LAWRGe 2023 Notes

LAWRGe Workshop

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Mirror symmetry and 3d topological quantum field theories (TQFTs)

Workshop website

Lecture 1: Mon Jun 12 11:05:22 2023

Preliminaries

TQFT: One starts with a cobordism, a manifold with boundary with disjoint union decomposition ∂W=M⊔N. Usual notions of framed, oriented, unoriented, etc apply to the boundary decomposition. We will mostly focus on oriented TQFTs via oriented cobordisms. This will be implicitly assumed for the rest of the talk.

You can glue cobordisms together along a common boundary component. Taking this gluing as morphism composition, the collection of cobordisms (with equal dimension) form a category Cobord,d−1, whose objects are closed oriented d−1-manifolds, and morphisms are diffeomorphism classes of compact cobordisms between them. Here, “closed” means compact without boundary. We can write morphisms like MW→N. Note: Taking the reverse orientation of W gives the opposite arrow N→M.

Taking the disjoint union of two cobordisms acts as the tensor product in this category. Placing W,V side-by-side in a diagram is the morphism W⊗V. This makes Cobord,d−1 a symmetric monoidal category.

Definition: A d-dimensional TQFT is a symmetric monoidal functor Z:Cobord,d−1→Vect.

The definition expresses a lot of things, such as Z(M1⊔M2)≅Z(M1)⊗Z(M2).

Remarks

• Given M a closed, oriented d−1-manifold, the value Z(M) is a vector space called the vector space of states of the field theory. Elements of the vector space are the states of the field theory.
  • Given M a closed d-manifold, can think of M∈Hom(∅,∅) as a cobordism of the empty manifolds. Under Z, Z(M)∈Hom(C,C) a linear map, which is just a number called the partition function.
• Given MW→N, Z(W):Z(M)→Z(N) corresponds to a transition amplitude, or S-matrix.

Extending up/down

Suppose we start with some closed d-manifold M which is a union of two manifolds with boundary Σ. The partition function Z(M) is a composite of morphisms C→Z(Σ)→C. This is the locality principle of (T)QFTs. The question is, can we cut Σ in a certain way to determine Z(Σ)?

Definition: Let Cobord,d−1,d−2 be a symmetric monoidal 2-category whose objects are d−2-manifolds, and 1-morphisms are d−1-dimensional cobordisms, and whose 2-morphisms are diffeomorphism classes of cobordisms between the cobordisms. The 2-morphisms are manifolds with corners.

• TODO pair of pants drawing

You can repeat this extension process all the way down to 0-manifolds at the bottom: a d-category Cobord whose k-morphisms are diffeomorphisms classes of cobordisms between the k−1-morphisms.

The above is the process of extending down.

Extending up is the process of removing the reliance on “diffeomorphism classes of” from the previous definition.

Let Bordord be an (∞,d)-category. This is a category with k<d-morphisms the same as in Cobord, and d-morphisms are done by d-dimensional iterated cobordisms

Definition: A fully extended d-dimensional TQFT is a symmetric monoidal functor Bordord→C where C is a symmetric monoidal (∞,d)-category.

In the case Z:Cobord→C, if we take some M a d−1-manifold, there is a natural cobordism from M→M by diffeomorphisms, so there is now an action of MCG(M) the mapping class group of M on the vector space Z(M). In physics, the vector space Z(M) is often not a vector space, but a chain complex in Ch. Since chain complexes have a notion of homotopy, we can view Ch as an ∞-category. The mapping class group action MCG(M) is an action on H∙(Z(M)). Then C∙(Diff(M)) as an algebra under the Pontjagin product acts in Z(M). There will be an exercise about what happens on M=S1, where Diffor(S1)∼S1, and MCG(S1)=⋆ a point.

Boundary Conditions

Given a TQFT Z:Cobord,d−1→Vec and M a d-manifold with boundary. A map Z(M):Z(∂M)→C induces a boundary condition Z∂ to Z. In particular, what this gives you is a vector Z∂(∂M)∈Z(∂M).

Given such a boundary condition, we can construct Z(M,Z∂ at ∂M)∈C as a composite CZ∂(∂M)→Z(∂M)Z(∂M)→C.

