European String Workshop, April 12, 2018 Partition Functions Of Twisted Supersymmetric Gauge Theories On Four-Manifolds Via u-Plane Integrals Gregory Moore Rutgers University Mostly review; new work with I. Nidaiev; work in progress with Jan Manschot. Now, this subject is not incredibly popular (although I think it is a good one), So progress takes place on geological time scales. So, here’s the outline of the talk…. European String Workshop, April 12, 2018

1 Introduction 2 Cambrian: Witten Reads Donaldson Carboniferous: SW Theory & u-plane Derivation Of Witten’s Conjecture. 3 Holocene: Three New Apps 4 5 Holomorphic Anomaly & Continuous Metric Dep. 6 Back To The Future

Donaldson Invariants Of 4-folds 𝑋: Smooth, compact, oriented, 𝜕𝑋=∅ , ( 𝜋 1 𝑋 =0) 𝑃→𝑋 : Principal SO(3) bundle. 𝑋 has metric 𝑔 𝜇𝜈 . 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 𝑝 & 𝜇 𝑆 ℘ 𝐷 𝑝 ℓ 𝑆 𝑟 ≔ ℳ 𝜇 𝑝 ℓ 𝜇 𝑆 𝑟 Say: Comparison with Freedman’s topological classification of four-manifolds shows there is a huge difference between the topological and smooth categories. Very deep. Independent of metric! ⇒ smooth invariants of 𝑋. Combined with Freedman theorem: Spectacular!

Witten’s Interpretation: Topologically Twisted SYM On X Consider 𝒩= 2 SYM theory on X for gauge group G Witten’s ``topological twisting’’: Couple to special external gauge fields for certain global symmetries. Result: Fermion fields and susy operators are differential forms; The twisted theory is defined on non-spin manifolds. And there is a scalar susy operator with 𝑄 2 =0 Formally: Correlation functions of operators in 𝑄-coho. localize to integrals over the moduli spaces of G-ASD connections (or generalizations thereof). Witten’s proposal: For 𝐺=𝑆𝑈 2 correlation functions of the special operators are the Donaldson polynomials.

Local Observables 𝜙∈ Ω 0 (𝑎𝑑𝑃⊗ℂ) ℘∈𝐼𝑛𝑣(𝔤) ⇒ 𝑈=℘ 𝜙 𝐻 ∗ (𝑋,ℂ) ℘∈𝐼𝑛𝑣(𝔤) ⇒ 𝑈=℘ 𝜙 𝜙∈ Ω 0 (𝑎𝑑𝑃⊗ℂ) Descent formalism 𝐻 ∗ (𝑋,ℂ) 𝐻 ∗ (𝐹𝑖𝑒𝑙𝑑𝑠𝑝𝑎𝑐𝑒;𝑄) 𝜇 Localization identity 𝐻 ∗ (ℳ) 𝑈(𝑝)=𝑇 𝑟 2 𝜙 2 8 𝜋 2 𝑈 𝑆 ∼ 𝑆 𝑇𝑟(𝜙𝐹+ 𝜓 2 ) 𝔤=𝔰𝔲(2)

Donaldson-Witten Partition Function 𝑍 𝐷𝑊 𝑝,𝑠 = 𝑒 𝑈(𝑝)+𝑈 𝑆 Λ 𝑋 𝑇𝑟 𝐹 2 =8 𝜋 2 𝑘 = 𝑘 Λ 1 2 dim ℳ 𝑘 ℳ 𝑘 𝑒 𝜇 𝑝 +𝜇 𝑆 Strategy: Evaluate in LEET ⇒ Witten (1994) introduces the Seiberg-Witten invariants. Major success in Physical Mathematics.

What About Other N=2 Theories? Natural 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 𝑅 symmetry and makes sense for any 𝒩=2 theory. 𝑍 𝒯 ≔ 𝑒 𝑈(𝑝)+𝑈 𝑆 𝒯 Also an interesting exercise in QFT to compute correlation functions of nontrivial theories in 4d.

