PoS(ICHEP2020)042 https://pos.sissa.it/ ∗ [email protected] Speaker ∗ I briefly review several important formal theorytheory developments that in were quantum reported field at theory ICHEP andnew conferences research string in area referred past to decades, as and physical or explain quantum howspecific mathematics. they context, To underlie illustrate I these a ideas discuss in some certainknots-quivers aspects correspondence. of topological string theory and a recently discovered Faculty of Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warsaw, Poland Walter Burke Institute for Theoretical Physics, CaliforniaPasadena, CA Institute 91125, of USA Technology, E-mail: Copyright owned by the author(s) under the terms of the Creative Commons © Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). 40th International Conference on High EnergyJuly Physics 28 – – ICHEP2020 August 6, 2020 Prague, Czech Republic (virtual meeting) Piotr Sułkowski Quantum fields, strings, and physical mathematics PoS(ICHEP2020)042 limit # , a current limit one can gauge theories. # Piotr Sułkowski , which has been . While rigorous ) # ( (* “own work now points towards expansion in quantum mathematics # knots-quivers correspondence or 2 [1], he presented his vision of the future of high energy . He then claimed that his limit are also intimately related to gauge theories with extended expansion were discovered by ’t Hooft, who hoped that this physical mathematics # # gauge theories may be mathematically well defined” → ∞) # “Stage 5, still further in the future, may finally give us a convergent field theory. ( “Theoretical perspectives” (* The plan of this note is as follows. In section2 I briefly review several important “formal theory Since 1950 most important developments in high energy physics, including those related to One such important theoretical idea is that of large This year’s International Conference on High Energy Physics (ICHEP) is exceptional for several physics, which in This is a fieldtheories in theory 2 or that 3 space-time is dimensions” mathematically as rigorous as presently known constructive field developments” in quantum field and string theory discussedvarious in past current ICHEP meetings, research which underlie areas.is In supposed section3 toI be. summarizeIn In what section4 section5 physicalII or discuss quantum explain a how mathematics crucial these role ideas of get these unified ideas in in the topological string theory. theoretical and mathematical structure of quantum field theoryduring and ICHEP string theory, conferences. have been reported Toimportant illustrate ICHEP at least summary some talks,research of which area these referred presented developments to as ideas let physical us that or recall quantum underlie mathematics. a later few development of the developed recently very actively. While the scopedefined, and it the is boundaries rooted of in this formal fieldthose developments that are in were not quantum reported so field in clearly theory the past andalso ICHEP string discuss meetings. theory, some in To aspects illustrate particular these of ideas topological incorrespondence. string a theory, specific as context, I well as recently discovered knots-quivers research direction that can be thought of as one representative of physical or quantum mathematics. 2. Formal theory developments and 40 ICHEP meetings Intriguing properties ofapproach large would lead to a1982, rigorous entitled formulation of quantum gauge theories. In his ICHEP talk in a conjecture: formulation of gauge theories isproved very still fruitful beyond in our various contexts. reach, Formatrix the example, models, it idea which led of can to powerful considering be methods thoughtintroduce the of of a solving large topological as random expansion 0-dimensional of matrix gauge model theories. amplitudes,their ribbon In solution diagrams, large in loop terms equations of and theetc. topological recursion Matrix (recently models further and their generalized large to Airy structures), reasons. It is the 40th ICHEPin Rochester conference, in taking 1950. place These 70 anniversaries years provide afterin a the great high first motivation meeting to energy summarize in physics some this from highlights developments. series the In last this 70 notediscussed years I and in follow to such a a contemplate separate planimpossible how to in they session present the underlie all at context the important of ICHEP current to contributions “formal the 2020. in theory research this area developments”, area. referred Of to Therefore as I course, focus on in results limited related space and time, it is Quantum fields, strings, and physical mathematics 1. Introduction PoS(ICHEP2020)042 , . 