Lectures on Mathematical Aspects of (twisted) Supersymmetric Gauge Theories

Abstract

Supersymmetric gauge theories have played a central role in applications of quantum field theory to mathematics. Topologically twisted supersymmetric gauge theories often admit a rigorous mathematical description: for example, the Donaldson invariants of a 4-manifold can be interpreted as the correlation functions of a topologically twisted $\mathcal {N}=2$ gauge theory. The aim of these lectures is to describe a mathematical formulation of partially-twisted supersymmetric gauge theories (in perturbation theory). These partially twisted theories are intermediate in complexity between the physical theory and the topologically twisted theories. Moreover, we will sketch how the operators of such a theory form a two complex dimensional analog of a vertex algebra. Finally, we will consider a deformation of the $\mathcal {N}=1$ theory and discuss its relation to the Yangian, as explained in [8, 9].

Appendix

1.1 A. Moduli Problems and Field Theories

Throughout this text, field theories are described in terms of elliptic moduli problems which in turn are encoded as elliptic $L_\infty$-algebras. These terms and their relations are constantly used. However, we only defined SUSY field theories in an informal way, so we will give some ideas and definitions here. For the full definitions and detailed explanations, see [7].

Let $M$ be a manifold. The ideal definition of a classical field theory would be to say that a classical field theory on $M$ is a sheaf of derived stacks (of critical loci, the derived spaces of solutions to the equations of motion) on $M$ equipped with a Poisson bracket of degree one (coming from the BV formalism). To simplify things, we make two observations.

• 1. If $X$ is a derived stack and $x\in X$, then $T_xX[-$ $1]$ has an $L_\infty$ structure, and this completely describes the formal neighborhood of $x$ [18, 22, 23, 29]. Thus, near a given section, a sheaf of derived stacks can be described by a sheaf of $L_\infty$-algebras.

• 2. If $X$ is a derived stack which is $n$-symplectic in the sense of [25] , then $T_xX$ has an anti-symmetric pairing (of degree $n$), so $T_xX[-$ $1]$ has a symmetric pairing (of degree $n-2$). One can show that the $L_\infty$-structure on $T_x X[-$ $1]$ can be chosen so that the pairing is invariant. More precisely, one can prove a formal Darboux theorem showing that formal symplectic derived stacks are the same as $L_\infty$-algebras with an invariant pairing.

From these observations it makes sense to define a perturbative classical field theory (perturbing around a given solution to the equations of motion) to be a sheaf of $L_\infty$-algebras with some sort of an invariant pairing, which we will define below. Moreover, we are interested in the situation where the equations of motion (or equivalently our moduli problem) are described by a system of elliptic partial differential equations, which lead to the following notion.

Definition 6

Anelliptic $L_\infty$ -algebra $\mathcal {L}$ on $M$ consists of

• a graded vector bundle $L$ on $M$, whose space of sections in $\mathcal {L}$,

• a differential operator $\mathrm {d}:\mathcal {L}\rightarrow \mathcal {L}$ of cohomological degree 1 and square 0, which makes $\mathcal {L}$ into an elliptic complex,

• a collection of polydifferential operators $l_n:\mathcal {L}^{\otimes n}\rightarrow \mathcal {L}$ which are alternating, of cohomological degree $2-n$, and which give $\mathcal {L}$ the structure of an $L_\infty$-algebra.

An invariant pairing of degree $k$ on an elliptic $L_\infty$-algebra $\mathcal {L}$ is an isomorphism of $\mathcal {L}$-modules

$$\mathcal {L}\cong \mathcal {L}^![-k],$$

which is symmetric, where $\mathcal {L}^!(U)=\varGamma (U, L^\vee \otimes Dens_M)$.

Remark 8

Note that the sheaf $\mathcal {L}^!$ is homotopy equivalent to the continuous Verdier dual, which assigns to $U$ the linear dual of $\mathcal {L}_c(U)$.

Such an invariant pairing yields an invariant pairing on the space $\mathcal {L}_c(U)$ for every open $U$ in $M$. The fact that the pairing on $\mathcal {L}_c(U)$ is invariant follows from the fact that the map $\mathcal {L}\rightarrow \mathcal {L}^![-$ $k]$ is an isomorphism of $\mathcal {L}$-modules.

From deformation theory, we know that there is an equivalence of $(\infty ,1)$-categories between the category of differential graded Lie algebras and the category of formal pointed derived moduli problems (see [18, 22, 23]). Here pointed means that we are deforming a given solution to the equations of motion. Thus, the following definitions make sense.

Definition 7

A formal pointed elliptic moduli problem with a symplectic form of cohomological degree $k$ on $M$ is an elliptic $L_\infty$-algebra on $M$ with an invariant pairing of cohomological degree $k-2$.

Definition 8

A perturbative classical field theory on $M$ is a formal pointed elliptic moduli problem on $M$ with a symplectic form of cohomological degree $-$1. The space of fields $\mathcal {E}$ of a classical field theory arises as a shift of the $L_\infty$-algebra encoding the theory, $\mathcal {E}=\mathcal {L}[1].$

The field theories we consider in this text all arise as cotangent theories.

Definition 9

Let $\mathcal {L}$ be an elliptic $L_{\infty }$-algebra on $M$ corresponding to a sheaf of formal moduli problems $\mathcal {M}_\mathcal {L}$ on $M$. Then the cotangent field theory associated to $\mathcal {L}$ is the classical field theory $\mathcal {L}\oplus \mathcal {L}^![-$ $3]$ (with its obvious pairing). Its moduli problem is denoted by $T^*[-$ $1]\mathcal {M}_{\mathcal {L}}$.

1.2 B. Supersymmetry

In supersymmetry, we have two gradings: one by $\mathbb {Z}/2\mathbb {Z}$ (=  fermionic grading), and one by $\mathbb {Z}$ (=  cohomological grading, “ghost number”). So one extends the definitions from Appendix A to this bi-graded ($=$  super) setting.

In this super-setting, we want all algebraic structures to preserve the fermion degree and have the same cohomological degree as in the ordinary setting. Thus, the differential of a super cochain complex is of degree $(0,1)$ and the structure maps of a super $L_\infty$ -algebra $L$, $l_n: L^{\otimes n}\rightarrow L$, are of bi-degree $(0,2-n)$, satisfying the same relations as in the ordinary case. The other notions from Appendix A carry over similarly.

Definition 10

A perturbative classical field theory with fermions on $M$ is a super elliptic $L_\infty$-algebra $\mathcal {L}$ on $M$ with an invariant pairing of bi-degree $(0,-$ $3)$, i.e. of cohomological degree $-$3 and fermionic degree 0.

Definition 11

A formal pointed super elliptic moduli problem with a symplectic form of cohomological degree $k$ on $M$ is a super elliptic $L_\infty$-algebra on $M$ with an invariant pairing of bi-degree $(0,k-2)$.

Now we can encode supersymmetry.

Definition 12

A field theory on $\mathbb {R}^4$ with $\mathcal {N}=k$ supersymmetries is a $\mathrm{Spin }(4)\ltimes \mathbb {R}^4$-invariant super elliptic moduli problem $\mathcal {M}$ defined over $\mathbb {C}$ with a symplectic form of cohomological degree -1; together with an extension of the action of the complexified Euclidean Lie algebra $\mathfrak {so}(4,\mathbb {C})\ltimes V_{\mathbb {C}}$ to an action of the complexified super-Euclidean Lie algebra $\mathfrak {so}(4,\mathbb {C})\ltimes T^{\mathcal {N}=k}$.

Given any complex Lie subgroup $G\subseteq \mathrm{GL }(k,\mathbb {C})$, we say that such a supersymmetric field theory has $R$ -symmetry group $G$ if the group $G$ acts on the theory in a way covering the trivial action on space-time $\mathbb {R}^4$, and compatible with the action of $G$ on $T^{\mathcal {N}=k}$.