Supergroup Chern-Simons Theory, Q-Series, and Resurgence

Supergroup Chern-Simons theory, q-series, and resurgence

200X.XXXXX w/ Francesca Ferrari

Pavel Putrov

ICTP, Trieste

July 27, 2020

1 / 19 Plan/Summary

I Proposal for the value of the index Zˆsl(m|n)[M 3] := Tr(−1)F qL0 of a system of M5-branes related by dualities to SU(M|N) CS on M 3 (L0 generates rotations of both C’s): ∗ 3 1 M-theory T M × C × C × Stime 3 1 N M5’s M × C × {0} × Stime 3 1 M M5’s M × {0} × C × Stime for a class of closed oriented 3-manifolds M 3. I Examples and properties of these q-series (resurgence). I Relation to the quantum invariants of [Costatino–Geer–Patureau-Mirand ’14] from non-semisimple category H of representations of Uq (sl(M|N)) (with modiﬁed trace) H when q is a root of unity (Uq (sl(2|1)) case worked out in [Ha ’18]).

Remark: The case of SU(N) and U(1|1) was considered in [Gukov-PP-Vafa ’16, Gukov-Mari˜no-PP’16, Gukov-Pei-PP-Vafa ’17, Gukov-Manolescu ’19, . . . ] 2 / 19 Brane realization of SU(N) CS

[Witten,. . . ]

N D3 N D4 N M5 on M3

a b £2 D U(1)q S',T on M S 1 on M3 R+ 3 R+ ' b flat £ £ £ S£1 connection NS5 D6

Coupling constant of the SU(N) SYM living on D3 branes is the analytically continued CS coupling: −k = τ. The fugacity 2πi q = e k .

3 / 19 Brane realization of SU(M|N) CS

[Vafa,Gaiotto-Witten,Witten-Mikhaylov,. . . ]

N D3 M D3 N D4 M D4 M3 N M5 M M5 a a0 b b0 £ S',T b M M ' b0 3 R+ 3 R+ 1 1 £1 £ £ M3 R+ S M R S S £ £ 3£ +£ NS5 D6

Remark: SU(N|M) Chern-Simons also appears in a relation to ABJM theory on S3 [Drukker-Trancanelli ’09, Mari˜no-Putrov’09]

4 / 19 U(1|1) Chern-Simons theory

U(1|1) CS = Reidemeister-Turaev torsion = SW invariants [Rozansky-Saleur,Meng-Taubes,Mikhaylov,. . . ] categoriﬁcation: Monopole/Heegaard Floer homology

Knots 3-manifolds 4-manifolds Group invariant categoriﬁcation invariant categoriﬁcation invariant U(1|1) Alexander knot Floer 3d SW Monopole Floer 4d SW polynomial homology invariants homology invariants SU(2) Jones Khovanov WRT ? ??? polynomial homology invariant

U(1|1) ⊂ SU(2|1) ⊃ SU(2)

5 / 19 Plumbed 3-manifolds

plumbing graph ¡ M3( ¡) = Dehn surgery on a a4 4 a9 a3 a9 a 3 a a a5 a6 5 6

a2 a7 a2 a7 a8 a1 a1 a8

1) cut out tubular neighborhoods of the link components a 2) glue back in solid tori using T i S ∈ SL(2, Z) transformations of the boundari tori. L - number of vertices (link components)   1, i1, i2 connected, L × L linking matrix Mi1i2 = ai, i1 = i2 = i,  0, otherwise.

∼ L L H1(M3) = Z /MZ 6 / 19 SU(2) homological blocks for plumbed 3-manifolds

plumbing graph ¡ M3( ¡) = Dehn surgery on a a4 4 a9 a3 a9 a 3 a a a5 a6 5 6

a2 a7 a2 a7 a8 a1 a1 a8

Homological blocks (“Zed-hats”) for SU(2) [Gukov-Pei-PP-Vafa ’17] (higher rank generalization [Park ’19]):

