Orthosymplectic superinstanton counting and brane dynamics

Abstract

We extend the study of superinstantons presented in (Kimura and Pestun in superinstanton counting and localization, 2019) to include orthosymplectic supergroup gauge theories, $B_{n_0|n_1}$, $C_n$, and $D_{n_0|n_1}$. We utilize equivariant localization to obtain the LMNS contour integral formula for the instanton partition function, and we investigate the Seiberg–Witten geometries associated with these theories. We also explore the brane configurations involving positive and negative branes together with O-planes that realize the orthosymplectic supergroup theories.

Appendices

A Supermathematics

1.1 A.1 Superlinear algebra

We will give a very brief introduction on supergroups first, following [26]. For more about supergroups and its mathematical meanings, we refer the readers to [1, 36, 75, 76].

A supervector space is a $\mathbb {Z}_2$-graded vector space consists of both commuting and anti-commuting variables

$$\begin{aligned} V=V_0 \oplus V_1. \end{aligned}$$

(A.1)

Its superdimension

$$\begin{aligned} \text {sdim} V=\sum _{\sigma =0,1}(-1)^\sigma \text {dim} V_\sigma =\text {dim} V_0-\text {dim} V_1 \end{aligned}$$

(A.2)

can be negative.

A linear map M between supervector space V and W can be written in terms of a supermatrix

$$\begin{aligned} M=\left( \begin{array}{ll} M_{00} &{}\quad M_{01} \\ M_{10} &{}\quad M_{11} \end{array}\right) \end{aligned}$$

(A.3)

with $M_{\sigma \sigma ^{\prime }} \in \text {Hom}\left( V_\sigma , W_{\sigma ^{\prime }}\right)$. The corresponding superdeterminant (or Berezinian) is

$$\begin{aligned} \text {sdet} M=\frac{\text {det}\left( M_{00}-M_{10} M_{11}^{-1} M_{01}\right) }{\text {det} M_{11}}. \end{aligned}$$

(A.4)

And the supertrace is

$$\begin{aligned} \text {str} M=\text {tr}_0 M-\text {tr}_1 M=\text {tr} M_{00}-\text {tr} M_{11}. \end{aligned}$$

(A.5)

Also, we define the supertranspositon of a supermatrix by

$$\begin{aligned} M^{\text{ st } }=\left( \begin{array}{ll} M_{00} &{} M_{01} \\ M_{10} &{} M_{11} \end{array}\right) ^{\text{ st } }=\left( \begin{array}{cc} M_{00}^{\text {t}} &{} M_{01}^{\text {t}} \\ -M_{10}^{\text {t}} &{} M_{11}^{\text {t}} \end{array}\right) \end{aligned}$$

(A.6)

where $A^{\text {t}}$ denotes the ordinary transpose operation. We can check that $\left( M_1 M_2\right) ^{\text{ st } }=M_2^{\text{ st } } M_1^{\text{ st } }$, but $\left( M^{\text{ st } }\right) ^{\text{ st } } \ne M$ in general.

The Hermitian conjugation of a supermatrix is defined as follows:

$$\begin{aligned} M^{\dag }={{\bar{M}}}^{\textrm{st}} \end{aligned}$$

(A.7)

which satisfies .

1.2 A.2 Supergroups and superalgebras

1.2.1 A.2.1 Classification of classical supergroups

Like the ordinary classification of semi-simple Lie groups (algebras), we can also classify classical supergroups (superalgebras). The results are roughly given by the following table [5]:

For details from the mathematical side, see [36].

1.2.2 A.2.2 Unitary supergroup

We define the squared norm

(A.9)

for $\Psi \in \mathbb {C}^{n|m}$. Then the A-type (unitary supergroup) $\text {U}(n|m)$ is defined as the isometry group with respect to the supervector space $\mathbb {C}^{n|m}$ such that

(A.10)

Thus, we have

(A.11)

The bosonic subgroup of $\text {U}(n|m)$ is given by $\text {U}(n) \times \text {U}(m)$, and the fermonic part is given by ${{\textbf {n}}} \times {{{\textbf {m}}}}$ and ${{{\textbf {n}}}} \times {{\textbf {m}}}$ bifundamental representations with the dimension 2nm.

