Wilson Surfaces for Surface Knots

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arXiv:1903.02853v1 [hep-th] 7 Mar 2019 isnSrae o ufc Knots Surface for Surfaces Wilson itnssmnlwr [ work seminal Witten’s aso on that. doing of ways in njs n ieso ihrteeaea es two least at are there higher dimension one just In notion? topological simple this generalizing knots mensional n ntcnqe fqatmfil hoy ihayknot any With theory. field quantum rely- of theory, techniques on gauge ing of instance an theory, Chern-Simons n a en –iesoa nta nembedding an as knot S 2–dimensional a define may one odne atrter n nttplg.A hw in shown As topology. knot theory, and theory string matter in condensed breaking formulation symmetry gravity, loop quantum chromodynamics, of quantum in ment inlmnfl,say manifold, sional nt r neetn ntplg swl si ag the- gauge in as [ well ory as topology in interesting are Knots Introduction 1 2–holonomies. crossed invariants to knot a application surface upon over yields that trace given of been has definition module Us- a determined. theory, been quandle have data ing base of change transforma- and 1–gauge tion under defined properties been covariance has its genus and arbitrary of knots higher surface holon- of the to omy 2–connection, applications flat a to For aiming theory. have Chern–Simons depth, theory in gauge studied higher been strict in invariants Holonomy Zucchini Roberto M-Theor in Structures Higher on Symposium Durham LMS/EPSRC aiod n a en –iesoa nta an as knot 2–dimensional of a embedding define oriented may closed one trivial non manifold, dimensional lowest the also td ennfl -adsraekosaeojcso in- [ of topologists objects by are investigation tense knots surface and separate 2- a meaningful. make study which properties However, special knots. very surface have 0 they genus just are 2–knots course, retdsrae hslast genus to leads This surface. oriented 2 into Since riaykosaeebdig of embeddings are knots Ordinary isnlns[ lines Wilson 1 ]. S 4 S hsyed h ocle –nt.Since 2–knots. called so the yields This . 1 stelws iesoa o rva sphere, trivial non dimensional lowest the is S ℓ 6 r eeati h nlsso confine- of analysis the in relevant are ] into S a, 3 7 S ∗ ,oecnsuyko oooyin topology knot study can one ], [ 4 2 where , , 3 .Cnoedfiehge di- higher define one Can ]. S 4 ℓ , 5 sagenus a is ℓ ]. S 1 ufc nt.Of knots. surface noa3–dimen- a into ℓ closed S 1 is hs xrsin sa hspitt efud nthis In found. be to point this at is expressions whose where W W h rbe a w parts: two has problem The ino h isnsurfaces Wilson defini- the a of for tion proposal a present shall we communication, r.T hsed n ed oascaewt n surface any with associate to knot needs one end, this To ory. invariants. clas- knot provide sic operators line Wilson of correlators Simons r [ theory gauge higher of instance version an theory, higher Chern–Simons appropriate of an in surfaces Wilson correlators knot computing of topology knot surface presumably or can 2– study one paradigm, Witten’s Following ogy. topol- knot higher and gravity quantum the- theory, gauge brane ory, higher of aspects perturbative non of study the a ∗ [ Picken [ ξ ae anyo u ok[ work our on mainly based h oko atn n ikn[ Picken and Caetano of includes work the transport parallel higher on endeavours Earlier trans- parallel of port. case distin- special but a being related holonomy are guished, holonomy and transport Parallel ii) i) 15 n soitsteWlo line Wilson the associates one , IA nvriyo oon,adIF,Blga Italy Bologna, INFN, and Bologna, of University DIFA, e-mail: author Corresponding R ( , Ξ isnsrae [ surfaces Wilson ( 16 ξ en ihrivrattraces invariant higher define en ufc ntholonomy knot surface define 10 ) Ξ ) = crie n adr [ Waldorf and Schreiber ] = R , isnsurface Wilson a 11 , ? 20 tr sarpeetto fteguegroup gauge the of representation a is ,uigaantcnqe fqatmfil the- field quantum of techniques again using ], R , h 21 Pexp ,Catre,Lhr n egpa[ Sengupta and Lahiri Chatterjee, ], ³ − 8 Z , ξ 9 A a unott erlvn in relevant be to out turn may ] í (1) . 12 W [email protected] , ( 13 Ξ 17 y nhge ag theory gauge higher in ) ]. 14 . ; – 19 ,Be n Schreiber and Baez ], ,FraMrisand Martins Faria ], G Chern– . 22 – 24

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F ae at based F p 2 A A M ]). → , , .Zchn:Wlo ufcsfrSraeKnots Surface for Surfaces Wilson Zucchini: R. B B F ) ( ( → Ξ p Σ A H ( ) ). n surface a and ξ B = −→ sbs aaadiooyin- isotopy and data base is ) p ( Ξ t G F n ufc knot surface a and A , ln standard along G H , M B ( ,(5) ), fteLecosdmod- crossed Lie the of . Σ 32 Ξ , .(6) ). 33 M G n ocn and Soncini and ] Σ , P , Ξ 1 H 30 M : sas obe to also is ) ι n Crans and ] p τ , a P stemming ⇒ 0 and 2 Ξ M τ based pto up ). b A cy- , B Proceedings . , ◦ e 3 1 2 3 γ the γ < of M is the given is the ⇒ ) M 1 ⇒ 2 y 2 Σ 0 p Σ 0 , γ 0 γ γ x P ( : → ◦ ⇒ 0 0 2 M such that 0 , the vertical M ofM such dh γ p Σ γ 2 : : and . (Thin) homo- : → are four curves γ , the horizontal → 1 3 1 1 3 ). 3 1 2 γ M Σ y γ < Σ p , rank ⇒ Ê 2 , 1/2. ◦ 1/2. ) 0 • Ê : 1 P x 2 : z 2 γ ⇒ ( → ≥ ≥ and , γ Σ 1 Σ 2 1/2, (17) 1/2, (15) y 0 : x 1 , γ Σ p 2 ≤ ≤ : x ≤ : ( = Σ 2 ) x for y the sets of thin homotopy , (19) 3 ). , z 1 Σ γ x 1 , , ,1) the sets of all curves and sur- ( p d H M defined by for y γ 2 1 y of two surfaces 1 ,1, 1 ◦ , = γ γ ) for 0 γ x , (18) M 1 1 , the vertical inverse of , x ) ( 1) y ⇒ 1 2 − 1 P 1 ( , the horizontal inverse of = Σ z ⇒ of two curves 1 rank p 0 γ γ − Π 1 , ) ) for 1, H , γ is the surface 0 1 defined by , is the surface y γ γ ). (14) ). (16) y y y d H = 0 − ,1) γ γ ⇛ 2 : , 2 0 y y ⇒ ) and γ : M ⇒ x , ,2 ,2 x x γ Σ ⇒ 0 (1, 1 Σ y ( 0 1 ), − 1 ◦ ⇒ x x x , , 0 Σ γ Σ 1 y M ( ( (2 (2 h H 1 Π 1 Σ 0 rank ⇒ , γ 1 2 1 2 − 1 : − ,1 : (1, , γ Σ Σ : 1 0 x ) x P Σ Σ Σ Σ h : ( , Σ ( (1 γ γ z ), Σ 0 0 : : = = = = Σ Σ , where x Σ p , • ◦ ) ) ) ) ( 3 ,0, , by 1 1 0 0 = = y y y y γ = = x − − , , , , ) ) p γ ( M ) x x x x y y Σ Σ ⇒ ( ( ( ( z = = , , ,0) , 1 1 1 1 ) 2 x x y y d H ( ( y Σ Σ Σ Σ , γ • ◦ ,0) Unfortunately, these operations are not nice enough; : • • ◦ ◦ x given by 1 1 x ( (0, 1 − 2 2 − 2 2 ( (0, 1 H H The homotopy is thin if topy of surfaces is an equivalence relation. Let us denote by rank h A homotopy H faces of that h Σ The homotopy is thin if in addition and homotopy classes of curves and with the same end-points, is a map H For a surface For two surfaces Σ surface by Σ composition of Σ For two surfaces composition of Σ surface For a surface Σ γ Σ associativity and invertibility fail tooperation hold are in in general. fact nice Th only up to homotopy. A homotopy h with the same end-points is a map h (Thin) homotopy of curves is an equivalence relation. : : p 1 Σ γ (10) (13) , → 0 γ p in M is : 1 p p ι is the curve → is the surface γ are manifolds, 0 γ p T : , the composition 2 of M is a map γ p 1 piecewise given by 1/2. In what follows, and γ 2 < → S p ǫ 1 1/2. ⇒ p < → ≥ 0 ) : , 0 1/2, (12) γ x x 2 ǫ ( p : γ ≤ M and any two curves 1 M, a curve : − , , where , (7) γ , (9) x Σ 1 ∈ 1 ∈ ǫ 1 1 T γ p = 1 1 p < > ◦ , the inverse curve of p , the unit surface of p → , 1 2 , → x x = 1 . (8) ,1) 0 0 γ p ) 0 1 p Ê x 1) for such that 0 y ( p p ǫ × − : of their domains. → ) for → Σ ). (11) = S 1 x x (1, 0 0 x : Ê γ M such that p Σ p given by (2 (2 f ), : (1) : − ,0) for ,1) for ). 