As an example, consider Z:Cobor2→C, for every closed 1-manifold N, Z∂(N)∈Z(N). We can consider Z∂(point):1→Z(point), where we can take Hom(1,Z(point)) as a category since C is a 2-category. This category contains all boundary conditions, as we are just picking out a vector.

Lecture 2: Mon Jun 12 13:04:01 2023

Summary of previous lecture

As a reminder, a TQFT is some kind of functor Z:Bordord→C which may or may not have some more enriched structure.

Mirror symmetry is an equivalence of two kinds of TQFTs.

2d Mirror Symmetry

Given a symplectic manifold M, there is a 2d TQFT Z2dA,M:Bordor2→Cat, called the 2-dimensional A-model. Given a complex algebraic manifold M, there is another 2d TQFT Z2dB,M:Bordor2→Cat called the 2-dimensional B-model. When evaluating the A-model at a point, you get a Fukaya category of M, and evaluating it at a circle, you get quantum/symplectic cohomology of M

In the complex algebraic case, the evaluation at a point is some derived category DbCoh(M) of coherent sheaves on M, and for the circle, some version ⨁p,qHp(M,Ωq) of Hodge cohomology. Mirror symmetry is an equivalence Z2dA,M≅Z2dB,M. 3d mirror symmetry says something analogous.

Gerstenhaber algebra

• Local operators

  Recall we wanted to make sense of the partition function with some boundary condition, or a defect. Currently, let the defect on the boundary be just a point. We want to say something akin to “everything away from the defect is the original TQFT, but at the defect it could be different”.

  Definition: A local operator in a TQFT Z is a vector O∈Z(Sd−1).

  Let M be a d-manifold, and consider a knot K inside M. The boundary of a neighborhood of the knot is Sd−2×S1, with neighborhood Sd−2×D. Z(MO) is a composite

  CZ(…K)→Z(Sd−2×S1)Z(complement)→C.

  Definition: The vector space of line operators is Z(Sd−2×S1). Imagine now we cut M along some submanifold N so it cuts the knot K. The “local line operators” give objects of Z(Sd−2). To a d−1-manifold N with defect points O∈N, Z(N) is a composite of

  CZ(…O)→Z(Sd−2)Z(N∖pt)→C.

  The upshot is that line operators give objects of Z(Sd−2) as a category of line operators.

  We can introduce algebraic structures on Z(Sd−1),Z(Sd−2).

  Say D is a d-dimensional ball, and consider an embedding of D⊔k↪D. To this embedding we can create a cobordism D:(Sd−1)⊔k→Sd−1. Thus, if we have a TQFT, from this induced cobordism, we get an operation Z(Sd−2)⊗k→Z(Sd−1) taking k local operators and giving a local operator Now consider the space of embeddings of k-balls Efrd(k) given the Whitney topology. If Z(Sd−1) is a chain complex, then we get C∙(Efrd(k))⊗Z(Sd−1)⊗k→Z(Sd−1) Then we get a co

  Efrd(k)→Hom((Sd−1)⊗k,Sd−1)Z→HomCh(Z(Sd−1)⊗k,Z(Sd−1))

  Now suppose we nest balls inside the embedded balls. We can combine all nested operations into Efrd as an operad. The upshot is that Z(Sd−1) is an Efrd-algebra.

  Let’s introduce Ed(k)=Embfr(D⊔k,D) the collection of framed embeddings.

  Proposition: Ed(k)≃Confk(Rd) the space of configurations of k points in Rd. For the framed version Efrd(k)≃SO(d)k×Ed(k).

  What now is H∙(Ed(k)fr,C)?

  Definition: a Pd-algebra is a chain complex A with graded commutative multiplication m:A⊗A→A and a Lie bracket {−,−}:A⊗A→A[1−d] satisfying the super-Leibniz rule.

  Such algebras have an operad. Let Pd(k) be the graded vector space of k-ary operations on a Pd-algebra. For example, Pd(1)=C, and Pd(2)=Cm⊕C{−,−}[d−1]. A P2-algebra is called a dg Poisson algebra.

  Proposition: H∙(Efrd(k))≅Pd(k) for d⩾2.

  The upshot is that if we start with some d-TQFT valued in chain complexes, and look at H∙(Z(Sd−1)), then this is a Pd-algebra.

  The example we will be interested in is d=3. We have H∙(Z(S2)) is a P3-algebra with Poisson bracket of degree −2. Recall that Z(Sd−2) is the category of line operators. This carries an Efrd−1-algebra structure.

