$k \to p-k$ and $m \to \binom{2p+1}{2} - m$ provided these numbers are positive. Sphere partition functions of these dual theories have not been compared: the usual method of comparing Higgs branch expressions fails due to the absence of FI parameter. Gauging a $U(1)$ flavor symmetry gives further dualities [30, 132], whose gauge group $U(1) \times USp(2k)$ (and its $\mathbb{Z}_2$ quotient) allow FI-theta parameters. Sphere partition functions have been compared for theories whose low-energy limit has a Calabi-Yau threefold target space. Presumably, integrating over the FI-theta parameters as in (5.22) should help prove that Hori duals have equal A-type sphere partition function.

## 5.4 $\mathcal{N}=(0,2)$ trialities

We describe dualities [118] between $\mathcal{N}=(0,2)$ theories with a gauge group $U(K)$ and (anti)fundamental matter. The theories are expected to flow at intermediate energy scales to NLSMs on bundles over Grassmannians [133] and each duality is due to an isomorphism between bundles over $Gr(K, N)$ and $Gr(N-K, N)$ [134].

We let $N_P, N_\Phi, N_\Psi$ denote the number of antifundamental chirals $P$, fundamental chirals $\Phi$, and antifundamental Fermi multiplets $\Psi$ (which could be mapped to fundamental ones by exchanging $E$ and $J$ interactions). The $SU(K)$ gauge anomaly $N_P/2+N_\Phi/2-N_\Psi/2-K$ due to fermions in the matter and vector multiplets must vanish, thus $K = (N_P + N_\Phi - N_\Psi)/2$. The $U(1)$ gauge anomaly $N_P K + N_\Phi K - N_\Psi K = 2K^2$ of these multiplets is cancelled$^{37}$ by adding two Fermi multiplets $\Omega_{1,2}$ with $U(1)$ charge $K$, i.e., in the determinant representation of $U(K)$. For convenience, we also include $N_P N_\Phi$ neutral Fermi multiplets $\Gamma$ with a J-term interaction $\Gamma P \Phi$.

Theories with $(N_P, N_\Phi, N_\Psi)$ equal to the same $(N_1, N_2, N_3)$ up to cyclic permutations, depicted by the quivers in Figure 1, are expected to flow to the same infrared fixed point. As evidence, we show that the elliptic genus$^{38}$ is invariant under cyclic permutations of $(N_P, N_\Phi, N_\Psi)$. The classical flavor symmetry is $S[U(N_P) \times U(N_\Phi) \times U(N_\Psi) \times U(2)]$ with holonomies $(\xi_i^P, \xi_i^\Phi, \xi_i^\Psi, \xi_i^\Omega)$ modulo gauge transformations, but mixed flavor-gauge anomalies reduce Abelian symmetries to a two-dimensional subgroup: this can be used for instance to fix the $U(1)_\Omega$ holonomy $\sum_i \xi_i^\Omega = -\sum_i \xi_i^P - \sum_i \xi_i^\Phi + \sum_i \xi_i^\Psi$ (we took $\Phi$ in the $\square$ of $SU(N_\Phi)$). Up to a constant, the elliptic genus is a sum of residues of

$$ \frac{\prod_{\ell=1}^{2} \theta(\xi_{\ell}^{\Omega} + \sum_{a} u_{a}) \prod_{a=1}^{K} \prod_{i=1}^{N_{\Psi}} \theta(\xi_{i}^{\Psi} - u_{a}) \prod_{j=1}^{N_{P}} \prod_{k=1}^{N_{\Phi}} \theta(\xi_{j}^{P} - \xi_{k}^{\Phi}) \prod_{a \neq b}^{K} \theta(u_{a} - u_{b}) \theta'(0)^{K} d^{K} u}{\prod_{a=1}^{K} \prod_{i=1}^{N_{P}} \theta(\xi_{i}^{P} - u_{a}) \prod_{j=1}^{N_{\Phi}} \theta(u_{a} - \xi_{j}^{\Phi})} $$

where $\theta(u) = \theta_1(\tau|u)/i\eta(q) = -\theta(-u)$ has zeros at $\mathbb{Z} + \tau\mathbb{Z}$, no poles, and $\theta'(0) = 2\pi i\eta(q)^2$. Several sets of poles give the same sum of residues: poles due to $P$ (at $\{u_a\} = \{\xi_j^P | j \in J\}$ for each set of $K$ distinct flavors $J \subset \{1, \dots, N_P\}$) with residue

$$ \frac{\prod_{\ell=1}^{2} \theta(\xi_{\ell}^{\Omega} + \sum_{j \in J} \xi_{j}^{P}) \prod_{j \in J} \prod_{i=1}^{N_{\Psi}} \theta(\xi_{i}^{\Psi} - \xi_{j}^{P}) \prod_{j \notin J} \prod_{k=1}^{N_{\Phi}} \theta(\xi_{j}^{P} - \xi_{k}^{\Phi})}{\prod_{j \in J} \prod_{i \notin J} \theta(\xi_{i}^{P} - \xi_{j}^{P})} ; \quad (5.26) $$