SU(2) With Matter 𝑀∈Γ( 𝑆 + ⊗ 𝐸 ⊕ 𝑁 𝑓𝑙 ) 𝒩= 2 ∗ :ℛ=(𝑎𝑑 𝑗 ℂ ⊕𝑎𝑑 𝑗 ℂ ∗ ) ℛ= 2 ⊕ 𝑁 𝑓𝑙 Mass parameters 𝑚 𝑓 ∈ℂ, 𝑓=1, …, 𝑁 𝑓𝑙 𝑀∈Γ( 𝑆 + ⊗ 𝐸 ⊕ 𝑁 𝑓𝑙 ) w 2 (P)= 𝑤 2 𝑋 𝒩= 2 ∗ :ℛ=(𝑎𝑑 𝑗 ℂ ⊕𝑎𝑑 𝑗 ℂ ∗ ) 𝑚∈ℂ 𝑀∈Γ( 𝑆 + ⊗ 𝐿 1 2 ⊗𝑎𝑑𝑃) 𝑤 2 𝐿 = 𝑤 2 𝑃 [Labastida-Marino ‘98]

UV Interpretation 𝐹 + =𝒟(𝑀, 𝑀 ) 𝛾⋅𝐷 𝑀=0 But now ℳ 𝑘 : 𝑍 𝑝,𝑆 = 𝑒 𝑈(𝑝)+𝑈 𝑆 𝒯 = 𝑘 Λ 1 2 dim ℳ 𝑘 ℳ 𝑘 𝑒 𝜇 𝑝 +𝜇 𝑆 But now ℳ 𝑘 : is the moduli space of: 𝐹 + =𝒟(𝑀, 𝑀 ) 𝛾⋅𝐷 𝑀=0 U(1) case: Seiberg-Witten equations. ``Generalized monopole equations’’ [Labastida-Marino; Losev-Shatashvili-Nekrasov]

Cambrian: Witten Reads Donaldson 1 Introduction 2 Cambrian: Witten Reads Donaldson 3 Carboniferous: SW Theory & u-plane Derivation Of Witten’s Conjecture. 4 Holocene: Three New Apps Now we turn to the carboniferous, the period when many of the resources we use today with such gay abandon were created. 5 Holomorphic Anomaly & Continuous Metric Dep. 6 Back To The Future

Coulomb Branch Vacua On ℝ 4 𝑆𝑈 2 →𝑈(1) by vev of adjoint Higgs field 𝜙: Order parameter: 𝑢= 𝑈 𝑝 ∈ℂ Coulomb branch: ℬ=𝔱⊗ℂ/𝑊≅ℂ 𝑎𝑑𝑃→ 𝐿 2 ⊕𝒪⊕ 𝐿 −2 Photon: Connection A on L 𝑈 1 𝑉𝑀: 𝑎, 𝐴, 𝜒, 𝜓,𝜂 𝑎: complex scalar field on ℝ 4 : Compute couplings in U(1) LEET. Then compute path integral with this action. Then integrate over vacua.

Seiberg-Witten Theory: 1/2 For G=SU(2) SYM coupled to matter the LEET can be deduced from a holomorphic family of elliptic curves with differential: 𝑑𝜆 𝑑𝑢 = 𝑑𝑥 𝑦 𝐸 𝑢 : 𝑦 2 =4 𝑥 3 − 𝑔 2 𝑥− 𝑔 3 𝑢∈ℂ 𝑔 2 , 𝑔 3 are polynomials in u, masses, Λ, modular functions of 𝜏 0 Δ≔4 𝑔 2 3 −27 𝑔 3 2 : polynomial in u Δ 𝑢 𝑗 =0 : Discriminant locus

