position at is corner lower-left whose and 1 k ,...,k,k i j i k − k , . . . , + ≤ min( = ) i ≤ k × )( − l i ( + 1 n i,j k i j,...,n ≤ 6 i i , 1 , ) arcs hr w arcsare matrices two where matrices, 6 i − 1. = G k h 1) hn ylf multiplication left by then, 0 = 1 1 , ( i ≤ ,N K, i l > i , k , . . . , det GL − ≤ k k j 1 F paewihdtrie a determines which -plane ( × ≤ j n , ≤ k 31 ) G ij Y arx fteleftmost the If matrix. ) ≤ i . n l h − l ( 1 − − ota h expression the that so sa is l ,n k, k , . . . , arxhsteform the has matrix hc stebiggest the is which , ftematrix the of j j j − 1 + 1 + k P k ) × and ) 6 i , ntebasis the In . SL l . 0. = arxwith matrix ( n, C with ) Y (118) (117) are Preliminary version July 10, 2013 cto l h nrznvrie fteiiilqie aea qa number equal an have arrows. quiver outgoing initial and the ingoing of vertices of unfrozen the all fication xenlndsaetefoe variables). (these frozen nodes the external four are to nodes connected external node central one has diagram quiver and frozen coordinates, by the labeled all node the rescale to We connects above. by quiver unfrozen, the to change by given is 32 igasw rsn eo r eie nti reference. this in derived are below present we 25]. diagrams [16, refs. of quivers the to respect with 9 8 e ssatwt oesml xmls o orpit in points four For examples. simple some with start us Let nodrt banteqiesi e.01,w edt aeoelast one make to need we 0.11, sec. in quivers the obtain to order In the for quiver initial the [16], ref. to According e e.[6 o h osrcino rsmnincutragba.Tequivers The algebras. cluster Grassmannian of reversed construction arrows the the for with and [16] quiver ref. the See of version flipped a presented are we Here f f f . . . kl 2 1    l l _? ? ? ? ? ? 8 ? ? ? ? h ? ? 1 ? ? k , . . . , ? ? ? / / ? · · · · · · · · · . . . i hspoue rznvariable frozen a produces This . / / f f f f 1 k . . . 23 13    l 3 _? ? ya non ro.Atrti modi- this After arrow. ingoing an by _? _? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 9 ? ? ? ? ? ? ? ? ? h eta oehstoarrows two has node central The ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? / / ? f f f k . . . 22 12    2 _? ? _? _? ? ? ? G ? ? ? ? ? ? ? ( ? ? ? ? ? ,n k, ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? lse algebra cluster ) ? ? ? h ? ? ? 1 ? ? ? ? ? ? k , . . . , / / ? ? ? 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(14)(23) (12)(34) ? ? ? ? / 34 23  respectively. , _? k ? G ? ? ≤ ? ( ? ,N K, ? ? X n ? 2 / ? ihu oso generality. of loss without / oriae o h central the for coordinates 45 15 12 24 34 33 )  O C o n G hc a qal be equally can which ( ,n k, n 14 = ) C CP A ree points ordered X n hscorre- This . -coordinate) coordinates. 1 G Thinking . ( n − A (120) (121) ,n k, co- X ). n Preliminary version July 10, 2013 uaea etx14t bantepicplpr fteqie hw at shown quiver of the diagram of Dynkin and the part below as principal left same the the the obtain is at to which quiver right, initial 124 an vertex with at start mutate can We diagram. 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Lie 1 ? } ? 1 ne uain ehv h transformation the have we mutations Under . ? ? sjs n etx h eta n.Ti steDni diagram Dynkin the is This one. central the vertex, one just is ? E ? ? ? 6 CP ? ? 134 124 234 uvri h ih ato h gr below. figure the of part right the in quiver   1 _? _? stesm steDni iga of diagram Dynkin the as same the is ? ? 10 ? ? ? ? ? ? ? ? CP ? ? h rnia ato h uvrfrconfigurations for quiver the of part principal The ? ? ? ? ⇒ ? ? / / ? 345 145 125 2   epeetblwteiiilqie.Atra After quiver. initial the below present we 4,2 = 23 345, _? _? → ? ? ? ? CP n ? ? ? ? ? ? 1) ti ayt e htb mutating by that see to easy is It (15). onsin points 3 + ? ? ? ? 2 ? ? / / ejs edt elc h aesby labels the replace to need just we , ? ? 456 156 126 ⇒ 4,etc. 145, A n CP CP lse lersapa in appear algebras cluster A 2 2 hr eoti a obtain we where , 1 D • • • . O ler edfie at defined we algebra 4 x ? o . ? ? → A ? ? ? 2 x ? ? − ykndiagram Dynkin ? ? A 1 ? . 