The u-plane Integral And Indefinite Theta Functions

The u-plane Integral And Indefinite Theta Functions Gregory Moore Rutgers University Simons Foundation, Sept. 8, 2017

Introduction 1/4 Much of this talk is review but I ll mention some new results with my student Iurii Nidaiev at the end: c. 1995: Mathematicians found mysterious relations of Donaldson invariants of 4-folds with b " # = 1 to modular forms: Göttsche; Göttsche &Zagier; Fintushel & Stern Can physics explain this??? 1988: Witten reformulates Donaldson theory in terms of supersymmetric Yang-Mills theory. 1994: Seiberg & Witten initiate the solution of the IR LEET of N=2 supersymmetric field theories.

Introduction 2/4 1994: Witten uses SW theory and his 1988 TFT interpretation to ``solve Donaldson theory in terms of the much more tractable SW invariants. 1997: Moore & Witten: Give a more systematic account, in the process explaining the mysterious `95 math results. Moreover, mock modular forms appear in the evaluation of the Donaldson invariants for CP " 2008: Malmendier & Ono, et. al. extend the connection of u-plane integrals for CP " to mock modular forms.

Introduction 3/4 2017: Very recently G. Korpas and J. Manschot revisited the u-plane integral, stressing the role of indefinite theta functions (and a modular completion.) Today: I will review some of these developments. But there has always been a nagging question first mentioned to me by Witten in the course of our work: Do other N=2 field theories, e.g. the superconformal ones, lead to new 4-manifold invariants?

Introduction 4/4 This was the motivation for my work in 1998 with Marcos Marino (also with Marino and Peradze) on u-plane integrals for other theories. Our conclusion was that Lagrangian theories will probably not lead to new four-manifold invariants, but superconformal points can still teach us new things about four-manifolds. At the end I ll present new results (with Iurii Nidaiev) on the partition function for the simplest AD theory on 4-manifolds with b " # > 0.

1 Introduction 2 3 4 Cambrian: Witten Reads Donaldson Carboniferous: SW Theory & u-plane Derivation Of Witten s Conjecture. Comments On The u-plane Integral 5 Other N=2 Theories 6 Holocene: AD3 Partition Function 6

Donaldson Invariants Of 4-folds X: Smooth, compact, X =, oriented, (π 3 X = 0) P X : Principal SU(2) or SO(3) bundle. Give X a metric and consider moduli space of instantons: M { A: F + F = 0} mod G Donaldson defines cohomology classes in M associated to points and surfaces in X : μ pt & μ S I pt l S K L μ pt l μ S K M Rational (often integer) valued. Independent of metric! smooth invariants of X.

Witten s Interpretation: Topologically Twisted SYM On X Consider N=2 SYM theory on X for gauge group G Choose an isomorphism of the principal SO 3 U bundle with that for Λ ",# (X) Set the R-symmetry connection = self-dual spin connection This defines topological twisting: All fermion fields and susy operators become differential forms. (In particular, the twisted theory is defined on non-spin manifolds.) One supersymmetry operator Q is scalar and Q " = 0 Formally: Correlation functions of Q-invariant operators localize to integrals over the finite-dimensional moduli spaces of G-ASD conns Witten s proposal: For G = SU 2 correlation functions of Q-invariant operators are the Donaldson polynomials.

Local Observables U Inv(g) U φ φ Ω p (adp C) H (X, Z) Descent formalism H (Fieldspace; Q) μ H (M) Localization identity g = su(2) U = Tr " φ " 8π " U S LTr(φF + ψ " ) n

Donaldson-Witten Partition t Z Is Function "u v#v n p, s = e w = Λ xy z {# } Λ"l#K p l l! r! I (pt l S K ) Mathai-Quillen & Atiyah-Jeffrey: Path integral formally localizes: M A/G Strategy: Evaluate in LEET: X compact Integrate over Q-invariant field configurations: This includes an integral over vacua on R z

1 Introduction 2 Cambrian: Witten Reads Donaldson 3 4 Carboniferous: SW Theory & u-plane Derivation Of Witten s Conjecture. Comments On The u-plane Integral 5 Other N=2 Theories 6 Holocene: AD3 Partition Function 11

Coulomb Branch Vacua On R z SU 2 U(1) by vev of adjoint Higgs field φ: Order parameter: u = U(φ) Coulomb branch: B = t C/W C adp L " O L x" Photon: Connection A on L U 1 VM: a, A, χ, ψ, η a : complex scalar field on R z : Do path integral of quantum fluctuations around a x = a + δa(x) What is the relation of a = a x to u? What are the couplings in the LEET for the U(1) VM?

