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Factorising the 3D Topologically Twisted Index

Alejandro Cabo-Bizet

TL;DR

The paper addresses the factorisation of the 3D topologically twisted index for $\mathcal{N}=2$ Chern-Simons-matter theory on $\mathbb{S}_2\times\mathbb{S}_1$, motivated by holographic links to $AdS_4$ black holes. It develops two localisation frameworks based on nilpotent supercharges $Q_{\epsilon}$ and $\tilde{Q}_{\epsilon}$, using a redefined gauge potential $\hat{A}$ and a $\mathbb{Z}_2$ mapping between the two pictures, to control reality conditions and contour choices. The index is shown to factorise into two blocks, $B_{\text{even}}$ and $B_{\text{odd}}$, connected by a kernel $\mathcal{G}$, reproducing the Jeffrey-Kirwan-residue structure and matching the known BZ result via two complementary routes: a real-path VP-II and a complex-path localisation (VP-I). A concrete toy model demonstrates the mechanism and reveals a large-$N$ scaling $Z_{U(N)}\sim e^{N^2 F}$ under mass deformation, illustrating how flux sums contribute nontrivially. Overall, the work provides a field-theoretic derivation of flux-summed TT indices via real contour localisation and clarifies the relation between real and complex localisation pictures with potential holographic implications.

Abstract

We explore the path integration -- upon the contour of hermitian (non-auxliary) field configurations -- of topologically twisted $\mathcal{N}=2$ Chern-Simons-matter theory (TTCSM) on $\mathbb{S}_2$ times a segment. In this way, we obtain the formula for the 3D topologically twisted index, first as a convolution of TTCSM on $\mathbb{S}_2$ times halves of $\mathbb{S}_1$, second as TTCSM on $\mathbb{S}_2$ times $\mathbb{S}_1$ -- with a puncture --, and third as TTCSM on $\mathbb{S}_2 \times \mathbb{S}_1$. In contradistinction to the first two cases, in the third case, the vector multiplet auxiliary field $D$ is constrained to be anti-hermitian.

Factorising the 3D Topologically Twisted Index

TL;DR

The paper addresses the factorisation of the 3D topologically twisted index for Chern-Simons-matter theory on , motivated by holographic links to black holes. It develops two localisation frameworks based on nilpotent supercharges and , using a redefined gauge potential and a mapping between the two pictures, to control reality conditions and contour choices. The index is shown to factorise into two blocks, and , connected by a kernel , reproducing the Jeffrey-Kirwan-residue structure and matching the known BZ result via two complementary routes: a real-path VP-II and a complex-path localisation (VP-I). A concrete toy model demonstrates the mechanism and reveals a large- scaling under mass deformation, illustrating how flux sums contribute nontrivially. Overall, the work provides a field-theoretic derivation of flux-summed TT indices via real contour localisation and clarifies the relation between real and complex localisation pictures with potential holographic implications.

Abstract

We explore the path integration -- upon the contour of hermitian (non-auxliary) field configurations -- of topologically twisted Chern-Simons-matter theory (TTCSM) on times a segment. In this way, we obtain the formula for the 3D topologically twisted index, first as a convolution of TTCSM on times halves of , second as TTCSM on times -- with a puncture --, and third as TTCSM on . In contradistinction to the first two cases, in the third case, the vector multiplet auxiliary field is constrained to be anti-hermitian.

Paper Structure

This paper contains 30 sections, 255 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The supercharges $Q_\epsilon$ and $\tilde{Q}_\epsilon$
  3. The supercharge in $I_N$: $Q_\epsilon$
  4. The supercharge in $I_S$: $\tilde{Q}_\epsilon$
  5. The $\mathbb{Z}_2$ transformation $\mathcal{ P}$: $Q_\epsilon \rightarrow \tilde{Q}_\epsilon$
  6. What is localisation computing?
  7. Two variational problems on $\mathbb{S}_2\times\mathbb{S}_1$
  8. The "complex" path of integration: The proper localising term
  9. The 3D TT Index on $\mathbb{S}_2\times \mathbb{S}_1$: "real" vs "complex" path
  10. Functional space of integration: boundary state on $\mathbb{S}_2\times I_N(S)$
  11. Wave function of boundary state
  12. On the $\mathcal{K}_{\blue even}$ and $\mathcal{K}_{\red odd}$ KK mode space
  13. 3D TT index on $\mathbb{S}_2$ times an interval
  14. Factorisation of the 3D TT Index: Variational problem VP-II
  15. Final formula
  16. ...and 15 more sections

Figures (3)

  • Figure 1: The $N$ roots of $P_N(z)$ for $\eta=e^{i v}=\frac{1}{8}$. For $N$ equal $50$ (Left) and 200 (Right) they are on $S_1$.
  • Figure 2: The real part of the components, $F_2(N)$ and $F_3(N)$ of the free energy \ref{['FEn1']} for $N$ ranging between $11$ and 211 at step 10.
  • Figure 3: The integration path to use in the $u$ complex plane is depicted in red in the figure above. The red (blue) point represents the pole associated with positively (negatively) charged matter multiplets. The position of these poles is determined by the flavour Wilson line along $\mathbb{S}_1$ that we denote as $v$. These poles have images that repeat with period $2 \pi \mathbb{Z}$. The integration along the two disconnected blue lines cancel out each other. The map $x:=e^{i u}$ sends the green line in the left to the $\mathbb{S}_1$ contour in the right, that encloses only the potential pole at the origin $x=0$($u=i \infty$). The integration along the green line equates then to the residue at the point $x=0$ ($u=i \infty$). The red line maps to the circumference of unit radius. The circumference of unit radius will enclose not only the pole at $x=0$ but also the poles associated to the presence of matter multiplets, namely the image of the red point in the figure in the left.