2 • CONTENTS i algebra. Lie (122) D 4 Preliminary version July 10, 2013 obtain agMlster.Tee h eeatGasana is Grassmannian relevant the There, theory. Yang-Mills G FOR ALGEBRA CLUSTER THE 0.12. If have from We arising type. algebras an infinite and is cluster of which is that subgraph algebra above mutations. cluster a of its seen contains the sequence of quiver then a diagram, the part after Dynkin of principal diagram affine part the Dynkin principal if a the variables), be if cluster to Further, made of be number can finite quiver a has it (i.e. details). more for [26] ref. an (see of form the into brought ie h sa lukrdtriat,w lofidmr opiae quan- complicated more find i.e. also relevant, we Pl¨ucker physically determinants, usual is the which type sides type finite infinite of of algebra algebras cluster cluster 2 different table. for In clusters anymore. [13]) of type ref. number (see finite of the not list are we algebras cluster the eight-point 167 ( n ,n k, aua usini htkn of kind what is question natural A hshssrkn mlctosfrsatrn mltdsin amplitudes scattering for implications striking has This nrf 1] oi n eeisysoe htacutri ffiietype finite of is cluster a that showed Zelevinsky and Fomin [12], ref. In in points eight for Finally, al :Tenme fcutr o lse lerso nt type. finite of algebras cluster for clusters of number The 2: Table G eobtain we 6 = ? ? n (3 ih2 with ) ? +2 1 ? G ? , ? ? (4 A )aeo nt ye nrf 2] ct a hw htalteother the all that shown has Scott [26], ref. In type. finite of are 8) ? 2 ? n n ? , n 127 467 126 367 267 +1 )= 7) +2  
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(781) 1457 i− 1678 ih 3456 ∩ ∩ ih 13 intersect. (123) CP i A 3456 . (123) CONTENTS i 3 coordinates + language. i i . . 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(see bras z z ij jk − z z kl il ignl nti raglto eemnsaqarltrladtherefore and quadrilateral a determines triangulation this in diagonals 3 h oso tutr sesett nesadfor understand to easiest is structure Poisson The hncnie opeetinuaino h oyo.Ec fthe of Each polygon. the of triangulation complete a consider Then oieta h ed ehv enuigbektecci symmetry cyclic the break using been have we seeds the that Notice fw i h diagonal the flip we If ehave We . b X ij ,j ,l k, j, i, i 0 = and − A b CP b ji ij 0 hr h reigi h aea h reigo h initial the of ordering the as same the is ordering the where oriae edt hne nedi snthr oshow to hard not is it Indeed change. to need coordinates r sthe is ( 1 r bandfrom obtained are ,j ,l k, j, i, ups diagonal a Suppose . b A arxo h lse.I snthr ocekthat check to hard not is It cluster. the of matrix = ) E oriae per.I ol eitrsigto interesting be would It appears. coordinates → { { X X hnteiiilcosrtoge oisinverse, its to goes cross-ratio initial the then r 24,ec hspoe h ylcsymmetry. cyclic the proves This etc. (234), i i 0 ( X , X , ,l ,j i, l, k, j j 0 } } X = = i hc mle httecross-ratio the that implies which ) b b and ij 0 ij E E X X n X eemnsaqarltrlwith quadrilateral a determines i i n ednthv ocoean chose to have don’t we and 0 points. X X b oriae nagvncluster. given a in coordinates ij j j 0 epciey hrfr the Therefore respectively. , , , X oriae.I senough is It coordinates. 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sides the are they if adjacent called are triangulation given a in oriae,i h matrix the if coordinates, 2 , osdrteΛ the Consider sa xml ecncnie etgntinuae ydiagonals by triangulated pentagon a consider can we example an As ecnso htfreeyterm every for that show can We − h oso rce ewe lse oriae a eoecmlctd san As complicated. become can coordinates cluster between bracket Poisson The 2 ftediagonal the if 1 = n by given is it language , n vnte eoti oecmlctdquantities. complicated more obtain we then even and ) h h 4 167 137 − , − )and 5) h ih ih E 2 n 347 467 × and − − − h 3 167 i i n n h , X o 2 4 b F − − b 0 2 × × ih 2 X on etcs2ad4adtenwcosrto are cross-ratios new the and 4 and 2 vertices joins ∧ = A B r o daetw set we adjacent not are h h h 234 3 n a 6 2  0 167 2 2 2 , ab r , − lse ler.Tee ehave we There, algebra. cluster = × × 4 (2 rjcino h ee-on eane ucin In function. remainder seven-point the of projection 7 i bfr h mutation). the (before 1 = h h ih × h h × X , 3 3 137 123 167 167 E 3 , , 234 a 6 1 , 4 4 1+ (1 , 4 i oe before comes ih ih × × ih ih 7 b o , x i ij ) ti ayt hwthat show to easy is It 1). 