P sl(2) 3L+ i Mii ˆ − 4 Zb (q) = q × Z dz Y i 2−deg(i) −2M ∆b v.p. (zi − 1/zi) ·Θb (z; q) ∈ q Z[[q]] 2πizi i ∈ Vertices |zi|=1 T −1 L X ` M ` Y ` −2M − 4 i Θb (z; q) := q zi `∈2MZL+b i=1 ˆsl(2) M is “weakly negative deﬁnite”, Zb 6= 0 for c 3 ∼ L L b ∈ Spin (M ) = {b ∈ Z /2MZ | bi = deg(i) mod 2} [Gukov-Manolescu ’19] 7 / 19 SU(2|1) homological blocks for plumbed 3-manifolds

plumbing graph ¡ M3( ¡) = Dehn surgery on a a4 4 Assume the plumbing graph is a a9 a3 a9 a 3 a a 3 3 a5 a6 5 6 tree and M = M (Γ) is a rational

a2 a7 a2 a7 a8 a1 homology sphere (b1 = 0) a1 a8 Z 2−deg(i) ˆsl(2|1) Y dyi dzi yi − zi Zb,c = × 2πiyi 2πizi (1 − yi)(1 − zi) Γ i∈Vert X mT M −1n Y mj nj q zj yj n=b mod M L j Z L m=c mod MZ L L ∼ 3 b, c ∈ Z /MZ = H1(M , Z) The integral means taking the constant term after expanding the rational function in y and z in the first line in the chamber corresponding to the choice of the contour Γ. Existence/uniqueness of Γ so that the result is a well defined ∆ T −1 element of q bc Z[[q]]?(∆bc = b M c = `k(b, c) mod 1) 8 / 19 SU(2|1) homological blocks for plumbed 3-manifolds (some technical details) Z 2−deg(i) ˆsl(2|1) Y dyi dzi yi − zi Zb,c = × 2πiyi 2πizi (1 − yi)(1 − zi) Γ i∈Vert X mT M −1n Y mj nj q zj yj n=b mod M L j Z L m=c mod MZ 1) The contour Γ such that the result is a well defined element of ∆ −1 q bc Z[[q]] exist iff −M is “weakly copositive”. Namely, there exists a vector αi = ±1, i ∈ Vert|deg6=2 such that the submatrix −1 Xij := −Mij αiαj , i, j ∈ Vert|deg>2, P is copositive (vi ≥ 0, ∀i implies i,j vivj Xij ≥ 0) and

−1 αiαj Mij ≤ 0, ∀i ∈ Vert|deg=1, j ∈ Vert|deg6=2, j 6= i.

2) If such Γ exist there are only two choices (for a generic plumbing), giving the same result. The choice of the contour is speciﬁed by the vector α above. 9 / 19 Invariance under Kirby moves

0 ∼ 0 3d Kirby moves Γ ∼ Γ moves s.t. M3(Γ) = M3(Γ ):

a 0 a2 a 1 1 a2 1 a 1 1 1 1§ § § 1§ §

a1+a2 a a1 a2 1

ˆsl(2|1) The “weak copositivity” condition on M and the q-series Zb,c are invariant.

10 / 19 Examples: 3-sphere

3 3 3 I M = S : H1(S , Z) = 0,

X qn X Zˆsl(2|1) = −1 + 2 = −1 + 2 d(m)qm = 1 − qn n≥1 m = −1 + 2(q + 2q2 + 2q3 + 3q4 + 2q5 + 4q6 + 2q7 + ...)

where d(m) is the number of divisors of m (Lambert series).

11 / 19 Examples: Lens spaces 3 ∼ 3 3 ∼ I M = L(p, 1) = S /Zp: H1(S , Z) = Zp,

ˆsl(2|1) X m/p Zbc = constbc + 2 d(m; p, b, c)q m where d(m; p, b, c) is the number of positive integer pairs (r, s) satisfying   r = b mod p s = c mod p  rs = m

Remarks: 1 1) Zˆsl(n)[L(p, 1)] are polynomials in q p 2) d(m; p, b, c) coincides with Euler characteristic of the moduli space of m SO(3) instantons on L(p, 1) × R propagating between ﬂat connections labelled by b ± c [Austin ’90]. 12 / 19 Examples: Seifert 3-manifolds w/ 3 exceptional ﬁber

ˆsl(n) For Seifet manifolds with 3 exceptional ﬁbers Zbc are linear combintations of the q-series of the form 2 X qαm +βm+γ F (q; α, β, γ; A, B) := . 1 − qAm+B m≥0 For example for Poincare homology sphere

-2 -2 -2 -2 -2 -2 -2

-2

Zˆsl(2|1) = −1 + 2q2 + 2q3 + 4q4 + 4q5 + 6q6 + 4q7 + ... 13 / 19 Resurgence 3 Assume H1(M , Z) = 0 for technical simplicity. Consider asymptotic expansion in ~ := − log q = −2πiτ → 0 ˆ X n Z = singular + cn~ n≥0 and deﬁne the Borel transform X cn B(ξ) := ξn Γ(n + 1) n≥0