𝐸 𝑢 𝑗 𝐸 𝑢 𝑢 𝑗 𝑢

Examples 𝑁 𝑓𝑙 =0: 𝑦 2 =(𝑥−𝑢)(𝑥− Λ 2 )(𝑥+ Λ 2 ) 𝑁 𝑓𝑙 =0: 𝑦 2 =(𝑥−𝑢)(𝑥− Λ 2 )(𝑥+ Λ 2 ) 𝒩= 2 ∗ : 𝑦 2 = 𝑖=1 3 𝑥− 𝛼 𝑖 𝛼 𝑖 =𝑢 𝑒 𝑖 𝜏 0 + 𝑚 2 4 𝑒 𝑖 𝜏 0 2 𝑁 𝑓𝑙 =4: 𝑦 2 = 𝑊 1 𝑊 2 𝑊 3 + 𝜂 12 𝑖=1 3 𝑅 4 𝑖 𝑊 𝑖 + 𝜂 24 𝑅 6 𝑊 𝑖 =𝑥−𝑢 𝑒 𝑖 𝜏 0 − 𝑅 2 𝑒 𝑖 𝜏 0 2 𝑅 2 ∼𝑇 𝑟 8 𝑖 𝔪 2 𝑅 4 𝑖 ∼𝑇 𝑟 8 𝑖 𝔪 4 𝑅 6 ∼𝑇 𝑟 8 𝑖 𝔪 6 Δ: 6th order polynomial in 𝑢 with ∼3× 10 3 terms

Local System Of Charges Electro-mag. charge lattice: Γ 𝑢 = 𝐻 1 ( 𝐸 𝑢 ;ℤ) has nontrivial monodromy around discriminant locus: Δ 𝑢 𝑗 =0 LEET: Requires choosing a duality frame: Γ 𝑢 ≅ℤ 𝛾 𝑒 ⊕ℤ 𝛾 𝑚 ⇒𝜏 𝑢 𝑆∼ 𝑋 𝜏 𝐹 + 2 +𝜏 𝐹 − 2 +⋯

LEET breaks down at 𝑢= 𝑢 𝑗 where 𝐼𝑚 𝜏 →0 𝑁 𝑓𝑙 =0 𝑁 𝑓𝑙 =1 𝒩= 2 ∗ 𝑁 𝑓𝑙 =4 LEET breaks down at 𝑢= 𝑢 𝑗 where 𝐼𝑚 𝜏 →0

Seiberg-Witten Theory: 2/2 LEET breaks down because there are new massless fields associated to BPS states 𝑢= 𝑢 𝑗 𝑈 1 𝑗 𝑉𝑀: 𝑎,𝐴,𝜒,𝜓,𝜂 𝑗 LEET u∈ 𝒰 𝑗 : + Charge 1 HM: 𝑀=𝑞⊕ 𝑞 ∗ , ⋯

Evaluate 𝑍 𝐷𝑊 Using LEET 𝑍 𝐷𝑊 𝑝,𝑆 = 𝑒 𝑈 𝑝 +𝑈 𝑆 Λ = 𝑘 Λ 0 1 2 dim ℳ 𝑘 ℳ 𝑘 𝑒 𝜇 𝑝 +𝜇 𝑆 𝑍 𝐷𝑊 = 𝑍 𝑢 + 𝑢 𝑗 𝑍 𝑗 𝑆𝑊 𝑢= Λ 2 𝑢=− Λ 2

u-Plane Integral 𝑍 𝑢 Can be computed explicitly from QFT of LEET Vanishes if 𝑏 2 + >1 . When 𝑏 2 + =1: 𝑍 𝑢 =∫𝑑u d u ℋ Ψ ℋ is holomorphic and metric-independent Ψ: Sum over line bundles for the U(1) photon. (Remnant of sum over SU(2) gauge bundles.) Ψ∼ 𝜆= 𝑐 1 𝐿 𝑒 −𝑖 𝜋 𝜏 𝑢 𝜆 + 2 −𝑖 𝜋𝜏 𝑢 𝜆 − 2 NOT holomorphic and metric- DEPENDENT

Initial Comments On 𝑍 𝑢 𝑍 𝑢 is a very subtle integral. It requires careful regularization and definition. (It is also related to integrals from number theory such as `` Θ−lifts’’ and mock modular forms…) But first let’s finish writing down the full answer for the partition function.