1 2 0 x → γ x γ γ γ − − γ , p ( (1 ( ( ( 0 0 = defined by = γ γ Ê f f of M, a surface p , γ . → is the curve : ) ) 0 γ = 1 = p = = 1 x x 2 = = γ p ) ) ( ( p M such that ) ) γ p ) 1 1 = ⇒ x y x x : , y = ( , ) γ γ , , ,0) 1 γ ◦ ◦ → Formally, curves and surfaces are defined as follows. → x x ◦ ◦ 1 1 : x γ − − ( 2 ( (0, ( ( 0 ( (0) − − 2 2 γ γ p f f I I p For any two points p has sitting instants if Ê Σ require that their parametrization hassmooth sitting map instants. A a map γ Σ For any two points p ι defined by for some number all maps will be tacitlyfor each assumed factor to have sitting instants Curve and surfaces can be combined through aural set operations of based nat- on the intuitivetion. idea We of begin concatena- by introducingcurves. the basic operations with For a point p, the unit curve of p is the curve For a curve γ For two curves γ of γ γ For a curve These turn out to bevertical. of two types, called horizontal and We introduce next the basic operations with surfaces. Proceedings ( ( h rtrlvn rpryo aalltasoti t con- its is transport parallel of property relevant first The 4 nue ytoeof those by induced –oncino sa is M on G–connection A aiod osdra riaygueter ntetriv- the principal on ial theory gauge ordinary an Consider manifold. oti ihrguetheory. gauge higher in transport parallel describe and gauge introduce ordinary we then in and Wetheory transport latter. parallel the reviewing of be properties begin the some and in parallel definition examine the of to detail case necessary special is it a Therefore, is transport. holonomy theory, gauge In transport parallel Higher 3 M. of 2–groupoids fundamental and path hs of those ruod fM. of groupoids hnvrdefined. whenever d θ F epciey h olwn eut r basic. are surfaces, results of following The classes respectively. homotopy and homotopy thin of sets F F o curve a For datum. o n on n n curves any and p point any For itnywt h prtoswt uvsdfie nSec- in defined curves tion with operations the with sistency d problem aalltasotrqie a requires transport Parallel hr u where F element M M θ θ θ θ x θ sfltif flat is ( ( ( ( u + , , ι γ γ γ Let P P p ( − 2 2 ) x 1 2 ) 1 1 . = 1 ◦ ) M = M [ ◦ G u γ θ ) u 1 ( : Π 1 , , ) F = eaLegopwt i algebra Lie with group Lie a be (1), x θ P G ) Ê and θ 1 ) ] F = 2 , ( r ruod,tept n fundamental and path the groupoids, are M − = γ M → γ θ G 1 F ( ) 0. − = γ fM h aalltasotalong transport parallel the M, of θ –bundle ) ∈ ( steuiu ouino h differential the of solution unique the is G ( ) M and γ − G 1 2 γ , (24) , ) endby defined P ∗ F 0 θ Π ( θ 1 M x ( 1 M γ ( and M M x , 1 P ) ), .(25) ). × 1 ihteoeain nue by induced operations the with g M G –valued G u , . cneto on –connection P (0) Π γ 0 2 2 , = r –ruod,the 2–groupoids, are M γ M 1 1 , ) G 1 γ ihteoperations the with –form (22) . 2 n has one , g n let and θ ∈ M Ω sinput as γ 1 M ( sthe is M ea be (23) (20) (21) , g ) . h eodrlvn rpryo aalltasoti its is transport parallel of property relevant second The aua n ovnet esalve h i 2–group Lie the view shall we convenient, and natural r ntetiilprincipal trivial the on ory let and a,gecdsantrltransformation natural a encodes g flat, Let technical the involved. though more much simple, are still details is construction idea the intuitive The of theory. gauge higher strict in also fined functors. transport parallel flat F rmtefnaetlgroupoid fundamental the from ag rnfraingecdsantrltransforma- natural a tion encodes g transformation gauge A interpretation. el- categorical an egant has also transport parallel of transformation Gauge If –ag rnfraini utaGvle mapping G–valued a just is transformation G–gauge Parallel A sense. appropriate property. the this also has in transport covariant gauge be h ag rnfr fteG–connection the of connection transform gauge The well– the in manner. connections known on act transformations Gauge o n curve any For un- properties transformation. gauge covariance der simple has transport Parallel aalltasotyed functor a yields transport parallel o n w hnyhmtpccurves homotopic thinly two any For optblt ihhmtp fcre sdfie nSec- in tion defined as curves of homotopy with compatibility rmtept groupoid path the from F aalltasotyed functor a yields transport Parallel interpreta- categorical elegant tion. an has transport Parallel When g g θ θ g ∈ θ θ ( = γ naporaefr fprle rnpr a ede- be can transport parallel of form appropriate An n ennflguetertccntuto should construction theoretic gauge meaningful Any ( Map( sflat, is K γ 1 2 F Ad ¯ ) ) . θ θ easrc i ru ihsrc i 2–algebra Lie strict with group 2 Lie strict a be = = g ⇒ M sflat, is F g M ( θ θ ( g F p eamnfl.Cnie ihrguethe- gauge higher a Consider manifold. a be ( , ) θ ¯ γ G g 1 − sflt too. flat, is θ 0 ) ) γ F ). d ( . fprle rnpr ucos When functors. transport parallel of 26 : θ g g p ( γ ) 0 − ) od lowhen also holds → g 1 .Zchn:Wlo ufcsfrSraeKnots Surface for Surfaces Wilson Zucchini: R. (27) . ( p p 0 ( 1 ) M − fM, of 1 , (28) . P K 1 M –2–bundle ( ) M fMit G o flat For BG. into M of F ¯ γ , F 0 ¯ P 0 γ θ θ , 0 0 ( : γ 1 ( : , M 1 γ M M r homotopic. are 1 M F ) n has one , ¯ , , fMit BG. into M of P 0 P × θ θ 0 1 1 ⇒ K M M steG– the is si is it As . ) F ) ¯ → → 0 g (26) θ θ BG BG of θ is k , Proceedings . : , : – ) 5 2 1 2 h Σ , Σ (41) (36) , from –fun- M ) ( 2 01 are ho- 1 γ H 1 , Ω of M into Σ ⇒ G ∈ , ) ( . For a flat 0 ) 00 B M Σ γ H 2 )), (40) : , 1 0 ) stands for the → 1 0 γ G ) P Σ and any surfaces ( from the path H , , Σ ( , 0 –form J M ) 2 Υ 1 γ , 2 G M γ ( θ ), (39) H 0 1 , , 1 F B 1 P P − , , G γ ) is flat and ( ))( , ) ), (38) 2 Σ M M γ B ( 1 ( 1 γ Υ ( Υ –valued , Σ and the further relations , P θ ( , → h θ θ ) ( F of M into B Υ F , ) ( ). M θ 25 ) . )( ( 3 H F m M 1 :( – ) ) , , (37) 2 M γ − ) 2 2 Υ 1 ) 2 , P G 0 − , and an Σ Σ θ P 23 ⇒ ) ( ( γ 0 –groupoid , ) ( ). (44) ( 2 ¯ Υ Υ 2 Σ M , , F 0 θ G ( γ 1 M θ θ , ), (42) ), (43) F : Σ Υ 1 P , F F ( ( , ). 2 00 01 θ P M Υ 0 , , , = = γ γ F m Σ ( ( M θ γ ) ) H ( θ θ M 1 1 = = F –gauge transformation is a pair of a G–valued 1 θ ( :( F F ) ) , 1 Σ Σ F = • ◦ Map( , relations –functor and = – • ◦ . 