<footnote>$^{37}$One could consider $SU(K)$ gauge theories and omit $\Omega_{1,2}$.</footnote>

<footnote>$^{38}$While [118] work in the NSNS sector we work in the RR sector; results are related by spectral flow.</footnote>
<figure><img src="image_2.png" /><figcaption>Figure 1: Quivers for three $\mathcal{N}=(0, 2)$ theories related by triality: $K_i = (N_1+N_2+N_3)/2 - N_i$ and $(N_P, N_\Phi, N_\Psi)$ is one of $(N_2, N_3, N_1)$, $(N_3, N_1, N_2)$, $(N_1, N_2, N_3)$. The table lists how the chiralities $P, \Phi$ and Fermis $\Psi, \Gamma, \Omega$ transform under the gauge group $U(K) = U(\frac{1}{2}(N_P + N_\Phi - N_\Psi))$ and the classical flavor symmetry $S[U(N_P) \times U(N_\Phi) \times U(N_\Psi) \times U(2)]$ which loses one Abelian factor due to a flavor-gauge anomaly.</figcaption></figure>

or poles due to $\Phi$ (at $\{u_a\} = \{\xi_j^\Phi | j \in J\}$ for $J \subset \{1, \dots, N_\Phi\}$) with residue

$$ \frac{\prod_{\ell=1}^{2} \theta(\xi_{\ell}^{\Omega} + \sum_{j \in J} \xi_{j}^{\Phi}) \prod_{j \notin J} \prod_{i=1}^{N_P} \theta(\xi_{i}^{P} - \xi_{j}^{\Phi}) \prod_{j \in J} \prod_{k=1}^{N_{\Psi}} \theta(\xi_{k}^{\Psi} - \xi_{j}^{\Phi})}{\prod_{j \in J} \prod_{i \notin J} \theta(\xi_{j}^{\Phi} - \xi_{i}^{\Phi})} . \quad (5.27) $$

Up to a sign, (5.26) is mapped to (5.27) under $(N_P, \xi^P) \to (N_\Phi, \xi^\Phi) \to (N_\Psi, \xi^\Psi) \to (N_P, \xi^P)$ and $J \to J^c$. One also has a shift $\xi_i^\Omega \to \xi_i^\Omega - \sum_j \xi_j^\Omega - \sum_j \xi_j^P$ fixed by the above constraint on $\sum \xi^\Omega$. Elliptic genera of theories in Figure 1 are thus equal, with $S[U(N_1) \times U(N_2) \times U(N_3) \times SU(2)]$ flavor symmetries identified. Another outcome of the calculation is that the elliptic genus vanishes if $K > N_P$ or $K > N_\Phi$ in any frame, which is equivalent to $K < 0$ in a dual frame. This suggests that supersymmetry is broken unless $(N_1, N_2, N_3)$ obey the triangle inequality.

Lack of space forces us to only mention variants of the Gadde-Gukov-Putrov triality. Gauging flavor symmetries leads to dualities between $\mathcal{N}=(0, 2)$ quiver gauge theories with bifundamental chiral/Fermi multiplets and Fermi multiplets in determinant representations; however, most quivers have either gauge anomalies or spontaneous supersymmetry breaking. Such quivers were obtained from brane brick models in [135]. As shown in [136], a twisted dimensional reduction of the 6d (2,0) theory on $S^2 \times \Sigma$ yields $\mathcal{N}=(0, 4)$ Lagrangians labelled by pants decompositions of $\Sigma$, and changes in pants decomposition give $\mathcal{N}=(0, 4)$ dualities. Similarly, twisted dimensional reductions on $S^2$ of 4d $\mathcal{N}=1$ and 4d $\mathcal{N}=2$ dualities yield two-dimensional (0, 2) or (0, 4) or (2, 2) dualities [137], in particular an $SU(K)$ variant of the (0, 2) triality above. All of these dualities are checked by comparing elliptic genera. The dimensional reduction is based on [138, 9] (see also [139, 140, 141]).

# 6 Conclusion

We have reviewed the main localization calculations in two dimensions on the sphere (section 2), other curved backgrounds (section 3) and the torus (section 4), and discussed applications
to mirror symmetry and gauge theory dualities (section 5). We now conclude this review by mentioning other developments.

