Russell universe

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#### Type theory
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#### Universes
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## Idea

Russell universes or universes à la Russell are types whose terms are types. In [[type theories]] without a separate type [[judgment]] $A \; \mathrm{type}$, only typing judgments $a:A$, what would have been type judgments are represented by typing judgments that $A$ is a term of a Russell universe $U$, $A:U$. Russell universes without a separate type judgment are used in many places in type theory, including in the [[HoTT book]], in [[Coq]], and in [[Agda]].

## Definition of a single Russell universe

If the type theory has a separate type judgment $A \; \mathrm{type}$, then one could define a single Russell universe in the type theory. We merely have

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash U \; \mathrm{type}} \qquad \frac{\Gamma \vdash A:U}{\Gamma \vdash A \; \mathrm{type}}$$

## Formal definition of a hierarchy of Russell universes

Dependent type theories typically come with a hierarchy of Russell universes, so that all types in the dependent type theory are elements of Russell universes. This is especially the case for dependent type theories without any separate type judgments at all, where types are necessarily defined as terms of Russell universes.

### Without a separate type judgment

One formal definition of a type theory with a hierarchy of Russell universes is as follows:

The type theory has [[judgments]]

* $\Gamma \; \mathrm{ctx}$, that $\Gamma$ is a [[context]]

* $i \; \mathrm{level}$, that $i$ is a universe level,

* $\phi \; \mathrm{prop}$, that $\phi$ is a [[proposition]],

* $\phi \; \mathrm{true}$, that $\phi$ is a [[true]] proposition,

and consists of the formal [[signature (in logic)|signature]] and [[inference rules]] of [[first-order theory|first-order]] [[Heyting arithmetic]] or [[Peano arithmetic]]. These rules ensure that there are an [[infinite]] number of indices, which are strictly ordered with [[strict total order]] $\lt$ and upwardly unbounded, where $i \lt s(i)$ is true for all indices $i$.

Now, we introduce the typing judgment $a:A$, which says that $a$ is a term of the type $A$. Instead of type judgments, we introduce a special kind of type called a **Russell universe** or **universe à la Russell**, whose terms are the types themselves. Russell universes are formalized with the following rules:

$$\frac{\Gamma \vdash i \; \mathrm{level}}{\Gamma \vdash U_i:U_{s(i)}} \qquad \frac{\Gamma \vdash i \; \mathrm{level} \quad \Gamma \vdash A:U_i}{\Gamma \vdash A:U_{s(i)}}\mathrm{cumul} \qquad \frac{\Gamma \vdash i \; \mathrm{level} \quad \Gamma \vdash A:U_i}{\Gamma \vdash \mathrm{Lift}_{i, j}(A):U_{s(i)}}\mathrm{lifting}$$

In addition, we have rules for contexts which state that one could add typing judgments to the list of contexts:

$$\frac{\Gamma \vdash i \; \mathrm{level} \quad \Gamma \vdash A:U_i}{(\Gamma, a:A) \; \mathrm{ctx}}$$

as well as rules saying that equality is preserved across universe levels:

$$\frac{\Gamma \vdash i \; \mathrm{level} \quad \Gamma \vdash j \; \mathrm{level} \quad \Gamma \vdash i = j \; \mathrm{true}}{\Gamma \vdash U_i \equiv U_j:U_{s(i)}}\mathrm{judgmental} \qquad \frac{\Gamma \vdash i \; \mathrm{level} \quad \Gamma \vdash j \; \mathrm{level} \quad \Gamma \vdash i = j \; \mathrm{true}}{\Gamma \vdash \mathrm{ap}_U^{i = j}:U_i =_{U_{s(i)}} U_j}\mathrm{typal}$$

