Comments on flavor symmetry breaking and three-dimensional   superconformal index

Ilmar Gahramanov

arXiv:2303.00482·hep-th·March 2, 2023

Comments on flavor symmetry breaking and three-dimensional superconformal index

Ilmar Gahramanov

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TL;DR

This paper studies the superconformal index of a 3D supersymmetric QCD with flavor symmetry breaking, showing how the index vanishes generally but reveals symmetry breaking at special fugacity values.

Contribution

It provides a method to derive the index for a theory with fewer flavors by integrating out a flavor from a higher-flavor theory, highlighting symmetry-breaking effects.

Findings

01

Index vanishes for generic fugacities

02

Special fugacity choices reveal flavor symmetry breaking

03

Method to obtain lower-flavor index from higher-flavor theory

Abstract

We investigate the superconformal index of a three-dimensional $S U (2)$ $N = 2$ supersymmetric QCD with $N_{f} = 4$ flavors. This theory confines with the breaking of a global symmetry. We obtain the index of this theory by integrating out a flavor from the theory with $N_{f} = 6$ flavors. The superconformal index of the theory vanishes for generic values of the flavor fugacities. However, the specific choice of fugacities allows us to describe the flavor symmetry-breaking phenomenon.

Equations52

m \in Z \sum \oint \frac{( q ^{1 + (n_{α} \pm n_{w} \pm m) /2} α ^{- 1} w ^{\mp} z ^{\mp} ) _{\infty}}{( q ^{n_{α} \pm n_{w} \pm m} α w ^{\pm} z ^{\pm} ) _{\infty}} \frac{( q ^{1 + (- n_{α} \pm n_{y} \pm m) /2} α y ^{\mp} z ^{\mp} ) _{\infty}}{( q ^{- n_{α} \pm n_{y} \pm m} α ^{-} y ^{\pm} z ^{\pm} ) _{\infty}} \frac{( 1 - q ^{m} z ^{\pm 2} )}{q ^{m}} \frac{d z}{4 π i z}

m \in Z \sum \oint \frac{( q ^{1 + (n_{α} \pm n_{w} \pm m) /2} α ^{- 1} w ^{\mp} z ^{\mp} ) _{\infty}}{( q ^{n_{α} \pm n_{w} \pm m} α w ^{\pm} z ^{\pm} ) _{\infty}} \frac{( q ^{1 + (- n_{α} \pm n_{y} \pm m) /2} α y ^{\mp} z ^{\mp} ) _{\infty}}{( q ^{- n_{α} \pm n_{y} \pm m} α ^{-} y ^{\pm} z ^{\pm} ) _{\infty}} \frac{( 1 - q ^{m} z ^{\pm 2} )}{q ^{m}} \frac{d z}{4 π i z}

= \frac{( q ^{1 + n_{α}} α ^{\mp 2} ; q ) _{\infty}}{( q ^{n_{α}} α ^{\pm 2} ; q ) _{\infty}} \frac{( q ^{1 \pm n_{y}} w ^{\mp 2} ; q ) _{\infty}}{( q ^{\pm n_{w}} w ^{\pm 2} ; q ) _{\infty}} (δ (θ + χ) δ_{n_{w} + n_{y}, 0} + δ (θ - χ) δ_{n_{w} - n_{y}, 0}),

I (q, {a_{i}}) = Tr [(- 1)^{F} e^{- β {Q, Q^{†}}} q^{\frac{1}{2} (Δ + j_{3})} i \prod a_{i}^{F_{i}}],

I (q, {a_{i}}) = Tr [(- 1)^{F} e^{- β {Q, Q^{†}}} q^{\frac{1}{2} (Δ + j_{3})} i \prod a_{i}^{F_{i}}],

I (q, {a_{j}}, {n_{j}}) = m_{i} \sum \frac{1}{∣ W _{m} ∣} \oint i = 1 \prod rank G \frac{d z _{i}}{2 π i z _{i}} Z_{gauge} (z_{i}, m_{i}; q) Z_{chiral} (z_{i}, m_{i}; a_{j}, n_{j}; q) .