1 1 = = 11 Υ ) 2 ) 2 , ) ) ∈ = 1 1 2 2 − − ) ) γ θ Σ γ ) γ H ¯ , higher parallel transport is likewise equivalent to , H I Σ Σ Σ Σ Σ F 10 11 1 , ) ( ( ( ( ( ( , 1 ⇒ Analogously to ordinary gauge theory, higher paral- γ γ γ Υ Υ Υ Υ Υ Υ Υ ⇒ G ( ( ( Σ G , , , , , , , ( ( , θ θ θ θ θ θ θ θ θ 10 0 θ B Here, with an abuse of notation, delooping of the strict Lie 2–groupLie crossed corresponding module to ( the lel transport is gauge covariant insense. the appropriate higher A map g the fundamental γ Σ For any two thinly homotopic surfaces F The same relations hold if F F a strict F groupoid motopic. Higher parallel transport has an elegantterpretation. 2–categorical in- Higher parallel transport is equivalent to a strict For any point p, any curves in the following sense. For any two thinly homotopic curves F ctor hold whenever defined, where in the last two identities F topy of curves and surfaces, as defined again in Section F F F γ 1–gauge transformations act on 2 connections. Third, higher parallel transport is compatible with homo- ( . – ∈ 2 g Υ (34) H 1 = G is con- ) and the are equiv- , (33) , (32) ∈ –form . For a sur- ,0) )), )) or H ) G 2 G to the curve E θ x y y 1 γ 1 ( 0 , , ( is flat if is the element E γ = θ = x x ) ( ( Aut( ) Σ = Υ as input datum. y , ) ,0) x y x y θ m y x ( Υ Υ −→ M ( –valued (1, ∗ ∗ v h u (1, G Σ Σ E )( )( t 1 1 ), ), − − −→ y y ) ) , , and a y y H x , x , ) ( ( x x g x y ( ( , joining the curve θ θ ). (35) v u ∗ ∗ 1 0 1 1 M = = ( − γ γ Σ − Σ ( ) 1 ) ) connection on )) 0. (29) 1 θ y Ω ⇒ ,0) y − F H = =− , ) =− x 0 , ∈ ) )) ( 1 y (1, x as the differential Lie crossed module 1 H is the unique solution of the two step γ G. ) corresponding to it. ( , G − v Υ Σ u − : θ 0. (30) ) h k ( ( ( ( x ) E ( ˙ ( t y → → Υ y x ˙ Σ ˙ of M, the parallel transport F , = , m m , E 2 ∂ θ 2 ) − γ ) x (0,1), (31) x 1 der –connection on M is a pair formed by a ] F ( ( Ê Υ y Ê ( − E 2 H defined by θ , such that , : ) u =− =− v t : –form , – ) b ) θ ) x y m ) = ∈ v 1 θ ( −→ ( y , y h = of M, the parallel transport along , [ ) ) , , , H b ) x E g 2 m 1 Σ Σ , x ( x 1 Σ y ( ( ( ( M γ + ∂ E ˙ G t + ( Υ Υ as a Lie crossed module ( v ( ( ( u , , , 2 −→ θ θ θ Υ θ 1 y y x x ∂ ∂ with u ∂ ∂ differential problem A valued K h Lie 2–algebra For a curve structed as done earlier for the G–connection port requires a ( face d alent: any solution of onethe other. is automatically a solution of Higher parallel transport has several remarkableties which proper- extend those of the ordinaryparallel case. transport First, higher along surfaces isalong compatible their with end-curves. that For a surface Ω d F where E F Second, higher parallel transportoperations is with consistent curves and with surfaces the defined in Section γ F The two forms of the differential problem for Analogously to the ordinary case, higher parallel trans- (vanishing fake curvature condition). Proceedings hr xt oino aalltasotfr1–gauge for transport parallel of notion a exits There G G sodnr aalltasot ag aalltransport parallel gauge transport, parallel ordinary As G gi sodnr aalltasot ag aalltrans parallel gauge transport, parallel ordinary as Again G G h esnwyw nrdcdgueprle rnpr is transport parallel gauge introduced we why reason The sthe is o curve a For 2–connections. to similarly transformations If h –ag rnfr fthe of transform 1–gauge The 6 where o n w hnyhmtpccurves homotopic thinly two any For curves any and p point any For Λ scnitn ihteoeain ihcre endin defined curves with Section operations the with consistent is hnvrdefined. whenever h lmn G element the d problem differential o n curve any For manner. trivial non a higher in of transport relation parallel covariance 1–gauge the in enters it that F oti optbewt oooyo uvsa endin defined as Section curves of homotopy with compatible is port g g , , x g g g g g g J J ( ( , u x θ Υ J , , , , , θ J J J J J θ ) ( ; ; ; ; ; , = θ θ θ θ θ ( = − x Υ γ ( ( ( ( ( 2 1 ) Ad ι γ γ γ γ ) m ) Λ d –connection u p ˙ 2 − ) 1 sflat, is = 2 2 x ) ( ( ) = 1 : ◦ . . Λ g x = g ◦ g = Ê γ ) ( ) Λ ( )( ( γ − θ 1 1 x G = p Υ (0), 1 H ) ) → ) fM h ag aalltasotalong transport parallel gauge the M, of 1 g g − γ − = m = ) − = ( , ) , , g − J J t : gg dg steuiu ouino h two–step the of solution unique the is H G ; ; ( , ( θ θ p J F G d γ ( θ g ( m 0 θ γ γ J ˙ , g ∗ , J ( → 0 ) g − , ; ( θ γ − − J θ .(53) ). , u ∈ ; x 1 J ) ( θ 1 2 m p − Υ ( γ ( − ie by given H ( b x x 1 [ γ 1 2 ) (Ad J )( ) ), t n has one , ) ˙ )) sflt too. flat, is − , m ( G J F 1 J ] γ ,(45) ), g ( θ g u − F , ∗ ( J ( γ θ (1) g ; ( θ θ ( G ) γ ( ) ( γ g x γ , − , = 2 ( γ ) H ) ))( p − − d 1 1 0 ) 1 , 1 g g G G – ) γ )( γ ,(51) ), − 2 (48) , 2 0 g γ 1 –connection − n has one , , , ∗ (54) . J Λ 1 γ ; J θ − 1 (1) x ( n has one , γ ( t ˙ x 1 ( = ) (49) )), ) (52) )). J ), 1 J H ). . ( θ (50) (47) (46) γ , Υ is ) - ( C min stp stentrlnto fmta deforma- mutual of notion natural the is isotopy Ambient F ust of subsets nt r mednso xdcoe oe manifold manifold model ambient closed an fixed into a of embeddings are Knots modifications 4 holonomy. knot in role no have parently ap The functors. transport parallel of transformations ral ag aalltasotdfie suoaua trans- pseudonatural a formation defines transport parallel Gauge in- categorical terpretation. a expected as has transport parallel Gauge rnpr ensapedntrltransformation pseudonatural a defines transport circle identified. are isotopy ambient –akn fCi onigp pointing a is C of C–marking A 2 Map( –akn fa retdmnfl sapointing a is M manifold oriented an of C–marking A na sind( assigned an on A for gauge by also symmetry. characterized gauge is theory gauge Higher ain f( of mations γ F o n w curves two any For sasot aiyF family smooth a is w akdC–knots marked Two ξ that such M into C cle iiyo marked of bility akdCko fMi embedding is M of C–knot marked A p ¯ G ( fntr.If –functors. 0 g M 0 kosaecrlsebde in embedded circles are –knots p ( , , θ J ⇒ G C H h ipetcoe oe aiodi h oriented the is manifold model closed simplest The θ C , ∈ Υ , , ) g ––ag rnfrain ecieguetransfor- gauge describe transformations )–2–gauge and – γ M H fM. of M , ⇒ C = J 1 Υ = ) . n has one , , p ( – F H ¯ Σ m 2 M 0 guetasomto sjs mapping a just is transformation –gauge M ) ) G g ( . . ¯ = , g J G g θ u apnsinto mappings but ( S , ( , , J p g G θ ; H ¯ , θ –knots 1 J , g C G Υ Υ )) ––guetasomtosdepending transformations gauge )–1– , : J –knots. , ¡ ) ffltprle transport parallel flat of ; F H G θ ¯ saflat a is γ θ .Zchn:Wlo ufcsfrSraeKnots Surface for Surfaces Wilson Zucchini: R. g ⇛ ––oncin( )–2–connection , z 0 Υ , ξ , J ∈ ; γ 0 θ G ⇒ , ¯ 1 ( Diff ξ Ω γ ˜ : 1 M g F 1 ¯ | p ) 2 θ r min stpci there if isotopic ambient are g F + hs nt r o simply not are knots Thus, . , , 0 cneto,gueparallel gauge –connection, Ω J ˜ θ ( θ J → M , , | Υ g θ M , C ; ( J θ ) M Σ Υ p z , ∈ nt ifrn yan by differing Knots . fguepseudonatu- gauge of 1 ) . G fprle transport parallel of fC. of C n n surface any and ξ ∈ g θ : , J , Ê C ; Υ θ forientation of , ( .Te encode They ). → γ 2 0 –functors. ftecir- the of M ) − 1 ¢ . G ¯ 0 g (55) (56) Ω , J Σ ; θ ∈ - : : Proceedings ) – 1 7 . C C M de- S ( are (63) ξ ξ 1 ,p γ γ , such 1 in S ◦ C ℓ 1 1 ξ M γ is smooth. ◦ p ξ ◦ and p γ 1 along its stan- → 1,...,2 − . 