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Free chiral protected-letter superconformal index

Physical scope

A conditional free, ungauged four-dimensional N=1 protected-letter/Fock trace on unit S3 with the explicit charge and convergence data stated here. It is not a continuum construction, gauge-projected index, empirical observation, or universal formula for interacting theories.

Example

Assume ${A C}_{ω}$ and use the right-handed index convention on the unit round $S^{3}$ from the earlier superconformal-index definition. Take an uncharged free four-dimensional $N = 1$ chiral model with trivial global gauge group, one complex scalar and one Weyl fermion in its stated chiral multiplet, no spatial boundary, and the directly supplied radial Hilbert-space construction; no path-integral measure is asserted. Its protected-letter data consist of one bosonic scalar letter $ϕ$ of charges $(E, J_{1}, J_{2}, R) = (1, 0, 0, \frac{2}{3})$ and one fermionic conjugate Weyl letter ${\overset{ˉ}{ψ}}_{\dot{+}}$ of charges $(\frac{3}{2}, 0, \frac{1}{2}, \frac{1}{3})$ . Its protected one-particle space is specified as follows. Let $N_{0} = {0, 1, 2, \dots}$ . For each $(m, n) \in N_{0}^{2}$ , let $b_{m n} = D_{p}^{m} D_{q}^{n} ϕ$ and $f_{m n} = D_{p}^{m} D_{q}^{n} {\overset{ˉ}{ψ}}_{\dot{+}}$ be orthonormal basis vectors. The commuting linear mode-shift maps $D_{p}, D_{q}$ act on their algebraic span by $D_{p} b_{m n} = b_{m + 1, n}$ , $D_{p} f_{m n} = f_{m + 1, n}$ , $D_{q} b_{m n} = b_{m, n + 1}$ , and $D_{q} f_{m n} = f_{m, n + 1}$ . They encode the two protected translation descendants in this specified Hilbert model; no continuum field-operator realization is asserted. Their increments of $(E, J_{1}, J_{2}, R)$ are $(1, \frac{1}{2}, \frac{1}{2}, 0)$ and $(1, - \frac{1}{2}, \frac{1}{2}, 0)$ , respectively. The mode charges are

\begin{aligned} (E, J_{1}, J_{2}, R) (b_{m n}) & = (1 + m + n, \frac{m - n}{2}, \frac{m + n}{2}, \frac{2}{3}), \\ (E, J_{1}, J_{2}, R) (f_{m n}) & = (\frac{3}{2} + m + n, \frac{m - n}{2}, \frac{1 + m + n}{2}, \frac{1}{3}), \\ (Δ, J_{+}, J_{-}) (b_{m n}) & = (0, m + \frac{1}{3}, n + \frac{1}{3}), \\ (Δ, J_{+}, J_{-}) (f_{m n}) & = (0, m + \frac{2}{3}, n + \frac{2}{3}) . \end{aligned}

The protected multiparticle Hilbert space is the completion of the span of orthonormal vectors $∣ r, s ⟩$ indexed by functions $r : N_{0}^{2} \to N_{0}$ and $s : N_{0}^{2} \to {0, 1}$ that vanish outside a finite subset of $N_{0}^{2}$ . Here $r_{m n}$ and $s_{m n}$ are the bosonic and fermionic occupation numbers; the fermion parity is $(- 1)^{\sum_{m, n} s_{m n}}$ , and the charge eigenvalues on $∣ r, s ⟩$ are the sums of the corresponding one-particle eigenvalues with multiplicities $r_{m n}, s_{m n}$ . Each charge is the maximal diagonal operator in this basis. This defines the symmetric bosonic and exterior fermionic Fock constructions and makes this the complete $\ker Δ$ sector: assume no additional zero-gap states or relations and no gauge projection. Any orthogonal complement in the full radial Hilbert space consists of positive-gap pairs satisfying the beta-independence theorem, its simultaneous gap/charge eigenbasis and reducing-domain conditions, and its trace-class conditions. For positive fugacities $0 < p, q < 1$ , the single-letter index and full index are

$i_{χ} (p, q) = \frac{(p q)^{1 / 3} - (p q)^{2 / 3}}{(1 - p) (1 - q)}, I_{χ} (p, q) = \prod_{m, n \geq 0} \frac{1 - (p q)^{2 / 3} p^{m} q^{n}}{1 - (p q)^{1 / 3} p^{m} q^{n}},$

where the double product is the limit of its finite rectangular partial products. These formulas describe the specified protected-letter model only; they assert no interacting or gauge-projected index.

Example references

Rastelli and Razamat, §2.3

Kinney et al., §3

Facts & Assumptions

Given: The unit-radius right-handed index convention and the free protected-letter/Fock data stated in the Example; $p, q \in (0, 1)$ are dimensionless.