I (q, {a_{j}}, {n_{j}}) = m_{i} \sum \frac{1}{∣ W _{m} ∣} \oint i = 1 \prod rank G \frac{d z _{i}}{2 π i z _{i}} Z_{gauge} (z_{i}, m_{i}; q) Z_{chiral} (z_{i}, m_{i}; a_{j}, n_{j}; q) .

I_{E} = m = - \infty \sum \infty \oint i = 1 \prod 6 \frac{( q ^{1 + (m + n_{i}) /2} / a _{i} z , q ^{1 + (n_{i} - m) /2} z / a _{i} ; q ) _{\infty}}{( q ^{(m + n_{i}) /2} a _{i} z , q ^{(n_{i} - m) /2} a _{i} / z ; q ) _{\infty}} \frac{( 1 - q ^{m} z ^{2} ) ( 1 - q ^{m} z ^{- 2} )}{q ^{m} z ^{6 m}} \frac{d z}{4 π i z}

I_{E} = m = - \infty \sum \infty \oint i = 1 \prod 6 \frac{( q ^{1 + (m + n_{i}) /2} / a _{i} z , q ^{1 + (n_{i} - m) /2} z / a _{i} ; q ) _{\infty}}{( q ^{(m + n_{i}) /2} a _{i} z , q ^{(n_{i} - m) /2} a _{i} / z ; q ) _{\infty}} \frac{( 1 - q ^{m} z ^{2} ) ( 1 - q ^{m} z ^{- 2} )}{q ^{m} z ^{6 m}} \frac{d z}{4 π i z}

i = 1 \prod 6 a_{i} = q and i = 1 \sum 6 n_{i} = 0 .

i = 1 \prod 6 a_{i} = q and i = 1 \sum 6 n_{i} = 0 .

I_{M} = \frac{1}{\prod _{i = 1}^{6} a _{i}^{n_{i}}} 1 \leq i < j \leq 6 \prod \frac{( q ^{1 + (n_{i} + n_{j}) /2} / a _{i} a _{j} ; q ) _{\infty}}{( q ^{(n_{i} + n_{j}) /2} a _{i} a _{j} ; q ) _{\infty}} .

I_{M} = \frac{1}{\prod _{i = 1}^{6} a _{i}^{n_{i}}} 1 \leq i < j \leq 6 \prod \frac{( q ^{1 + (n_{i} + n_{j}) /2} / a _{i} a _{j} ; q ) _{\infty}}{( q ^{(n_{i} + n_{j}) /2} a _{i} a _{j} ; q ) _{\infty}} .

m = - \infty \sum \infty \oint i = 1 \prod 6 \frac{( q ^{1 + (m + n_{i}) /2} / a _{i} z , q ^{1 + (n_{i} - m) /2} z / a _{i} ; q ) _{\infty}}{( q ^{(m + n_{i}) /2} a _{i} z , q ^{(n_{i} - m) /2} a _{i} / z ; q ) _{\infty}} \frac{( 1 - q ^{m} z ^{\pm 2} )}{q ^{m} z ^{6 m}} \frac{d z}{4 π i z}

m = - \infty \sum \infty \oint i = 1 \prod 6 \frac{( q ^{1 + (m + n_{i}) /2} / a _{i} z , q ^{1 + (n_{i} - m) /2} z / a _{i} ; q ) _{\infty}}{( q ^{(m + n_{i}) /2} a _{i} z , q ^{(n_{i} - m) /2} a _{i} / z ; q ) _{\infty}} \frac{( 1 - q ^{m} z ^{\pm 2} )}{q ^{m} z ^{6 m}} \frac{d z}{4 π i z}