1.2.3 A.2.3 Orthosymplectic supergroup

Consider the real supervector space $\mathbb {R}^{n|2m}$. The squared norm is defined by:

$$\begin{aligned} |\Psi |^2=\Psi ^{\text {t}} \Omega \Psi =\text {str}\left( \Omega \Psi \Psi ^{\text {t}}\right) \end{aligned}$$

(A.12)

with the skew-symmetric form

$$\begin{aligned} \Omega =\left( \begin{array}{cc} \mathbb {1}_n &{} 0 \\ 0 &{} \mathbb {1}_m \otimes {{\textbf {j}}} \end{array}\right) , \quad {{\textbf {j}}}=\left( \begin{array}{cc} 0 &{} 1 \\ -1 &{} 0 \end{array}\right) \end{aligned}$$

(A.13)

Then, the isometry group of $\mathbb {R}^{n|2m}$ is given by

$$\begin{aligned} |\Psi |^2=|O \Psi |^2, \quad O^{\text{ st } } \Omega O=\Omega . \end{aligned}$$

(A.14)

Thus, we have

$$\begin{aligned} \text {OSp}(n|m)=\left\{ O \in \text {GL}\left( \mathbb {R}^{n|2m}\right) \mid O^{\text {st}} \Omega O=\Omega \right\} \end{aligned}$$

(A.15)

The bosonic part of ${{\,\textrm{OSp}\,}}(n|m)$ is thus given by $\text {O}(n) \times \text {Sp}(m)$, and the fermonic part is given by ${{\textbf {n}}} \times {\textbf {2m}}$ bifundamental representation with the dimension 2nm.

1.2.4 A.2.4 Superalgebras

Superalgebra is also graded

$$\begin{aligned} {\mathfrak {A}}={\mathfrak {A}}_0 \oplus {\mathfrak {A}}_1 \end{aligned}$$

(A.16)

with the supercommutator

$$\begin{aligned}{}[a, b]=a b-(-1)^{|a||b|} b a \end{aligned}$$

(A.17)

for $a,b\in {\mathfrak {A}}$. Here, we denote the parity of an element $x \in V_\sigma$ by $|x|=\sigma$ for $\sigma =0,1$, which is called even/bosonic for $\sigma =0$, and odd/fermionic for $\sigma =1$. The supercommutator also satisfies the super-version of Jacobi identity

$$\begin{aligned}{}[a,[b, c]]=[[a, b], c]+(-1)^{|a||b|}[b,[a, c]] . \end{aligned}$$

(A.18)

For example, we denote

$$\begin{aligned} \mathfrak {s l}_{n | m}=\left\{ a \in {\mathfrak {A}}={\mathfrak {A}}_0 \oplus {\mathfrak {A}}_1, \text {dim} {\mathfrak {A}}_0=n, \text {dim} {\mathfrak {A}}_1=m | \text {str} a=0\right\} , \end{aligned}$$

(A.19)

which corresponds to the A-type supergroup $A_{n-1|m-1}$.

B Orientifold planes

In order to realize the Hanany–Witten construction of BCD-type gauge theory, we need to introduce the orientifold planes (or O-plane), see [77, 78]. A very pedagogical review is [79, Problem 15.4]. Here, we only briefly introduce the effect of O4-planes.

For a BCD-type gauge theory, the brane configuration is:

The world volume of an O-plane reflects the supersymmetry generators, and thus results in the space-time reflection $\left( x^4, x^5, x^7, x^8, x^9\right) \rightarrow \left( -x^4,-x^5,-x^7,-x^8,-x^9\right)$ and the gauging of world sheet parity $\Omega$.

Without the O4-plane, the worldvolume of coincident D4-branes will make up a $\text {U}(n)$ gauge theory. If all D4-branes are coincident and lying on the O4-plane, the open string modes between the D4-branes and their mirror image under O4-plane will give rise to an ${{\,\textrm{SO}\,}}\left( 2n \right)$ or an $\text {Sp}\left( n\right)$ gauge theory, depending on the choice of whether $\Omega ^2=\pm 1$ [80]. And for the case of ${{\,\textrm{SO}\,}}\left( 2n+1\right)$, an extra (half) D4-brane should be placed at on the O4-plane.

The brane contents of a BCD-type gauge theory are listed as follows:

in which “−” means extending direction and “$\bullet$” means the brane is pointlike in this direction. Notice the D4$^+$-brane is finite along $x^6$ direction. In our BCD-type supergroup gauge theory, we also need to consider D4$^-$-branes in addition to ordinary D4$^+$-branes. The extra brane contents are: