Secondary Products in Supersymmetric Field
Theory
Mathew Bullimore
with D. Ben-Zvi, C. Beem, T. Dimofte, A. Neitzke
Part I : Introduction
Introduction I
I would like to talk about some aspects of TQFTs that arise from
‘topological twisting’ a supersymmetric quantum field theory.
The story is well-known to mathematicians through the formalism of the
Cobordism Hypothesis and derived algebraic geometry.
Our aim was to extract a key structure that emerges from this formalism
and understand it concretely in familiar examples.
This is the structure of ‘higher products’.
Introduction II
An important idea in this talk is ‘topological descent’.
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O(2)<latexit sha1_base64="kz3m+Y0VmIxO5EMLyI4YR1fUaJw=">AAAB+nicbVDLSsNAFL3xWesr1aWbwSLUTUlKQZcFN+6sYB/QxjKZTtqhk0mYmSgl9lPcuFDErV/izr9x0mahrQcGDufcyz1z/JgzpR3n21pb39jc2i7sFHf39g8O7dJRW0WJJLRFIh7Jro8V5UzQlmaa024sKQ59Tjv+5CrzOw9UKhaJOz2NqRfikWABI1gbaWCX+iHWY4J5ejO7Tyu189nALjtVZw60StyclCFHc2B/9YcRSUIqNOFYqZ7rxNpLsdSMcDor9hNFY0wmeER7hgocUuWl8+gzdGaUIQoiaZ7QaK7+3khxqNQ09M1kFlQte5n4n9dLdHDppUzEiaaCLA4FCUc6QlkPaMgkJZpPDcFEMpMVkTGWmGjTVtGU4C5/eZW0a1XXqbq39XKjntdRgBM4hQq4cAENuIYmtIDAIzzDK7xZT9aL9W59LEbXrHznGP7A+vwBxBSTnQ==</latexit><latexit sha1_base64="kz3m+Y0VmIxO5EMLyI4YR1fUaJw=">AAAB+nicbVDLSsNAFL3xWesr1aWbwSLUTUlKQZcFN+6sYB/QxjKZTtqhk0mYmSgl9lPcuFDErV/izr9x0mahrQcGDufcyz1z/JgzpR3n21pb39jc2i7sFHf39g8O7dJRW0WJJLRFIh7Jro8V5UzQlmaa024sKQ59Tjv+5CrzOw9UKhaJOz2NqRfikWABI1gbaWCX+iHWY4J5ejO7Tyu189nALjtVZw60StyclCFHc2B/9YcRSUIqNOFYqZ7rxNpLsdSMcDor9hNFY0wmeER7hgocUuWl8+gzdGaUIQoiaZ7QaK7+3khxqNQ09M1kFlQte5n4n9dLdHDppUzEiaaCLA4FCUc6QlkPaMgkJZpPDcFEMpMVkTGWmGjTVtGU4C5/eZW0a1XXqbq39XKjntdRgBM4hQq4cAENuIYmtIDAIzzDK7xZT9aL9W59LEbXrHznGP7A+vwBxBSTnQ==</latexit><latexit sha1_base64="kz3m+Y0VmIxO5EMLyI4YR1fUaJw=">AAAB+nicbVDLSsNAFL3xWesr1aWbwSLUTUlKQZcFN+6sYB/QxjKZTtqhk0mYmSgl9lPcuFDErV/izr9x0mahrQcGDufcyz1z/JgzpR3n21pb39jc2i7sFHf39g8O7dJRW0WJJLRFIh7Jro8V5UzQlmaa024sKQ59Tjv+5CrzOw9UKhaJOz2NqRfikWABI1gbaWCX+iHWY4J5ejO7Tyu189nALjtVZw60StyclCFHc2B/9YcRSUIqNOFYqZ7rxNpLsdSMcDor9hNFY0wmeER7hgocUuWl8+gzdGaUIQoiaZ7QaK7+3khxqNQ09M1kFlQte5n4n9dLdHDppUzEiaaCLA4FCUc6QlkPaMgkJZpPDcFEMpMVkTGWmGjTVtGU4C5/eZW0a1XXqbq39XKjntdRgBM4hQq4cAENuIYmtIDAIzzDK7xZT9aL9W59LEbXrHznGP7A+vwBxBSTnQ==</latexit><latexit sha1_base64="kz3m+Y0VmIxO5EMLyI4YR1fUaJw=">AAAB+nicbVDLSsNAFL3xWesr1aWbwSLUTUlKQZcFN+6sYB/QxjKZTtqhk0mYmSgl9lPcuFDErV/izr9x0mahrQcGDufcyz1z/JgzpR3n21pb39jc2i7sFHf39g8O7dJRW0WJJLRFIh7Jro8V5UzQlmaa024sKQ59Tjv+5CrzOw9UKhaJOz2NqRfikWABI1gbaWCX+iHWY4J5ejO7Tyu189nALjtVZw60StyclCFHc2B/9YcRSUIqNOFYqZ7rxNpLsdSMcDor9hNFY0wmeER7hgocUuWl8+gzdGaUIQoiaZ7QaK7+3khxqNQ09M1kFlQte5n4n9dLdHDppUzEiaaCLA4FCUc6QlkPaMgkJZpPDcFEMpMVkTGWmGjTVtGU4C5/eZW0a1XXqbq39XKjntdRgBM4hQq4cAENuIYmtIDAIzzDK7xZT9aL9W59LEbXrHznGP7A+vwBxBSTnQ==</latexit>
Starting from a local operator O, we can construct extended operators
O(1), O(2), . . .