2 # = , and ” N . Since ) . 1 “May the ( topological × 10 Piotr Sułkowski “String Theory: " “Yet field theory also supersymmetric Yang- 4 “some examples of recent = N . Among others, he mentioned [2]. Apart from developments in string . P. Townsend summarized developments [4], admitting that [5] that he finished with the wish theory generalizes to heterotic-type II duality” limit in gauge and string theories. 3 2 is a matrix in the Lie algebra of some group, say “Recent developments in non-perturbative quantum # = 8 , you obtain a gauge invariant and generally covariant )  : # “very recent work, Witten has opened up the subject in  [6]. Among others, he listed 9  8  “summarize some of the main basic results of the years 94-96, supergravity has been an active area of research [...]” 3 2 8 + 9 =  : , which shows the enthusiasm concerning these topics at that time. N m 8  “Superstrings and Unification” limit. It is hard to imagine modern theoretical high energy physics without ( super Yang-Mills theories in the planar limit [...]. 2. In another line of # Tr 4 “Recent Progress in Field Theory” 8 9: = Gn N 3 3 talk [3], S. Ferrara did in which the gauge invariant observables cannot depend on the metric and thus must ∫ “Superstring and supermembrane theory” : = . Now, if you construct the Wilson loop around a curve, or knot in three dimensions you get ) ( . Witten demonstrated that these invariants coincide with [...] the Jones polynomials” Formal aspects of high energy physics were also summarized in two talks in ICHEP 2000. The last historical talk that we mention is by A. Sen in ICEHP 2010, entitled Furthermore, in ICHEP 1996 in Warsaw, in From several talks mentioned above we can immediately identify important formal develop- Another important idea bringing together physics and mathematics was presented by D. Gross “M-theory and strings may undergo a further unification in twelve dimensions (F-theory)” “The Seiberg-Witten solution of rigid : # ( (* M. Dine presented “Witten proved the equivalence of differenttype string theories IIA in at higher strong dimensions coupling andthat with the 11D duality supergravity of at largeNote radius that (M-theory all these on developments rely in a crucial way on extended supersymmetry. has limitations. We probably needproblems to of go black beyond holes, quantum the fieldground states theory cosmological of if constant M-theory we and problem, are what the selectsin to principles among his understand them” which the talk determine the next 20 years be as fruitful!” Basic Facts and Recent Developments” developments in quantum field theories.of 1. operators in We now have exact results for anomalous dimensions field theory” in the context of string theory and its non-perturbative regime” development, many exact results for on-shell S-matrix elements in then relations between gauge theorytheir analysis and has knot grown into theory an have important been research generalized direction, as to we string also review theory in and what follows. that Mills theories have been derived usingGaiotto on-shell and methods his [...]. collaborators developed 3. toolssuperconformal In to field a construct theories remarkable and series in study of a four[...] papers whole dimensions 4. new – Possible class finiteness some of of without even a Lagrangian description a topological knot invariant, i.e. a number that depends only on the topology of the knot and on Quantum fields, strings, and physical mathematics supersymmetry in various dimensions (e.g.topological string in Dijkgraaf-Vafa theory. theory), Chern-Simons Furthermore,properties theory, the of and the vast large field of AdS/CFT correspondence also relies on field theory be topological invariants. The gauge field and developments that rely on the analysis of large action. It is generally covariant without containing the metric explicitly thus it is a in his ICHEP 1988 talk theory, in his talk he