  • E2-category is a braided monoidal category
  • E1-category is a monoidal category
  • Efr2 is called a balanced monoidal category

  In the case d=3, Z(S1) the category of line operators is a braided, balanced monoidal category.

Lecture 3: Mon Jun 12 16:07:19 2023

Goals

• What is supersymmetry?
• What is a topological twist?

Field Theories

Regardless of how we model a (quantum) field theory, it will always involve dg (differential graded) linear algebra. i.e. some E a graded vector space over C and differential d (satisfying d2=0 of degree 1) E could model:

• quantum states
• classical/quantum observables
• classical fields

If g is a Lie algebra, we can equip E with a g-action.

Example

• Field theory on R4, where g=so(1,3) (the Lorentz algebra)
• Poincare algebra so(1,3)⋊R4
• Complexified gC=so(k,C)⋊C4, in case we don’t want to worry about choice of signature

Supersymmetric Field Theories

Let n be a dimension. Defintion: A field theory (E,d) is an n-dimensional supersymmetric theory with supersymmetry Σ if E carries an additional Z/2-grading (so in total Z×Z/2-graded) and d has bidegree (1,0) wrt this grading and (E,d) has an iso(n,C)=so(n,C)⋊Cn action that lifts to an action of the super Poincare algebra associated to Σ. Some things to unpack here.

Definition: A super-Lie algebra g is a Z/2-graded vector space with Lie bracket [−,−] such that it is graded-antisymmetric: [x,y]=(−1)|x||y|+1[y,x], and satisfies the graded Jacobi identity.

Σ is a spinorial representation of so(n,C), which means it is a finite sum of spin or semi-spin representations.

A super-Poincare algebra is a super-Lie algebra with underlying graded vector space iso(n,C) as its even part, and Σ as its odd part. i.e. iso(n,C)⊕ΠΣ. This algebra is equipped with the brackets

• usual bracket on the even part
• action of so(n) on Σ in between the direct summands
• so(n,C)-equivariant map Γ:Sym2Σ→Cn which is nondegenerate

For example in n=3, so(3,C)≅sl(2,C), and Spin(3,C)≅Sl(2,C). The spinorial representations look like S⊗W=SN, where W is isomorphic to CN. Then Γ:Sym2(S⊗W)→C3≅Sym2S. But then the domain is ≅Sym2(C)⊗Sym2(W). Thus Γ is given by specifying a linear map g:Sym2(W)→C that is nondegenerate, i.e. an inner product on W. The super Poincare algebra given by (W,g) is called the N=dimW 3d complex super Poincare algebra.

The group O(W) (the orthogonal group) acts on the super Poincare algebra. These are called R-symmetries.

Twisting

Let (E,d) be acted on by siso(n|Σ), the super Poincare algebra with odd part Σ. Let Q∈Σ, and suppose [Q,Q]=0. Let α(Q)∈End(E) be the action of Q.

Definition: The twist of (E,d) of Q, (E,d+α(Q))=(EQ,dQ). Recall the degree of d was originally (1,0), while the degree of α(Q) is (0,1). So the new differential dQ is only naturally Z/2-graded, and not a priori Z-graded.

Suppose that E itself carries a Lie bracket and the action of siso(n|Σ) as an inner action, i.e. we have a homomorphism H:siso(n|Σ)→E, with the action of X∈siso(n|Σ) by [H(X),−]. Also suppose that X is Q-exact in siso(n|Σ), i.e. X=[Q,Q′] for some Q′. Then H(X)=H([Q,Q′])=[H(Q),H(Q′)] is exact wrt dQ=d+[H(Q),−]. Thus it vanishes in dQ-cohomology.

Definition: If the map [Q,−]:Σ→Cn is surjective, we say that Q is topological. The twisted theory (EQ,dQ) is called a topological twist.

Defintion: Nilp={Q∈Σ∣[Q,Q]=0}. and PNilp=(Nilp∖0)/C×.

Lecture 4: Mon Jun 12 18:02:13 2023

Gauge Theories

3d N=4 theories became a subject of interest in the mid 90s as a dimension reduction of 4d N=2 theories. Something that came up from this study was the discovery of 3d mirror symmetry.