I O(p) can be wrapped on cycles in Hp(MD,Z).
I This plays an important role in mathematical applications -
Donaldson theory, Gromov-Witten theory
Introduction III
Here we use descendent operators to define a ‘secondary product’ on
local operators.
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O(D1)2
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I In D-dimensions, the secondary product O1,O2 is defined by
surrounding O1 with the descendent OD−12 wrapping SD−1.
I This defines a kind of ‘Poisson bracket’ of degree 1−D.
I Some examples known to physicists but not systematically explored.
Part II : General Setup
Twisted Superalgebra
I will consider a twisted supersymmetric theory in D-dimensions with
superalgebra
Q2 = 0 [Q,Qµ] = iPµ [Qµ, Qν ] = 0
where Q / Qµ transform as scalar / vector with respect to
Spin(D)′ ⊂ Spin(D)×GR .
I Graded commutator [a, b] := ab− (−1)F (a)F (b)ba where F is Z2
fermion number.
I I assume F lifts to a Z grading with F (Q) = 1 and F (Qµ) = −1
(combination of flavour and R-symmetry).
Topological Operators
Topological operators are annihilated by the scalar supercharge
QO = 0 .
Their correlation functions have two important properties:
I They depend only on Q-cohomology classes,
〈Q(O1)O2 · · · 〉 = 〈Q(O1O2 · · · ) 〉 = 0 .
I They are independent of position,
∂µ〈O1(x)O2(y) · · ·〉 = 〈 ∂µO1(x)O2(y) · · ·〉= 〈QQµO1(x)O2(y) · · ·〉= 〈Q(QµO1(x)O2(y) · · · )〉= 0
Q-cohomology
This motivates introduction of the Q-cohomology
A := Im(Q)/Ker(Q) .
I I denote cohomology classes [O(x)] = [O].
I They are independent of position by same argument as before
∂µ[O(x)] = [∂µO(x)] = [Q(QµO(x))] = 0 .
I The Q-cohomology inherits Z-grading by F ,
A =⊕p∈ZAp .
The Primary Product
In dimension D ≥ 2, there is a unique product
∗ : A⊗A → A
defined by
[O1] ∗ [O2] := [O1(x1)O2(x2)] .
This has the properties
I It is will defined: independent of x1, x2.
I Graded commutative: O1 ∗ O2 = (−1)F1F2O2 ∗ O1
I Associative: O1 ∗ (O2 ∗ O3) = (O1 ∗ O2) ∗ O3
We claim that A also inherits a Poisson structure...
Topological Descent I
The first step is to consider the descent construction:
I Start from a topological operator O(x).
I Define sequence of descendent p-form operators
O(p)(x) :=1
p!Qµ1· · ·Qµp
O(x) dxµ1 ∧ · · · ∧ dxµp
I They obey the descent equations
QOp(x) = dO(p−1)(x)
as a consequence of [Q,Qµ] = iPµ.
Example: QO(1) = Q(QµO) dxµ = Q,QµO dxµ = iPµO dxµ = dO
Topological Descent II
We can integrate the descendent over a p-chain γ
O(γ) :=
∫γ
O(p)(x) .