Action L τ F # " + τ F x " + da Im τ d a + LEET: Constraints of N=2 SUSY General result on N=2 abelian gauge theory with t u 1 u 1 : Action determined by a family of Abelian varieties and an ``N=2 central charge function : A t C D Z: Γ C Γ H 3 A; Z dz, dz = 0 Duality Frame: Γ Γ K Γ a = Z α a I, Z β = F a τ a I, a

Seiberg-Witten Theory: 1/2 For G=SU(2) SYM A is a family of elliptic curves: E ² : y " = x " x u + Λz 4 x Z γ = λ u : Invariant cycle A: λ = dx y u C x u a u = λ º Choose B-cycle: τ a Action for LEET LEET breaks down at u = ±Λ " where Im τ 0

Seiberg-Witten Theory: 2/2 LEET breaks down because there are new massless fields associated to BPS states u = Λ " u = Λ " LEET u U w ½: U 1 I VM: (a I, A I, χ I, ψ I, η I ) + Charge 1 HM: M = q q, Similar story for LEET for u U xw ½

t Apply SW LEET To Z Is t Z Is "u v#v n p, s = e w = Λ xy z {# } Λ"l#K p l l! r! I (pt l S K ) t Z Is p, s = Z ² + Z w ½ + Z xw ½ u = Λ " u = Λ "

u-plane Integral Z ² Can be computed explicitly from QFT of LEET Vanishes if b " # > 1. When b " # = 1: { Z ² = da daá Â du " da Ã Δ Å e "u²#n½ Æ ² Ψ Δ = u Λ " u + Λ " Contact term: T u = È² È " E" τ 8 u Ψ: Sum over line bundles for the U(1) photon.

Photon Theta Function Ψ = e ÉÊË È² È ½ n Ì ½ } y x3 " ÍÎÍ Ï #Ð ½,Z 1 Ó ½ Ô ÍxÍ Ï e x ØÙ ØÚ n Í Ê e x ÑÒÁÍ Ì ½ x ÑÒÍ Ê ½ dτ daá (λ # + 1 4πy S # du da ) τ = x + i y 2λ p is an integral lift of ξ = w " P Metric dependent! λ = λ # + λ x

Metric Dependence b " # = 1 H " X has hyperbolic signature All metric dependence enters through period point ω ω = ω ω " = 1 Define: λ # = λ ω H " (S " S " ) II 3,3 e = PD(S " pt) f = PD pt S " 2 1 0 ω = 1 2 (v e + 1 v f ) - 1

Initial Comments On Z ² Z ² is a very subtle integral. It is similar to (sometimes same as) a Θ lift. The integrand is related to indefinite theta functions and for some manifolds it can be evaluated using mock modular forms. We will return to all these issues. But first let s finish writing down the full answer for the partition function.

Contributions From U w ½ Path integral for U 1 I VM + HM: General considerations imply: } SW λ e "Ñ Í Í Ï R Í p, S Í 3 " ç ½ #Ð ½,Z R Í p, S = Res[ da I 3# È Í " a I e 2pu#S2 T u #i du da D S λ C u λ 2 P u E u { ] d λ = 2λ " c " u = Λ " + Series a I 4 c " = 2χ + 3 σ C,P,E:Universal functions. In principle computable.

Deriving C,P,E From Wall-Crossing d Z dg ² = Tot deriv = du. + du. + du. øù û Z ² piecewise constant: Discontinuous jumps across walls: Δ û Z ² : W λ : λ # = 0 λ = λ p + H " X, Z Precisely matches formula of Göttsche! Δ ±w ½Z ² : W λ : λ # = 0 λ = 1 2 w "(X) + H " X, Z Δ w ½Z ² + ΔZ w ½ = 0 C u, P u, E u w ½ xw ½

Witten Conjecture t Z Is a formula for all X with b " # > 0. For b " # X > 1(and Seiberg-Witten simple type) the ``Witten conjecture : we derive p, s = 2 ½ x{ ý ( e 3 " n½ #"u } SW λ e "Ñ Í Í Ïe "n Í + e x3 " n½x"u } SW λ e "Ñ Í Í Ïe x" n Í ) χ þ = Í χ + σ 4 c " = 2χ + 3 σ Example: X = K3, w " P = 0: Z = sinh( n½ " + 2 p) Major success in Physical Mathematics. Í 24