234 347 × 234 234 2 6 6 and ∧ sdfie sflos w diagonals Two follows. as defined is ylcpruain f1 of permutations cyclic + , , X 1 7 7 X i i i i i n b − × × o o a y 1 − 2 2 ) . 1 1 = − − ∧ {− i i h 1 n o o 7 X , r n − (1 2 2 x × − F } ∧ ∧ h h , h 2 1  167 137 3 127 11 n n EF h , hnlsigtedaoasat diagonals the listing when {− ∧ , 5 2 4 − − b b ih ih , × 0 ih × nfc,w n htthese that find we fact, In 0. = x 5), = 347 467 h h h h h onn etcs1ad3. and 1 vertices joining 345 { 6 3 123 167 246 146 234 and y x , , X } 1 X 7 4 x , i i 2 i b − ih ih ih ih ih × o × b ∈ 2 1 y 2 = 567 456 345 567 456 } . 1 5 ∧ eogt t This it. to belong Λ = , i b n r 2 2 o EF , i i i i i x (1 B − 2 × o o 2 1 , . . . , x 2 , 2 2 h h CONTENTS = 2 3 2 347 345 b itdabove listed h Poisson The . , i fe the After . 3 − o , 7 ih ih 2 .I two If 1. ) Now 4). ,y x, . 456 467 E X (131) (132) we ) and i i a 0 o = 2 Preliminary version July 10, 2013 φ is field section base some this over in work will discussed we constructions following the algebra In Lie [27]. the ref. notions for mathematical source The of Glossary .1 NOTIONS MATHEMATICAL OF GLOSSARY .1. nvra neoigalgebra enveloping universal .1.5 Definition by denoted algebras Lie two .1.1 Definition optdin computed nvra neoigalgebra enveloping universal elements the Let by generated follows. as coset, a that: algebra enveloping universal .1.2 Definition edefine we 1 If ento .1.4 Definition for .1.3 Definition ler ydfiigapoutby product tensor a their defining form by can we algebra and spaces vector also are product they then field) same the B ([ ,B A, then , teeeit map a exists there 1. if 2. ,y x, ,c a, hs.Tecniinthat condition The phism.  ooopim hnteei nqehomomorphism unique a is there then homomorphism, fascaieagba uhthat such algebras associative of )=[ = ]) ([ ∈ r soitv then associative are A ,y x, A A A sayascaieagbawt ntand unit with algebra associative any is T ⊗ ⊗ )= ]) V T n and φ V L B B ( x is , hl h rce [ bracket the while = svco ed.Ti esrpoutcnb aeit an into made be can product tensor This fields. vector as φ a nt1 unit has ) ,d b, Uieslevlpn ler,atraiedefinition) alternative algebra, enveloping (Universal Uieslevlpn algebra) enveloping (Universal Leagbahomomorphism) algebra (Lie φ , Tno rdc falgebras) of product (Tensor ( L Tno algebra) (Tensor | V x , ( ) ∈ y φ ⊗ · · · ⊗ L )] ( 0 B y ( 0 {z saLeagbahmmrhs fi slna and linear is it if homomorphism algebra Lie a is n a ) I o all for n xedn tt h whole the to it extending and −  ⊗ etetosddielo h esralgebra tensor the of ideal two-sided the be A : x A φ b ⊗ V U L V T U )( ( U ⊗ } ⊗ U y with , 1 ( ( c ( ) → L L B y ( L B φ ,y x, ⊗ L of ) . = sa soitv ler ihui such unit with algebra associative an is ) ( fteLealgebra Lie the of ) − ·  = ) . x sascaie If associative. is d , U ⊕ ). · ( = ) y saLeagbahmmrhs reads homomorphism algebra Lie a is If ] ( 0 ∈ L n ∞ T L ⊗ L/I. T scmue in computed is =0 α V hc saLeagbahomomor- algebra Lie a is which ) 0 sdfie as defined is L V x T ac = oeta h rce [ bracket the that Note . savco pc vrafield a over space vector a is − n = ) β V. ⊗ [ ◦ ,y x, K . . (  . h esragbaof algebra tensor The . map A bd . If ie i algebra Lie a Given ,for ], α ) A , : ,B A, , L B L L → φ ,y x, 0 A . r lers(over algebras are : aeuis1 units have a edfie as defined be can A ⊗ L ∈ K saLealgebra Lie a is B → β . L ylinearity. by : L hnthe Then . U 0 between , ( L ) L A · . (134) (135) (133) , → the , · The and is ] L T V K 39 A , , Preliminary version July 10, 2013 U then homomorphism, bra .1.1. Lemma 40 nue nascaieagbahmmrhs : ∆ homomorphism algebra associative an induces x algebras enveloping universal of morphism .1.1. Theorem homomorphism algebra φ Define U Proof.  