alsoa explained totally that new in direction. Itthree-dimensional turns general out relativity! that two-dimensional If conformal you fieldterm: take theory as is a connected three-dimensional with action the Chern-Simons PoS(ICHEP2020)042 . Piotr Sułkowski “Physmatics” “review the remarkably fruitful . limit, etc., which underlie the field of # 4 by myself and others is not meant to detract from the Mathematical Physics For example, during his summary talk at Strings conference in 2014 [7], G. Moore explained In a similar vein, M. Atiyah, R. Dijkgraaf and N. Hitchin in [8] The terms physical mathematics and quantum mathematics have been around for some time. Finally, in [9] E. Zaslow characterizes such an activity and developments as “after 40-odd years of a flowering of intellectual endeavor a new field has emerged with its own distinctive character, its own aims andsubject values, as its Physical own Mathematics. standards of [...]the proof. The more I use traditional like of to the refer term to “Physical the Mathematics” in contrast to that They are not rigorouslyquite defined often and – their thoughto meaning not considerations may always motivated depend – by a theystring physics, bit theory, are conducted and on used relying in on a interchangeably. the physicalconjectural) context arguments, formalism statements which In and in of lead mathematics. a a to quantum Quite (unexpected, often broad user; field non-trivial, suchphysical duality statements sense, and theory relating arise often two from they systems, or the which analysis refer have of independent mathematical some of descriptions. these A systems duality leads thenfrom to completely new different relations theories. between Itwas various is presented mathematical instructive on to objects several recall other that how occasions. originate the idea of physical mathematics venerable subject of Mathematical Physics butby rather to questions delineate and a smaller goals subfield thattheory, characterized are and supersymmetry, often (and motivated, more onmatter recently the physics), by and, physics the on side, notion the of by mathematicsLie topological quantum algebras side, (and phases often gravity, groups), in involve topology, string deep geometry, condensed and relationsmore even to traditional analytic relations infinite-dimensional number of theory, physics to in algebra, addition groupprinciples to theory, the and is analysis. the [...] goal oneinsight of of leads the to understanding guiding a the significant new ultimate resultjust in foundations as mathematics, of profound that is and physics. considered notable aprediction success. as of [...] an It a is experimental a peak If confirmation success orfour-dimensional from a manifolds trough. a is physical a laboratory vindication For of just example, a as the theoretical satisfying as discovery the of discovery of a a new new and particle.” powerful invariant of 3. Physical and quantum mathematics Quantum fields, strings, and physical mathematics ments in quantum field theory, superstring theory,and and non-perturbative M-theory, effects, such as topological understanding invariance, of large physical dualities mathematics. In particular, extended supersymmetrythese is developments an – important on ingredient one in hand mostthe it of enables other to hand keep it non-perturbative may effects be under used control, to and impose on topological invariance in field or string theories. interactions between mathematics andgeneral quantum trends physics and in highlighting thegeometry, several past invariants of examples, decades, knots and pointing such four-dimensional out as topology” some the counting of curves in algebraic PoS(ICHEP2020)042 , , 2 3 } 8 (1) (2) ( B = ∗ 6 & { ) Chern- N = ) & # that can be ( are encoded limit, indeed G ) (* # & Piotr Sułkowski ( 6  limit the open A-model encodes classical triple # ) & ( 0  , ,   and open parameters ) ) & & ( G,& 6 (  characterizing its 2-cycles, and = 2 8 ( − C 1 6 4 2 B − = B 6 = 6 0 8 = 0 ∞ 5 & 6 = Õ ∞ = Õ   exp exp free energies and, mathematically, they encode Gromov- , and geometric transitions = ). The deformed conifold is simply a cotangent bundle 6 = # 8 I limit arises in the context of an equivalence of open and closed closed open / / # . The process of replacing one of these