= \frac{1}{\prod _{i = 1}^{6} a _{i}^{n_{i}}} 1 \leq i < j \leq 6 \prod \frac{( q ^{1 + (n_{i} + n_{j}) /2} / a _{i} a _{j} ; q ) _{\infty}}{( q ^{(n_{i} + n_{j}) /2} a _{i} a _{j} ; q ) _{\infty}}

a_{5} a_{6} = q, n_{5} + n_{6} = 0

a_{5} a_{6} = q, n_{5} + n_{6} = 0

a_{1} = α w, a_{2} = α w^{- 1}, a_{3} = β y, a_{4} = β y^{- 1},

a_{1} = α w, a_{2} = α w^{- 1}, a_{3} = β y, a_{4} = β y^{- 1},

n_{1} = n_{α} + n_{w}, n_{2} = n_{α} - n_{w}, n_{3} = n_{β} + n_{y}, n_{4} = n_{β} - n_{y} .

n_{1} = n_{α} + n_{w}, n_{2} = n_{α} - n_{w}, n_{3} = n_{β} + n_{y}, n_{4} = n_{β} - n_{y} .

a_{5} a_{6} α^{2} β^{2} = q, 2 n_{α} + 2 n_{β} + n_{5} + n_{6} = 0 .

a_{5} a_{6} α^{2} β^{2} = q, 2 n_{α} + 2 n_{β} + n_{5} + n_{6} = 0 .

\int_{T}

\int_{T}

\times \frac{( q ^{1 + \frac{n _{5} + n _{6}}{2}} ( a _{5} a _{6} ) ^{- 1} ; q ) _{\infty}}{( q ^{\frac{n _{5} + n _{6}}{2}} a _{5} a _{6} ; q ) _{\infty}} \frac{( q ^{1 + \frac{n _{α} \pm n _{w} + n _{5}}{2}} ( α w ^{\pm 1} a _{5} ) ^{- 1} ; q ) _{\infty}}{( q ^{\frac{n _{α} \pm n _{w} + n _{5}}{2}} α w ^{\pm 1} a _{5} ; q ) _{\infty}} \frac{( q ^{1 + \frac{n _{α} \pm n _{w} + n _{6}}{2}} ( α w ^{\pm 1} a _{6} ) ^{- 1} ; q ) _{\infty}}{( q ^{\frac{n _{α} \pm n _{w} + n _{6}}{2}} α w ^{\pm 1} a _{6} ; q ) _{\infty}}

\times \int_{T} \frac{d y}{2 π i y} f (y) y^{- 2 n_{y}} \frac{( q ^{1 + \frac{n _{α} + n _{β} \pm n _{w} \pm n _{y}}{2}} ( α β w ^{\pm 1} y ^{\pm 1} ) ^{- 1} ; q ) _{\infty}}{( q ^{\frac{n _{α} + n _{β} \pm n _{w} \pm n _{y}}{2}} α β w ^{\pm 1} y ^{\pm 1} ; q ) _{\infty}} j = 5 \prod 6 \frac{( q ^{1 + \frac{n _{β} \pm n _{y} + n _{j}}{2}} ( β y ^{\pm 1} a _{j} ) ^{- 1} ; q ) _{\infty}}{( q ^{\frac{n _{β} \pm n _{y} + n _{j}}{2}} β y ^{\pm 1} a _{j} ; q ) _{\infty}} .

y_{in}

y_{in}

y_{o u t}

\int_{T}

\int_{T}

\times \frac{( q ^{1 + \frac{n _{5} + n _{6}}{2}} ( a _{5} a _{6} ) ^{- 1} ; q ) _{\infty}}{( q ^{\frac{n _{5} + n _{6}}{2}} a _{5} a _{6} ; q ) _{\infty}} j = 5 \prod 6 \frac{( q ^{1 + \frac{n _{α} \pm n _{w} + n _{j}}{2}} ( α w ^{\pm 1} a _{j} ) ^{- 1} ; q ) _{\infty}}{( q ^{\frac{n _{α} \pm n _{w} + n _{j}}{2}} α w ^{\pm 1} a _{j} ; q ) _{\infty}} (I (w) + I (w^{- 1}) + I (C))

I (w)

I (w)