Two-dimensional gauge theory descriptions of several non-critical strings in 6d SCFTs have been tested by comparing topological vertex results to the elliptic genus of a string. As discussed in Contribution [124], this was done for M-strings [123, 142, 143, 144] (namely M2-branes suspended between M5-branes), E-strings [145, 146] (M2-branes suspended between M5- and M9-branes), little strings [147] (strings on type IIA or IIB NS5-branes), and for a class of 6d $\mathcal{N}=(1, 0)$ SCFTs engineered from F-theory [148, 149].

Nekrasov's instanton partition function of 4d $\mathcal{N}=2$ theories such as $SU(N)$ super-Yang-Mills can be reproduced by an appropriate (R-preserving) $S^2$ partition function [150].<fnref n="39" /> The 4d theory is engineered by $N$ fractional D3-branes on $\mathbb{C}^2/\mathbb{Z}_2$ and its instantons by $k$ D($-1$)-branes. Blowing up the singular point yields a D1-D5 brane system, described in the gauge theory limit by a 2d $\mathcal{N}=(2, 2)$ $U(k)$ GLSM on the blown-up sphere $\mathbb{CP}^1$. In the zero-radius limit, its sphere partition function reproduces the equivariant volume of the ADHM moduli space of $k$ instantons, while for non-zero radius it captures genus zero Gromov-Witten invariants of the ADHM moduli space [16, 152]. This construction was later used to extract spectra of hydrodynamic quantum integrable systems [153, 154].

Another appearance of integrable models in relation to 2d localization is that elliptic genera give solutions to Yang-Baxter equations [155, 156]. In this context, the Yang-Baxter equation amounts to the invariance of the genus under $\mathcal{N}=(2, 2)$ Seiberg-like dualities.

We have already mentioned that twisted dimensional reductions of 4d $\mathcal{N}=1$ theories on a sphere yield 2d $\mathcal{N}=(0, 2)$ theories and that 4d dualities become 2d dualities. The 2d $\mathcal{N}=(0, 2)$ elliptic genus is thus a limit of a $T^2 \times S^2$ partition function [9]. Similarly, the partition function of 3d $\mathcal{N}=2$ theories on Lens spaces, described in Contribution [157], reduces to an $S^2$ partition function when the circle fiber of the Lens space shrinks to zero size (see also [158, 159]). The $S^2$ partition function also appears when localizing 4d $\mathcal{N}=2$ theories on $S^4$ on their Higgs branch [160, 161].

In four-dimensional supersymmetric gauge theories, a class of surface operators can be constructed by coupling a two-dimensional theory to the bulk fields supersymmetrically (see also Contribution [162]). Superconformal indices of coupled 2d/4d systems were computed in [88] (see [163, 164] for related 5d calculations) and led to discovering the node-hopping duality (see also [165]): coupling the same 2d $\mathcal{N}=(2, 2)$ theory to different fields in a 4d $\mathcal{N}=2$ quiver theory gives the same surface operator at low energies. The AGT correspondence relates the node-hopping duality to crossing symmetry in a 2d CFT [128]: the $S^4$ partition function of the 4d theory is identified with a Toda CFT correlator, adding a surface operator on $S^2 \subset S^4$ corresponds to inserting a (degenerate) vertex operator in the correlator, and different choices of couplings correspond to different OPEs of that vertex operator with others. Kähler parameters (the 4d gauge coupling and 2d FI/theta parameters) correspond to positions of vertex operators in the correlator and are expected to transform non-trivially

<footnote n="39">³⁹In a different approach [151], for 4d $\mathcal{N}=2$ theories engineered by string theories on a Calabi-Yau three-fold $X$, the Seiberg-Witten Kähler potential can be obtained as that of $X$, itself derived from the $S^2$ partition function of a GLSM flowing to an NLSM on $X$.</footnote>
under node-hopping. The map could be found by comparing $S^2 \subset S^4$ partition functions of these 2d/4d systems, but instanton-vortex partition functions which appear when localizing [166] are unknown. It would be interesting to derive them.

## Acknowledgements

We are grateful to Nikolay Bobev, Cyril Closset, Stefano Cremonesi, Nima Doroud, Richard Eager, Jaume Gomis, Kentaro Hori, Sungjay Lee, Daniel S. Park, Mauricio Romo, Yuji Tachikawa, Ran Yacoby, Alberto Zaffaroni and Peng Zhao for collaborations and insightful discussions over the years on some of the material presented here. F.B. is supported in part by the Royal Society as a Royal Society University Research Fellowship holder, and by the MIUR-SIR grant RBSI1471GJ "Quantum Field Theories at Strong Coupling: Exact Computations and Applications".

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