algebras Lie em 11 eseta hyidc homomorphisms induce they χ that see we .1.1, lemma ler homomorphism. algebra χ Since steiett.Tesm od o (Id for holds same The identity. the is all for map a define We non-ambiguous. and commutative is  nue homomorphism a induces on jection ( 0 0 h map the ( ( π ( 7→ : : L L u ( Let ealta ehave we that Recall ehave We [ Since ehv homomorphism a have We trmist hwthat show to remains It L U x 0 ) 0 ) ). ψ x )) χ ( K → → L u ⊗ ( : ε ewl sals nioopimbetween isomorphism an establish will We ⊗ L ⊗ u 0 U ∈ ) : A 1 + 1 o any for ) U 1 + 1 L ( U → eaLeagbaoe field a over algebra Lie a be ,x x, L  L L ( saLeagbahmmrhs,then homomorphism, algebra Lie a is ( L 0 ∼ = : , ) hsi i ler ooopimad ytelma.1.1 lemma the by and, homomorphism algebra Lie a is This . L L U L 0 u ⊗ 0 ) , ) ( for 0 = ] L ⊗ ⊗ ( iial,if Similarly, . 0 If ε L → L → ∈ U ecnietf 1 identify can we ⊗  If x 0 ,L L, 0 ⊕ ( nteruieslevlpn ler.W lodnt by denote also We algebra. enveloping universal their in L u ( U K L ensaLeagbahmmrhs.B em 11it .1.1 lemma By homomorphism. algebra Lie a defines Id) π ,L L, 0 L ∈ ( 0 0 ) ( n hnetn tt h whole the to it extend then and L by 0 0 x ). U → ◦ r w i lersand algebras Lie two are ⊕ ) where )), U  U 0 ( ( ε : ( L r i lers hnw aetefloigiso- following the have we then algebras, Lie are x ( x U 1 and 1 = (1) L φ φ L L L ), ⊗ 0 ( φ n by and ) ) ∈ : ⊕ L nue nascaieagbahomomorphism algebra associative an induces → u U → 1 + 1 and L ⊕ 0 L L ( ∈ L U L 0 A ) L saLeagbaand algebra Lie a is π ( and ⊗ U ⊕ . ⊕ ∼ = 0 χ L ) ⊗ stepoeto on projection the is ( ψ , x and ) L L L r nes foeanother. one of inverse are U  x with 0 0 0 0 ε x ( .Teeoe h product the Therefore, ). ⊗ ) 1 = ) ( L h map the 0 → x → K ε ) for 0 = ) ∈ )  h map The . ⊗ x ( 0 ◦ U U u o( so , : L ⊗ ∆. U ( ( L ⊗ 0 L L egtthat get we , ( ,x x, L ) ) → u  ⊗ 0 ε ⊗ φ 0 0 ) = ) φ ⊗ . : U x U U : U U L ∆) χ nue nassociative an induces ( L ∈ ( ( ( L ( 0 χ L L L L L A : → L 0 → L L ( ◦ ,wihebdthe embed which ), 0 ∈ 0 U ) ). ⊗ ⊕ u endby defined ) : ∆ sa ler and algebra an is → ) n xeda an as extend and and → ( U L. L χ L L L ( 0 0 0 0 ) ( L U and ) U U u saLealge- Lie a is ylinearity. by χ → CONTENTS π ( ⊕ 0 ( ( ( ) 0 L L L u , stepro- the is χ U L ) ) ) ) ( χ 0 ⊗ ( ⊗ → u .From ). U L 0 ( ) ( U U u χ ⊕ (136) (138) (137) U L x 0 0 ( ( = ) L ( ( ) L L u L → 0 ⊗ ), ). ), 0 ) ) Preliminary version July 10, 2013 ento .1.9 Definition homomorphism. algebra 1 an = is ∆(1) used have we where K (∆ .1.8 Definition then homomorphisms, algebra are .1.7 Definition ( satisfying counit : ∆ map ob h usaeof subspace the be to x .1.6 Definition NOTIONS MATHEMATICAL OF GLOSSARY .1. L h utpiaino gr on multiplication The ε U .1.2. Lemma gr the that implies which x .1.10 Definition gr so grading rmtv,then primitive, ([ hnw have we Then . ( ⊗ (∆ ∈ ,y x, and L ⊗ xml fcascaiecmlilcto.Take comultiplication. coassociative of Example edefine We If 1 + 1 L ) ⊗ Id) ⊗ ,y x, ]) scle rmtv f∆( if primitive called is gr U 1) U ∈ ( n ◦ C ⊗ L ( ∈ ◦ U (Id = ∆ L U t nvra neoigalgebra, enveloping universal its ) → ∆( x ( ) L 1 L Then, . U L then , uhthat such x = ) C x ( ngr In . Let x (∆ = ) L Gae i algebra) Lie (Graded (Coassociativity) ⊗ ⊗ (Bialgebra) ∈ (Coalgebra) sacmuaiealgebra. commutative a is ) ε U Piiieelement) (Primitive ⊗ 1 L ⊗ C L U n ε ∆) ⊗ . ( 0 U U ( Id) eaLeagbawt comultiplication with algebra Lie a be L aldcmlilcto n map a and comultiplication called ( x 2 0 ⊗ L 1 + 1 ( ) U ( ) U ◦ L ∆( /U = ) L U ε , ◦ m 1)( eeae ypout fa most at of products by generated ) ∆. ( ) 2 ( L ( ( d=(Id = Id = ∆ x n L L ⊂ y − x h H ftepeiu qainvanishes equation previous the of RHS the ) . = ) scmuaie h aehlsfrhigher for holds same The commutative. is K ⊗ ) sidcdb h utpiainon multiplication the by induced is ) . ) 1 ⊗ x ⊗ If U ( U , ∈ · Let = ) x L 1 ,wihflosfr h odto ht∆ that condition the form follows which 1, U 1 + 1 C ( ) ⊗ U x n L , . C ( ( x ⊗ sa ler n oler n ∆, and coalgebra a and algebra an is ) C 1 + 1 L outpiain∆i osoitv if coassociative is ∆ comultiplication A L · · · ⊗ ) sabialgebra. a is n ehave we and ) 1 + 1 . 1 eavco pc vrafield a over space vector a be ⊗ ( ⊂ . Let 1 + 1 gr L U x ⊗ ie bialgebra a Given = ) ⊗ U n ∆( = ) U ( ε n L L 1 ) ⊗ ( + L ) ⊗ ◦ m K ⊗ eaLeagbaoe h field the over algebra Lie a be  x = ) · · · ⊂ ( ,i aldacoalgebra. a called is ∆, x ⊕ L x : if . x ) L (1 =  ) . ⊕ ( x ⊗ → L n , ε ∈ ) gr ( ∆(1) + 1 x , ⊗ U x n L ) uhta ∆( that such ( ε ∆) U L If .  ( y .Define ). ( : L ) ◦ L C ) n − : ∆ nelement an , ∆( x . ⊗ → lmnsof elements ε ∈ x ( x x ) U K U , U ) = ( ε ( U ( L called K ( L (139) (142) (140) (143) (141) L x y n ) A . = ) ). ) = ) ( L → 41 is ε ) Preliminary version July 10, 2013 exists ee set dered .1.14 Definition ( h qiaec relation equivalence The ∼ −→ lim φ gop,rns oue,agba)idxdb nodrdset ordered an by indexed algebras) A modules, rings, (groups, .1.13 Definition given that, is name the series the .1.12 Definition L decomposition .1.3. Lemma call and all ( by make ento .1.11. Definition of elements of on Multiplication Set with 42 x ii X i , i → Id = a edefine We Let A seuvln oalisiae ne h maps the under images its all to equivalent is = M ab n ,b a, i A A X A ) k ntedcmoiinof decomposition the in fthe of c X X j X | ∈ = ( + L A ∈ ehmmrhssdfie o all for defined homomorphisms be /I I L i noa algebra. an into X east eset We set. a be I and , ` X [ , sastwt ata order partial a with set a us bc uhthat such h reLeagbaof algebra Lie free the sagae ler swell. as algebra graded a is n ∞ 1 A ,L L, A ) so , =1 a a i h ideal The M X ( + stedson no of union disjoint the as = φ X X ob h etrsaeo nt omllna combinations linear formal finite of space vector the be to ,[[ ], ik M X Drce atal ree set) ordered partially (Directed n Drc Limit) (Direct P NloetLealgebra) Lie (Nilpotent h utpiainon multiplication The . Let ca nelement An . = 2 X ,L L, ) = a i φ b scnaeaino o-soitv words. non-associative of concatenation is n I for , a ij ≤ X ∼ nohmgnoscomponents homogeneous into ] ⊂ I L , ∈ ◦ 1 k X X sdfie ytkn,frany for taking, by defined is endaoei rdd hti for is that graded, is above defined φ × A −→ lim L i ,ec eoe eoeetal.Terao for reason The eventually. zero becomes etc. ], and ,b c b, a, ∈ n 1 jk X h ieroeao ad operator linear the , I X := = a A o all for L etetosddielgnrtdby generated ideal two-sided the be 2 ic h ideal the Since . edefine We . j X p i . X w + a = ∈ ≤ Let q = X Then . = ∈ A G k n . A i i X . M X A X ≤ A eset We . A p x ≤ /I i i j × i  scle o-soitv word. non-associative a called is . eafml fagbacobjects algebraic of family a be X uhta o any for that such ouoa qiaec relation equivalence an modulo ≤ i ∼ i algebra lie A X M 2 ≤ k q ossso l expressions all of consists . X . edfietedrc limit direct the define We . j ihtepoete that properties the with , xed yblnaiyto bilinearity by extends . φ I ij ietdprilyor- partially directed A sgae,tequotient the graded, is ). a x snilpotent. is a i n ∈ then , L A snloetif nilpotent is i ,j i, CONTENTS , a I x Let . i a ∈ ∈ ∼ n I I ∈ aa φ there , n a and (145) (144) (146) ij I φ ( and ij x for ab i ) : Preliminary version July 10, 2013 CP .1.16 Definition h ipettp fcosrtoi h rs-ai ffu ons( points four of cross-ratio the is cross-ratio geometry of type projective simplest The of Elements .2 algebras. nilpotent of limit inverse an as written be can h ie ( lines the ( points is limit inverse that properties gop,rns oue,agba)idxdb ietdprilyordered partially directed a by indexed set algebras) modules, rings, (groups, ffu onsb aiga rirr line arbitrary an taking points by points four of con- to line. situations projective complicated a more on reduce points to four try of will figurations we following the In .1.15 Definition GEOMETRY PROJECTIVE OF ELEMENTS .2. hrfr,w a akaottecosrtoo orlnsin lines four of cross-ratio the about talk can we Therefore, is 1 h rs-aiso orlns( lines four of cross-ratios The in point a duality, By I ftepit aehv oriae ( coordinates have have points the If . Let . a ,b ,d c, b, a, = ,β ,δ γ, β, α, φ ρ ←− lim ij i ∩ ∈ I : α iue2 h rs-ai ffu ie in lines four of cross-ratio The 2: Figure on ) A A φ , i ii j b a = (Pro-nilpotent) ) → IvreLimit) (Inverse = Id = ρ n ρ A sidpneton independent is ~a ∩ i | ∈ β A r ehmmrhssdfie o all for defined homomorphisms be ( i , Y and , ,β ,δ γ, β, α, i ∈ c r CP α I ( = ,b ,d c, b, a, A 2 b ρ i φ si orsodnewt iein line a with correspondence in is ∩