Contributions From 𝒰 𝑗 𝑐 2 =2𝜒+3 𝜎 Path integral for 𝑈 1 𝑗 VM + HM: General considerations imply 𝑍 𝑗 𝑆𝑊 = 𝜆∈ 1 2 𝑤 2 𝑋 + 𝐻 2 𝑋,ℤ 𝑆𝑊 𝜆 𝑒 2𝜋𝑖 𝜆⋅ 𝜆 0 𝑅 𝜆 𝑝,𝑆 Special coordinate: 𝑎 𝑗 𝑢= 𝑢 𝑗 + 𝜅 𝑗 𝑎 𝑗 +𝒪( 𝑎 𝑗 2 ) 𝑅 𝜆 𝑝,𝑆 =𝑅𝑒𝑠[ 𝑑 𝑎 𝑗 𝑎 𝑗 1+ 𝑑 𝜆 2 𝒆 𝟐𝒑𝒖+ 𝑺 𝟐 𝑻 𝒖 +𝒊 𝒅𝒖 𝒅 𝒂 𝒋 𝑺⋅𝝀 𝑪 𝒖 𝝀 𝟐 𝑃 𝑢 𝜎 𝐸 𝑢 𝜒 ] C,P,E:Universal functions. In principle computable. 𝑑 𝜆 = 2𝜆 2 − 𝑐 2 4 𝑐 2 =2𝜒+3 𝜎

Deriving C,P,E From Wall-Crossing 𝑑 𝑑 𝑔 𝜇𝜈 𝑍 𝑢 =∫𝑇𝑜𝑡 𝑑𝑒𝑟𝑖𝑣= ∞ 𝑑𝑢 …. + 𝑢 𝑗 𝑑𝑢 … 𝑍 𝑢 piecewise constant: Discontinuous jumps across walls: Δ 𝑗 𝑍 𝑢 +Δ 𝑍 𝑗 𝑆𝑊 =0⇒ 𝐶 𝑢 , 𝑃 𝑢 , 𝐸 𝑢 Then for 𝑏 2 + >1 , 𝑍 𝑢 =0, but we know the couplings to compute 𝑍 𝑆𝑊 ! 𝑍 𝑢 is the tail that wags the dog.

Witten Conjecture ⇒ a formula for all X with 𝑏 2 + >0. For 𝑏 2 + 𝑋 >1 we derive ``Witten’s conjecture’’: 𝑍 𝐷𝑊 𝑝,𝑆 = 2 𝑐 2 − 𝜒 ℎ ( 𝑒 1 2 𝑆 2 +2𝑝 𝜆 𝑆𝑊 𝜆 𝑒 2𝜋𝑖 𝜆⋅ 𝜆 0 𝑒 2𝑆⋅𝜆 + 𝑒 − 1 2 𝑆 2 −2𝑝 𝜆 𝑆𝑊 𝜆 𝑒 2𝜋𝑖 𝜆⋅ 𝜆 0 𝑒 −2𝑖𝑆⋅𝜆 ) 𝜒 ℎ = 𝜒+𝜎 4 𝑐 2 =2𝜒+3 𝜎 Example: 𝑋=𝐾3: 𝑍= sinh ( 𝑆 2 2 +2 𝑝)

Generalization To 𝑁 𝑓𝑙 >0 𝑍 𝑝;𝑆; 𝑚 𝑓 = 𝑗=1 2+ 𝑁 𝑓𝑙 𝑍(𝑝,𝑆; 𝑚 𝑓 ; 𝑢 𝑗 ) 𝑏 2 + >1 𝑍 𝑝,𝑆; 𝑚 𝑓 ; 𝑢 𝑗 = 𝛼 𝜒 𝛽 𝜎 𝜆 𝑆𝑊 𝜆 𝑒 2𝜋𝑖 𝜆⋅ 𝜆 0 𝑅 𝑗 (𝑝,𝑆) X is SWST ⇒ 𝑅 𝑗 (𝑝,𝑆) is computable explicitly as a function of 𝑝,𝑆, 𝑚 𝑓 ,Λ, 𝜏 0 from first order degeneration of the SW curve.