threefolds by another one is . Furthermore, it turns out that in the large 6 # B characterizes boundary conditions, which can be interpreted as being . On the other hand, open topological string partition function takes form 6 " with Kähler parameters . In particular, the leading contribution # = C " " parametrized by 4 C depend both on closed Kähler parameters branes wrapping a lagrangian cycle [11]. It is in this sense that we can consider in ) are referred to as genus 0 # ) = & G,& ( 2 4 ( I 6 = limit. Note that various observables in Chern-Simons theory, in the large ( +  # In topological strings, large Topological string theory is a simplified version of string theory, whose playground is a Above we reviewed several highlights presented during past ICHEP conferences. In this section The simplest and prototype example of a geometric transition arises when open and closed ... + 2 1 I imposed by large Simons theory, where have topological expansion analogous to (1) or (2); in particular, contributions supersymmetric sigma-model, whose appropriate twistsin impose A-model topological or invariance B-model and versionimportant result of ingredient the of physical theory mathematics, [10]. alsoAs plays in a In ordinary fundamental this string role theory, sense in one extended topological canCalabi-Yau threefold supersymmetry, strings. consider closed as and an open version of topological strings. For a thought of as characterizing (topological) branes. versions of the A-model theory onsetup, two open appropriately A-model chosen Calabi-Yau theory manifolds. turns In appropriate out to have an effective description in terms of Calabi-Yau threefold (of 6 real dimensions). Its definition involves a two-dimensional we briefly describe how these phenomena,in originally topological discovered string independently, theory, play which a itself crucial is role an important playground of4.1 physical mathematics. Topological string theory, large Quantum fields, strings, and physical mathematics 4. A playground – topological string theory where where in ribbon diagrams of genus topological strings are considered for twoone, versions which of a are conifold two geometry: different deformed deformations and of resolved a singular conifold (defined by a complex equation closed topological string partition function has the following expansion in the string coupling intersection numbers of theory on one manifold can be interpretedthe as above closed mentioned version of lagrangian the cycle theorythe replaced on ’t another by Hooft manifold, a with coupling 2-cycle, whosealso Kähler referred parameter to is as given a by geometric transition, and it provides a nice example of an open-closed duality. Witten invariants of PoS(ICHEP2020)042 . , 3 A # # # B (3) ( ( @ 6 , we 4 = 3 = branes ( = ' ∗ 0 # 0 ) depends on , and B Piotr Sułkowski ; in particular, in or colored Jones 6 # 3 , identified with a c8 4 ) + ) ( 2 : . = ) # ( 0, @ @ ( exp . The resolved conifold 0, @ ' (* 3 in (2). ( % = ( A is transformed into resolved & % @ 3 ( ) ≡ 0, @ ( , A along a knot of interest. In this case A ( , colored polynomials reduce to ordinary G branes on % 3 )  ( # . Open opological strings on the deformed = Chern-Simons theory on 1 0, @ ) ( P ' A # ( % 6 0 = is an open Kähler parameter, ∞ (* A Õ denotes a representation of G = ' open . From gauge theory perspective, the parameter ) / , with a 2-cycle ,@ 1 2 Chern-Simons gauge group, while P @ , and appropriate observables in these two theories are equal [12]. ) # . The color = # are described by B : ( 3 6 ) → 0 ( ( 1 = (* ' C % = ) ) ⊕ O(− @ of size 1 branes on ( 1 ' P # O(− Note that the geometric transition provides an interesting relation, in the spirit of physical math- As mentioned in section2, Chern-Simons theory with Wilson loop observables computes It turns out that the relation between knot invariants and gauge theory observables can be Furthermore, polynomial knot invariants, such as Jones or HOMFLY-PT polynomials, turn out polynomials encodes the rank of the ematics, between two mathematical fields:such low-dimensional as topology Witten-Reshetikhin-Turaev invariants, (which that provides characterize invariants, 3-manifolds wrapped by polynomial knot invariants, such as colored HOMFLY-PT polynomials this setup, topological string partitionUpon function