,β ,δ γ, β, α, . . ik a γ = ) nalgebra An i Let , = ρ = ) β = d φ r = φ A ij ( O ρ ij z z ,b ,d c, b, a, z i a erltdt h cross-ratio the to related be can ) ρ ρ ◦ bc ab ( a n seult h rs-ai of cross-ratio the to equal is and a eafml fagbacobjects algebraic of family a be γ ∩ c z , n optn h intersection the computing and z φ z j da cd δ jk ) b , z , hn h rs-ai fthe of cross-ratio the Then, . . A o all for ∀ δ c ) i z , . scle rnloeti it if pronilpotent called is ≤ d ,te hi cross-ratio their then ), ,i j i, j, i d CP ≤ j ∈ 2 CP . i ≤ I ≤ o 2 k j hnthe Then . sefi.2). fig. (see ,b ,d c, b, a, ihthe with , (147) (149) (148) CP in ) 43 2 . Preliminary version July 10, 2013 rdcdb ocao.W aetesxpit obe to points six the take We Goncharov. by troduced 4. fig. See ueial,ti rpertoi ie by given is ratio triple this Numerically, points the of cross-ratio the be to ( conic X a given However, belong cross-ratio. their define ( oiswihcnanthem. contain which conics are which lines four ( have line the we with There, intersecting by 5. fig. blue: in and situation dashed the first sider where inter- common their by determined lines four of point cross-ratio section The 3: Figure 44 b XA OC ( = 12 nteconic the on e snwdsustetil ai fsxpit in points six of ratio triple the discuss now us Let ftefu points four the If points of pairs by defined are lines the If ttrsotta hsrtohssvrlgoerclitrrttos Con- interpretations. geometrical several has ratio this that out turns It n oi sdtrie yfiepit.Gvnfu onsteei nifiiyof infinity an is there points four Given points. five by determined is conic Any r ), ,( ), AX ( ,β ,δ γ, β, α, h 12 δ YZ XY r XB ( ( = ) to ,β ,δ γ, β, α, ∩ ,( ), ( OD C BY i hnw a en hi rs-ai sflos ikapoint a pick follows: as cross-ratio their define can we then , O spootoa oteoine rao h rage∆( triangle the of area oriented the to proportional is = ) r XC 3 ,a nfi.3 hntecosrtoo h orlnsis lines four the of cross-ratio the then 3, fig. in as ), n nte on nec no them. of on each on point another and ), C ( ,B C B, A, hn yCals hoe h rs-ai ftelines the of cross-ratio the theorem Chasles’ by Then, . = ) a r c n ( and ) α ( = ,b ,d c, b, a, A r ( = , A ( B ,b ,d c, b, a, ; and A XD , ,Y Z Y, X, CB C ( = ) , d sidpneto h point the on independent is ) ), AX D b ( = ) ( = β ontbln oaln ecntgenerically can’t we line a to belong not do O = ) ,i ie by given is ), A CZ | ( = ,B ,D C, B, A, , C B B h h | ) ABY ABX B, , Cb ∩ C O ( ( , ), AX AX C D c ih ih γ BCZ ) BCY wt epc oteconic the to respect (with .Tercosrto obtained cross-ratio, Their ). ) α ≡ ( = ∩ ( = C ( h h BY D OBC OAB ih ih Cc uhthat such OA CAX CAZ ), ) ,Z A, , d A CP ), ih ih δ , OCD ODA i i X β B 2 ( = . , ) hc a in- was which ( = n sdefined is and . CONTENTS C A Cd i i , , OB , X B ,Y Z Y, X, ,where ), , , ), Y C (150) (151) (152) γ , , C Z D = ). ). . Preliminary version July 10, 2013 γ h orsodn gr sfi.6 fw eoeby denote we If 6. fig. is figure corresponding The ( line the on points of cross-ratio a as expressed ratio, Triple 5: Figure points of cross-ratio The 4: Figure GEOMETRY PROJECTIVE OF ELEMENTS .2. h nescinpit are points intersection The ( line the r ( 0 r α ( = ( u,isedo osdrn h nescin ftelns( lines the of intersections the considering of instead But, o ecnrpa h rvospoeue ecmuetecross-ratio the compute We procedure. previous the repeat can we Now 0 ,b ,d c, b, a, β , AC 0 γ , AX 0 ), δ , = ) δ 0 ycnieigteitreto ih( with intersection the considering by ) saoe ecncnie h nescinwt h ie( line the with intersection the consider can we above, as ) 0 ( = r ( Ad ,β ,δ γ, β, α, a C D a d c b 0 0 0 0 0 ,w have we ), = = = = = C r δ γ β α d b ( = ) ∩ ∩ α ∩ ∩ B 0 ( ( β , ( ( BY BY BY BY r 0 ( γ , a Y ( = ) ( = ) 0 = ) = ) b , X 0 δ , A 0 c , 0 ( = ) , b B, CZ CA 0 B ( = d , , 0 ) A ) = ) C A X AX B ∩ ∩ , | = ,X C, X, B, ( D ( BY BY ) c ∩ ihrsett h conic the to respect with C ) ( ) . BY , α CZ ( 0 BY ) Z ( = , A .Teintersection The ). ) AB ∩ ,β ,δ γ, β, α, ( ), CZ β 0 )) ( = . with ) AX AX BY (154) (156) (155) (153) (157) 45 C ), ). ). . Preliminary version July 