A large family of quantum field theories are the non-linear gauged sigma models labeled by (G,X) where G is a compact Lie group and X is a smooth hyper-Kahler manifold with G action. (Sigma model here is just a general term involving a theory of maps into manifolds).

Some specializations of this concept:

1. A non-gauged sigma model has G=1 and X hyper-Kahler. The B twist is called Rozansky-Witten theory
2. Let X be a point and any G. This is pure 3d N=4 gauge theory
3. Linear gauge theory, or a linear gauged sigma model (GLSNs) We take here X to be a hyper-Kahler (quaternionic) vector space. The action of G then means X is a quaternionic representation of G. This is what people mean when they say 3d N=4 gauge theories a. Cotangent matter, so W=T∗V=V⊕V∗ where V is a complex representation of G

Hyper-Kahler manifolds

Definition: A hyper-Kahler manifold X is a Riemannian manifold (X,γ) with a P1 of Kahler structures compatible with γ. Typically, P1 is viewed as a the sphere in R3 with basis I,J,K. The associated complex structures associated to I,J,K satisfy I2=J2=K2=−1=IJK, which also implies IJ=K and JK=I, and KI=J. All of this implies that a generic αI+βJ+γK where √α2+β2+γ2=1 is also −1. There are also associated Kahler forms ωI,ωJ,ωK, which are real nondegenerate two forms of Hodge type (1,1) in its complex structure such that the metric γ(v,w)=ωI(v,Iw), same for J,K.

ALso, ΩI=ωJ+iωK has type (2,0) in I and a nondegenerate holomorphic 2-form. i.e. a holomorphic symplectic form. Similarly, ΩJ,ΩK are holomorphic symplectic forms cyclically permuting I,J,K. Oftentimes we fix one such complex structure I with data γ,I,ωI,ΩI. Algebraic geometers choose a complex structure and throw everything else out. In particular, there is no information about the metric.

Definition: A hyper-Kahler vector space is a vector space with real dimension divisible by 4 (equiv. complex dim divisible by 2). The main example is W=C2n. In a fixed complex structure I, there are holomorphic coordinates (X1,…,Xn,Y1,…,Yn). In these coordinate the metric is

γ=n∑i=1(|dXi|2+|dYi|2)=|d→X|2+|d→Y|2.

The Kahler metric is

ωI=i2(d→X∧d¯→X+d→Y∧d¯→Y).

We note we can view the metric as a Hermitian inner product on C2n and Ω as a symplectic form on C2n. Then

Aut(W;I,γ,Ω)={g∈GL(2n,C)∣g†γg=γ,gTΩg=Ω}=U(2n)∩Sp(2n,C)=USp(n).

Then a linear 3d N=4 gauge theory is G,W≅C2n, and ρ:G→USp(n).

Lecture 5: Tue Jun 13 10:34:03 2023

Idea: if you have a compact Lie group G and a quaternionic representation W of G, you can associate 3d TQFTs Z3dA,Z3dB. If (G,W) is a 3d mirror to another pair (G′,W′), then the A model Z3dA and B model Z3dB′ will be equivalent

The Gauged Gromov-Witten Invariant

• 2d Case

  To start with an easier case, consider a 2d A-model: Let G be a compact Lie group with an identification g≅g∗, and W a unitary G-representation (or more generally, W is some hyper-Kahler manifold with a G-action). Recall a unitary G-representation is a Hermitian vector space W and a homomorphism G→U(W). Then Z3dA,W//G is a 2d TQFT, with Z3dA,W//G(Σ) the gauged Gromov-Witten invariant of W, which counts the solutions of some PDE on Σ, where Σ is some Riemann surface. (The vanilla GW invariant is when G is trivial.) The PDE in question is called the symplectic vortex equation. Before we write it, let’s introduce a function

  μ:W→g∗v↦12(xv,v)

  where x∈g. μ is a moment map for the G-action on W.

  Choose a principal G-bundle P→Σ and a connection ∇ a connection on P, and a smooth section φ∈Γ(Σ,PG×W). The symplectic vortex equations are

  ¯∂φ=0∈Ω0,1(Σ,PG×W)⋆F+μ(φ)=0∈Ω0(Σ,adP).

  We recall adP=PG×g→Σ is the adjoint bundle. The gauged GW invariant counts solutions of these equations modulo gauge transformations. The dimension of the moduli space of solutions is

  d=(2−2g)(dimCW−dimG)+2deg(PG×W).