If γ is closed (∂γ = 0) then O(γ) is topological,
QO(γ) =
∫γ
dO(p−1) =
∫∂γ
O(p−1) = 0 .
I The Q-cohomology class [O(γ)] depends only on the homology class
[γ] ∈ Hp(MD,Z).
I Such classes play an important role in mathematical applications:
Donaldson theory, Gromov-Witten theory.
I Does not product anything new on MD = RD but...
The Secondary Product I
Descendent operators may be used to define a ‘secondary product’
, : A⊗A → A
defined by O1,O2 := [O1(SD−1x )O2(x) ] .
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O1(SD1x )
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I SD−1x is a sphere surrounding the point x.
I Cohomology class independent of radius of SD−1x .
The Secondary Product II
The secondary product has the following properties:
I It has degree 1−D with respect to the Z-grading.
I It has definite symmetry,
O2,O1 = (−1)F1F2+DO1,O2 .
I It is a graded derivation over the primary product,
O1,O2 ∗ O3 = O1,O2 ∗ O3 + (−1)(F1+D−1)F2O2 ∗ O1,O3 .
I It obeys the Jacobi identity.
This endows A with the structure of ‘PD-algebra’.
More Sophisticated Viewpoint
Consider a pair of operators inserted at points x1 6= x2.
The configuration space is homotopically
C2(RD) ∼ SD−1 .
There is a product for each homology class
Hp(SD−1,Z) =
Z if p = 0 → primary product ∗Z if p = D − 1 → secondary product , ∅ otherwise
Part III : Two Dimensions
The B-Twist
The N = (2, 2) supersymmetry on M2 = R2 has four supercharges Q+,
Q−, Q+, Q− obeying
[Q+, Q+] = 2iPz [Q−, Q−] = 2iPz .
I I will consider the combinations
Q := Q+ + Q− Qµ =
(Q−
Q+
)
I They transform as a scalar / vector with respect to the diagonal
subgroup U(1)′ ⊂ U(1)× U(1)A.
I They are known as the B-type supercharges.
Example: Chiral Multiplet
In the B-twist a single chiral multiplet consists of
I Scalar boson φ
I Scalars fermions η, ζ
I 1-form fermion χ
S =
∫dφ ∧ ∗dφ+ ζ ∧ dχ+ χ ∧ ∗dη
We choose the Z-grading by U(1)V .
φ φ η ζ χ
U(1)V 0 0 1 1 −1
The supersymmetry transformations are (Q = Qµdxµ)
Qφ = 0 Qζ = 0 Qη = 0 Qφ = η Qχ = dφ
Qφ = χ Qζ = − ∗ dφ Qη = dφ Qφ = 0 Qχ = 0 .
Primary Product
Topological operators are polynomials in φ, φ, η, ζ modulo Qφ = η, which
are simply polynomials in φ, ζ.
Two geometric interpretations:
I We haveA = H0,•
∂(C,∧•TC)
under the identification
Q ∼ ∂ η ∼ dφ ζ ∼ ∂
∂φ.
I Recalling the Z-grading
φ ζ
U(1)V 0 1
this is functions on the shifted cotangent bundle T ∗[1]C.
Secondary Product I
In dimension D = 2, the secondary bracket has degree −1.
From the Z-grading
φ χ
U(1)V 0 1
the only possible non-vanishing bracket is ζ, φ ∼ 1.
To compute the bracket, we descend once on ζ.
I The first descendent is ζ(1) = Qζ = − ∗ dφ .I It’s exterior derivative is an equation of motion,
dζ(1) = −d ∗ dφ =δS
δφ.
Secondary Product II
Here is the computation of the bracket,
ζ, φ =
∮S1x
ζ(1)φ(x)
=
∫D2
x
dζ(1)φ(x)
=
∫D2
x
(−d ∗ dφ)φ(x)
=
∫D2
x
δS
δφφ(x) = 1 .
I Schouten-Nijenhuis bracket on holomorphic polyvector fields on
X = C (the unique extension of the Lie bracket).
I Poisson structure on shifted cotangent bundle T ∗[1]C.
General Kahler Target
Consider a supersymmetric sigma model to a Kahler target X.
