2000 character limit reached
Coloured invariants of torus knots, $\mathcal{W}$ algebras, and relative asymptotic weight multiplicities (2401.15230v2)
Published 26 Jan 2024 in math.QA, math-ph, math.GT, math.MP, and math.RT
Abstract: We study coloured invariants of torus knots $T(p,p')$ (where $p,p'$ are coprime positive integers). When the colouring Lie algebra is simply-laced, and when $p,p'\geq h\vee$, we use the representation theory of the corresponding principal affine $\mathcal{W}$ algebras to understand the trailing monomials of the coloured invariants. In these cases, we show that the appropriate limits of the renormalized invariants are equal to the characters of certain $\mathcal{W}$ algebra modules (up to some factors). This result on limits rests on a purely Lie-algebraic conjecture on asymptotic weight multiplicities which we verify in some examples.
- G. E. Andrews. The theory of partitions. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1998. Reprint of the 1976 original.
- T. Arakawa. Vanishing of cohomology associated to quantized Drinfeld-Sokolov reduction. Int. Math. Res. Not., (15):730–767, 2004.
- T. Arakawa. Representation theory of 𝒲𝒲\mathscr{W}script_W-algebras. Invent. Math., 169(2):219–320, 2007.
- T. Arakawa. Two-sided BGG resolutions of admissible representations. Represent. Theory, 18:183–222, 2014.
- T. Arakawa. Associated varieties of modules over Kac-Moody algebras and C2subscript𝐶2C_{2}italic_C start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT-cofiniteness of W𝑊Witalic_W-algebras. Int. Math. Res. Not. IMRN, (22):11605–11666, 2015.
- T. Arakawa. Rationality of W𝑊Witalic_W-algebras: principal nilpotent cases. Ann. of Math. (2), 182(2):565–604, 2015.
- C. Armond and O. T. Dasbach. Rogers-Ramanujan type identities and the head and tail of the colored Jones polynomial. arXiv:1106.3948 [math.GT].
- A vector partition function for the multiplicities of 𝔰𝔩kℂ𝔰subscript𝔩𝑘ℂ\mathfrak{sl}_{k}\mathbb{C}fraktur_s fraktur_l start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT blackboard_C. J. Algebra, 278(1):251–293, 2004.
- Higher depth false modular forms. Commun. Contemp. Math., 25(7):Paper No. 2250043, 53, 2023.
- M. Brion and M. Vergne. Residue formulae, vector partition functions and lattice points in rational polytopes. J. Amer. Math. Soc., 10(4):797–833, 1997.
- Sphere packings, lattices and groups, volume 290 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, New York, 1988. With contributions by E. Bannai, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov.
- Characters and fusion rules for W𝑊Witalic_W-algebras via quantized Drinfel’d-Sokolov reduction. Comm. Math. Phys., 147(2):295–328, 1992.
- W. Fulton and J. Harris. Representation theory, volume 129 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1991. A first course, Readings in Mathematics.
- S. Garoufalidis and T. T. Q. Lê. Nahm sums, stability and the colored Jones polynomial. Res. Math. Sci., 2:Art. 1, 55, 2015.
- The SL3subscriptSL3\mathrm{SL}_{3}roman_SL start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT colored Jones polynomial of the trefoil. Proc. Amer. Math. Soc., 141(6):2209–2220, 2013.
- S. Garoufalidis and T. Vuong. A stability conjecture for the colored Jones polynomial. Topology Proc., 49:211–249, 2017.
- K. Hikami and S. Sugimoto. Torus links T2s,2tsubscript𝑇2𝑠2𝑡T_{2s,2t}italic_T start_POSTSUBSCRIPT 2 italic_s , 2 italic_t end_POSTSUBSCRIPT and (s,t)𝑠𝑡(s,t)( italic_s , italic_t )-log VOA, to appear.
- J. E. Humphreys. Introduction to Lie algebras and representation theory, volume 9 of Graduate Texts in Mathematics. Springer-Verlag, New York-Berlin, 1978. Second printing, revised.
- V. G. Kac. Infinite-dimensional Lie algebras. Cambridge University Press, Cambridge, third edition, 1990.
- V. G. Kac and M. Wakimoto. Modular invariant representations of infinite-dimensional Lie algebras and superalgebras. Proc. Nat. Acad. Sci. U.S.A., 85(14):4956–4960, 1988.
- V. G. Kac and M. Wakimoto. On rationality of W𝑊Witalic_W-algebras. Transform. Groups, 13(3-4):671–713, 2008.
- S. Kanade. Characters of logarithmic vertex operator algebras and coloured invariants of torus links. arXiv:2305.17543 [math.QA].
- S. Kanade. Coloured 𝔰𝔩r𝔰subscript𝔩𝑟\mathfrak{sl}_{r}fraktur_s fraktur_l start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT invariants of torus knots and characters of 𝒲rsubscript𝒲𝑟{\mathcal{W}}_{r}caligraphic_W start_POSTSUBSCRIPT italic_r end_POSTSUBSCRIPT algebras. Lett. Math. Phys., 113(1):Paper No. 5, 21, 2023.
- S. Kanade and A. Ayyer. In progress.
- H. R. Morton. The coloured Jones function and Alexander polynomial for torus knots. Math. Proc. Cambridge Philos. Soc., 117(1):129–135, 1995.
- Lie groups and algebraic groups. Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, 1990. Translated from the Russian and with a preface by D. A. Leites.
- M. Rosso and V. Jones. On the invariants of torus knots derived from quantum groups. J. Knot Theory Ramifications, 2(1):97–112, 1993.
- B. Sturmfels. On vector partition functions. J. Combin. Theory Ser. A, 72(2):302–309, 1995.
- J. Tarski. Partition function for certain simple Lie algebras. J. Mathematical Phys., 4:569–574, 1963.
- M. Wakimoto. Lectures on infinite-dimensional Lie algebra. World Scientific Publishing Co., Inc., River Edge, NJ, 2001.
- W. Yuasa. The zero stability for the one-row colored 𝔰𝔩3𝔰subscript𝔩3\mathfrak{sl}_{3}fraktur_s fraktur_l start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT Jones polynomial. arXiv:2007.15621 [math.GT].
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