Return To The u-plane Integral { Z ² = da daá Â du " da Ã Δ Å e "u²#n½ Æ ² Ψ Ψ = e ÉÊË È² È ½ n Ì ½ } y x3 " ÍÎÍ Ï #Ð ½,Z 1 Ó ½ Ô ÍxÍ Ï e x ØÙ ØÚ n Í Ê dτ daá (λ # + e x ÑÒÁÍ Ì ½ x ÑÒÍ Ê ½ i 4πy S # du da )

SW Curve And Modular Forms -1/2 E ² : y " = x " x u + Λz 4 x u C P p = 1 2 (1, 1 u) 2P p = (0,0) 3P p = 1 2 (1, 1 u) 4P p =

SW Curve And Modular Forms -2/2 E ² : y " = x " x u + Λz 4 x u C u-plane Modular curve for Γ p (4) u = 1 2 θ " z + θ y z θ " θ y " da du = 1 2 θ "θ y

Defining The Integral Near each cusp expand in q-puiseux series Ingredients involve expansions in q Ë % in arbitrarily high positive and negative powers. Í has the right period: Only powers of q Ë ' survive. Expand in power series in p l S K bound on negative powers Then near each cusp check all potentially divergent terms are killed by the integral over the phase.

Z ² Wants To Be A Total Derivative Ψ 3 = } d dτ Í Z ² = dτdτ H τ Ψ 3 E ρ Í + e x ÑÒÍ½ x n È² È #" Ñ ÍxÍ Ï ç ½ K E r = L e x"ñ ½ dt??? ρ + Í = yλ # i 4π y S du # da??? Must be independent of τ but can depend on λ, ω, τ. Choice is subtle: Want both convergence and modular invariance

Relation To Korpas & Manschot KM modify the 2-obsevable: U S U S + Q Z ² + Z ² + Ï = dτdτ H τ d dτ Θ,+,+ Ï Θ,+,+ Ï = } E r Í + E r Í + Ï e.. Í r + Í yλ # + 1 2π y S # Im du da Modular completion of indefinite theta function of Vigneras, Zwegers, Zagier Actually, there is no need to modify U S Just use ρ Í +. Gives a different modular completion.

Wall-Crossing Formula Gives a cleaner way to re-derive the old WCF for Z ² Z ² + Z ² + Ï = } dτ 1 H τ Θ4+,+Ï ²1u1 ² 2 u 1 {, Λ ", Λ " } for SU 2, N 7 = 0 Contribution of the cusp u 1 : } e Ñ ÍxÍ du da Ï ç ½ 1 sign λ ω sign λ ω p dτ 1 du Í 5 ½ " Δ Åe "u²#n½æ È² x n Í x Í½ e È 2 " q 1 Ï

Three Techniques For Explicit Evaluations Of Z ² Put ω p in a vanishing chamber Use unfolding/method of orbits Sometimes we can find a convergent modular invariant completion of the anti-derivative and write the integral as a total derivative.

Vanishing Chamber I p l S K = 0

Donaldson Invariants For S " S " H " X, Z II 3,3 Z ns = 0 Z ² = cnst. g h cot S e 2h Ï Z ² = cnst. g h cot S f 2h Ï g = u " 1 e "u²#n½ Æ h = da du

Donaldson Invariants For CP " No chambers. S = S # & Z ns = 0. Can choose w " P = 0 or 1 Case w " P = 0 (Moore & Witten) Zagier s weight (3/2,0) form for Γ p 4

Donaldson Invariants For CP " Act with differential operators to modify the shadow by powers of E 9 " Case w " P = 1 (Malmendier & Ono, 2008 et. seq. ) Also considered massless matter; Noted that the mock modular form for N 7 = 0 same as in Mathieu Moonshine.

What About Other N=2 Theories? Central Question: Given the successful application of N=2 SYM for SU(2) to the theory of 4-manifold invariants, are there interesting applications of OTHER N=2 field theories? Topological twisting just depends on SU 2 U symmetry and makes sense for any N=2 field theory. Z T e v#v n T The u-plane integral makes sense for ANY family of Seiberg-Witten curves.