### With a type judgment for each universe

One could also define a hierarchy of Russell universes [[à la Coquand]], in that the type theory has a type judgment for each universe $U$. Using the dependent type theory with no separate type judgment, instead of having only one term judgment $a:A$, for level $i$ and type $A:U_i$, we instead have an infinite number of [[type]] [[judgments]], one type judgment $A \; \mathrm{type}_i$ for every level $i$, indicating that $A$ is a type with level $i$, in addition to the [[term]] judgments $a:A$. Then, one has the following rules for Russell universes à la Coquand:

$$\frac{\Gamma \vdash i \; \mathrm{level}}{\Gamma \vdash U_i \; \mathrm{type}_{s(i)}} \qquad \frac{\Gamma \vdash i \; \mathrm{level} \quad \Gamma \vdash A \; \mathrm{type}_i}{\Gamma \vdash A \; \mathrm{type}_{s(i)}}\mathrm{cumul} \qquad \frac{\Gamma \vdash i \; \mathrm{level} \quad \Gamma \vdash A \; \mathrm{type}_i}{\Gamma \vdash \mathrm{Lift}(A) \; \mathrm{type}_{s(i)}}\mathrm{lifting}$$

$$\frac{\Gamma \vdash i \; \mathrm{level} \quad \Gamma \vdash A \; \mathrm{type}_i}{\Gamma \vdash A:U_i} \qquad \frac{\Gamma \vdash i \; \mathrm{level} \quad \Gamma \vdash A:U_i}{\Gamma \vdash A \; \mathrm{type}_i}$$

In addition, we have rules for contexts which state that one could add typing judgments to the list of contexts:

$$\frac{\Gamma \vdash i \; \mathrm{level} \quad \Gamma \vdash A \; \mathrm{type}_i}{(\Gamma, a:A) \; \mathrm{ctx}}$$

One could derive from these rules the above rules for Russell universes and context extension

$$\frac{\Gamma \vdash i \; \mathrm{level}}{\Gamma \vdash U_i:U_{s(i)}} \qquad \frac{\Gamma \vdash i \; \mathrm{level} \quad \Gamma \vdash A:U_i}{\Gamma \vdash A:U_{s(i)}}\mathrm{cumul} \qquad \frac{\Gamma \vdash i \; \mathrm{level} \quad \Gamma \vdash A:U_i}{\Gamma \vdash \mathrm{Lift}(A):U_{s(i)}}\mathrm{lifting}$$

$$\frac{\Gamma \vdash i \; \mathrm{level} \quad \Gamma \vdash A:U_i}{(\Gamma, a:A) \; \mathrm{ctx}}$$

### With a single separate type judgment

It is also possible to define the hierarchy of Russell universes with a single separate type judgment, such that every single type is in a Russell universe. The advantage of doing so is that one doesn't need to define the theory of universe levels before defining the type theory; one could instead simply define the natural numbers inside of the type theory itself, along with the hierarchy of Russell universes:

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathbb{N} \; \mathrm{type}} \qquad \frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash \mathrm{Level}(A):\mathbb{N}}$$

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma, n:\mathbb{N} \vdash U(n) \; \mathrm{type}} \qquad \frac{\Gamma \vdash A \; \mathrm{type}}{\Gamma \vdash A:U(\mathrm{Level}(A))}$$

$$\frac{\Gamma \vdash n:\mathbb{N} \quad \Gamma \vdash A:U(n)}{\Gamma \vdash A \; \mathrm{type}} \quad \frac{\Gamma \vdash n:\mathbb{N} \quad \Gamma \vdash A:U(n)}{\Gamma \vdash \mathrm{Level}(A) \equiv n:\mathbb{N}}$$

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash 0:\mathbb{N}} \qquad \frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathrm{Level}(\mathbb{N}) \equiv 0:\mathbb{N}}$$

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma, n:\mathbb{N} \vdash s(n):\mathbb{N}} \qquad \frac{\Gamma \; \mathrm{ctx}}{\Gamma, n:\mathbb{N} \vdash \mathrm{Level}(U(n)) \equiv s(n):\mathbb{N}}$$

In particular, every type $A$ has a universe level, which is a natural number, and the universe level of $\mathbb{N}$ is zero and the universe level of $U(n)$ given natural number $n$ is the successor $s(n)$.