10, 2013 onsare ( points line the on points of cross-ratio a as expressed ratio, Triple 6: Figure 46 e g o emtia ersnain fw en h lines the define we If representation. geometrical a for 7 fig. See β ( ht( that symmetry the by implied also is r this that Notice aezrsadplsas poles and zeros same ( ntefis case first the In A CZ 3 00 ( ( | B ,C A C, B, ( = ,X C, X, B, ehv hrfr hw that shown therefore have We e snwso htteivrat( invariant the that show now us Let ) | i A, A Bb .Tescn he-rce aihsif vanishes three-bracket second The 0. = | ( ,X C, X, B, CZ 00 ; ), ,Z X Z, Y, ( γ ) BY ∩ 00 ( ( = AX ) ( ,C Y C, B, ∩ BY B ). BC ( b ) CZ ′ = ,Y C, , a d c b ) 00 00 00 00 ), ∩ a )=( = )) = = = = ′ r δ ( r olna n hrfr ( therefore and collinear are 3 00 CZ = ) ( α δ β γ ,B C B, A, ( = 0 0 0 0 Y ∩ ∩ ∩ ∩ B )vnse when vanishes )) X r ( Bd ( | ( ( ( CZ A, CZ CZ CZ α 00 00 ( β , ; ,w have we ), ( = ) CZ = ) ( = ) ( = ) ,Y Z Y, X, 00 γ , ) C, ∩ BY AX AB 00 A A ( d δ , AX ′ | .Fr h ento,w know we definition, the Form ). ,X C, X, B, 00 ) ) ) = ) ∩ ∩ ∩ ) ,Y C, , h ( C ( ( ABX CZ h CZ r CZ BCY ( a ) 00 ) ( ) r ( = ) . b , Z , BY , 3 i BY ( i 00 c ,B C B, A, or 0 = c , or 0 = ′ ) C ) 00 ∩ d , ∩ | B, ( 00 ( CZ CONTENTS = ) CZ ( ; h h α AX AC ,Y Z Y, X, CAZ 00 )hsthe has )) = ) r ( = ( ( ) α BY ∩ i 0 BY (161) (158) (160) (159) (163) BA ( β , C 0. = BY = ) ) 0 so γ , ∩ ). ), ) 0 ,Z A, , δ , (162) 0 ) . ) . Preliminary version July 10, 2013 te rs-aista a ecntutdfo hs v onso ( on points five these the from for three-brackets 5): constructed of fig. be terms (see can in that expressions cross-ratios the other find to us motivates This h ( have we rmfiepit ( points five From way. same the in h ( line the on points of cross-ratio a as expressed ratio, Triple 7: Figure GEOMETRY PROJECTIVE OF ELEMENTS .2. h ueao of numerator the if vanishes CZ AC CAZ oieta nfi.5 ehv v ons( points five have we 5, fig. in that Notice .Snealteetiso h he-rce r olna,w n that find we collinear, are three-bracket the of entries the all Since ). ( BY i ehv that have we 0 = ) h AC ∩ h ABX ( CZ ( BY ) i r r r r z i ) r r 3 ( ( 1 ( or 0 = ( X ( ,X ,d A, X, a, ( ,b ,A X, b, a, ,X ,d A, X, b, z , . . . , i .W aesonta ( that shown have We 0. = ∩ B ,b ,d X, b, a, =1 ,B C B, A, ,b ,d A, b, a, 5 ( ( CZ − Y 1) 5 A h X ) in ) BCY i i ; { ∈ = ) = ) ( = ) = ) ( = ) ,Y Z Y, X, r = ( ( CP z CZ 1 r r h i h , . . . , C B ACC 3 3 A 1 or 0 = ( ( | ), | ,B C B, X, C B, A, ,X ,Z A, X, B, × ecnpoueadlgrtmidentity dilogarithm a produce can we ,Y ,A X, Y, C, .I re ofidteplsw reason we poles the find to order In ). A h C BXY ,B X, z b d i i ′′ z , . . . , ∈ .I h eodcs,when case, second the In 0. = h CAZ ( C ; ; CZ ih × ,Y Z Y, X, ,Y Z Y, A, ,b ,c d c, X, b, a, = ACZ ) ,C Y, ) 5 . and ) , ) c i Z } ′′ A hc stesm as same the is which 0 = 2 | i × 0 = b ) ) ,X C, X, B, ′′ , , Z P a . i nteln ( line the on ) ′′ ≡ , ( BY ( BY ) ) ∩ ∩ ( CZ ( AX CZ (167) (164) (168) (166) (165) (169) CZ AX ) 47 )) ∈ ). ). ) Preliminary version July 10, 2013 a eitrrtdgoerclya v ons(3 points five as geometrically interpreted be can dilog- the of one is example, useful For is zero. which is identities identity arithm trilogarithm 40-term the of 48 5) nteln (34). line the on (56)) a eitrrtdgoerclya v ons(1 points five as geometrically interpreted be can (1 points five as geometrically interpreted be can 5) nteln (12). line the on (56)) (12). line the on (56)) − −   − hskn fiette r sflt hc htthe that check to useful are identities of kind This hspoie emti ro o h olwn ioaih identity dilogarithm following the for proof geometric a provides This   − −  − h h h − A 1 1 h 1 h × × × 1 h × h h + BXY 2 2 × ,B X, 156 123 h , , 2 123  h + 3 3 2 , 156 3 , × × ih ih h h   3 XBY XBA × ih ih × 234 456 − − 4 4 ih × 456 , , ACZ 4 ,C Y, 234 h h h h 5 5 4 ,   126 123 i i 125 124 5 , × × ih i ih − − 5 × i BCY 6 6 BCZ × ih ih ih ih i × h h h h i i 124 125 125 124 6 134 156 134 156   6 i Z i  2 2 i  ih ih ih ih + ih ih ih ih ih ih 2 −  − 2 134 156 234 256  CXZ CXA 356 346 456 345 2  +  + − − −  ih ih ih ih  i i i i h h h h  123 125  345 456 456 345 h h − h h 