\times \frac{( q ^{1 + n_{α} + n_{β} + n_{w} - n_{y}} ; q ) _{\infty}}{( q ; q ) _{\infty}} \frac{( q ^{1 - n_{w} + n_{y}} ( α β ) ^{- 2} ; q ) _{\infty}}{( q ^{n_{α} + n_{β}} ( α β ) ^{2} ; q ) _{\infty}} \frac{( q ^{1 + n_{α} + n_{β} - n_{y}} w ^{2} ; q ) _{\infty}}{( q ^{- n_{w}} w ^{- 2} ; q ) _{\infty}}

\times i = 5 \prod 6 \frac{( q ^{1 + \frac{- n _{α} + 2 n _{y} + n _{i}}{2}} ( α β ^{2} w a _{i} ) ^{- 1} ; q ) _{\infty}}{( q ^{\frac{2 n _{β} + n _{α} + n _{w} + n _{i}}{2}} α β ^{2} w a _{i} ; q ) _{\infty}} \frac{( q ^{1 + \frac{2 n _{β} + n _{α} + n _{i} + n _{w} - 2 n _{y}}{2}} ( α w ^{- 1} a _{i} ) ^{- 1} ; q ) _{\infty}}{( q ^{\frac{- n _{w} - n _{α} + n _{i}}{2}} α w ^{- 1} a _{i} ; q ) _{\infty}}

I (w^{- 1})

I (w^{- 1})

\times \frac{( q ^{1 + n_{α} + n_{β} - n_{y}} w ^{- 2} ; q ) _{\infty}}{( q ^{n_{w}} w ^{2} ; q ) _{\infty}} \frac{( q ^{1 + n_{y}} ( α β w ^{- 1} ) ^{- 2} ; q ) _{\infty}}{( q ^{n_{α} + n_{β} - n_{w}} ( α β w ^{- 1} ) ^{2} ; q ) _{\infty}} \frac{( q ^{1 + n_{α} + n_{β} - n_{w} - n_{y}} ; q ) _{\infty}}{( q ; q ) _{\infty}}

\times i = 5 \prod 6 \frac{( q ^{1 + \frac{- n _{α} + 2 n _{y} + n _{i}}{2}} ( α β ^{2} w ^{- 1} a _{i} ) ^{- 1} ; q ) _{\infty}}{( q ^{\frac{2 n _{β} + n _{α} - n _{w} + n _{i}}{2}} α β ^{2} w ^{- 1} a _{i} ; q ) _{\infty}} \frac{( q ^{1 + \frac{2 n _{β} + n _{α} + n _{i} - n _{w} - 2 n _{y}}{2}} ( α w a _{i} ) ^{- 1} ; q ) _{\infty}}{( q ^{\frac{n _{w} - n _{α} + n _{i}}{2}} α w a _{i} ; q ) _{\infty}}

I (C)

I (C)

\frac{( q ^{1 - m /2} z ^{- 1} ; q ) _{\infty}}{( q ^{- m /2} z ; q ) _{\infty}} = \frac{q ^{m /2}}{( - z ) ^{m}} \frac{( q ^{1 + m /2} z ^{- 1} ; q ) _{\infty}}{( q ^{+ m /2} z ; q ) _{\infty}}, m \in Z .

\frac{( q ^{1 - m /2} z ^{- 1} ; q ) _{\infty}}{( q ^{- m /2} z ; q ) _{\infty}} = \frac{q ^{m /2}}{( - z ) ^{m}} \frac{( q ^{1 + m /2} z ^{- 1} ; q ) _{\infty}}{( q ^{+ m /2} z ; q ) _{\infty}}, m \in Z .