Furthermore, every type $A$ of level $n$ lifts to another type $\mathrm{Lift}(A)$ of level $s(n)$, such that $\mathrm{Lift}(A)$ is a [[negative copy]] of $A$:

$$\frac{\Gamma \vdash n:\mathbb{N} \quad \Gamma \vdash A:U(n)}{\Gamma \vdash \mathrm{Lift}(A):U(s(n))}$$

$$\frac{\Gamma \vdash n:\mathbb{N} \quad \Gamma \vdash A:U(n)}{\Gamma, x:A \vdash \mathrm{LiftEl}(A)(x):\mathrm{Lift}(A)}$$

$$\frac{\Gamma \vdash n:\mathbb{N} \quad \Gamma \vdash A:U(n)}{\Gamma, y:\mathrm{Lift}(A) \vdash \mathrm{LiftEl}(A)^{-1}(y):A}$$

$$\frac{\Gamma \vdash n:\mathbb{N} \quad \Gamma \vdash A:U(n)}{\Gamma, x:A \vdash \mathrm{LiftEl}(A)^{-1}(\mathrm{LiftEl}(A)(x)) \equiv x:A}$$

$$\frac{\Gamma \vdash n:\mathbb{N} \quad \Gamma \vdash A:U(n)}{\Gamma, y:\mathrm{Lift}(A) \vdash \mathrm{LiftEl}(A)(\mathrm{LiftEl}(A)^{-1}(y)) \equiv y:\mathrm{Lift}(A)}$$

Next are the rules for [[function types]], which are necessary to define [[type families]] as elements of a type for [[dependent product types]] and the induction principle of the [[natural numbers type]]. Here, [[function types]] are indexed by the universe level $n$, since the function type indexed by $n$ are only definable for $U(n)$-small types, aka types with level $n$.

$$\frac{\Gamma \vdash n:\mathbb{N} \quad \Gamma \vdash A:U(n) \quad \Gamma \vdash B:U(n)}{\Gamma \vdash A \to_{n} B:U(n)}$$

$$\frac{\Gamma \vdash n:\mathbb{N} \quad \Gamma \vdash A:U(n) \quad \Gamma \vdash B:U(n) \quad \Gamma, x:A \vdash b(x):B}{\Gamma \vdash \lambda x:A.b(x):A \to_{n} B}$$

$$\frac{\Gamma \vdash n:\mathbb{N} \quad \Gamma \vdash A:U(n) \quad \Gamma \vdash B:U(n) \quad \Gamma \vdash b:A \to_{n} B}{\Gamma, x:A \vdash b(x):B}$$

$$\frac{\Gamma \vdash n:\mathbb{N} \quad \Gamma \vdash A:U(n) \quad \Gamma \vdash B:U(n) \quad \Gamma, x:A \vdash b(x):B}{\Gamma, x:A \vdash (\lambda x:A.b(x))(x) \equiv b(x):B}$$

$$\frac{\Gamma \vdash n:\mathbb{N} \quad \Gamma \vdash A:U(n) \quad \Gamma \vdash B:U(n) \quad \Gamma \vdash b:A \to_{n} B}{\Gamma \vdash \lambda x:A.b(x) \equiv b:A \to_{n} B}$$

Similarly, the rules for dependent function types are as follows:

$$\frac{\Gamma \vdash n:\mathbb{N} \quad \Gamma \vdash A:U(n) \quad \Gamma \vdash B:\mathrm{Lift}(A) \to_{s(n)} U(n)}{\Gamma \vdash \Pi(n, A, B):U(n)}$$

$$\frac{\Gamma \vdash n:\mathbb{N} \quad \Gamma \vdash A:U(n) \quad \Gamma \vdash B:\mathrm{Lift}(A) \to_{s(n)} U(n) \quad \Gamma, x:A \vdash b(x):B(\mathrm{LiftEl}(A)(x))}{\Gamma \vdash \lambda x:A.b(x):\Pi(n, A, B)}$$

$$\frac{\Gamma \vdash n:\mathbb{N} \quad \Gamma \vdash A:U(n) \quad \Gamma \vdash B:\mathrm{Lift}(A) \to_{s(n)} U(n) \quad \Gamma \vdash b:\Pi(n, A, B)}{\Gamma, x:A \vdash b(x):B(\mathrm{LiftEl}(A)(x))}$$