0 1 0 C S M = M that obeys γ p M , \ 0 p ,p → –knots have homo- ξ can be a marked M with the property 0 : γ ; C C C 1 . : M → M γ , the curve ξ of M, i Mi S ξ ζ is fixed. : closed oriented surface are thinly homotopic. –knot holonomy, it will be = Mi 1 S into –knots. The next to simplest C S Φ ζ ℓ γ S Si of C M is a marked C–knot with re- ◦ such that with finite derivatives and non ζ 1 , one can associate a curve 1 | C ◦ –knots which are continuous but C ξ ξ → p M p of M; Φ C γ p , C \ ◦ M : ◦ M → 1 ξ –knots are marked. p − –knot 0 1 C intersect only at p = C M –cycles. The cuts are the images of spiky γ ) ) p , b S C 0 : –marking | ( p ξ 1 ( C γ γ M Mi Φ p ζ two freely ambient isotopic marked C–knots, there – and 1 = a ξ ) that that: , C We can now introduce The results just expounded are standard. Our aim is 0 p ξ –knots, generalized ( ξ closed manifold is a genus knot in more than one way. The correspondingrelated curves are in the expected manner. If the embedding zero first derivatives at both ends of C Freely ambient isotopic marked fined in the same way as above and smoothFor anyway. every spiky marked C–knot and is smooth on C finding their generalization to surfacesee, knots. this As task we is not shall completely straightforward. Problems occur for higher genus knots. We shall proposein a due solution course. To this end,notions. we need to introduce further topic curves up to conjugation. If is a curve spect to two distinct C–markings p Note that spiky of C and M, there exists a curve An S–marking of M consists of the following elements:i) a C–marking p such that ii) a set of spiky C–knots homotopic. Notice that the “compose rightmostused first” here convention and in is the following for curve composition. not smooth at the marked point. A spiky C–knot is an embedding necessary to cut thedard model manifold iii) the iv) there is an embedding To construct higher genus The same embedding of Again, the C With any spiky , 0 C at ξ γ ξ γ (58) (61) γ 0 and ≥ ) inC such x ( ; C α p (1,0). A com- Ê x = . d . , of orientation → ǫ 1 2 S C Ê − p p 1 ∈ : > C , z γ x is fixed here. ) C M there is associated a curve ( + of 1 for –knots have homotopic cur- ξ –marking with respect to two distinctC– C = –knot holonomy, we need pa- C C is given by 1 p ) Diff C 1 ξ x , S ( ∈ is an orientation preserving diffeo- 0 , (57) . (62) is an open interval in α ξ 0 0 be the circle standardly embedded z ) C → ξ ξ ), (59) 1 p C ◦ ◦ of M are freely ambient isotopic if there )) (60) Ê \ S ϑ p 1 1 1 x : and \ –knot as detailed next. ( C F F = 1 ǫ M ), with the –marking S C C [0,1] is a function such that = = ,sin π ( given by → C < γ C πα has a number of nice properties. ,p . –knots. This is achieved by assigning a curve 1 1 1 ϑ 0 C x − → ξ M ξ ξ (2 C M I 1 γ M C . [0,2 p : –knot can now be constructed. p S Ê γ C . Let s , , are ambient isotopic marked C–knots, then ∈ : C (cos C I = γ → = | 0 for 1 M M ϑ = ) α = ◦ through ξ C ) C M ) = id id ξ M 2 γ morphism. , x p ϑ ) are homotopic in the sense of Section 0 In order to compute ( p ( = is independent of the choice of the compatible curve = = : Ê x ( 1 ξ 1 1 ( z 0 0 ξ ξ ξ ξ S S s up to thin homotopy. Note that the α F preserving diffeomorphisms such that F γ If γ to any marked rametrized It should be possible to alter marking data changing F preserving diffeomorphisms such that ii) A compatible curve in C is a curve in γ Example γ With every marked C–knot marked most by a (thin) homotopy. Two marked C–knots marking p is a smooth family F γ patible curve that i) I Ambient isotopic marked The curve ves A curve furnishing a natural parametrization of a given where where Proceedings sacmail uv in curve compatible a As h udmna oyo of polygon fundamental The ene parametrized need we nlgul to Analogously min stp stentrlnto fmta deforma- mutual of notion natural the is isotopy Ambient marked optbesraei sasurface a is S in surface compatible A aein face it uv oec marked each to curve a ciate hntefnaetlplgno stecregvnby τ given curve the is S of polygon fundamental the Then γ let and manifold C–marked a as S View constructions. sequent )D i) that standard cutting results that 2–fold open connected ply if inn ufc oaymarked any to surface a signing sasot aiyF family smooth a is akdSko fMa sembedding is an M of S–knot marked A w akdS–knots marked Two Ξ iiyas fmarked of also bility F 8 ii) Example F that such diffeomorphisms preserving of notion the that Note Ξ that such M into S surface S S htof that S S Si kosaesrae meddin embedded surfaces are –knots 0 z ( ℓ ◦ 2 ( p = = S p ( ζ = ϑ S oanin domain Σ diffeomorphism. Si = M β id ) , γ S S S ϕ = ) 0 M = ζ S S | ℓ Si D = = ) , C S mrigof –marking p , srqie nodrt soit ufc oeach to surface a associate to order in required is Let . ζ τ = S − a p –knot. M Σ Mi S htis that 1 : and – (cos M ◦ S D Ξ , = ◦ . − , S 1 S ι 1 α p = ( F → = Ê ϑ S S S C z ℓ . F b sin S \ 2 α S S kos ocompute to –knots, ◦ 1 cce.I ly ai oei h sub- the in role basic a plays It –cycles. ; − 2 ∪ Sr ζ \ ◦ 1 eteshr meddin embedded sphere the be Mi · ◦ ··· i ϑ ∪ ◦ S Ξ M ζ z = Ξ (1 ◦ i –knots. Si 0 ζ ∈ S = β 0 γ , when S Si ( − β mrigof –marking , S C ξ Diff Ξ kos hsi civdb as- by achieved is This –knots. ζ Sr S ℓ ( cos C Mi )) C S 1 1 , − ◦ ) srqie nodrt asso- to order in required is r min stpci there if isotopic ambient are sa pnsml connected simply open an is S β + 1 . α sa retto preserving orientation an is ϕ ◦ M ( Sr C stebudr ftesim- the of boundary the is M S ◦ ,(68) ), ℓ ko,acmail sur- compatible a –knot, S α = = S ) –knot. S ◦ z , S γ M 1 . η − S Sr . 