𝜇 𝑗 ∼ 𝑔 2 3 Δ ′ ​ 𝑢 𝑗 𝑦 2 =4 𝑥 3 − 𝑔 2 𝑢,𝑚 𝑥 − 𝑔 3 𝑢,𝑚 𝑅 𝑗 𝑝,𝑠 = 𝜇 𝑗 𝜒 ℎ 𝑑𝑢 𝑑𝑎 𝜒 ℎ +𝜎 exp 2𝑝 𝑢 𝑗 + 𝑆 2 𝑇 𝑢 𝑗 −𝑖 𝑑𝑢 𝑑𝑎 𝑗 𝑆⋅𝜆 𝑦 2 =4 𝑥 3 − 𝑔 2 𝑢,𝑚 𝑥 − 𝑔 3 𝑢,𝑚 Assume a simple zero for Δ as 𝑢→ 𝑢 𝑗 Choose local duality frame with 𝑎 𝑗 →0 Nonvanishing period: 𝑑 𝑎 𝑗 𝑑𝑢 and 𝑑 𝑎 𝑗,𝐷 𝑑𝑢 →𝑖 ∞: 𝑑 𝑎 𝑗 𝑑𝑢 2 ​ 𝑢 𝑗 ∼ 𝑔 2 𝑔 3 ​ 𝑢 𝑗 𝜇 𝑗 ∼ 𝑔 2 3 Δ ′ ​ 𝑢 𝑗

1 Introduction 2 Cambrian: Witten Reads Donaldson 3 Carboniferous: SW Theory & u-plane Derivation Of Witten’s Conjecture. 4 Holocene: Three New Apps 5 Holomorphic Anomaly & Continuous Metric Dep. 6 Back To The Future

Application 1: Case Of 𝒩= 2 ∗ Using these methods analogous formulae were worked out for N=2*, by Labastida-Lozano in 1998, but only in the case when 𝑋 is spin. They checked S-duality for the case 𝑏 2 + >1 The generalization to 𝑋 which is NOT spin is nontrivial and involves new effective interactions not in the literature. 𝑢− 𝑢 1 − 𝑐 1 𝔰 2 8 𝑒 −𝑖 𝜕 𝑎 𝐷 𝜕𝑚 𝜆⋅ 𝑐 1 𝔰

Application 2: S-Duality Of 𝑁 𝑓𝑙 =4 𝑍(𝑝,𝑆; 𝜏 0 ) is expected to have modular properties: 𝑗=1 6 𝜇 𝑗 𝜒 ℎ 𝜅 𝑗 𝜒 ℎ +𝜎 𝑒 𝑝 𝑢 𝑗 +𝑖 𝜅 𝑗 𝜆⋅𝑆+⋯ 𝜅 𝑗 = 𝑔 3 𝑢 𝑗 𝑔 2 𝑢 𝑗 1 2 𝜇 𝑗 = 𝜂 −24 𝜏 0 𝑔 2 𝑢 𝑗 3 𝑘 𝑢 𝑗 − 𝑢 𝑘 −1 Need to say: That’s without taking into account the alpha and beta coefficients. These will be overall powers of \eta(\tau_0) and might change the net weight. Sum over 𝑗 gives symmetric rational function of 𝑢 𝑗 ⇒ 𝑍 will be modular:

= 𝑐 𝜏 0 +𝑑 𝜒+3𝜎 𝑐 𝜏 0 +𝑑 𝜒+3𝜎 𝑍(𝑝,𝑆; 𝜏 0 ;𝔪) 𝛾= 𝑎 𝑏 𝑐 𝑑 ∈𝑆𝐿(2,ℤ) 𝑍 𝑝+ 𝑐 𝑐 𝜏 0 +𝑑 𝑆 2 𝑐 𝜏 0 +𝑑 2 , 𝑆 𝑐 𝜏 0 +𝑑 2 ; 𝑎 𝜏 0 +𝑏 𝑐 𝜏 0 +𝑑 ;𝛾⋅𝔪 = 𝑐 𝜏 0 +𝑑 𝜒+3𝜎 𝑐 𝜏 0 +𝑑 𝜒+3𝜎 𝑍(𝑝,𝑆; 𝜏 0 ;𝔪) (Neglecting 𝛼,𝛽 coefficients.)

