Topological quantum field theories (TQFT) are the latest chapter in the long and fruitful discourse between mathematics and physics. We use these results in Chapter 6 to describe two categories of surface operators in GL.
Supersymmetry
Twisting
While the original actions S are not invariant under U0(n), there exists an invariant action Se identical to the original action S on the Euclidean space Cn.
BRST Cohomology and Metric Independence
As we will see in the following sections, the categories and 2-categories that appear in TQFT have an additional structure. Note that the composition axiom gives VAA and Vf f the structure of an associative algebra, which we will call the endomorphism algebra.
The Category of Boundary Conditions in Two-Dimensional TQFTs
Consider two local observables OAB and OBC sitting at the appropriate intersections of intervals with boundary conditions A, B and C (see Figure 3.2). Composition of morphisms comes from contraction of the interval B (a trivial operation in a TQFT), fusion of the local observables.
The 2-Category of Two-Dimensional TQFTs
The composition of morphisms comes from contraction of the world sheet by TQFTY (a trivial operation in a TQFT), fusion of the defect lines. The objects in this category are fault lines, and the morphisms are local observables at the intersection of two fault lines.
The 2-Category of Boundary Conditions in Three-Dimensional TQFTs
Furthermore, we can compact S1 (a trivial process in a TQFT) to obtain a two-dimensional TQFT with the appropriate boundary condition. Therefore, the category of line observers in a three-dimensional TQFT is equivalent to the category of boundary conditions in the two-dimensional TQFT obtained by compaction S1.2.
The 2-Category of Surface Observables in Four-Dimensional TQFTs
A-Model
The fermionic fieldsχ,χare scalars on Σ, valued in the pullback of the holomorphic tangent bundle TX and the antiholomorphic tangent bundle TX, respectively. The fermionic fieldsρ,ρare a (0,1)-form, (1,0)-form on Σ, valued in the pullback of the holomorphic tangent bundle T
B-Model
The fermionic field ρ is a 1-form on Σ, valued in the withdrawal of the holomorphic tangent bundle TX,. The fermionic fieldsψandχare scalars on Σ, valued in the withdrawal of the antiholomorphic tangent bundleT
Mirror Symmetry
Conversely, we can construct a line observable given an antiholomorphic, flat superconnection∇=∂+Aon aZ-graded smooth vector bundleEoverX,. The isomorphism of line observables implies Kontsevich's homologous mirror symmetry conjecture [23] (with an appropriate generalization of the Fukaya category).
A-Type Gauge Theory
First consider two parallel defect lines with a piece of the A-type 2D gauge theory between them. We would like to show that this is equivalent to the trivial defect line in the A-type 2D gauge theory.
B-Type Gauge Theory
To define them, it is necessary to choose a finite-dimensional graded representation of V from Gand to consider the holonomy of A in the representation of V. This means that we can include the holonomy of A in the representation of V in the path integral.
Gauged B-Model
The fermionic fieldsη and β are scalars unM valued in the retraction of the tangent bundleT X, η∈Γ σ∗T X. The fermionic fieldsψen χ are 1-forms unM valued in the retraction of the tangent bundleT X, ψ∗Γ σ∗Γ Xσ Ω1. It is convenient to introduce an auxiliary a scalarP and auxiliary 1-form ˜P valued in the retraction of the tangent bundleT X,.
Rozansky-Witten Model
The fermionic fieldsχ are a 1-form onM valued in the retract of the holomorphic tangent bundle T X,. Local observables in the Rozansky-Witten model are elements of the BRST cohomology on smooth functionals of scalar fields,σi,σ¯i andη¯i. Line observables in the Rozansky-Witten model are in one-to-one agreement with the boundary condition for the two-dimensional theory obtained by compression on a circle [18].
B-Type Gauge Theory
Local observables in this TQFT are gauge-invariant functions ofρ, which correspond to elements in the external algebra Λ•(g) invariant with respect to the additive action. The supertrace of the holonomy of N along a curve γ in M is therefore a BRST-invariant, gauge-invariant loop operator in topological gauge theory. To see this, we can use the dimensional reduction trick and identify the category of line operators in the 3d theory with the category of branes in the 2d theory obtained by compacting the 3d theory onto a circle.
Gauged A-Model
The subcategory formed by line operators described above is the G-equivariant derived category of DG-modules over the DG-algebra Λ•(g) (with zero differential). The space of morphisms between the line operators is the cohomology of δ on the G-invariant part of HomC(V1, V2⊗Λ•(g)). More generally, one can also consider Gukov-Witten type line operators for which the conjugation class of the holonomy of Ais is fixed. also in the additional representation).
Gauged Rozansky-Witten Model
Linear A-Model
The linear A model is constructed by twisting the SU(2)L rotational symmetry of the N = 2 linear σ model with the diagonal subgroup of the SU(2)R×SU(2)X symmetry (see Table 6.1 for the charges of fields in theN = 2 linearσ model). We construct the action for the linear A model by writing the action of the N = 2 linearσ model (6.1) covariantly in terms of the rotated fields. It is not difficult to construct an action equivalent to the original linear A-model action (6.3) that imposes the appropriate constraint on the auxiliary field (6.4) and respects the BRST symmetry (6.5).