The topological algebra in the B-model is the Dolbeault cohomology of
holomorphic polyvector fields
A =⊕p,q
H0,p
∂(X,∧qTX)
I The Z-grading by U(1)V is p+ q
I Due to the absence of instanton corrections, the primary product
coincides with the wedge product of polyvector fields.
I The secondary product , is Schouten-Nijenhuis bracket.
Part IV : Three Dimensions
3d N = 4 Theories
We have supercharges QAAα transforming in the tri-fundamental
representation of
SU(2)× SU(2)H × SU(2)C .
There are two topological twists:
I H-twist: SU(2)′ ⊂ SU(2)× SU(2)H
I C-twist: SU(2)′ ⊂ SU(2)× SU(2)C
Specification of the supercharges Q, Qµ requires a further choice of
complex structure on the Coulomb / Higgs branch.
Today I will consider the C-twist. [Rozansky-Witten]
Hypermultiplet
After performing the C-twist, a single hypermultiplet consists of
I Complex scalar bosons XI
I Scalar fermions ηI
I 1-form fermions χI
where I = 1, 2 index C2 with holomorphic symplectic form ΩIJ .
S =
∫dXI ∧ ∗dXI + ΩIJχ
I ∧ dχJ + ηId ∗ χI
The supersymmetry transformations are (Q = Qµdxµ)
QXI = 0 QXI = ηI QηI = 0 QχI = dXI
QXI = χIµ QXI = 0 QηI = dXI QχI = ΩIJdXJ .
Primary Product
The topological operators are polynomials in XI , XI ηI modulo the
relation QXI = ηI , which are simply polynomials in XI .
Two geometric interpretations:
I We have A = H0,•(T ∗C) under the identification
ηI ∼ dXI Q ∼ dXI∂XI.
I Introducing the Z-grading
X1 X2 η1, χ1 η2, χ
2
U(1)′H 2 0 1 −1
(a combination of R-symmetry and flavour symmetry) this is
functions on the shifted cotangent bundle T ∗[2]C.
Secondary Product I
In dimension D = 3, the secondary product has degree −2.
The only possible non-vanishing bracket is X1, X2 ∼ 1 or more
invariantly XI , XJ ∼ ΩIJ .
Let us compute the descendents of XI :
I The first descendent is (XI)(1) = QXI = χIµ.
I The second descendent is (XI)(2) = QχI = ΩIJ ∗ dXJ .
I Finally, the exterior derivative is an equation of motion,
d(XI)(2) = ΩIJd ∗ dXJ = ΩIJδS
δXJ.
Secondary Product II
We now compute the secondary product as follows
XI , XJ =
∫S2x
(XI)(2)XJ(x)
=
∫D3
x
d(XI)(2)XJ(x)
= ΩIK∫D3
x
δS
δXKXJ(x)
= ΩIJ
I This is tantamount to the holomorphic symplectic structure on the
Higgs branch MH = T ∗C.
I Alternatively, keeping track of the Z-grading, it is the Poisson
structure on the shifted cotangent bundle T ∗[2]C.
Supersymmetric Gauge Theory
A 3d N = 4 gauge theory has two important moduli spaces of vacua
1. Higgs branch MH
2. Coulomb branch MC
which are both holomorphic symplectic varieties.
I C-twist : ( A , , ) coincide with the coordinate ring of MH and
its holomorphic Poisson bracket.
I H-twist : ( A , , ) coincide with the coordinate ring of MC and
its holomorphic Poisson bracket.
The physical construction of the holomorphic symplectic structure is new!
Part V : Conclusions
Topological Defects
The formalism presented here can be generalised to ‘higher products’ of
extended topological operators.
This is a shadow of full machinery of Cobordism Hypothesis.
One example: line operators form a braided tensor category.
S1
L M L M
RLM
The braiding is an example of a ‘secondary product’ of line operators.
Omissions
I Examples of higher products involving extended operators.
I Ω-deformation and deformation quantisation of the secondary
bracket , .
I Topological twists in four dimensions and connections to geometric
Langlands.
Future Directions
I Systematic exploration of higher products of extended operators.
I Construction of ED−k-monoidal structure of categories of
k-dimensional operators.
I The role this plays in ‘generalised global symmetries’.
[Gaiotto-Kapustin-Seiberg-Willet]
I Higher products on ‘holomorphically twisted’ supersymmetric field
theories. [Costello]