Application 3: AD3 Partition Function Consider 𝑁 𝑓𝑙 =1. At a critical point 𝑚= 𝑚 ∗ two singularities 𝑢 ± collide at 𝑢= 𝑢 ∗ and the SW curve becomes a cusp: 𝑦 2 = 𝑥 3 [Argyres,Plesser,Seiberg,Witten] Two mutually nonlocal BPS states have vanishing mass: 𝛾 1 𝜆→0 𝛾 2 𝜆→0 𝛾 1 ⋅ 𝛾 2 ≠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 𝑓 =1 SW curve in the scaling region: 𝑦 2 = 𝑥 3 −3 Λ 𝐴𝐷 2 𝑥+ 𝑢 𝐴𝐷 Λ 𝐴𝐷 2 ∼ 𝑚− 𝑚 ∗ Call it the ``AD3-Family over the 𝑢 𝐴𝐷 −plane’’

AD3 Partition Function - 1 Work with Iurii Nidaiev The 𝑆𝑈 2 𝑁 𝑓𝑙 =1 u-plane integral has a nontrivial contribution from the scaling region 𝑢 ± → 𝑢 ∗ lim 𝑚→ 𝑚 ∗ 𝑍 𝑢 − 𝑑𝑢 𝑑 𝑢 lim 𝑚→ 𝑚 ∗ 𝑀𝑒𝑎𝑠𝑢𝑟𝑒 𝑢, 𝑢 ;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. (No noncompact Higgs branch.) 2. The partition function is a sum over all Q-invariant field configurations. 3. Scaling region near 𝑢 ∗ governed by AD3 theory. lim 𝑚→ 𝑚 ∗ 𝑍 𝑆𝑈 2 , 𝑁 𝑓𝑙 =1 ``contains’’ the AD partition function Extract it from the scaling region. Our result: 𝑍 𝐴𝐷3 = lim Λ 𝐴𝐷 →0 𝑍 𝑢 𝐴𝐷3−𝑓𝑎𝑚𝑖𝑙𝑦 + 𝑍 𝑆𝑊 𝐴𝐷3−𝑓𝑎𝑚𝑖𝑙𝑦 Claim: This is the AD3 TFT on 𝑋 for 𝑏 2 + >0 .

AD3 Partition Function: Evidence 1/2 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 ℎ − 𝑐 2 −3≤0 b.) 𝜆 𝑆𝑊 𝜆 𝑒 2𝜋 𝑖 𝜆⋅ 𝜆 0 𝜆 𝑘 =0 0≤𝑘≤ 𝜒 ℎ − 𝑐 2 −4 Then the limit 𝑚→ 𝑚 ∗ exists It is now a rigorous theorem that SWST ⇒ SCST

AD3 Partition Function: Evidence 2/2 For 𝑏 2 + >0 recover the expected selection rule: 𝑈(𝑝) ℓ 𝑈 𝑆 𝑟 ≠0 only for 6ℓ+𝑟 5 = 𝜒 ℎ − 𝑐 2 5 Consistent with background charge for AD3 computed by Shapere-Tachikawa. Explicitly, if 𝑋 of 𝑆𝑊𝑆𝑇 and 𝑏 2 + >1:

Explicitly, if 𝑋 of 𝑆𝑊𝑆𝑇 and 𝑏 2 + >1: 𝑍 𝐴𝐷3 = 1 𝔅! 𝐾 1 𝐾 2 𝜒 𝐾 3 𝜎 𝜆 𝑒 2𝜋𝑖 𝜆⋅ 𝑤 2 𝑆𝑊 𝜆 ⋅ ⋅{ 𝜆⋅𝑆 𝔅−2 (24 𝜆⋅𝑆 2 +𝔅 𝔅−1 𝑆 2 } 𝔅= 𝜒 ℎ − 𝑐 2 =− 7 𝜒+11 𝜎 4 Surprise! p drops out: U(p) is a ``null vector’’

1 Introduction 2 Cambrian: Witten Reads Donaldson 3 Carboniferous: SW Theory & u-plane Derivation Of Witten’s Conjecture. 4 Holocene: Three New Apps 5 Holomorphic Anomaly & Continuous Metric Dep. 6 Back To The Future