1 ∈ ◦ scmail with compatible is Σ S (66) , Ê ◦ ko holonomy –knot S Ξ β forientation of , : S : ι 1 p S S ◦ → S α → ln the along S Ê fthe of M 1 τ 3 S as such (67) (65) (64) where where where surface compatible A (0,0,1). ufc unsigantrlprmtiaino given a of parametrization natural a furnishing surface A h surface The where where where risa eoeand before as erties a oyo of polygon tal c marked eventually all boundary. square’s it the sweep approximating and square the of corner one from edt dniyacrein curve a identify to need Σ taogthe along it aaerzdfml fcoe uvson curves closed of family parametrized c Σ S itdblwEuto ( Equation below listed fi,sweep it, of Example the on side. pole other north the to converge finally and contracting lson cles η marking pnufligtetorus the unfolding Upon function. Dirac ̺ by given ξ 2 1 T ( T T T S ( ( s 2 2 2 2 2 x x , ( ( ( ( ( t , , ϑ ϑ ϑ x x ) y y , 1 ) ) , = ) ) ϑ α y , g ϑ α y = = r = = ϑ S ) β ) g t S ∈ : Let . p (1 ((1 2 2 = ∈ : ( < = ̺ ̺ ko a o ecntutd oti n,we end, this To constructed. be now can –knot Ê ) w [0,2 T Ê β hc pigfo h ot oeo n side one on pole north the from spring which (4 (4 S (0, = + S 2 ³ S sfie and fixed is 1 + ) → a S T Σ α α (1 = r 2 (cos → 2 = S and – 2 r π π S S ( ( ( cos iaig ecigtesuhpl n then and pole south the reaching dilating, 01 safnto noigteproperties the enjoying function a is [0,1] (2 (1 πα )cos + x x = 2 ), .Acmail surface compatible A ). . 1/(exp( 01 safnto ihtesm prop- same the with function a is [0,1] ) ), π ecie aaerzdfml fcir- of family parametrized a describes s T + ϑ ϕ − ( α ϑ c − 2 y 1 r 1 1 1, ,0, ( ϑ b ∈ (1 00 and ,0,0) − .Zchn:Wlo ufcsfrSraeKnots Surface for Surfaces Wilson Zucchini: R. ),2 ( t etetrsebde in embedded torus the be y ̺ − ,(1 x cce h surface the –cycle, )(2 α )) [0,2 sin + r , : 2 β 60 πα ( y sin − sin Ê y s r + − w )),2 )) ϑ ). ̺ cos π ( M r × s T ) ϑ x (4 sin − ϑ )sin ,wt the with ), + ϑ − ) (69) )), 2 1 [0,1] π ), htmthstefundamen- the matches that α ̺ 1 (1 ϑ Σ )with 1) c t noasquare a into , ϕ (4 ( 2 ) 2 ϑ S x + ϑ ,(70) ), ´ ( ,1 2 α 2 x (71) , ) ,0), r : → − ( , − ∈ Ê x cos y 2, ) sin ))) 01 stefunction the is [0,1] 2 [0,2 − β α → ϑ 3, S 2 Σ ( > 2 y ϑ –marking S π Σ T ), I α )), steFermi– the is 0 (1 2 2 2 ,wt the with ), T r ( sgvnby given is : y hc spring which 2 I sin − Ê )), 2 ecie a describes cos 2 ycutting by Ê ϑ → 2 3 ϕ ), as T p )), 2 S 2 is S = – Proceedings , , ◦ s. i 9 ∈ S of 0 1 ′ | S z 1 1 z and ζ Ξ k Ξ ◦ ♯ ) ) of i i , z is kept ′ Σ 1 0 k 0 S ◦ are freely are freely M S ◦ such that ◦ ζ Ξ 1 ζ , 1 1 } , then there , M 1 i − M 1 0 Ξ S 1 1 of ∆ . , ∆ S γ M p M ) of i 0 , z ζ = 0 ′ p ∆ , 0 are ambient iso- , I Si i ( Ξ . , 1 are thinly homo- M 0 ),( ζ 1 0 | i M 0 S i , , ζ 0 1 S ) are fixed. 1 lying in the image 0 ) M S ∆ p S the normalized sur- ◦ γ ζ M Ξ i S { Si I ♯ 1 p ζ 0 ∆ ◦ z ζ ∆ of M are said to be ζ ′ S , Ξ and F , ◦ Σ z M , of freely ambient iso- = 0 to 0 ζ ) M are simultaneously F 1 S k 1 , p –knots have homotopic | S i i ′ } M 0 ◦ 0 p 1 S = i Ξ 1 p Ξ are homotopic. S S ♯ 0 ∆ → i M → Ξ , with respect to distinct S– ζ 1 S Ξ p 0 Σ ζ 0 ( ζ –knots yield homotopic nor- 0 ◦ Ξ , S 1 , , –knots have homotopic nor- ◦ M ♯ ′ S Ξ 1 1 M 0 and ( : ζ ◦ S 0 Ξ Σ k S and 1 M p ◦ such that , , –marking ( C , Ξ of ) − p : p 0 z 0 Ξ 1 1 . (76) { S 0 1 p ( , z S Ξ 0 S γ 1 of S and M and there is an ambient Ξ M ♯ , , F p M γ p ) ) Ξ ) p , I Σ ( i i 0 and ∆ –markings ( ◦ 0 1 = z 0 | S ) 1 M –knot surfaces. k → M 1 Ξ , of orientation preserving diffeomor- M are homotopic. 0 F ♯ ◦ S p 0 ζ ζ S 1 Ξ ( , Ê , | Σ = M p z 1 M 0 , by a (thin) homotopy. ′ Ξ ( 1 p ♯ 0 ∈ ∆ M 1 M F : Ξ k Ξ Σ p ♯ Ξ p = 1 of S shifting ( , = ( ) Σ , are ambient isotopic marked S–knots, then the 0 γ ) z , z | ) 0 , 0 ) i 1 S –knots, it should be possible to alter the marking Ξ 1 ♯ M M p Ξ S C M ( Σ such that ( p ζ , id z ( , 0 ) + are homotopic. z k 1 = S S Ξ 1 ◦ such that ( 0 γ p I phisms such that F Notice that above the normalized surfaces updance to with reference conjugation knots. under concor- Suppose the reference marked S–knots Note that the markings ambient isotopic. If the marked S–knots topic marked S–knots areist called ambient concordant isotopies if F there ex- faces t. F fixed. Two pairs topic, then for every marked S–knot changing If the reference marked S–knots markings Two marked S–knots malized marked Freely ambient isotopic marked Diff ambient isotopic concordantly with malized surfaces. If normalized surfaces freely ambient isotopic if there is a smooth family F Before stating theproperty. next For two result, we recall the following is curve a there is an orientation preserving ambient isotopy isotopy k S Ξ If the embeddings the reference and considered marked S–knotto with two respect distinct S–markings then there is a curve Ξ topic. Ambient isotopic reference Ambient isotopic marked As for ( 0 to = (73) (74) Ξ S up to ℓ C γ are differ- is the sur- M Ξ 1 p , (72) and ι M α –knot has nice S 6= Mr ◦ S η Σ 1 M γ ◦ τ M –knot “ratio” of S = ℓ S β ⇒ , pick a reference mar- ◦ M , the source and target Mr ◦ M α M are equal. In gauge the- Ξ 1 β p ◦ − ι , 1 S : M , there is associated a surface ℓ M Mr p Ξ ξ M Ξ α –knot holonomy. However, for γ Σ –knot β ◦ S → denotes vertical surface composi- S ◦ ◦ = , the source and target of the as- are equal as well. In higher gauge 1 ◦ • ξ M 1 − . M 1 –knot holonomy. For a genus − Mr p S given by p S –marking of ι M : τ α C ℓ and M . , (75) β ◦ , the source and target of the associated C ]). ξ ). M p ⇒ M ι Ξ –knot M γ 2 (e. g. Hosokawa’s and Kawauchi’s surface Ξ Ξ α p 35 ∆ given by ι Σ M C [ ◦ 0 marked ···◦ ⇒ acting as a normalizing knot. = M p • Σ and ◦ 4 = ι M • 1 that is . ∆ > M : S S M M 1 = τ − S p : M ℓ τ S − S ι ∆ –knot can now be constructed. Ξ –knot τ Σ ℓ : Mi ℓ ⇒ S S M M , Σ ζ ◦ M Ξ 0 Σ γ Σ M ♯ β Ξ –knot p = is independent from the choice of ι Σ = = S = = : , with S Ξ Ξ is as a surface characterizing the ℓ M ♯ ♯ Ξ Ξ Mi M ∆ τ Then, the fundamental polygon of the markingcurve of M is the The normalized surface of a marked S–knot γ View M as a C–marked manifold and let tion (cf. Section Ξ properties. Σ Σ where face Σ Σ marked if For a marked Note that thin homotopy. With every marked S–knot sociated curve ory, this ensures nice ambient isotopyance and properties gauge of covari- surface theory, this also ensures nice ambient isotopycovariance and properties gauge of marked a genus of the associated surface ent. This is likely to be a problemgauge for covariance ambient properties isotopy of and holonomy. We have aposal pro- for the solution of this difficulty. For given unknots in ked The normalized surface of a marked An intuitive way of thinking of the normalized surface of A surface furnishing a natural parametrization of a given Proceedings elet We C C C C ie by given G hspoet eeaie sflos i the Fix follows. as generalizes property This h oooyo marked a of holonomy The sacurve a is If conjugation. to up marked is knot aknsp markings F hr xssacurve a exists there If F F If topy. h oooyo akdC–knot marked a of holonomy The knot. the to associated curve hieo h optbecurve compatible the of choice o n akdC–knot marked any For parametrization. ocnuain ebgnwt eiwn o hsi done is for this how reviewing with begin We up conjugation. knots to of invariants holonomy constructing is aim Our where respectively. 