  Remarks

  • If G is trivial, no principal bundle is here, the second equation drops out, and the first equation just says that φ is a holomorphic map.
  • If W=0, then the first equation drops out, and the second equation just says that the curvature of the connection is zero. In general there are few flat connections compared to non-flat connections, but it is a theorem in this case that the moduli space of flat G-bundles is identified with the space of holomorphic GC-bundles, the complexified G-bundles

• 3d case

  Again in the 3d case there will be a pair of equations. Fix again a compact Lie group G, W a quaternionic representation of G. Recall that Spin(3)=SU(2)→SO(3), which can be thought of as unit quaternions. If we have u∈H a unit quaternion, it acts on ImH=R3 by conjugation. Let S=H the spin representation by action u⋅x=ux. There is a natural map c:V⊗S→S where V=\imH. This is a map of Spin(3)-representations called Clifford multiplication.

  A spin structure on a Riemannian 3-manifold N is a Spin(3)-bundle P→N with an identification PSpin(3)×V≅TN compatible with metrics. The spinor bundle associated to the spinor representation is SN=PSpin(3)×S. Spin structures are obstructed by w2(N)∈H2(N,Z/2) the second Stiefel-Whitney class. The set of spin structures has a free transitive action by H1(N,Z/2).

  • Dirac operator

    Let φ∈Γ(N,Sn⊗HW). Given the Riemannian structure, we have the LV connection on P, so we can take the covariant derivative ∇φ∈Ω1(W,SN⊗HW). We can rewrite the latter collection as Γ(N,PSpin(3)×(V⊗HS⊗HW)) which has the Clifford multiplication mapping to

    Γ(N,PSpin(3)×(S⊗HW))=Γ(N,Sn⊗HW).

    This composite is called the Dirac operator \slashed∇φ. The equation \slashed∇φ=0 is the Dirac equation.

    Now let W=0. Let P be a principal G-bundle, ∇ a connection on P, and σ∈Γ(N,adP). The Bogomolny equation is

    ⋆F+∇σ=0∈Ω1(N,adP).

    It is a fact that if N is closed, ∇σ=0.

  • Sieberg-Witten Equations

    Now lets introduce the Seiberg-Witten equation. The Seiberg-Witten equation happens usually when G=U(1),W=H. Let’s introduce SpinG(3)=(Spin(3)×G)/±1, i.e. −1∈G acts by −1 on W.

    A relevant notion is the structure SpinC(4)=SpinU(1)(t)=(SU(2)×U(1))/±1=U(2). Set ¯G=G/±1, ¯P=PGׯG. The data that go into the Seiberg-Witten equations consists of

    • a principal SpinG(3)-bundle P→N with identification PSpinG(3)×V≅TN
    • a connection ∇ on ¯P along with the LC connection on P
    • σ∈(N,adP)
    • φ∈Γ(N,PSpinG(3)×W).

    The actual equations are then

    \slashed∇φ=0⋆F+∇σ+μ(φ)=0

    μ is the correct analogue of the moment map as in the 2d case.

    One fact is that the linearization is elliptic, and the index of the elliptic operator is 0.

  • The U(1) Case

    Consider the case where W=H and G=U(1). In this case the spin structure is SpinC on N, which has a free transitive action by h∈H2(N,Z). Then for c∈SpinC we get a c+h∈SpinC, and SWN(c) counts the solutions.

Lecture 6: Tue Jun 13 12:44:10 2023

Recall we constructed the Seiberg-Witten invariant of a manifold, which counts the solutions to a system of PDEs. Fix some c0∈SimC on N, let H=H1(N,Z)≅H2(N,Z). We can build a “total” invariant SWN,c0=∑h∈H~SWN(c0+h)h∈Z(H). (cf taking the sum of Stiefel-Whitney numbers) If we mod out by H, we can get a number independent of the choice of spin structure.

This all holds for the A-model

Topological Interpretation

Now what is Z3dB,W///G(N)?

• Reidemeister Torsion

  Historical note: There is a class of 3-manifolds called lens spaces L(p,q), which ones are homotopy equivalent? Which ones are homeomorphic? There exist examples of homotopy equivalent lens spaces that are not homeomorphic, distinguished by their R torsion.