\int_{T} f (y) I_{E} \frac{d y}{2 π i y}

\int_{T} f (y) I_{E} \frac{d y}{2 π i y}

\times (f (w q^{(n_{w} - n_{y}) /2}) δ_{n_{w} + n_{y}, 0} + f (w^{- 1} q^{(- n_{w} - n_{y}) /2}) δ_{n_{w} - n_{y}, 0})

m \in Z \sum \oint \frac{( q ^{1 + (n_{α} \pm n_{w} \pm m) /2} α ^{- 1} w ^{\mp} z ^{\mp} ) _{\infty}}{( q ^{n_{α} \pm n_{w} \pm m} α w ^{\pm} z ^{\pm} ) _{\infty}} \frac{( q ^{1 + (- n_{α} \pm n_{y} \pm m) /2} α y ^{\mp} z ^{\mp} ) _{\infty}}{( q ^{- n_{α} \pm n_{y} \pm m} α ^{-} y ^{\pm} z ^{\pm} ) _{\infty}} \frac{( 1 - q ^{m} z ^{2} ) ( 1 - q ^{m} z ^{- 2} )}{q ^{m}} \frac{d z}{4 π i z}

m \in Z \sum \oint \frac{( q ^{1 + (n_{α} \pm n_{w} \pm m) /2} α ^{- 1} w ^{\mp} z ^{\mp} ) _{\infty}}{( q ^{n_{α} \pm n_{w} \pm m} α w ^{\pm} z ^{\pm} ) _{\infty}} \frac{( q ^{1 + (- n_{α} \pm n_{y} \pm m) /2} α y ^{\mp} z ^{\mp} ) _{\infty}}{( q ^{- n_{α} \pm n_{y} \pm m} α ^{-} y ^{\pm} z ^{\pm} ) _{\infty}} \frac{( 1 - q ^{m} z ^{2} ) ( 1 - q ^{m} z ^{- 2} )}{q ^{m}} \frac{d z}{4 π i z}

= \frac{( q ^{1 + n_{α}} α ^{\mp 2} ; q ) _{\infty}}{( q ^{n_{α}} α ^{\pm 2} ; q ) _{\infty}} \frac{( q ^{1 \pm n_{y}} w ^{\mp 2} ; q ) _{\infty}}{( q ^{\pm n_{w}} w ^{\pm 2} ; q ) _{\infty}} (δ (θ + χ) δ_{n_{w} + n_{y}, 0} + δ (θ - χ) δ_{n_{w} - n_{y}, 0})

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TopicsParticle physics theoretical and experimental studies · Quantum Chromodynamics and Particle Interactions · Black Holes and Theoretical Physics

Full text

**Comments on flavor symmetry breaking and three-dimensional superconformal index

**

Ilmar Gahramanov

1* Department of Physics, Bogazici University, 34342 Bebek, Istanbul, Turkey

2Steklov Mathematical Institute of Russ. Acad. Sci., Gubkina str. 8, 119991 Moscow, Russia

3* Department of Mathematics, Khazar University, Mehseti St. 41, AZ1096, Baku, Azerbaijan

4* Institute of Radiation Problems ANAS, B.Vahabzade 9, AZ1143 Baku, Azerbaijan

We investigate the superconformal index of a three-dimensional $SU(2)$ ${\mathcal{N}}=2$ supersymmetric QCD with $N_{f}=4$ flavors. This theory confines with the breaking of a global symmetry. We obtain the index of this theory by integrating out a flavor from the theory with $N_{f}=6$ flavors. The superconformal index of the theory vanishes for generic values of the flavor fugacities. However, the specific choice of fugacities allows us to describe the flavor symmetry-breaking phenomenon.

1 Introduction

Supersymmetric duality relates the infrared limits of different supersymmetric gauge theories. The basic example of such duality [1] is a four-dimensional $SU(N_{c})$ “electric” gauge theory with $N_{f}$ flavors of quarks in fundamental and antifundamental representations which possesses a dual description in terms of $N_{f}$ “magnetic” flavors of quarks charged under $SU(N_{f}-N_{c})$ gauge group. For $N_{f}=N_{c}$ , the infrared theory is confined, and the classical constraint on the meson and baryon fields is modified by one instanton effect, leading to chiral symmetry breaking. In case $N_{c}=2$ the global symmetry $SU(4)$ is broken111The same happens for the two colors non-supersymmetric QCD with the two flavors which enjoys the extended chiral symmetry $SU(4)$ instead of $SU(2)\times SU(2)$ , see e.g. [2, 3], see, e.g. [4, 5] for the three-dimensional case. to its maximal vector-like subgroup $SP(4)$ by the chiral condensate [6]. The chiral symmetry breaking for four-dimensional ${\mathcal{N}}=1$ supersymmetric gauge theories was considered on the level of the superconformal indices in [7] by Spiridonov and Vartanov.