136 123 i i CBX CXA 123 125 2 2  h h 145 124 − + ih ih i i i i 2 ih ih ih ih   245 234 −   234 346 145 134 2 2 ih ih ih ih − −  − + 234 345 CZB CAZ i i h h h h i i h h i i    125 124 126 123  BCY YX BY  2 i i − − , , , 2 2 +  4 2 2 − ih ih ih ih − i i h h h h , , ,  2 135 235 256 156  (15) (12) (12)  234 256 234 256 ih  + ih − 2 − − − BXA BAC ih ih ih ih  h h ih ih ih ih h h ∩ ∩ ∩ 125 123  h h 456 456 345 345 125 123 − B 456 345 356 346 126 156 (34) (34) (34) h h 2 h h ABY ABX ih ih 125 156 i i i i i i ih ih ∧ i i i i ih ih 234 256    , , ,   134 156 C (12) (12) (12) 356 236 CONTENTS 2 2 2 ih ih 2 2 ∗ ih 0 = 0 = 0 = ih ih ih 456 245 (173) 0 = (172) 0 = ih ih projection i i BCZ BCY 356 345 ∩ ∩ ∩  356 345 (34) (45) (36) , . , i i 2  i i i i (171) (170) (174)  2 ih  ih , , , − 2 (34) (12) (12) CAX 2 CAZ − + ∩ ∩ ∩ i i  2 Preliminary version July 10, 2013 erclitrrtto.Truhtefiepit ,2 ,5 assaunique a passes 6 5, 4, 2, 1, points conic five the Through interpretation. metrical as written geometrically more be can (3 points five as geometrically interpreted be can BIBLIOGRAPHY ihrsett h conic the to respect with below written too, identity simpler another, is type there of but sufficient, terms are the For identity. type trilogarithm of terms of (34). line the on (56)) ae yCals hoe,ta ( that theorem, Chasles’ by have, −{ becomes identity SecrJ Bloch. J. Spencer [3] al- Cluster Zelevinsky. Andrei and Fomin, Sergey Berenstein, Arkady Simon [2] Cachazo, Freddy Bourjaily, L. Jacob Arkani-Hamed, Nima [1] Bibliography hc steuulfr ftedlgrtmiett,weetecross-ratios the where identity, ( lines dilogarithm the the of of cross-ratios form are usual the is which { (1 − ( uiul,ti ipeloigiett a lgtymr bcr geo- obscure more slightly a has identity simple-looking this Curiously, It all. at 3 point on depend not does it because special is identity This h dniisaoeaeteiette eddt hwtevanishing the show to needed identities the are above identities The X | ers i:Uprbud n obebua cells. bruhat double and bounds 2005. 126(1):1–52, Upper iii: gebras. sym. scattering n=4 for planar integrand in all-loop amplitudes The Trnka. Jaroslav and Caron-Huot, ahmtclScey rvdne I 2000. RI, Providence, Society, Mathematical curves elliptic of tions 2654)  C | − 2456) h rs-ai (1 cross-ratio The . h h 124 126 } 2 } + ih ih 2 + 156 145 { { (2 ( h ⊗ ∗ X i i | 1456)  | 1456) 2 + ihrrgltr,algebraic regulators, Higher 123 }   2 C } oue1 of 11 volume , − i − + | u ecnpc nte point another pick can we But . 2 64 stecosrtoo h ons(2 points the of cross-ratio the is 2654) −{ X h h ntepoeto to projection the in h h { 124 126 125 156 (4 ) ( 1), ( X | ih ih 1652) ih ih X | 256 245 X 1256) 456 245 | JHEP 64 (1 = 2654) ) ( 2), i i } i i   2 R oorp Series Monograph CRM } 2 X + 2 2 + − 1101 2011. 1101:041, , − ) ( 4), { { (5  h ⊗ ∗  ( X − | − , 1246) X | 4 | 64.Te h previous the Then 2654). h h 1246) h h , 124 146 ) ( 5), 126 156 B K 124 (12) 2 ter,adzt func- zeta and -theory, } ih ih ⊗ ih ih i 2 X } ∩ 456 245 456 246 + h aeidentities same the 2 C (34) 6). −{ ∗ { i i i i (6 ueMt.J. Math. Duke  ( fte40-term the of  , X X | (25) 2 2 1542) + | C ∈ 1245) 0 = American . ∩ (34) . } , n we and 6 2 } , (176) (175) (177) 2 0 = 5 , 0 = , (34) 49 4) . , , ∩ Preliminary version July 10, 2013 Kota hn lerso trtdpt nerl n fundamental and integrals path iterated of Algebras Chen. Kuo-tsai [5] Vaii .Fc n lxne .Gnhrv lse ensembles, Cluster Goncharov. B. Alexander and Fock V. Vladimir [10] and polygons From Rhodes. R. John and Gangl, Herbert Duhr, Claude application an [9] symbols: and coproducts algebras, Hopf Duhr. Claude [8] two-loop The Smirnov. A. Vladimir and Duhr, Claude Duca, Vittorio [7] analytic An Smirnov. Vladimir and Duhr, Claude Duca, del Vittorio [6] values. zeta multiple motivic of decomposition the On Brown. F. [4] 50 Sre oi n nriZlvnk.Cutragba.i:Fnt type Finite ii: algebras. Cluster Zelevinsky. Andrei and Fomin Sergey [12] Foundations. i: algebras. Cluster Zelevinsky. Andrei and Fomin Sergey [11] McalGkta,McalSaio n lkVisti.Cluster Vainshtein. 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