[A1]

Assume ${A C}_{ω}$ , as required by the Hilbert, basis, and trace suppliers. The Axiom of Countable Choice ( ${A C}_{ω}$ )

[A2]

The free model supplies the orthonormal one-particle modes and mode-shift maps stated in the Example, together with the finite-support bosonic/fermionic occupation basis, additive charges, and parity. These occupation states exhaust $\ker Δ$ ; there is no gauge projection or further zero-gap relation. Any orthogonal complement consists of positive-gap pairs satisfying the beta-independence theorem, its joint eigenbasis/reducing-domain conditions, and its trace-class assumptions. This is exact model input for the example, not a theorem about a general SCFT.

[F1]

Four-dimensional $N = 1$ superconformal index: $Δ = E - 2 J_{2} - \frac{3}{2} R$ , $J_{+} = J_{1} + J_{2} + R / 2$ , and $J_{-} = J_{2} - J_{1} + R / 2$ . A fugacity pair is admissible when the full diagonal weighted heat operator extends to a bounded trace-class operator for every $β > 0$ ; the index is then its signed supplied-basis diagonal sum. Equality with the protected-sector trace is the separate theorem [F2].

[F2]

Beta independence of the trace-class superconformal index: under exact positive-energy pairing and the weighted trace-class hypothesis, the full superconformal heat supertrace equals its absolutely convergent weighted trace on $\ker Δ$ for every $β > 0$ .

Verification

Given: [A1]–[A2], [F1]–[F2], and the charge assignments in the Example.

1.1F1A2givenalgebra

For the scalar letter, $Δ = E - 2 J_{2} - \frac{3}{2} R = 1 - 0 - 1 = 0$ and $J_{+} = J_{-} = 1 / 3$ , so its signed fugacity weight is $(p q)^{1 / 3}$ .

1.2F1A2givenalgebra

For ${\overset{ˉ}{ψ}}_{\dot{+}}$ , $Δ = \frac{3}{2} - 1 - \frac{1}{2} = 0$ and $J_{+} = J_{-} = \frac{1}{2} + \frac{1}{6} = 2 / 3$ ; its fermion parity contributes the negative weight $- (p q)^{2 / 3}$ . Each $D_{p}$ adds one to $J_{+}$ and zero to $J_{-}$ , while each $D_{q}$ adds zero to $J_{+}$ and one to $J_{-}$ , and both preserve $Δ = 0$ .

2.1step 1.1step 1.2algebra

Summing the scalar and fermion towers gives $i_{χ} (p, q) = \sum_{m, n \geq 0} ((p q)^{1 / 3} - (p q)^{2 / 3}) p^{m} q^{n}$ . Since $0 < p, q < 1$ , the geometric sums are $\sum_{m, n \geq 0} p^{m} q^{n} = ((1 - p) (1 - q))^{- 1}$ , yielding the displayed single-letter formula.

3.1A2step 2.1algebra

For a bosonic letter of weight $x$ , all occupations contribute $1 + x + x^{2} + \dots = (1 - x)^{- 1}$ ; for a fermionic letter, the two allowed occupations contribute $1 - x$ . Applying these factors to each derivative mode produces the rectangular partial product in the Example. For finite mode sets this is an ordinary product of occupation sums; passage to the infinite product requires absolute convergence.

4.1A2step 2.1step 3.1algebra

Put $a = (p q)^{1 / 3} < 1$ , $b = (p q)^{2 / 3} < 1$ , $x_{m n} = a p^{m} q^{n}$ , and $y_{m n} = b p^{m} q^{n}$ . Their sums are $a / ((1 - p) (1 - q))$ and $b / ((1 - p) (1 - q))$ . For a finite rectangular mode set $S$ , the sum of absolute occupation weights supported in $S$ is $P_{S} = \prod_{(m, n) \in S} (1 + y_{m n}) / (1 - x_{m n})$ . Since $- \log (1 - x) \leq x / (1 - a)$ for $0 \leq x \leq a$ and $\log (1 + y) \leq y$ for $y \geq 0$ , one has $P_{S} \leq \exp ((a / (1 - a) + b) / ((1 - p) (1 - q)))$ . Every finite family of finite-support occupation states is supported in some such $S$ , so its absolute weight sum is bounded by this same finite constant. Thus the entire occupation-weight family is absolutely summable. Increasing rectangles exhaust all its states, justifying the signed trace/product identity from step 3.1. Moreover, $∣ (1 - y_{m n}) / (1 - x_{m n}) - 1 ∣ \leq (x_{m n} + y_{m n}) / (1 - a)$ , so the displayed product is absolutely convergent.

5.1A1A2F1F2step 4.1algebra∎

On the protected occupation basis let $Ξ_{0}$ have the positive diagonal weights just summed. It is bounded, and its finite-rank diagonal truncations have operator-norm error at most the omitted weight sum, which tends to zero. Hence it is compact, with summable singular values given by these nonnegative diagonal weights, and therefore trace class. The heat factor is one on this sector for every $β > 0$ . Its orthogonal direct sum with the complementary weighted heat operator supplied by [A2] is bounded trace class: the two diagonal weight families are summable, and finite-rank truncations again converge in operator norm. This proves full admissibility in [F1], using [A2]'s independent complementary-sector assumptions. Now [F2] applies and identifies the protected trace with the full superconformal heat supertrace, independent of $β$ .

Free chiral protected-letter superconformal index

Example

Example references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources