1. StringMath2018, Tohoku Univ. June 20, 2018
Partition Functions Of Twisted Supersymmetric Gauge Theories On Four-Manifolds Via u-Plane Integrals
Gregory Moore
Rutgers University
Mostly review. Includes new work
with Iurii Nidaiev & 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….
StringMath2018, Tohoku Univ.
June 20, 2018

2. 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

3. 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!

4. 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.

5. 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)

6. Donaldson-Witten Partition Function
𝑍 𝐷𝑊 𝑝,𝑠 = 𝑒 𝑈(𝑝)+𝑈 𝑆 Λ
𝑋 𝑇𝑟 𝐹 2 =8 𝜋 2 𝑘
= 𝑘≥0 Λ 4𝑘 ℳ 𝑘 𝑒 𝜇 𝑝 +𝜇 𝑆
Strategy: Evaluate in LEET ⇒ Witten (1994)
introduces the Seiberg-Witten invariants.
Major success in
Physical Mathematics.

7. 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.

8. SU(2) With Matter On 𝑋 𝑀∈Γ( 𝑆 + ⊗ 𝐸 ⊕ 𝑁 𝑓𝑙 )
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 ⊗𝑎𝑑𝑃)
𝑐 1 𝐿 ≡ 𝑤 2 𝑋
[Labastida-Marino ‘98]

9. UV Interpretation 𝐹 + =𝒟(𝑀, 𝑀 ) 𝛾⋅𝐷 𝑀=0 𝑍 𝑝,𝑆 = 𝑒 𝑈(𝑝)+𝑈 𝑆 𝒯
𝑍 𝑝,𝑆 = 𝑒 𝑈(𝑝)+𝑈 𝑆 𝒯
= 𝑘 Λ 4𝑘 ℳ 𝑘 𝑒 𝜇 𝑝 +𝜇 𝑆 ℰ
But now ℳ 𝑘 : is the moduli space of:
𝐹 + =𝒟(𝑀, 𝑀 )
𝛾⋅𝐷 𝑀=0
``Generalized monopole equations’’
[Labastida-Marino; Losev-Shatashvili-Nekrasov]
U(1) case: Seiberg-Witten equations.

10. Cambrian: Witten Reads Donaldson1
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

11. 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.

12. 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 : polynomial in u
Δ 𝑢 𝑗 =0 : Discriminant locus

13. 𝐸 𝑢 𝑗
𝐸 𝑢
𝑢 𝑗 𝑢

14. 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 𝑚 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× terms

15. 𝑆𝐿 2,ℤ →𝐴𝑢𝑡 𝔰𝔭𝔦𝔫 8 ≅ 𝑆 3

16. 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 +⋯

17. 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

18. 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 ∗ , ⋯

19. Evaluate 𝑍 𝐷𝑊 Using LEET
𝑍 𝐷𝑊 𝑝,𝑆 = 𝑒 𝑈 𝑝 +𝑈 𝑆 Λ
= 𝑘 Λ 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

20. 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

21. 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.
It is very nearly an integral of a total derivative…
But first let’s finish writing down the
full answer for the partition function.

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

23. 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.

24. Witten Conjecture
⇒ a formula for all X with 𝑏 >0. For 𝑏 2 + 𝑋 >1 we derive ``Witten’s conjecture’’:
𝑍 𝐷𝑊 𝑝,𝑆 = 2 𝑐 2 − 𝜒 ℎ ( 𝑒 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= Λ 6 sinh ( Λ𝑆 Λ 2 𝑝)

25. 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 = 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.

26. 𝑍 𝑝,𝑆; 𝑚 𝑓 ;Λ =
𝛼 𝜒 𝛽 𝜎 𝑗=1 2+ 𝑁 𝑓𝑙 𝜇 𝑗 𝜒 ℎ 𝜈 𝑗 𝜒 ℎ +𝜎 𝑒 2 𝑝 𝑢 𝑗 + 𝑆 2 𝑇 𝑗 𝐹 𝑗 𝑆
𝜇 𝑗 = 𝑔 2 𝑢 𝑗 3 / 𝑡≠𝑗 ( 𝑢 𝑗 − 𝑢 𝑡 )
𝜈 𝑗 ∼ 𝑔 2 𝑢 𝑗
𝐹 𝑗 𝑆 = 𝜆 𝑆𝑊 𝜆 𝑒 𝑖 𝜋𝜆⋅ 𝑐 1 𝔰 cos 𝜈 𝑗 𝜆⋅𝑆
𝑦 2 =4 𝑥 3 − 𝑔 2 𝑢, 𝑚 𝑗 𝑥 − 𝑔 3 𝑢, 𝑚 𝑗

27. 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

28. Application 1: S-Duality Of 𝑁 𝑓𝑙 =4
𝑍(𝑝,𝑆; 𝜏 0 ) is expected to have modular properties:
𝛼 𝜒 𝛽 𝜎 𝑗=1 6 𝜇 𝑗 𝜒 ℎ 𝜈 𝑗 𝜒 ℎ +𝜎 𝑒 2 𝑝 𝑢 𝑗 + 𝑆 2 𝑇 𝑗 𝐹 𝑗 𝑆
𝜇 𝑗 = 𝑔 2 𝑢 𝑗 3 / 𝑡≠𝑗 ( 𝑢 𝑗 − 𝑢 𝑡 )
𝜈 𝑗 ∼ 𝑔 2 𝑢 𝑗
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:

29. = 𝑐 𝜏 0 +𝑑 𝑤 𝑍(𝑝,𝑆; 𝜏 0 ;𝔪) 𝛾= 𝑎 𝑏 𝑐 𝑑 ∈𝑆𝐿(2,ℤ) 𝑤= 2𝜒+3 𝜎 2
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 𝑤 𝑍(𝑝,𝑆; 𝜏 0 ;𝔪)
𝑤= 2𝜒+3 𝜎 2

30. Application 2: Collision Of Mutually Local Singularities
Interesting things when mass parameters 𝑚 𝑗
take values so that 𝑢 𝑗 collide. 𝑢
- Λ 0 2
Λ 0 2

31. Application 2: Collision Of Mutually Local Singularities
Interesting things when mass parameters 𝑚 𝑗
take values so that 𝑢 𝑗 collide. 𝑢
- Λ 0 2
Λ 0 2

32. 𝑚 𝑗 =𝑀+𝛿 𝑚 𝑗 𝑒 𝑖 𝜋 𝜏 0 ∼ Λ 0 4 𝑚 1 𝑚 2 𝑚 3 𝑚 4 𝑊 𝑒𝑓𝑓 ∼ 𝑓 𝛿 𝑚 𝑓 𝑄 𝑓 𝑄 𝑓
𝑒 𝑖 𝜋 𝜏 0 ∼ Λ 𝑚 1 𝑚 2 𝑚 3 𝑚 4
𝑚 𝑗 =𝑀+𝛿 𝑚 𝑗
𝑊 𝑒𝑓𝑓 ∼ 𝑓 𝛿 𝑚 𝑓 𝑄 𝑓 𝑄 𝑓
⇒𝑍 should be 𝑆𝑈 𝑁 𝑓𝑙 −equivariant integral over
moduli space of 𝑈 1 −multimonopole equations:
𝐹 + = 𝑓 𝑀 𝑓 𝑀 𝑓 𝛾⋅𝐷 𝑀 𝑓 =0

33. 𝐽( 𝑆 ) 𝑓 𝑡≠𝑓 𝛿 𝑚 𝑓 −𝛿 𝑚 𝑡 − 𝜒 ℎ 𝑒 𝛿 𝑚 𝑓 𝑝
In the limit 𝛿 𝑚 𝑗 →0 the dominant term in 𝑍 is:
𝐽( 𝑆 ) 𝑓 𝑡≠𝑓 𝛿 𝑚 𝑓 −𝛿 𝑚 𝑡 − 𝜒 ℎ 𝑒 𝛿 𝑚 𝑓 𝑝
𝐽 𝑆 = 𝑒 𝑆 2 /2 𝜆 𝑒 𝑖 𝜋𝜆⋅ 𝑐 1 𝔰 𝑆𝑊 𝜆 cos 𝜆⋅ 𝑆
This agrees with computations of Dedushenko, Gukov and Putrov of the equivariant integrals
on 𝑈(1)-multimonopole space.

34. 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 ∗

35. 𝑢 AD3 From 𝑆𝑈 2 𝑁 𝑓𝑙 =1 SW curve in the scaling region:
Λ 1 2 𝑢
𝑚→ 3 2 Λ 1 = 𝑚 ∗
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’’

36. 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

37. 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 AD3 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 .

38. AD3 Partition Function: Evidence 1/2
Existence of limit is highly nontrivial. It follows
from ``superconformal simple type sum rules’’ :
Theorem [Marino,Moore,Peradze, 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
Feehan & Leness

39. 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:

40. 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’’

41. Should We Be Surprised? Already for 𝑁 𝑓𝑙 =0 and SWST it is also
true that 𝑈 𝑝 2 − Λ is a null vector.
Without a good physics reason why
these should be null vectors one
suspects that there are standard
4-manifolds not of SWST.

42. 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

43. 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 )

44. ∮d u 𝑑 𝑑 𝑢 ℋ 𝑢 Θ 𝜔 =∮𝑑 𝑢 ℋΨ
No!
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 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

45. 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

46. 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.

47. 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.

48. This has never been properly derived from a path integral argument.
But we just derived completely analogous results for the 𝑁 𝑓𝑙 =4 theory
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.

49. 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

50. Future Directions 𝑍∈ 𝐻 ∗ (𝐵𝐷𝑖𝑓𝑓 𝑋 )
There is an interesting generalization to invariants for families of four-manifolds.
Couple to ``topologically twisted supergravity’’
𝑍∈ 𝐻 ∗ (𝐵𝐷𝑖𝑓𝑓 𝑋 )
Generalization to all theories in class S:
Many aspects are clear – this is under study.

51. u-plane for class S: General Remarks
UV interpretation is not clear in general.
These theories might give new 4-manifold invariants.
The u-plane is an integral over the base ℬ of a
Hitchin fibration with a theta function associated
to the Hitchin torus. It will have the form
𝑍 𝑢 = ℬ 𝑑𝑢 𝑑 𝑢 ℋ Ψ
ℋ is holomorphic and metric-independent
𝚿: NOT holomorphic and metric- DEPENDENT
``theta function’’

52. Class S: General Remarks
ℋ= 𝛼 𝜒 𝛽 𝜎 det 𝑑 𝑢 𝑗 𝑑 𝑎 𝑖 1− 𝜒 2 Δ 𝑝ℎ𝑦𝑠 𝜎 8
Δ 𝑝ℎ𝑦𝑠 a holomorphic function on ℬ with first-order zeros at the loci of massless BPS hypers
𝛼,𝛽 will be automorphic forms on
Teichmuller space of the UV curve 𝐶
𝛼, 𝛽 are related to correlation functions for fields in the (0,2) QFT gotten from reducing 6d (0,2)

53. Class S: General Remarks
Ψ∼ 𝜆 𝑒 𝑖 𝜋𝜆⋅𝜉 𝑒 −𝑖 𝜋 𝜏 𝜆 + , 𝜆 + −𝑖 𝜋𝜏 𝜆 − ⋅ 𝜆 − +⋯
Γ⊂ 𝐻 1 Σ;ℤ
𝜆∈ 𝜆 0 +Γ⊗ 𝐻 2 𝑋;ℤ
Lagrangian sublattice
𝜉∈Γ⊗ 𝐻 2 (𝑋;ℝ)
If 𝜉=𝜌⊗ 𝑤 2 𝑋 𝑚𝑜𝑑 2 then WC from interior of ℬ will be cancelled by SW invariants
⇒ No new four-manifold invariants…

54. 6d (2,0) 𝑋 2d (2,0) with chiral bosons
Ψ comes from a ``partition function’’ of a level 1 SD 3-form on 𝑀 6 =Σ×𝑋
Quantization: Choose a QRIF Ω on 𝐻 3 𝑀 6 ;ℤ
Natural choice: [Witten 96,99; Belov-Moore 2004]
Ω 𝑥 = exp 𝑖 𝜋 𝑊𝐶𝑆 𝜃∪𝑥; 𝑆 1 × 𝑀 6
With this choice: 𝜉=𝜌⊗ 𝑤 2 𝑋

55. Case Of SU(2) 𝒩= 2 ∗
Using these methods analogous formulae were worked out for 𝒩= 2 ∗ , by Moore-Witten and Labastida-Lozano in 1998, but only in the case when 𝑋 is spin.
L&L checked S-duality for the case 𝑏 2 + >1
The generalization to 𝑋 which is NOT spin is nontrivial: The standard expression from
Moore-Witten and Labastida-Lozano is
NOT single-valued on the u-plane.

56. This is not surprising: The presence of external
𝑈 1 𝑏𝑎𝑟𝑦𝑜𝑛 gauge field 𝐹 𝑏𝑎𝑟𝑦𝑜𝑛 ∼ 𝑐 1 𝔰 means
there should be new interactions:
𝑒 𝜅 1 𝑢 𝑐 1 𝔰 2 + 𝜅 2 𝑢 𝜆⋅ 𝑐 1 𝔰
Shapere & Tachikawa
Holomorphy, 1-loop singularities,
single-valuedness forces:
𝑢− 𝑢 1 − 𝑐 1 𝔰 𝑒 −𝑖 𝜕 𝑎 𝐷 𝜕𝑚 𝜆⋅ 𝑐 1 𝔰

57. Surprise!! ANY family of Seiberg-Witten curves ? It doesn’t work!
Correct version appears to be non-holomorphic.
With Jan Manschot we have an alternative
which is currently being checked.
Does the u-plane integral make sense for
ANY family of Seiberg-Witten curves ?

58. どうもありがとうございます