the is geometric equal transition, to deformed conifold Chern-Simons with conifold partition with function on before the transition), and Gromov-Wittenclosed theory topological (which string theory provides after a the mathematical transition). description of 4.2 From Chern-Simons theory to knot homologies and the Chern-Simons level Young diagram. For the fundamental representation have to include anopen additional topological string brane, amplitudes reduce which to Chern-Simons intersects invariants amplitudes, of and this indeed knot. reproduce A knot configurationtransition, with which such produces an a resolved additional conifold with brane an can extraknot. also brane that undergo In also a this captures geometric properties case, of open the string partition function, which afterpolynomials the which transition are encodes computed open by Gromov- Chern-Simons theory (before the geometric transition) where only symmetric representations arise, Quantum fields, strings, and physical mathematics where we can consider topologicalis branes a wrapped bundle on theconifold with lagrangian orHOMFLY-PT Jones polynomials. In what follows we focus onlabeled symmetric by representations Young diagrams made of one row, and denote also lifted to topological string theory. To this end, in the open A-model theory on to have much deeper meaning andhomological structure. theories Namely, associated they to arise knots. asis Euler A Khovanov characteristics prominent homology, of example which of various such categorifies homologicalEuler Jones character construction of polynomial Khovanov homology). (i.e. There are Jonespolynomial many other polynomial knot knot arises invariants; homologies associated as some to an of other while them in are some quite other abstract cases and(without we providing difficult can their to at explicit construct construction), most as in predict is practice, certain the expected case properties for HOMFLY-PT homology. of such homologies is identified with the closed Kähler parameter of the resolved conifold Witten invariants according to (2), takes form of a generating function of colored HOMFLY-PT PoS(ICHEP2020)042 (5) (4) is a ; subspace of 2 R Piotr Sułkowski of the M5-brane. 1 ( , and the 11th dimension , , 4 R V, 9 that counts plane partitions. V,:, 9 ; ;# # ) − ) 9 ) 2 ; + / ; @ 1 @ − 9 V − + ; & 1 ( @ 1 − V = ∞ ; 1 & ( , it reads : Î 1 G G = ∞ ; Ö = − , the contribution from the product over 1 9 ) 7 9 ( Ö @ spacetime directions and extra 1 ( ) = ∞ 2 ; Ö " " ( and R 2 Ö  V Ö ∈ 9,V,: V = . In particular, a lagrangian brane that engineers a knot in a = 1 ) ( ) & ( &,G ( closed open / . For a fixed / B 6 4 = @ are conjecturally integer Gopakumar-Vafa invariants that count BPS states of closed V, 9 # . In addition, we include in this setup topological branes mentioned above that are identified Amusingly, homological knot invariants can be also related to high energy physics and various As mentioned above, an important aspect of topological string theory is its relation to knot BPS states in such M-theory systems are represented by M2-branes, and corresponding partition To be more specific, this BPS interpretation constrains the form of closed topological string 1 , and wrap the M-theory circle ( 4 There is an analogous expression forassuming open that topological it string depends partition on function a (2). single open For parameter simplicity, constructions in string theory, in particularit to is the postulated that brane homological system spaces mentioned should abovesystem, be [13]. as identified we review with In in a spaces this little of more case, BPS detail below. arisingspaces While in from explicit this identification this of brane BPS viewpoint or is homological difficultexample and computation often of still impossible, superpolynomials, this which physicalof are approach knot enables Poincare homologies for and (rather do than capture Euler) certainknots can characteristics homological be information. computed Superpolynomials using for various various approaches:theory, refined relation topological vertex, to refined Chern-Simons DAHA, orother relations expected between structural gauge theory properties and string of theory knot and knots homologies. homologies,4.3 for example There see are [14]. Topological string also theory and