2007-2008: IAS school of Mathematics

6d perspective:
Rozansky-Witten theory with target
Higgs bundles

K-theoretic (“4d”) Coulomb branches6d perspective:
Rozansky-Witten theory with target

m w1w2

6d perspective:
Rozansky-Witten theory with target
wild
ramification

Fibonacci MTC

6d perspective:
Rozansky-Witten theory with target•at generic• spherical DAHA
•space of BFN triples (cf.

)

Quantum topology

gauge theory,
ramification, quasimaps, …

Geometric representation
theoryCoulomb branches,affine Grassmannians

Theorem: Using the R-matrix for Verma modules,
for all links of unknots (plumbings), torus links,
positive braid links, fibered knots up to 10 Xs,
and homogeneous braid links the two-variable series
is well defined and invariant under the required braid
moves (cf. Reidemeister moves).
S.G., D.Pei, P.Putrov, C.Vafaxycomplex
weights

S.G., C.Manolescu
S.Park (2020, 2021)
J.ChaeA.Gruen:Surgery formula:

Theorem [Lickorish, Wallace]:
Every connected oriented closed 3-manifold arises byperforming an integral Dehn surgery along a link in .

Surgery formula:orcf.

for WRT invariants

Conjecture:“conformal weight”

Character of a logarithmic Vertex Algebra
M.Cheng, S.Chun, F.Ferrari, S.G., S.Harrison
M.Cheng, S.Chun, B.Feigin, F.Ferrari, S.G., S.Harrison, D.Passaro
S.Sugimoto:* cf. sheaf counting on complex surfaces

Corollary (surprise #1):

John Cardy

Surprise #2:

BPS quiver

DT theory, curvecounting, Hall algebras,knot-quiver
correspondence, …

BPS spectrum

P.Kucharski, M.Reineke, M.Stosic, P.Sulkowski
T.Ekholm, A.Gruen, S.G., P.Kucharski, S.Park, M.Stosic, P.Sulkowski

Quantum groups

Kazhdan-Lusztig

Vertex Algebras
2d CFT

Integrable latticemodelsH.Bethe (1931):
A.Zamolodchikov, Al.Zamolodchikov (1979)
A.Zamolodchikov (1989)
Al.Zamolodchikov (1990)
F.Smirnov (1990)
N.Reshetikhin, F.Smirnov (1990)

Yangian symmetry,
Bethe ansatz equation, …

Rogers-Ramanujan

S.Kerov, A.Kirillov, N.Reshetikhin (1986)
A.Kirillov, N.Reshetikhin (1988)
R.Kedem, B.McCoy (1993)
R.Kedem, T.Klassen, B.McCoy, E.Melzer (1993):
S.Dasmahapatra, R.Kedem, T.Klassen, B.McCoy, E.Melzer (1993)
R.Kedem, B.McCoy, E.Melzer (1993)
A.Berkovich, B.McCoy, A.Schilling, S.Warnaar (1997)
E. Frenkel, A. Szenes (1993):W.Nahm, A.Recknagel, M.Terhoeven (1993)
B.Feigin, E.Feigin, M.Jimbo, T.Miwa, E.Mukhin (2009)

M.Finkelberg (ICM - 2018): Zastava characters, relation to monopole formulae?

should admit a definition via
Conjecture:moduli spaces in gauge theory *S.G., D.Pei, P.Putrov, C.Vafa

Kapustin-WittenPDEs* Compactification and choice of chamber: K-theory /multiplicative / caloron version
Early clues: S.Chun, S.G., S.Park, N.Sopenko
More detailed analysis: S.G., P.-S.Hsin, D.Pei (to appear soon)should admit a definition via
Conjecture:moduli spaces in curve counting
S.G., D.Pei, P.Putrov, C.Vafa
genus g, with n boundary components

Some evidence: L.Diogo, T.Ekholm:

3d-3dK-theory
quasimaps

Target:Rozansky-Witten
theoryforcomplex

Domain:3d N = 2

3d N = 4

Complex
coadjoint orbit

ramification
P.Kronheimer
C. Simpson
H. Konno
H. Nakajima:S.G., E.Witten

Equivariant cohomology of based quasimaps toflags produces a universal Verma module for
A.Braverman, M.Finkelberg
B.Feigin, M.Finkelberg, I.Frenkel, L.Rybnikov
M.Bullimore, T.Dimofte, D.Gaiotto, J.Hilburn, H.Kim
J.Hilburn, J.Kamnitzer, A.Weekes
Z.Zhou:G.Naisse, P.Vaz
R.Rouquiercomplexweights

circle action on the domain
highest weightaction on the target3d-3dK-theory
quasimaps

Target:Rozansky-Witten
theoryforcomplex

Domain:3d N = 2

3d N = 4

3d TQFT

commutative ringA raviolo would be betterline operators

3d TQFT
associated toat generic q

Hilbert series ofA raviolo would be betterfor3d TQFT
associated toat generic q

Hilbert series offorxaction (tri-holomorphic)t“stretching fibers” (holomorphic)q“loop rotation”

Expect:“”forinfinite-rank vector bundle (smooth) overaffine Grassmannian

formal disk
formal punctured diskx

action (tri-holomorphic)t“stretching fibers” (holomorphic)q“loop rotation”

Expect:“”forinfinite-rank vector bundle (smooth) overaffine Grassmannian
S.G., P.-S.Hsin, H.Nakajima, S.Park, D.Pei, N.Sopenko

Rozansky-Witten TQFT with target“Rozansky-Witten invariants via formal geometry”“Rozansky-Witten invariants via Atiyah classes”

Examples: Lens spaces

The raviolo subvariety plays the role of the “affineGrassmannian Steinberg variety”
Theorem:admits an associative convolution product,commutative at
A.Braverman, M.Finkelberg, H.Nakajima
Theorem: with the convolution product,
N.Chriss, V.Ginzburg
G.Lusztig

cf.

R.Bezrukavnikov, M.Finkelberg, I.Mirkovićquantized K-theoretic Coulomb branches:
spherical nil-DAHA
M.Finkelberg, A.Tsymbaliuk

forM.Varagnolo, E.Vasserot
A.Braverman, M.Finkelberg, H.Nakajima:
S.G., P.Koroteev, S.Nawata, D.Pei, I.Saberi

cf. geometric Satake correspondenceQuestions:• Categorification of spherical DAHA?
•cf. S.G., E.Witten• Relations between

andcf. M.Finkelberg

andat generic ?
Mirror symmetry, Hikita conjecture, ...
• R-matrix for quantum toroidal algebras

A.Negut:• Perfect crystal bases of Verma modules
S.-J.Kang, M.Kashiwara, K.Misra
S.-J.Kang, M.Kashiwara, K.Misra, T.Miwa, T.Nakashima, A.Nakayashikitopology
enumerative
geometrymathematicalphysicsgauge
theoryvertex algebra

quantum groups