5 holonomy. knot higher constructing of task aalltasotfntr(f Section (cf. functor transport parallel F 10 M p M θ θ θ θ C ko oooyi ute nain ne min iso- ambient under invariant further is holonomy –knot ko oooyi needn ftewyagiven a way the of independent is holonomy –knot ko oooyi loguecvrata desired. as covariant gauge also is holonomy –knot ko oooyi needn rmtecoc of choice the from independent is holonomy –knot ξ ξ ξ | . ute,w fix we Further, . ( ( ( 1 C 0 0 ξ ξ ξ C eyn nteaoersls ecnnwtcl the tackle now can we results, above the on Relying of ( , , samre –ntwt epc otodsic C– distinct two to respect with C–knot marked a is 1 1 ) ξ –knots ξ ξ and – ) ) = ) C 1 1 γ = = G = F ξ r min stpcmre –nt fM then M, of C–knots marked isotopic ambient are r reyabetiooi akdCkos then C–knots, marked isotopic ambient freely are u lo w distinct two allow but F F F θ eaLegopand group Lie a be : γ θ θ ( p θ γ 1 ( ( C ( M γ ξ ξ γ : 0 0 ), 1 1 p S p , → ). ) ) F M F ko holonomy –knot θ M θ p 0 ( | 0 0 M → ξ γ C ( 0 n p and ξ 1 ) uv of curve –markings p ) F : F M θ p θ ( 1 M ( γ γ yn in lying C 1 ξ 0 1 ) F , 1 → ) − p , − θ 1 ξ 1 C . θ p γ eaflat a be c.Section (cf. . M ( C ko sbitoto the of out built is –knot M C ξ p 1 ) –markings 1 C fC. of ξ fCadM hnthere then M, and C of ξ sidpneto the of independent is ( fMsc that such M of C and steeeetF element the is 3 ) ). uhthat such G p cneto on –connection 4 M n F and ) p of C M C –marking 0 , and p θ θ M sthe is ( (78) (77) 1 ξ C M ) of ∈ – , elt( let We h atthat fact The hs unlike Thus, F hs F Thus, o akdS–knot marked a For consequence. a ( flat h oooyo akdS–knot marked a of holonomy The tively. o vr akdS–knot marked every For parametrization. S kernel. trivial non has map target mod- whose crossed ules for only trivial non and Abelian damentally Let F has one g, transformation F ∈ S etdaoea oe,w lutaeteconstruction the illustrate we of model, a as above sented where rnemarked erence S C. of nsummary, In fterfrnemre S–knots marked reference the If needn n stp nain pto up invariant isotopy and independent h hieo h optbesurface compatible the of choice the F S–knot marked any for then topic, If topy. S F t Z ( ko oooyi needn rmtecoc of choice the from independent is holonomy –knot mrig ( -markings ko oooyi nain ne hneo h ref- the of change a under invariant is holonomy –knot ko oooyi ute nain ne min iso- ambient under invariant further is holonomy –knot θ θ θ θ g H ie by given H F Ξ θ , , , , S Υ Υ Υ Υ . θ et sn h ramn of treatment the using Next, ( 0 ξ ko holonomy. –knot | ( ( , ξ 0 G steprle rnpr 2–functor. transport parallel the is Υ , Ξ Ξ ) ( eamre –nto .Te,frayG–gauge any for Then, M. of C–knot marked a be Ξ ( Ξ , Σ = ) 1 Ξ H θ 1 ) = G ♯ ) , )) g Ξ Υ = ––oncinpi on pair )–2–connection r min stpcmre –nt,then S–knots, marked isotopic ambient are = , F ( ( = H : p F Ξ θ F ι , θ M 1 p eaLecosdmdl n ( and module crossed Lie a be ) θ Υ C ) p C , G , M Υ Σ ( = Υ ) ko holonomy, –knot S Σ F ko oooyis holonomy –knot ( . | S ♯ , ⇒ 1 Ξ 1 θ Ξ ♯ ζ ( kosi h olwn sense. following the in –knots Ξ H ( Ξ Si 0 ξ ι : .(82) ). ) p ) .(81) ). n ( and ) unless = ι g M .Zchn:Wlo ufcsfrSraeKnots Surface for Surfaces Wilson Zucchini: R. p Ξ ( M F p stenraie ufc of surface normalized the is , θ M ⇒ , Υ Ξ ) ( ker p − Σ ι F , p 1 M M . M , θ t ) ζ , ∆ − a h olwn crucial following the has Υ 6= SMi S 1 M ( F M Ξ C {1 ko oooyi fun- is holonomy –knot Ξ 0 C Ξ θ ko oooypre- holonomy –knot ) of ) , H utemr,w fix we Furthermore, . , Σ mrigadgauge and –marking Υ ∆ sidpnetfrom independent is steeeetF element the is } ( S ute,F Further, . M Σ fSadcurve and S of S 1 Ξ r min iso- ambient are G ,(79) ), and –conjugation. M θ , respec- , θ Υ , Υ Ξ ea be ) ( Ξ θ and (80) ( γ ) Ξ C ∈ ) Proceedings ) ξ J C ⇒ γ 11 , 0 (91) g of ξ ( –mar- γ : M C p Σ –knot in- –connec- , its holon- C G ξ and , (90) C 1 p − . (87) ) 1 –knot –knot holonomy is − M be a flat C ) C p 1 θ ( γ and a surface ofC and M, then there g ( ) 1 θ 1 a marked C–knot. If )) (89) γ such that F M M ( . (88) ) ξ ) – and Ξ θ 1 p 0 ( –markings S C − –knot holonomy are F ξ ,p Υ ( ) ( , C 1 that for a S → )) ξ 1 θ θ C ξ 0 5 γ F F ( γ ( M )) θ ))( θ p ; F Σ J ) –conjugation, that is ( : M , – and ξ and p Υ g 1 . p G , ( C ( θ 3 0 0 γ curve in G | ( g 1 is the gauge parallel transport along F θ M 1 ( t ( − ) F ) t of M such that M ) ξ –marking and isotopy invariant and gauge a ) m ,p 1 ) M p 1 1 γ 0 C ( ξ –gauge transformation, then γ = p γ γ ( ( C θ ( ( 1 ) ; → . Further, we let ◦ θ θ θ J – g ) is , 1 again be a Lie group and ) 0 Ξ F . F ξ g ξ ( M are freely ambient isotopic marked C–knots, then aF = –knot is marked up to conjugation. ( M H γ = G Υ G , respectively, be given. = ) 1 θ , J p C ) be a marked S–knot and , ◦ ≡ ) ξ ξ ∈ F g : ξ G ( ◦ M ) 1 , is a marked C–knot with respect to two distinct C– , ( ( 1 Ξ a θ θ 1 To summarize, ξ ξ 0 1 J J ( ( | − ξ , , γ ξ –knot holonomy is again independent of the way a g g θ θ θ 1 king and gauge independent and isotopythe invariant appropriate up form to of crossed module conjugation.shall We analyze this point intion. greater depth in the next sec- 6 Invariant traces Having applications to knotat topology a in construction mind, of we holonomy invariants. aim This requires tion on and for omy F independent up to gauge covariant in the appropriate sense. Let and F is a defined in Section where G F is markings p If given If F γ there exist a curve F There is a well established way of extracting variants from knot holonomy. working out invariant traces. Also in higher gauge theory, C We let We have seen in Section , ), i ∈ i of 0 ) 0 S ξ 1 (85) and ζ ( M M ◦ θ ζ ) , i z p ) is not 0 0 k , ξ are freely ( M 0 M ◦ such that –marking θ -markings ζ 1 p } M , then there , C M F S i 1 0 p Ξ 1 ∆ S , M M 0, 0 ζ = p ∆ , = i ( Ξ 1 , 0 ) S 0 , S lying in the image ) p Υ ζ M i { ( 1 0 ◦ is the elementF ˙ t ∆ S z M of M such that is independent of the –marking ( ξ to ζ M are simultaneously k ofC. , p 1 S –markings ) } ξ 0 ◦ )). (83) i ξ C S C 0 γ 0 ( → → γ Ξ )). (84) S p θ Ξ ( ζ 0 ( ⇒ Ξ S , 0 ( | 0 M 0 , F : 0 ξ | S Υ and ξ such that p , Υ γ ) p θ , : 1 Ξ : ). (86) { 0 θ , F 0 S 1 of S and M and there is an ambient M F Σ ξ M γ p ) ( p ))( ( i are freely ambient isotopic reference θ 1 ))( ∆ 1 z 1 F γ is not flat unless 1 → k M ( γ )) ◦ 0 ( θ . ζ M θ , θ Σ F M ∆ 1 M ( ( M F but allow distinct p ( , Υ ) ∆ M , : 0 m ξ θ S p m = 1 of S shifting γ ( M F ) of = ( ) γ , ( i = z ) ∆ θ ) 0 1 t but allow two distinct are two ambient isotopic marked C–knots of M, 1 ) i S F 1 ) of M 1 –knot holonomy has however weaker properties = p Ξ Ξ S C ζ ( ξ = ( ( ) Si such that ζ C , z 1 1 ) 1 , , ζ | | 1 ) of , 0 k ξ ξ 1 Υ Υ S ( ( . , , M S S ξ ◦ ( –knot holonomy in the same way as before. –knot holonomy is still independent of the choice of C θ θ θ θ –knot holonomyis independent of the