Three-dimensional $\mathcal{N}=2$ supersymmetric gauge theories have rich structures [8, 9, 10, 11, 12], such as duality, confinement, and chiral symmetry breaking, similar to four-dimensional $\mathcal{N}=1$ supersymmetric gauge theories222The three-dimensional $\mathcal{N}=2$ theory has the same number of supercharges as a four-dimensional $\mathcal{N}=1$ theory. In fact, the three-dimensional $\mathcal{N}=2$ theories discussed in this work are the dimensional reduction of the corresponding four-dimensional $\mathcal{N}=1$ theories [13, 7, 14, 15].. In this work, we consider the flavor symmetry-breaking phenomenon for the three-dimensional $SU(2)$ $\mathcal{N}=2$ theory with $N_{f}=4$ flavors by using the superconformal index technique. Similar to the case of the three-dimensional $S^{3}$ sphere partition function case [7] the superconformal index of such a theory involves the Dirac delta functions which reflect the presence of symmetry breaking

[TABLE]

where we use the usual $q$ -Pochhammer symbol $(z;q)_{\infty}=\prod_{i=0}^{\infty}(1-zq^{i})$ and shorthand notations $f(z^{\pm})=f(z)f(\frac{1}{z})$ . On the left-hand side of equality (1) we have the three-dimensional $\mathcal{N}=2$ superconformal index of a theory with $SU(2)$ gauge group and $N_{f}=4$ chiral fields with the naive flavor group $SU(2)\times SU(2)$ . However, the true flavor group is $(SU(2)\times SU(2))_{diag}$ and the index has non-zero support only on the corresponding subset of flavor fugacities. This is precisely the manifestation of chiral symmetry breaking in confining theories similar to the three-dimensional squashed-sphere partition functions case [7]. Note that the expression (1) plays a role of the local inversion relation in integrable lattice models [16, 17, 18].

One can consider the derivation of (1) as another algebraic approach to the inversion relation.

1.1 Superconformal index

We consider supersymmetric index of a three–dimensional ${\mathcal{N}}=2$ superconformal field theory333For more detailed introduction, see e.g. [19, 16, 20]. which can be defined as a partition function on $S^{2}\times S^{1}$ with the periodic boundary condition of the fields along $S^{1}$ [21, 22, 23, 24]

[TABLE]

where the trace is taken over the Hilbert space of the theory on $S^{2}$ . The $Q$ is a certain supersymmetric charge in three-dimensional ${\mathcal{N}}=2$ superconformal algebra which satisfies the anti-commutation relation $\{Q,Q^{\dagger}\}=\Delta-R-j_{3}$ , where $\Delta$ is the energy, $j_{3}$ is the third component of the angular momentum on $S^{2}$ , $R$ is the R-charge. The operator F plays a role of the fermion number, $F_{i}$ is the charge of global symmetry commuting with supercharge $Q$ , and $a_{i}$ is the corresponding fugacity.

The superconformal index can be computed exactly by the supersymmetric localization technique [25] and it takes the form of the following matrix integral444We skip the contribution of the Chern–Simons term, since in this work we consider theories without this term.

[TABLE]

Here $Z_{\text{gauge}}$ and $Z_{\text{chiral}}$ stand for the contributions from the vector and chiral multiplets, respectively, $|W_{m}|$ is the dimension of the Weyl group, the sum is over monopole configurations, and the integral is over gauge fugacities. The form of the index is completely determined if one knows group-theoretical data, i.e. symmetries and corresponding representations of a theory. The reader interested in the precise formulation of the three-dimensional superconformal index can refer to the papers [24, 20].