  Let N be a finite connected CW complex with basepoint x, R a commutative ring, k a field, and R→k a homomorphism. Say L is a free, finite rank R-module which is a representation ρ:π1(N)→GLR(L) Say ˜N→N is a universal cover, and lift the CW structure to ˜N. e.g. we can lift the 1-cell structure of S1 to R by taking the points to the Z-lattice, and the 1-cell lifts to the intervals between the Z-points.

  Consider C∙(˜N,Z) the chain complex of cells of the universal cover endowed with the action of π1(N). Thus this is a chain complex of Z[π1(N)]-modules. Define C∙(N,L)=C∙(˜N,Z)⊗Z[π1(N)]L twisting the coefficients by the representation. Assume C∙(N,L)⊗Rk is acyclic, i.e. the homology is trivial

  Goal: To define a torsion τ(N,L)∈k×/∼ with ∼ to be defined.

  You could think of this as a “determinant” of d in the chain complex.

  Fact: there exists a homotopy h:C∙(N,L)⊗Rk→C∙+1(N,L)⊗Rk such that dh+hd=id called the contracting homotopy.

  Consider d+h:Ceven(N,L)⊗Rk→Codd(N,L)⊗Rk. Fact: It is an isomorphism. We want to compute its “determinant”. First we should give bases for C∙(N,L)⊗Rk. Choose

  • R-basis of L
  • ordering of d-cells of N for all d
  • a Turaev spider

  Definition: A Turaev spider on N is a path from x to every cell in N.

  Given these choices,

  C∙(˜N,Z)=⨁d-cellsZ(π1(N)).

  Adding in the basis of L gives a basis of the R-module C∙(N,L)=C∙(˜N,Z)⊗Z[π1(N)]L. This implies the det(d+h)∈k×/(R×det(H1(N,Z))). Varying the first two choices pushes us around by a factor of R×, while changing the choice of Turaev spider pushes us around by the determinant of a map induced from the homology group. This det(d+h) is called the Reidemeister torsion, and is well defined up to these choices.

• Milnor Torsion

  Assume H=H1(N,Z) has no torsion of rank r. Take R=Z[H]=Z[t±11,…,t±1r], and k=Q(H)=Q(t1,…,tr). Let L=R and ρ:π1(N)→H1(N,Z)→L. Let Z(H)×=±H. We get that τ(N)∈Q(H)/±H called the Milnor torsion.

  Example:

  • For S1, H1(S1)=Z, so τ(S1)=11−t.
  • τ(S1×S2)=τ(S1)=11−t.

Connection to SW Invariants

Two Turaev spiders S1,S2 are equivalent if S1−S2=0 in H1(N,Z). A theorem of Turaev says that if N is a closed oriented 3-manifold, there is a natural isomorphism between {spiders} and {SpinC structures}. On the former there is a natural action of H1(N,Z) and the latter has an action by H2(N,Z).

Let N be a closed oriented 3-manifold and choose a SpinC structure c, also giving a spider on N. In this case, there exists a refined torsion (due to Turaev) τc(N)∈Q(H)/±1.

Theorem: Assume b1(N)>1 where N is a closed oriented 3-manifold. Choose a SpinC structure c. Then SWN,c=τc(N)∈Z(H)/±1.

Recall that SWN=Z3dA,H///U(1)(N). In fact, τ(N)=Z3dB,H(N). We can generalize this relationship to more general targets, where V is a unitary G-representation, and W=V⊗CH “cotangent matter”. Then Z3dB,W///G(N) is also related to torsion. The theorem then provides an equivalence between the 3d TQFTs. This is an example of mirror symmetry of (U(1),H)↔(∙,H).

Lecture 7: Tue Jun 13 15:44:00 2023

Review

The goal of this lecture is write down fields and susy actions. As a reminder of yesterday, hyper-Kahler manifolds are Riemannian manifolds who have a P1 worth of Kahler structures parameterized by the sphere S2 in coordinates I,J,K. The group USp(n)=U(n)∩Sp(n) acts as isometries of W=C2n. Gauge groups in these field theories will act as subgroups of this group. In order to write down equations of motion for fields, we need a notion of a moment map.

Moment Maps

Definition: A continuous HK G-action on a HK W comes with a P1-worth of moment maps: functions W→g∗ such that V=ωIdμJ=ω−1JdμK=⋯ where V is the vector field generating the G-action. Like with coordinates, there are moment maps for each complex structure μI,μJ,μK. In the case of a vector space W, these moment maps are always quadratic.