way–knot a is given marked knot up to conjugation. p p p p M Suppose marked S–knots. If the marked S–knots ambient isotopic concordantly with isotopy k is a curve F Ξ S S If the embeddings the reference and considered marked S–knotto with two respect distinct S–markings then there is a curve Ξ F In this higher gauge theoretic set–up, one can define also The holonomy of a marked C–knot F is defined. than in ordinary gauge theory. parametrization. For any marked C–knot choice of the compatible curve ambient isotopy invariant. If then there is a surface Since, however, F This property generalizes as follows. Fix the This This property generalizes as follows. Fix the ( ( ( C G given by C Proceedings elt( let We ( ooti ntivrat,oenestae nain un- invariant traces needs one invariants, knot obtain To Assume ih( with ih( with hs nain rcscnb sdt extract to used be can traces invariant These C W W W conjugation h traces The e ( der relation. sures sfollows. as a ( flat u for 12 an ihRarpeetto fGpoie –ntinvariant. C–knot a provides G of representation a R with pt ( to up S S nSection In given. be tively, ytmtcwyt build to way systematic F F h isnln n surface and line Wilson The ordinary the in as holonomy case. knot from invariants knot tr tr tr tr h isnline Wilson The u mrigadiooyivratadgueindependent gauge and invariant isotopy and –marking ( -markings θ θ ′ ko oooyivrat samdl epooea propose we model, a as invariants holonomy –knot R R R R R R R , , ( = Υ U , , , , S ξ ( , , , et aigtepoeuejs eiwdt construct to reviewed just procedure the taking Next, S S S S θ S S u ( ) –knot | | G | | aua G , , ( ) Ξ b b f f θ θ ≡ , ξ a µ u ∈ ( ( | , ( ( , U G , Υ b ) ) U u aua m , G H , H aF G ( | = ≡ G A U , ),( ) G , ξ f − ) H ––oncinpi on pair )–2–connection ( –ojgto.T hsed n ol proceed could one end, this To )–conjugation. ) = µ ( ) , × tr m a = 1 ),( , θ Ξ H ∈ Ξ a = 5 )–conjugation H t tr − )( ( R Z H H , Z ( ( ehv lose htfra for that seen also have we , , ξ ) G 1 u h holonomy the , eaLecosdmdl n ( and module crossed Lie a be ) ikrepresentations Pick . H U tr ( R a A p G A ) = t F . ′ , a )( × r opc ihb-nain armea- Haar bi-invariant with compact are ( S R ), , S d ) d )) θ U tr A − | , ∈ , F b µ ( S µ H ζ = 1 ξ )) R | , θ ′ G Si H b G t ),( ) (92) )), , , ( . Υ S tr tr ( ( = ( ( n ( and ) × F | A x ( X G U a f R R Ξ tr θ ,(93) ), )tr H ( , , , )tr , ( S S F ′ R )(94) )) A H ξ | | = . θ f , f S S ) (100) )), ) S R , –ojgto sdfie by defined is )–conjugation ( ( Υ ( | ko oooyinvariants. holonomy –knot m U U ∈ b ( m p ( ut ( Ξ G ) ( (99) ) u M ( a F x ) (101) )). ,(98) ), r nain under invariant are ( × , )( θ X )( ζ ( U SMi H U ) (96) )), ξ and ) M (95) ) ) (97) )), n sa equivalence an is and R of ) utemr,w let we Furthermore, . , S F of S θ , and Υ G C ( , Ξ –knot H is ) M C θ Set . , and – respec- , Υ C ea be ) ( ξ and – G and , H S – ) htmtesi o ( not is matters What htmtesi o h ru tutr of structure group the not is matters What with with tecs fitrs o ufc knots). surface for interest of case (the hr sapolmwt hswyo rceig The proceeding. of way this with problem a is There G ( 13 etdpitdqadecosdmdl ( module crossed quandle pointed mented h action the tr( a r: tr map If obnto fodnr rcstr traces ordinary of combination uainsrcuecdfidi h ojgto pointed conjugation the of quandle in codified structure jugation representations reducible one unl sastGwt iayoperation binary a with G set a is quandle pointed A nodnr ag hoywt ag group gauge with theory gauge ordinary In rcsmyb rva.Frisac,if instance, For trivial. be may traces invariant. S–knot and C– a provide o n a any for osntdpn on depend not does sapi fmp tr maps of pair a is o a for iha with a a a a a a A tr tr tr G ⊲ ⊲ ⊲ ⊲ ⊲ ⊲ b b ≻ × f G ]: , a ( ( ( H nhge ag hoywt ag rse module crossed gauge with theory gauge higher In U u u u 1 ( G a a a A ⊲ b sacmatLegop hnt eue oalinear a to reduces tr then group, Lie compact a is G a a ∈ ,asmlrpito iwi prpit.Ataepair trace A appropriate. is view of point similar a ), ⊲ ⊲ = : ≻ : : → : u ⊲ = = = , , , = = G u u b u U u a ) G ut c aua aua . n itnuse element distinguished a and G , m 1 = ) ∈ ∈ , ) ∈ c ) ) G = → = = G G = ( ( tr( and G ∈ G 1 , a A tr tr , htis that , .Frhr h a a map the Further, G. ( − − tr C . )( , ) u a A b b 1 1 U f .(103) ). ( ( ⊲ , nain ne h action the under invariant (107) , G ( u u U U (109) ) ⊲ b ,(111) ), ,(110) ), ∈ .(112) ). ) b a ⊲ : H .Zchn:Wlo ufcsfrSraeKnots Surface for Surfaces Wilson Zucchini: R. u = G G ( htis that , n tr and a , a → H ⊲ tefbtiscnuainaug- conjugation its but itself ) C c R (105) ) tr , R of , f S G f ( : . U R H ⊲ ) soitdwt h ir- the with associated → = · : t tr G C ( 1 S H → G ( nain under invariant U ) ∈ sinvertible is G G = G for ) G uhthat such G rc sa is trace a , G u t con- its but , H tr , U [ ) R ∈ 30 , (108) (102) (104) (106) Sb ker , ⊲ ( 31 u t ) , : Proceedings → 13 g (123) . Sup- × ) yield 3 g ξ ( M ) furnish ): θ · , ξ , ( R · θ , W R , . . . . Therefore, is called level. W 3 is its Lie algebra. ): S k ) , is not flat and con- ]. g 2 7 20 θ : S ( Z × π 1 2 S ´ ] θ ), (125) , g θ [ is equipped with a properly nor- 3 1 The coefficient HOMFLY polynomial; Kauffman polynomial... Jones polynomial; (cf. Equation wn( g + · 0 (122) ⇒ ⇒ θ ⇒ θ k = Seifert, e. g. F F F π ,d 3 ]) ) is not ambient isotopy invariant. How- 2 = = θ = y ξ ³ M ( − , R R R 3 θ ) 0. (124) z . , is the winding number of g. is the gauge group and θ ), ), M g R ,[ = ) Z n n x ∈ ] g G W π θ CS( z + k , 4 , ) SU(2), SO( SU( θ = y a G–connection. y [ wn( , ) = = = = 2 1 θ ], x ) θ x g θ In order to compute surface knots invariants in quan- G G G + , so that z θ genuine knot invariants. tum field theory, oneSimons needs theory, a 2-Chern–Simons higher theory. version Weposal of have for such Chern– a a model. pro- There are howeverlems unsolved prob- to be discussed below. knot invariants, for instance: Chern–Simons Wilson loop correlators In the Chern–Simons path integral, ever, the theory somehow localizes onof the flat connections moduli even space though it istopological not field a theory. cohomological This has been provenand by Witten Beasley for sequently malized invariant non singular bilinear form ( The Chern–Simons action is invarianttransformations g under only a modulo G–gauge CS( Chern–Simons theory is a Schwarz typege topological theory gau- on a closed 3-dimensional manifold where Quantum gauge invariance holds if the level k isChern–Simons integer. correlators of Wilson loop 7 Higher Chern–Simons theory needs Chern–Simons theory. This haslong been time known since for Witten’s 1988 a paper [ pose that The Chern–Simons field equationsflatness are condition equivalent of to the d with Suppose further that The Chern–Simons action is