1.2 Duality for 3d $SU(2)$ $\mathcal{N}=2$ SQCD with $N_{f}=6$ flavor

As an example, we will be studying the global symmetry breaking in three-dimensional $\mathcal{N}=2$ supersymmetric QCD with $SU(2)$ gauge group and $N_{f}=4$ flavors of quarks. To obtain such theory one can take first the $s$ -confining theory555Three-dimensional s-confining theories were discussed in several works, e.g. [26, 27, 28, 29] and the superconformal indices for such theories were presented in [30, 20]. with the gauge group $SU(2)$ and $N_{f}=6$ flavors. In the case of $SU(2)$ gauge group quarks and antiquarks transform in the same way under the action of the gauge group, i.e. the fundamental and antifundamental representations are equivalent, and therefore the flavor group enhanced to $SU(6)$ symmetry. The superconformal index of the theory can be expressed in the following form

[TABLE]

with the balancing conditions

[TABLE]

The variables $a_{i}$ and $z$ are the fugacities associated with the $SU(6)$ flavor group and $SU(2)$ gauge group, respectively. In [31], it was shown that the superconformal index of this theory has an extended $SO(12)$ flavor symmetry when coupled to four-dimensional multiplets with specific boundary conditions.

It is known that there is magnetic dual theory without gauge degrees of freedom and with fifteen chiral multiplets in the totally antisymmetric tensor representation of the same flavor group. The superconformal index of the magnetic theory is

[TABLE]

Since these theories are IR dual, the superconformal indices need to be in agreement [32, 20]. The result is given by the following q-hypergeometric identity

[TABLE]

with the balancing conditions (5). This integral identity was proven in [20] and studied in [20, 16, 33].

The identity (7) can be written as a star-triangle relation (the non-operator form of the Yang-Baxter equation) [16, 18] which is a sufficient condition for integrability for the Ising-type lattice spin models. The Bailey pairs and the operator form of the Yang-Baxter equation related to this integral identity also were constructed in [16] (see, also [34]).

Note that the basic hypergeometric integral identity (7) is new in the literature and needs detailed study and classification [35]. Such identities (containing summation and integration) recently appear in supersymmetric gauge theory computations [19, 36, 14, 37, 38, 39, 20, 33, 40, 41] and integrable models of statistical mechanics [16, 42, 43, 44, 45, 34].

2 Breaking of flavor symmetry

Our goal will be to reduce the number of flavors from $N_{f}=6$ to $N_{f}=4$ case for the integral identity (7). One can perform the reduction of superconformal indices (7) by taking the following limit666Note that in work [30], the authors take another limit $a_{5}a_{6}=q^{1/2}$ and obtain the s-confining theory without breaking $SU(4)$ flavor symmetry. In [16], authors considered superconformal indices of dual theories with $SU(4)$ flavor symmetry without superpotential term (without balancing condition).

[TABLE]

Generically this limit gives zero on both sides, however for special choices of fugacities $a_{i}$ one can deal with such integrals. Let us denote

[TABLE]

and the corresponding discrete parameters

[TABLE]

In terms of new variables, the balancing conditions read as

[TABLE]

To proceed further we need to consider the following integral: multiply the integral (4) by a holomorphic function near the unit circle $f(y)$ and integrate over the variable $y$ . Then using the integral identity (7) we obtain the following expression777Here we assume that $\alpha^{2}\neq 1$ .

[TABLE]

The first term in the integrand of (12) has the following sequences of poles

[TABLE]

where $i,j\in\mathbb{Z}_{\geq 0}$ and the in-poles converge to zero $y=0$ and out-poles go to infinity. In order to calculate the integral (12) we deform the counter $\mathbb{T}$ such that only the residues of the in-poles are picked up and there are no singularities lying on the counter $C$ . Then the residue calculation leads to the result

[TABLE]

where

[TABLE]

In the limit $\alpha\beta\to 1$ and $n_{\alpha}+n_{\beta}\to 0$ , the poles at $\alpha\beta w^{\pm 1}q^{\pm n_{w}-n_{y}}$ and $w^{\pm 1}q^{\pm n_{w}-n_{y}}/{\alpha\beta}$ start to pinch $\mathbb{T}$ . The integral (18) vanishes in this limit. As a result, we obtain888In order to cancel out many terms we used the equality [46, 16]

$\frac{(q^{1-m/2}z^{-1};q)_{\infty}}{(q^{-m/2}z;q)_{\infty}}=\frac{q^{m/2}}{(-z)^{m}}\frac{(q^{1+m/2}z^{-1};q)_{\infty}}{(q^{+m/2}z;q)_{\infty}},\quad m\in{\mathbb{Z}}.$

(19) .