In a fixed complex structure I with ωI,ΩI, we can split the moment map into real part μR=μI and μC=μJ+iμK. It will turn out that Ω−1dμC is a holomorphic vector field that complexifies the action we started with: a complexified holomorphic GC-action on W. In general this does not preserve the metric, and it acts on W only as a complex symplectic vector space.

• In the case of cotangent matter, W=T∗V, ρ:G→U(n)→USp(n) by g↦diag(g,g∗).

Let Vn denote the n-dimensional complex irreducible representation of SU(2). Denote ¯Vn its dual/conjugate, meaning SU(2) acts by g as ¯g.

Fact: ¯Vn≅Vn because for all g∈SU(2), ¯g=ϵgϵ−1 where \epsilon =\begin{pmatrix}0 & -1 \\1 & 0 \\end{pmatrix}. Let ea a∈{±} denote a weight basis for V2∈C2. Here e+=diag(1,0) and e−=diag(0,1). The matrix diag(eiθ,e−iθ) acts on e± as e±iθ. Let ϵ:V2∼→¯V2. Then ea=ϵabeb is a weight basis for ¯V2.

Note V2⊗V2≅V3⊕V1, The map that goes from the former to V1 can be though of as ϵ. The other intertwiner map that goes from the former to V3 is called σ, the matrix elements of which are the Pauli matrices.

Supersymmetry

Recall the 3d N=4 susy algebra is 3+8-dimensional, even and odd dimension respectively, and is a representation of SU(2)E=Spin(3)E. It is a representation of SU(2)E×SU(2)H×SU(2)C and has bracket

{Qa,a′α,Qbb′β}=ϵabϵa′b′σμαβ∂μ

with all other brackets vanishing. The H and C stand for “Higgs” and “Coulomb”. The E stands for Euclidean. The susy algebra lives inside an extension of itself by SU(2)E by semidirect product which itself sits inside the extension by SU(2)E×SU(2)H×SU(2)C via semidirect product.

3d mirror symmetry is nothing but the swapping of the last two copies of SU(2) in the formulation.

Fields

Fix a gauge theory labelled by G and a hyper-Kahler vector space W, and ρ:G→USp(W). We want to describe gauge theory on R3. Classically, gauge theory is a space of sections of various bundles on R3 together with differential equations of motion such that the solutions to the equations of motion have an action of the susy algebra extended at least by SU(2)E via semidirect product.

A quantum field theory with 3d N=4 susy is the same data except the equations of motion are replaced with some function of distribution to be used in a path integral.

Fix a trivial principal G-bundle E on R3. There is a connection A=Aμdxμ on E living in sections of g⊗VE3,R.

Lecture 8: Tue Jun 13 18:01:16 2023

Fix a gauge theory with data G,W,ρ:G→USp(W).

Let A=Aμdxμ be a connection on a principal G-bundle as a section of g⊗Ω1(R3). So that Aμ∈g⊗RV3,RE. We also have dA=d+A(=∇), F=d2A, Φm′∈gR⊗RVC3,R, λaa′α∈Π(g⊗VE2⊗VC2⊗VH2), with the VE2 being the spin part.

Fermions are called gaugeinos (fermion related to the gauge boson)

We also have Zai∈VH2⊗CW such that ¯Zai∈ϵabΩijZbj. Also Ψa′i′α∈Π(VE2⊗VC2⊗CW). Here, Π means parity shift in the sense of a supervector space.

Schematic SUSY

The supercharges act on matter by QA∼λ, QΦ∼λ, Qλ∼F+dAΦ+[Φ,Φ]+μ(Z), QZ∼Ψ, QΨ∼∂AZ+ΦZ.

∂αβ=σμαβ∂μ as a map VE2→¯V2E. Also ∂αβ:=(σμ)αβ∂μ. It is convenient to rewrite

Aαβ:=σμαβAμΦa′b′:=σa′b′m′Φm′μ(Z)ab=σabmμm

in SU(2)E,SU(2)C,SU(2)H, respectively.

The differential operator on VE2⊗CVC2 as

Dα,a′β,b′=(∂+A)αβδa′b′+δαβΦa′b′