given by CS( g ([ To compute knot invariants in quantum field theory, one with G, G, ∈ irre- → (116) (118) (115) (120) (121) G S 1 G , × R G : are compact, ⊲ G is given with H ) with , → G (a map respect- 97 G G → ), ( × H is invertible. H H 96 : : , respectively? → ≻ α H ) (114) H , ), (113) : ). (119) B A · G b ⊲ ⊲ ⊲ ⊲ reduce to linear combinations of a a a ( f ( ( G is invertible. ⊲ of the form ( H and distinguished elements ⊲ ≻ H hold. , tr ) ) ) f ), (117) → b | ∈ A H. b A → A S , G ( H, B ∈ ⊲ ⊲ R : ⊲ , B H α is the quandle crossed module analog of ∈ H a H, the relations a B a × ( ( ⊲ , ( ), ⊲ , tr ≻· ∈ ∈ α ) is given such that b G, the map a = A A = a 1 | = G, A , A H ( ) . S ) ∈ ) , G, A A = ∈ 1 , = H,G a α B G, A A R b ∈ ) H, A ) is the quandle counterpart of the Peiffer identity. b G G, A B = = ∈ = = and , ≻ ⊲ A ⊲ → b ∈ ∈ ∈ , A H H such that ⊲ a G A A b ⊲ ) ⊲ 118 ( 1 ( H ( 1 ∈ ≻ ⊲ A a × ≻ ⊲ ⊲ ⊲ ⊲ ( ( H H G A H a a a a For a α For A α does a trace pair tr ing traces tr For a 1 are fulfilled. Further, a quandle morphism Finally, an augmentation map the following properties. For a with a with a For any a i) G is aii) pointed quandle, H is a pointedand quandle the following requirements are satisfied. The relations An augmented pointed quandle crossedof module setsG, is a H pair endowed with three operations ducible representations of 1 the group crossed module target morphisms.lar ( In particu- Notice that 1 The following question is still open. If Proceedings with ( ( ( A nte nxetdfaueo h oe ocrs1– concerns model the of feature unexpected Another hsi ut ie u tsgasaptnilproblem potential a signals it but nice, quite is This n ( ing pair- bilinear singular non invariant normalized properly rse ouei aacd hti dim is that balanced, is module crossed uethat sume rse oue suefrhr( further Assume module. crossed oueadthat and module ([ ( with CS by given is action 2–Chern–Simons The –hr–ioster saShaztp topological manifold 4-dimensional type closed a Schwarz on theory a gauge is theory 2–Chern–Simons The level. to analog cient odto,wihmksi genuine a it makes which condition, once, at Thus, d d aifistevnsigfk uvtr odto ( condition curvature fake vanishing the satisfies Ω ento fthe of definition knots face surfaces Wilson of insertion the integral, path 2–Chern–Simons the in condition curvature fake h -hr–iosato sivratunder invariant is action 2-Chern–Simons The invariance. gauge way. basic o h osrcino –hr–ioster safull ( a Since as theory. field theory quantum 2–Chern–Simons of construction the for zsa a one flat a character- as which izes condition, curvature vanishing the and 14 ag transformation gauge CS θ 30 t ˙ Υ θ y 2 ( , G X 2 , 2 ( Υ + ) + x M ( ( ): , · ), θ ) g 2 ], ( , H 2 m 1 x , b 4 en flat a being θ , · CenSmn edeutosaeeuvln to equivalent are equations field –Chern–Simons Y J : ) X [ , Υ , –rcneto sjs ar( pair a just is 2–preconnection ) θ , ( θ y h ) Υ θ ) ) , , g .( ). − g + , θ ∈ = ) × Υ , ] ( J a H Ξ ( κ g t θ − θ ) ˙ h x ( ( , 2 = Υ , G , Y t ntept nerli rbeai,a the as problematic, is integral path the in −→ → ˙ Υ Z ( m X ( ) b .(128) 0. θ , ), t M Υ ( a ) . H = , , ( Ê Y W 4 X ) Υ y ( G h 2 ) CS G ³ = )( ) ) uhthat such d R ∈ −→ , G = aifistevnsigfk curvature fake vanishing the satisfies −→ X , ,(127) 0, θ peoncin[ –preconnection S 2 H m t ˙ h , , ( + 0, )) ( θ H θ ) oeta hsrqie htthe that requires this that Note . g , 2cneto c.Equation (cf. –2–connection Υ g , = 1 2 Aut( , ––oncini nadto it addition in if )–2–connection Υ ( J [ −→ Ξ 0 ) θ .(129) ). m b , eursta odto na in condition that requires ) , θ H θ der ] , stegueLecrossed Lie gauge the is ) − Υ ( osntoe h zero the obey not does ) 2 1 h g t ˙ sisdfeeta Lie differential its is ) , ( h ( Υ G 34 seupe iha with equipped is ) ), , θ W , H Υ g , 33 Υ R ) ´ = –2–connection, , (126) , ]. ) S , dim ∈ θ κ , Υ Ω 2 ( sacoeffi- a is 1 ( Ξ h G ( 29 . M fsur- of ) M , H ). 4 ( 4 , 29 ) As- . g –1– ) ) × ), l retto rsrigdiffeomorphisms preserving orientation All C Acknowledgements. hssget that suggests This inpeevn diffeomorphisms preserving tion rnfraino htw r isn l h topologi- the all ( trivial missing non are cally we that or transformation rnfrain ( transformations tdigpl–ak fkosmyb interesting. be may knots of pull–backs Studying this. for reasons further are intrigu- Here possibility. an ing remains surface theory Wilson 2–Chern–Simons of in correlators insertion as obtaining of invariants possibility knot the surface issues, open these of spite In problem. open an still zdsurfaces ized circle the of ovrey o ihrgnssurface genus higher a for Conversely, the lie yhpteiigta ihral( all either that ex- be hypothesising by can sur- finding plained This disappointing model. somewhat Chern–Simons and ordinary prising the quantiza- in level cause as to tion such number winding kind higher some of by shift no is there theory, 2–Chern–Simons In the theory. mathematical well–developed a exists there which about References knots higher to route theoretic field A knots, higher ory, words. Key discussions. useful many for and subject the ooi oid to motopic MCG covariance interesting group class have mapping the under may properties theory gauge higher ngeneral in 3 .R Lickorish, R. W. [3] 2 .C Adams, C. C. [2] 1 .H Kauffman, H. L. [1] –knot C S –knots + –knots pigr 1997. Springer, ahmtclScey rvdne 2004. Providence, Society, knots, Mathematical of theory mathematical the to tion igpr,1991. Singapore, ( S ξ ) h curves the , = Diff ihrgueter,hge hr-iosthe- Chern-Simons higher theory, gauge Higher C Ξ ξ S Σ and osqety o a for Consequently, . r oooi oid to homotopic are and ♯ + Ξ ( G S , g h ntbo:a lmnayintroduc- elementary an book: knot The f , )/Diff Σ f , H ∗ S .Zchn:Wlo ufcsfrSraeKnots Surface for Surfaces Wilson Zucchini: R. J ∗ nitouto oko theory, knot to introduction An ♯ etakR iknfrhsitrs in interest his for Picken R. thank We nt n physics, and Knots ξ r ml nieodnr gauge ordinary unlike small are ) ko nainscmue using computed invariants –knot ––ag rnfrain.Ti is This transformations. )–1–gauge γ Ξ f ∗ aetesm holonomy. same the have ξ Ξ , onthv h aeholonomy same the have not do 0 γ ( r o hnyhmtpcand homotopic thinly not are S f ∗ ,(130) ), ξ r hnyhmtpcand homotopic thinly are f C S ∈ osqety o a for Consequently, . –knot Diff ol Scientific, World S o l orienta- all not , + G ( Ξ S , h normal- the , of ) American H f )–1–gauge ∈ S Diff r ho- are + ( C ) Proceedings 15 16 ]. (1979) (2015) 14 16 40 ]. 1310.4705 [ 1112.2819 Commun. Math. [ ]. (2014) 79 On 2-holonomy, (2016) 052301 Proposals for unknot- (2014) 85 Wilson surfaces and Crossed modules of 1410 57 d semistrict higher Homol. Homot. App. math.QA/0409602 16 Osaka J. Math. 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