If Zˆ originates from a “path-integral of analytically continued CS theory” (for ordinary compact Lie groups studied by [Witten,Kontsevich,Costin-Garoufalidis,Gukov-Mariño-PP,. . . ]) we expect the series for B(ξ) to have finite radius of convergence, to be analytically continuable beyond that, with possible singularities at ξ = 4π2S where S mod 1 is the CS invariant of a flat connection (for the complexified even subgroup). Is it the case for Zˆsl(2|1)? 14 / 19 Resurgence for S3

For S3 the asymptotic expansion is explicitly known (e.g. [Bettin-Conrey ’13]):

X qn Zˆsl(2|1) = −1 + 2 ∼ 1 − qn n≥1 −1 + 2πi/ log(i ) − γ X B2 ~ − 2 ~ − 2 2n 2n−1 2 (2n)! (2n) ~ ~ n≥1

2 The Borel transform B(ξ) has singularities at ξ = 4π S, ∀S ∈ Z (lifts of CS functional of trivial ﬂat connection).

sl(n) 3 Remark: Zˆ [S ] is an entire function in ~.

15 / 19 Resurgence for Poincare homology sphere

For Seifet manifolds with 3 exceptional ﬁbers it is suﬃcient to analyze asymptotic expansion of

2 X qαm +βm+γ F (q; α, β, γ; A, B) = . 1 − qAm+B m≥0

for general parameters α, β, γ, A, B. This can be done using Euler-Maclaurin summation formula. Its Borel tranform has 2 K2 (2Kα/A)2 singularities at ξ = 4π S, S = 4α and S = − 4α , K ∈ Z. Example: M 3 =Poincare homology sphere. Many “fake” singularities cancel out between contributions from diﬀerent F ’s. 1 49 The singularities with S = 0, 120 , 120 mod 1 do not. They correspond to CS functional of SU(2) ⊂ SU(2|1) ﬂat connections.

16 / 19 H Quantum invariants associated to Uq (sl(2|1)) Using the standard Witten-Reshetikhin-Turaev approach to topological invariants for quantum sl(2|1) leads to trivial invariants as quantum dimensions vanish. Instead one should use notion of modiﬁed trace [Geer–Patureau-Mirand–Turaev ’09, Geer–Kujawa–Patureau-Mirand ’11] and construction of [Costantino–Geer–Patureau-Mirand ’14] applied to the non-semisiple category of representations (satisfying certain H 4πi conditions) of Uq (sl(2|1)) at q = e k , odd k [Ha ’18]. The result is an invariant 3 Nk(M , ω) of closed oriented 3-manifolds M 3 colored by 1 ω ∈ H1(M 3, / × / ) \ H1(M 3, / × / ) ∪ ... C Z C Z 2Z Z C Z obtained via surgery on a framed link (For each link components one sums over representations with weights in C × C equal to ω([meridian]) modulo Z × Z) 17 / 19 Relation between the q-series and “non-semi-simple quantum invariants”

Conjecture: for a rational homology sphere M 3

3 T (2ω1) Nk(M , ω) = 3 × `|H1(M , Z)| X 2πik·`k(β,γ)+4πi(b−ω2)(γ)+2πi(c−(ω1+ω2))(β) sl(2|1) × e ·Zˆ | 4πi b,c q→e k 3 β,γ∈H1(M ,Z) 1 3 b,c∈H (M ,C/Z)

1 3 where ω ∈ H (M , C/Z × C/Z) \ ... (as before) and T is the Reidemeister torsion. 1 3 1 3 Remark: For a rational homology sphere H (M , C/Z) =∼ H (M , Q/Z) and the linking pairing

3 3 `k : H1(M , Z) ⊗ H1(M , Z) → Q/Z 3 ∼ 1 3 provides an isomorphism H1(M , Z) = H (M , C/Z).

18 / 19 Some open questions and future directions

I In simple examples all coeﬃcients are positive (possibly except the constant term). Is it true in general? Explanation?

I Categoriﬁcation of the q-series similar to U(1|1) case? Might be easier to understand than for SU(2). psl(n|n) 3 I Relation between Zˆ [M ] and PSU(N) instanton 3 counting on M × R ? I Manifolds with (torus) boundaries.

19 / 19