BPS counting homologies, which can beof identified BPS with spaces states of arise.10-dimensional BPS superstring states. To theory identify or Let in them,dimensional M-theory. us space, one Focusing recall whose should 6 on how dimensions the embed are these latter identified a(e.g. spaces with system, a we topological deformed threefold Calabi-Yau consider considered or string 11- above resolved setup conifold), 4 in dimensions the form full “spacetime” Quantum fields, strings, and physical mathematics functions encode multiplicities of such BPS states. Moreby precisely, closed M2-branes BPS states that are wrap represented 2-cycles insidein Calabi-Yau]. [15 systems On the with other M5-branes; hand,boundaries open such are BPS states open attached arise to BPSrepresented M5-branes states by effective [16]. are supersymmetric represented quantum Furthermore, by gauge theories it open in is M2-branes directions expectedM5-brane whose normal normal that to to such Calabi-Yau Calabi- space, i.e. systems are conifold is identified as a 3-dimensional part of the full M5-brane. partition function (1) as follows is as parts of M5-branes, whichR extend along 3 dimensions inside Calabi-Yau, span where M2-branes and generalization of the MacMahon function Yau space. For systems with M5-branes, such an effective theory may also live in 3 dimensions of PoS(ICHEP2020)042 .  (7) (6) [17, nodes repre- ; < . 9 ; < ,...,3 and generating 1 Piotr Sułkowski 3 @ Ω 1 + 9 , ) < 1 3 < (− G  , the product over 2 : ··· )/ 1 1 − 3 1 9 G and 9 +( < ; 3 3 8 knots-quivers correspondence @ ) 3 as elements of a “quiver matrix” 9, V
@ < 8, 9 ; 3 8, 9 <  @ G 1  = . ) < 8, 9 ··· ; ···( Í 1 1 ) 3 1 3 2 &@ G ) / 1 @ − @ ; − 8 1 @ ( 1 ( (− 0  . We refer to = < ∞ 9 ; 1 = ∞ ; Ö Î Õ ,...,3 Z = and 1 ∈ 3 9 8 Ö ∞ 0 ) = ≠ @ ) ) ; < < & ( Ö encoded in these expressions are indeed integer. Until recently these ,...,3 1 3 ( ,...,G 1 V,:, 9 = G # ( ) . This series has the following product decomposition  < < % and V, 9 arrows between nodes count open BPS states and are referred to as Ooguri-Vafa or (in case of knots) ,...,G # ,...,G 1 9,8 1 G G (  are non-negative integers, and it depends on a motivic parameter  V,:, 9 = 8 % 3 # 8, 9 The formulas (4) and (5) are important examples of nontrivial string-theoretic conjectures. In this final section we present how various historical developments from2 section and other From the physical perspective, a quiver that encodes interactions of elementary BPS states To proceed, let us provide a few quantitative results. Suppose we have a quiver with  parameters In the past twomultiplicities decades it has been verified in numerous examples that closed and open BPS ideas mentioned above merge together in a recently discovered arises in the effective descriptionquantum mechanics. of Moreover, the the nodes system of in a quiverinterpretation. question, in various in systems In of terms interest the also of have case a aof direct of supersymmetric HOMFLY-PT homology knots, quiver [18]. the nodes Fromcertain are elementary the in discs, viewpoint whose one-to-one of counting correspondence topological is with captured strings, by generators the Gromov-Witten theory nodes [19, represent 20]. Quantum fields, strings, and physical mathematics where Labastida-Marino-Ooguri-Vafa (LMOV) invariants. For fixed checks were conducted only upmultiplicities. to However, the some knots-quivers order, correspondence,to and discussed prove in that in certain the consequence infinite next only families section,that of for there enables open is a BPS still finite no invariants general are set mathematicalfrom integer. definition of a of Furthermore, BPS BPS mathematical let numbers perspective, us discussed the stress above. expressions Nonetheless, (4) and (5) mean thatdeep mathematical closed statements, and in open line Gromov- with the philosophy of physical mathematics. 