$$\frac{\Gamma \vdash n:\mathbb{N} \quad \Gamma \vdash A:U(n) \quad \Gamma \vdash B:\mathrm{Lift}(A) \to_{s(n)} U(n) \quad \Gamma, x:A \vdash b(x):B(\mathrm{LiftEl}(A)(x))}{\Gamma, x:A \vdash (\lambda x:A.b(x))(x) \equiv b(x):B(\mathrm{LiftEl}(A)(x))}$$

$$\frac{\Gamma \vdash n:\mathbb{N} \quad \Gamma \vdash A:U(n) \quad \Gamma \vdash B:\mathrm{Lift}(A) \to_{s(n)} U(n) \quad \Gamma \vdash b:\Pi(n, A, B)}{\Gamma \vdash \lambda x:A.b(x) \equiv b:\Pi(n, A, B)}$$

Finally, we have for each universe level $n:\mathbb{N}$ a [[natural numbers type]] $\mathrm{Nat}(n)$ such that $\mathrm{Nat}(0) \equiv \mathbb{N}$ and $\mathrm{Nat}(s(n)) \equiv \mathrm{Lift}(\mathrm{Nat}(n))$. In addition, each $\mathrm{Nat}(n)$ has element $\mathrm{zero}(n):\mathrm{Nat}(n)$ and function $\mathrm{succ}(n):\mathrm{Nat}(n) \to_{n} \mathrm{Nat}(n)$, defined via lifting the elements of $\mathrm{Nat}(n)$ across universe levels. Finally, each $\mathrm{Nat}(n)$ satisfies the induction principle of the [[natural numbers type]] over the universe $U(n)$.

* Formation rules for natural numbers types:

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma, n:\mathbb{N} \vdash \mathrm{Nat}(n):U(n)}$$

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathrm{Nat}(0) \equiv \mathbb{N}:U(0)} \qquad \frac{\Gamma \; \mathrm{ctx}}{\Gamma, n:\mathbb{N} \vdash \mathrm{Nat}(s(n)) \equiv \mathrm{Lift}(\mathrm{Nat}(n)):U(s(n))}$$

* Introduction rules for natural numbers types:

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma, n:\mathbb{N} \vdash \mathrm{zero}(n):\mathrm{Nat}(n)} \qquad \frac{\Gamma \; \mathrm{ctx}}{\Gamma, n:\mathbb{N} \vdash \mathrm{succ}(n):\mathrm{Nat}(n) \to_{n} \mathrm{Nat}(n)}$$

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathrm{zero}(0) \equiv 0:\mathrm{Nat}(0)} \qquad \frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \mathrm{succ}(0) \equiv s:\mathrm{Nat}(0) \to_{0} \mathrm{Nat}(0)}$$

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma, n:\mathbb{N} \vdash \mathrm{zero}(s(n)) \equiv \mathrm{LiftEl}(\mathrm{Nat}(n))(\mathrm{zero}(n)):\mathrm{Nat}(s(n))}$$

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma, n:\mathbb{N}, m:\mathrm{Nat}(s(n)) \vdash \mathrm{succ}(s(n)) \equiv \mathrm{LiftEl}(\mathrm{Nat}(n))(\mathrm{succ}(n)):\mathrm{Nat}(s(n)) \to_{s(n)} \mathrm{Nat}(s(n))}$$

* Elimination rules for natural numbers types:

$$\frac{
\begin{array}{c}
\Gamma \vdash n:\mathbb{N} \quad \Gamma \vdash C:\mathrm{Lift}(\mathrm{Nat}(n)) \to_{s(n)} U(n) \quad \Gamma \vdash c_0:C(\mathrm{LiftEl}(\mathrm{Nat}(n))(\mathrm{zero}(n))) \\
\Gamma \vdash c_s:\Pi\left(n, \mathrm{Nat}(n), \lambda x:\mathrm{Nat}(n).C(\mathrm{LiftEl}(\mathrm{Nat}(n))(x)) \to_{n} C(\mathrm{LiftEl}(\mathrm{Nat}(n))(\mathrm{succ}(n, x)))\right) \\
\end{array}}{\Gamma \vdash \mathrm{ind}_\mathrm{Nat}(n, C, c_0, c_s):\Pi\left(n, \mathrm{Nat}(n), \lambda x:\mathrm{Nat}(n).C(\mathrm{LiftEl}(\mathrm{Nat}(n))(x))\right)}$$