𝑍 𝑢 Wants To Be A Total Derivative 𝑍 𝑢 =∫𝑑u d u ℋ 𝑢 Ψ Ψ= 𝜆 𝑑 𝑑 𝑢 ℰ 𝑟 𝜆 𝜔 , 𝑏 𝜆 𝑒 −𝑖 𝜋𝜏 𝜆 2 −𝑖 𝑆⋅ 𝑑𝑢 𝑑𝑎 +2 𝜋𝑖 𝜆− 𝜆 0 ⋅ 𝑤 2 ℰ 𝑟,𝑏 = 𝑏 𝑟 𝑒 −2𝜋 𝑡 2 𝑑𝑡 𝑟 𝜆 𝜔 = 𝑦 𝜆 + − 𝑖 4𝜋 𝑦 𝑆 + 𝑑𝑢 𝑑𝑎 Lower limit 𝑏 𝜆 must be independent of u but can depend on 𝜆,𝜔,𝑢. [Korpas & Manschot; Moore & Nidaiev ] ? 𝑍 𝑢 = ∫𝑑u d u 𝑑 𝑑 𝑢 ( ℋ 𝑢 Θ 𝜔 )

No! 𝑍 𝑢 =∮d u 𝑑 𝑑 𝑢 ℋ 𝑢 Θ 𝜔 =∮𝑑 𝑢 ℋΨ has nontrivial periods in general! BUT: The difference for two metrics CAN be evaluated by residues! 𝑍 𝑢 𝜔 − 𝑍 𝑢 𝜔 0 =∫𝑑𝑢 𝑑 𝑢 𝑑 𝑑 𝑢 (ℋ Θ 𝜔, 𝜔 0 ) Θ 𝜔, 𝜔 0 = 𝜆 ℰ 𝑟 𝜆 𝜔 −ℰ 𝑟 𝜆 𝜔 0 𝑒 −𝑖 𝜋𝜏 𝑢 𝜆 2 +⋯ 𝑟 𝜆 𝜔 = 𝑦 𝜆 + − 𝑖 4𝜋 𝑦 𝑆 + 𝑑𝑢 𝑑𝑎 Modular completion of indefinite theta function of Vigneras, Zwegers, Zagier

Holomorphic Anomaly & Metric Dependence For theories with a manifold of superconformal couplings, 𝜏 𝑢 → 𝜏 0 when 𝑢→∞ Θ 𝜔, 𝜔 0 𝜏 𝑢 ,… → Θ 𝜔, 𝜔 0 𝜏 0 ,… Contour integral at ∞⇒ 𝐹𝑜𝑟 𝑏 2 + =1, many ``topological’’ correlators: Nonholomorphic in 𝜏 0 Depend continuously on metric -- albeit sotto voce ] [Moore & Witten, 1997

Special Case Of ℂ ℙ 2 No walls: 𝜔∈ 𝐻 2 𝑋,ℝ ≅ℝ, so we only see holomorphic anomaly. ⇒ Path integral derivation of the holomorphic anomaly.

Vafa-Witten Partition Functions VW twist of N=4 SYM formally computes the ``Euler character’’ of instanton moduli space. (Not really a topological invariant. True mathematical meaning unclear, but see recent work of Tanaka & Thomas; Gholampour, Sheshmani, & Yau.) Physics suggests the partition function is both modular (S-duality) and holomorphic. Surprise! Computations of Klyachko and Yoshioka for 𝑋=ℂ ℙ 2 show that the holomorphic generating function is only mock modular. But a nonholomorphic modular completion exists.

This has never been properly derived from a path integral argument. . But we just derived completely analogous results for the 𝑁 𝑓𝑙 =4 and 𝒩= 2 ∗ theories from a path integral, suggesting VW will also have continuous metric dependence. Indeed, continuous metric dependence in VW theory for SU(𝑟>1) has been predicted in papers of Jan Manschot using rather different methods. We hope that a similar path integral derivation can be found for the 𝒩=4 Vafa-Witten twisted theory.

1 Introduction 2 Cambrian: Witten Reads Donaldson 3 Carboniferous: SW Theory & u-plane Derivation Of Witten’s Conjecture. 4 Holocene: Three New Apps Holomorphic Anomaly & Continuous Metric Dep. 5 6 Back To The Future

Future Directions Lots of details to clean up Comparison with localization results of Gukov et. al. There is an interesting generalization to invariants for families of four-manifolds. Generalization to all theories in class S: Many aspects are clear – this is under study. Does the u-plane integral make sense for ANY family of Seiberg-Witten curves ?