[TABLE]

Here $\delta_{i,j}$ is the Kronecker delta. Now we denote $y=e^{2\pi\textup{i}\theta},w=e^{2\pi\textup{i}\chi}$ and pass to the integration over real variable $\theta$ . Because of the arbitrariness of the function $f(y)$ , we can give to the function $I_{E}$ a distributional sense and rewrite the expression (20) in the following form

[TABLE]

where $\delta(\theta)$ is the periodic Dirac delta function, $\delta(\phi+1)=\delta(\phi)$ . The appearance of the Dirac delta functions describes the symmetry breaking of the flavor group to the $\left(SU(2)\times SU(2)\right)_{diag}$ .

The equality (21) has an interpretation as the inversion relation999Inversion relation is a simple relation as a star-triangle relation within the framework of integrable lattice spin models of statistical mechanics, see, e.g. [47, 48, 49, 50]. In the context of S-matrix for particle physics in $1+1$ dimensions it has the meaning of the unitarity relation. for exactly solvable lattice spin model introduced in [16, 18]. In particular, this observation can be used [51] for a better understanding of the correspondence [52, 42, 53, 54] between supersymmetric gauge theories and integrable models of statistical mechanics.

3 Concluding remarks

We have shown that it is possible to obtain a three-dimensional $\mathcal{N}=2$ s-confining theory with flavor symmetry breaking by analyzing the superconformal index of the theory. There are several potential directions to extend our work. It would be interesting to obtain the superconformal index with delta functions via localization technique if it is possible.

Though studied only $SU(2)$ supersymmetric gauge theory with $N_{f}=4$ flavors the approach discussed here also applies to gauge theories with other flavor groups. We would expect analogous behavior of the superconformal index for the general $SU(N)$ theory, however, it is computationally complicated to work with the multivariate analog of the basic hypergeometric integral identity (7).

The four-dimensional s-confining theories are classified [55, 56, 57]. The chiral symmetry is always broken in such theories if one reduces flavor symmetries to $N_{f}=N_{c}$ . The corresponding three-dimensional analysis [26, 58, 28, 59] for such theories need to be performed.

Finally, there are many integrable lattice models [60, 17, 61, 62, 63, 64] obtained recently via exact results from supersymmetric gauge theories. A similar approach can be used for obtaining corresponding inversion relations. We hope to elaborate on this as well as on other topics elsewhere.

Acknowledgements

The work originated from stimulating discussions with Vyacheslav Spiridonov, I thank him for his help, suggestions, and contributions to the ideas presented here. I would like to thank Keita Nii, Mustafa Mullahasanoglu, Erdal Catak, Yasin Şale, Merve Kaftan, and Yağız Eren Kiremit for the discussions. This work has been supported by the Russian Science Foundation grant number 22-72-10122 2 (https://rscf.ru/en/project/22-72-10122). I am grateful to the Institut des Hautes Etudes Scientifiques, IHES (Bures-sur-Yvette, France), for the warm hospitality, where part of this work was carried out. The visit to IHES was supported by Nokia Fund in 2023.

Bibliography64

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TL;DR

Contribution

Findings

Abstract

Peer Reviews

Videos

Taxonomy

1 Introduction

1.1 Superconformal index

1.2 Duality for 3d SU(2)SU(2)SU(2) N=2\mathcal{N}=2N=2 SQCD with Nf=6N_{f}=6Nf​=6 flavor

2 Breaking of flavor symmetry

3 Concluding remarks

Acknowledgements

1.2 Duality for 3d $SU(2)$ $\mathcal{N}=2$ SQCD with $N_{f}=6$ flavor