5. Knots-quivers correspondence 18]. As its name indicates, itand provides a a relation quiver between representation two branches theory. of mathematics:and However, its characterizes knot physical a theory interpretation wide has class broader ofdiscussed consequences topological above, string whose theories multiplicities and openNamely, are it BPS encoded turns states out in in that brane generating these systems finite functions BPS number states of of (typically certain the infinite elementary number form BPSis of states, (5)[ encoded them) and19]. in the are way a bound in symmetric states whichconnected of quiver. such a by bound Recall states arrows. that are a A formed quiver quiverone is is node a symmetric to diagram if, another that one for consists is all of the pairs several same of nodes, as nodes, the the number number of arrows of in arrows the from opposite direction. where sents the quantum dilogarithm and To such a quiver one associates a generating series [21] Witten invariants satisfy certain integrality relations – so in this sense a physical insight leads to PoS(ICHEP2020)042 8 G Piotr Sułkowski are expressed as , in “Proceedings, and parameters in  V,:, 9 # . Since the latter invariants are 9 ; < ,...,3 1 3 Ω , in “Proceedings, 30th ICHEP: Osaka, Japan, is introduced). In case of knots, represented by 9 , in “Proceedings, 24th ICHEP: Munich, Germany, G , in “Proceedings, 21st ICHEP: Paris, France, July (1982) no.C3, 755-764. 43 (in the exponent). Roughly, these invariants are Betti numbers of certain . For other consequences of the relation to quivers see [17–20, 22]. 8 9 This work has been supported by the TEAM programme of the Foundation G ; < “Theoretical perspectives” “Superstrings and unification” “Recent developments in nonperturbative quantum field theory” “The Utility of quantum field theory” ,...,3 1 3 Ω 26-31, 1982”, J. Phys. Colloq. August 4-10, 1988”, PUPT-1108. 28th ICHEP: Warsaw, Poland, July 25-31, 1996”, [arXiv:hep-th/9610085 [hep-th]]. July 27 – August 2, 2000”, [arXiv:hep-ph/0010035 [hep-ph]]. The knots-quivers correspondence has been verified for all knots up to 6 crossings, certain We can present now the relation between quivers and topological string amplitudes. Namely, The above identification has deep consequences. First, it implies that the corresponding prod- [2] D. J. Gross, [3] S. Ferrara, [1] G. ’t Hooft, [4] M. Dine, Quantum fields, strings, and physical mathematics In quiver representation theory, thisinvariants product decomposition encodes motivic Donaldson-Thomas (so that a dependence on a single parameter proved to be integer, it must be so alsoture for of open BPS their invariants, integrality which in proves a quite long-standing aviewpoint, conjec- general colored setting. HOMFLY-PT Furthermore, polynomials in are casequiver of expressed moduli knots, in spaces, from mathematical which terms can ofidentification be numerical interpreted of invariants (3) as of a with novel (6)nomials type implies is of that determined categorification. the by a whole Moreover, finite the infinitethe number family identification of of of parameters, colored i.e. HOMFLY-PT poly- the quiver matrix infinite families of torus knots and twistfamily knots, of rational arborescent knots, knots and [22]. more generally Providing forto its a yet general large infinite proof more for general all branes knots,manifestations as and of well Calabi-Yau this manifolds, as correspondence generalization are revealed important toconvince date challenges. the are reader, already Nonetheless, they quite also all inspiring. nicely illustrate As the we ideas hoped of to Acknowledgments physical and quantum mathematics. representations of a quiver in question, and it is proven that theyas are found non-negative in integers. [17, 18 ], open topologicalin string the partition form function given of in the2) ( and quiver (5) generating can series be expressed (6) with an appropriate identification of parameters for Polish Science co-financed by theFund European (POIR.04.04.00-00-5C55/17-00). Union under the European Regional Development References uct expansion (5) is identified with (7). It follows that open BPS invariants (in general complicated) branes inHOMFLY-PT polynomials the (3) that resolved conifold, is identified thisbrane with is partition (6). functions a in Moreover, generating more one general function can manifolds; Calabi-Yau branes for of also a in identify simple colored toric class (6) threefolds of with without Aganagic-Vafa compact 4-cycles such an identification is found in [19]. combinations of motivic Donaldson-Thomas invariants PoS(ICHEP2020)042 , , Adv. , Phys. (2000), 577 (2019), 124 01 , AMS/IP Stud. 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