N=2 Theories Lagrangian theories: Compact Lie group G, quaternionic representation R with G-invariant metric, τ p < H 1 u 7 =K1 m Lie(G 7 ) G 7 = Z G O R Class S: Theories associated to Hitchin systems on Riemann surfaces. Superconformal theories Couple to N=2 supergravity

Lagrangian Theories Superconformal Theories Class S Theories

SU(2) With Fundamental Hypers R = N 7 2 2 Spin 2N 7 G 7 Mass parameters m 7 C, f = 1,, N 7 Must take ξ = w " X Seiberg-Witten: E ² y " = x y + a " x " + a z x + a? Z ² Moore & Witten a @ : Polynomials in Λ, m 7, u has exactly the same expression as before but now, e.g. da/du depends on m 7 New ingredient: D has 2 + N 7 cusps u A : Δ = < u u 1 1

For example, for SU 2, N 7 = 1 with large mass:

Analog Of Witten Conjecture b " # > 1 "#C DE Z p; s; m 7 = } Z(p, s; m 7 ; u B ) BÎ3 Z(p, s; m 7 ; u B ) = α { βf } SW λ e "Ñ Í Í Ï Í R B (p, s) R B p, s = κ B { ý È² È X is SWST { ý # exp 2p ub + S " T u B i È² È B S λ u = u B + κ B q B + O(q B " ) Everything computable explicitly as functions of the masses from first order degeneration of the SW curve.

R B p, s = κ B { ý È² È { ý # exp 2p ub + S " T u B i È² È B S λ u = u B + κ B q B + O(q B " ) y " = 4x y g " u, m x g y u, m Assume a simple zero for Δ as u u B Choose local duality frame with a B 0 Nonvanishing period: È I and È I,J È² È² i : da B du " K g " K κ B g y " ²I g y ²I Δ L K ²I

Superconformal Points Consider N 7 = 1. At a critical point m = m two singularities u ± collide at u = u and the SW curve becomes a cusp: y " = x y [Argyres,Plesser,Seiberg,Witten] Two mutually nonlocal BPS states have vanishing mass: λ 0 Ë λ 0 ½ γ 3 γ " 0 Physically: No local Lagrangian for the LEET : Signals a nontrivial superconformal field theory Appears in the IR in the limit m m

AD3 From SU 2 N 7 = 1 SW curve in the scaling region: y " = x y " 3 Λ ºI x + u ºI " Λ ºI m m

AD3 Partition Function - 1 Work with Iurii Nidaiev The SU 2 N 7 = 1 u-plane integral has a nontrivial contribution from the scaling region u ± u lim Z ² Ldu duá lim Measure u, uá; m 0 Limit and integration commute except in an infinitesimal region around u Attribute the discrepancy to the contribution of the AD3 theory

AD3 Partition Function - 2 1. Limit m m exists. 2. The partition function is a sum over all Q-invariant field configurations. 3. Scaling region near u governed by AD3 theory. lim Z nv ",C DÎ3 ``contains the AD partition function Extract it from the scaling region. Our result: Z ºIy = lim w NJ p Z ºIyx7 É ºIyx7 É ² + Z ns Claim: This is the AD3 TFT on X for b " # > 0.

AD3 Partition Function: Evidence 1/2 1. Existence of limit is highly nontrivial. It follows from ``superconformal simple type sum rules : Theorem [MMP, 1998] If the superconformal simple type sum rules hold: a.) χ þ c " 3 0 b.) SW λ e "Ñ Í Í Ïλ @ = 0 Í 0 k χ þ c " 4 Then the limit m m exists It is now a rigorous theorem that SWST SCST

AD3 Partition Function: Evidence 2/2 2. For b " # > 0 recover the expected selection rule: U l U S K 0 only for 6l + r 5 = χ þ c " 5 3. For b " # = 1 correlation functions have continuous metric dependence a hallmark of a superconformal theory (M&W SU 2, N 7 = 4) Explicitly, if X of SWST and b " # > 1: Z ºIy = 1 r! K 3K " { Ky } e "Ñ Í Í ÏSW λ λ S Kx" (24 λ S " + S " ) Í r = χ þ c " 2

Vafa-Witten Partition Functions Discussions with G. Korpas & J. Manschot VW twist of N=4 SYM formally computes the ``Euler character of instanton moduli space. (Not really a topological invariant.) Physics suggests the partition function is modular (S-duality) and holomorphic. Surprise! Computations of Klyachko and Yoshioka show that the holomorphic generating function is only mock modular. This has never been properly derived from a physical argument. We are trying to fill that gap.