A-Model
It will be convenient to introduce an auxiliary bosonic fieldP which is a 1-form onM valued in the retraction of the tangent bundleTRN,. The bosonic fieldτ is an antiselfdual 2-form onM valued in the retraction of the tangent bundleT X,. We construct the action for the A-model by making the linear A-model action (6.6) covariant and adding the appropriate curvature terms.
N = 4 SYM Theory
Geometric Langlands Theory
In this section we show that the twisted GL theory att=uncompacted in a circle is equivalent to a measured version of the Rozansky-Witten model. The asymptotic form of the G-invariant in the target space can be read from the L4 term of the action and is proportional to. Furthermore, the G-invariant K¨ahler form in the target space can be derived termL2 of the action and is proportional to.
Specifically, in the case of the RW model with target T∗[2]C∗, the category of boundary line operators VX0X0 is equivalent to Db(Coh(C∗)). This gives us a putative description of the 2-category of surface operators in the overall 4d gauge theory. From the 3d point of view, h0 determines the limit value of scalarh= exp(−2πτ) in the RW sector.
Indeed, fixing a semisimple conjugation class of exp(−2π(A4+iφ4)) is the same as fixing a semisimple conjugation class of the limiting holonomy of the complex compound A+iφ in the 4d gauge theory. In other words, it is the GC-equivariant derivative category of the graded algebra⊕pSympgwhere depth component is in cohomological degree 2p. By analogy with the Rozansky-Witten model, we propose that the most general surface operator att=i (or equivalently, the most general boundary condition in the 3d theory) can be defined as a module category of the monoidal category of boundary line operators for the respective boundary condition X0 .
This category is the brane category in the 2d TQFT obtained by compacting the 3d TQFT in an interval, with the boundary conditions at the two ends given by XandX0. Let's start with the Dirichlet condition which states that the restriction of A on the boundary is trivial. This means that the gauge symmetry is continuous in the limit and the restriction of the form 1 ?F vanishes.
Electric-Magnetic Duality
On the other hand, the category of boundary line operators in the Rozansky-Witten model is Db(Coh(C∗)), as explained in [18]. On the B side, reduction on an interval yields a B model with targetC∗×C∗, therefore the category of boundary line operators is Db(Coh(C∗×C∗)). On the A side, reduction yields an A model of measure T∗U(1)×T∗U(1), therefore the category of boundary line operators is the Fukaya–Floer category.
His action on the category of line operators amounts to a T-duality on T2 (duality acts trivially on the business sector). So this is the category of line operators sitting on the Gukov-Witten surface operator. This works trivially on the category of line operators since the fieldσ is not involved in the duality.
Observables
Having constructed the holomorphic theory, we now need to find physical observables on closed K¨ahler surfaces.1 From supersymmetry variations, we see that the gauge-invariant physical observables in the SQCD sector of the theory are the sticky ball, mesons, and their descendants . . The gauge-invariant physical observables in the SYM sector of the theory are left chiral gluons and their descendants. If we restrict the path integral to the trivial bundle U(1)5 and eliminate the constant mode of the auxiliary field H0, there are two physical gauge-invariant observers.
Correlation Functions
For closed K¨ahler surfaces with h2,0(X)>0, we can arrive at a simpler modulus space by adding twisted mass terms for the matter fields. However, these expressions change the modulus space of bosonic fixed points, and add an additional constraint equation. Choosing the twisted mass to be nondegenerate outside the canonical divisor, the moduli space of solutions to equations and (7.27) is simply the moduli space of instantonsM,.
N = 1 Chiral Model
Holomorphic Chiral Model
Observables
Our next task is to construct physical observables on closed Kühler surfaces.2 From the supersymmetry variations we see that the physical observables in the chiral model sector are the mesons, the auxiliary scalar φX and their descendants. The gauge invariant physical observables in the SYM sector of the theory (constraining the path integral to the trivial U(1)5 bundle and eliminating the constant mode of the auxiliary field H0) are the gluinos.
Correlation Functions
To evaluate the path integral over twisted chiral model fields, we use the following localization technique: the saddle point approximation around fixed points of any Grassmann symmetry is exact [41]. For K¨ahler surfaces withh2,0(X)>0, we can simplify this modulus space by making the following perturbation of the superpotential. For closed K¨ahler surfaces withh2,0(X)>1 there are generically no solutions to the perturbed saddle point equations.
Seiberg Duality
By performing the Gaussian path integral over quadratic fluctuations and summing the contribution of each saddle point, we find that the non-vanishing correlation functions (up to normalization of the observables) on simply-connected K¨ahler surfaces meth2,0( X) = 1 is. By normalizing the measure of the holomorphic chiral model by Λ−4χ, we find that correlation functions in these theories match exactly under the duality map, providing a highly non-trivial test of Seiberg's conjecture.