* Computation rules for natural numbers types:

$$\frac{
\begin{array}{c}
\Gamma \vdash n:\mathbb{N} \quad \Gamma \vdash C:\mathrm{Lift}(\mathrm{Nat}(n)) \to_{s(n)} U(n) \quad \Gamma \vdash c_0:C(\mathrm{LiftEl}(\mathrm{Nat}(n))(\mathrm{zero}(n))) \\
\Gamma \vdash c_s:\Pi\left(n, \mathrm{Nat}(n), \lambda x:\mathrm{Nat}(n).C(\mathrm{LiftEl}(\mathrm{Nat}(n))(x)) \to_{n} C(\mathrm{LiftEl}(\mathrm{Nat}(n))(\mathrm{succ}(n, x)))\right) \\
\end{array}}{\Gamma \vdash \mathrm{ind}_\mathrm{Nat}(n, C, c_0, c_s, \mathrm{zero}(n)) \equiv c_0:C(\mathrm{LiftEl}(\mathrm{Nat}(n))(\mathrm{zero}(n)))}$$

$$\frac{
\begin{array}{c}
\Gamma \vdash n:\mathbb{N} \quad \Gamma \vdash C:\mathrm{Lift}(\mathrm{Nat}(n)) \to_{s(n)} U(n) \quad \Gamma \vdash c_0:C(\mathrm{LiftEl}(\mathrm{Nat}(n))(\mathrm{zero}(n))) \\
\Gamma \vdash c_s:\Pi\left(n, \mathrm{Nat}(n), \lambda x:\mathrm{Nat}(n).C(\mathrm{LiftEl}(\mathrm{Nat}(n))(x)) \to_{n} C(\mathrm{LiftEl}(\mathrm{Nat}(n))(\mathrm{succ}(n, x)))\right) \\
\end{array}}{\Gamma, x:\mathrm{Nat}(n) \vdash \mathrm{ind}_\mathrm{Nat}(n, C, c_0, c_s, \mathrm{succ}(n, x)) \equiv c_s(\mathrm{ind}_\mathrm{Nat}(n, C, c_0, c_s, x)):C(\mathrm{LiftEl}(\mathrm{Nat}(n))(\mathrm{succ}(n, x)))}$$

Something similar could be done for [[Coquand universes]].

## Analogues in set theory

There are analogues of cumulative Russell universes in [[set theory]].

### With a single set judgment

One formal definition of a set theory with cumulative Russell universes is as follows:

The set theory has [[judgments]]

* $\Gamma \; \mathrm{ctx}$, that $\Gamma$ is a [[context]]

* $\kappa \; \mathrm{level}$, that $\kappa$ is a universe level,

* $\phi \; \mathrm{prop}$, that $\phi$ is a [[proposition]],

* $\phi \; \mathrm{true}$, that $\phi$ is a [[true]] proposition,

and consists of the formal [[signature (in logic)|signature]] and [[inference rules]] of [[first-order theory|first-order]] [[Heyting arithmetic]] or [[Peano arithmetic]]. These rules ensure that there are an [[infinite]] number of indices, which are strictly ordered with [[strict total order]] $\lt$ and upwardly unbounded, where $\kappa \lt \kappa^+$ is true for all indices $\kappa$.

Now, we introduce a single [[set]] [[judgment]] $A \; \mathrm{set}$ which says that $A$ is a set, as well as the membership relation $a \in A$, which says that $a$ in the set $A$. We introduce a special kind of set called a **cumulative Russell universe** or **cumulative universe à la Russell**, which formalized with the following rules:

$$\frac{\Gamma \vdash \kappa \; \mathrm{level}}{\Gamma \vdash V_\kappa \; \mathrm{set}} \qquad \frac{\Gamma \vdash \kappa \; \mathrm{level}}{\Gamma \vdash V_\kappa \in V_{\kappa^+} \; \mathrm{true}} \qquad \frac{\Gamma \vdash \kappa \; \mathrm{level} \quad \Gamma \vdash A \; \mathrm{set} \quad \Gamma \vdash A \in V_\kappa \; \mathrm{true}}{\Gamma \vdash A \in V_{\kappa^+} \; \mathrm{true}}\mathrm{cumul}$$

### With a separate set judgment for each set theory

There are also analogues of cumulative Russell universes [[à la Coquand]] in [[set theory]]. Instead of having a single set theory, one has a whole collection of set theories which embed into each other, with indices indicating which level the set theory lies on.

One formal definition of a set theory with cumulative Russell universes [[à la Coquand]] is as follows:

The set theory has [[judgments]]

* $\Gamma \; \mathrm{ctx}$, that $\Gamma$ is a [[context]]

* $\kappa \; \mathrm{level}$, that $\kappa$ is a level of set theory,

* $\phi \; \mathrm{prop}$, that $\phi$ is a [[proposition]],

* $\phi \; \mathrm{true}$, that $\phi$ is a [[true]] proposition,

and consists of the formal [[signature (in logic)|signature]] and [[inference rules]] of [[first-order theory|first-order]] [[Heyting arithmetic]] or [[Peano arithmetic]]. These rules ensure that there are an [[infinite]] number of indices, which are strictly ordered with [[strict total order]] $\lt$ and upwardly unbounded, where $\kappa \lt \kappa^+$ is true for all indices $\kappa$.

This allows us to add an infinite number of [[set]] [[judgments]], one set judgment $A \; \mathrm{set}_\kappa$ for every level $\kappa$, indicating that $A$ is a set with level $\kappa$, as well as an infinite number of [[membership relations]] $x \in_\kappa A$, one for each set judgment $\mathrm{set}_\kappa$. Then, one has the following [[inference rules]] for cumulative Russell universes [[à la Coquand]]:

$$\frac{\Gamma \vdash \kappa \; \mathrm{level}}{\Gamma \vdash V_\kappa \; \mathrm{set}_{\kappa^+}} \quad \frac{\Gamma \vdash \kappa \; \mathrm{level} \quad \Gamma \vdash A \; \mathrm{set}_\kappa}{\Gamma \vdash A \; \mathrm{set}_{\kappa^+}}$$

$$\frac{\Gamma \vdash \kappa \; \mathrm{level} \quad \Gamma \vdash A \; \mathrm{set}_\kappa}{\Gamma \vdash A \in_{\kappa^+} V_\kappa \; \mathrm{true}} \qquad \frac{\Gamma \vdash \kappa \; \mathrm{level} \quad \Gamma \vdash A \; \mathrm{set}_{\kappa^+} \quad \Gamma \vdash A \in_{\kappa^+} V_\kappa \; \mathrm{true}}{\Gamma \vdash A \; \mathrm{set}_\kappa}$$

This says that each $V_\kappa$ is a [[set]] which satisfies a [[reflection principle]].

## See also

* [[Tarski universe]]

* [[Coquand universe]]

* [[hierarchy of universes]]

* [[two-level type theory]]

* [[reflection principle]]

## References

The notion is due to:

* [[Per Martin-Löf]] (notes by [[Giovanni Sambin]]): p. 48 in: _Intuitionistic type theory_, Lecture notes Padua 1984, Bibliopolis, Napoli (1984) [[pdf](https://archive-pml.github.io/martin-lof/pdfs/Bibliopolis-Book-retypeset-1984.pdf), [[MartinLofIntuitionisticTypeTheory.pdf:file]]]

For more see the references at *[[type universe]]*.

[[!redirects Russell universe]]
[[!redirects Russell universes]]

[[!redirects à la Russell]]

[[!redirects universe à la Russell]]
[[!redirects universes à la Russell]]


Russell's paradox

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#### Type theory
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# Russell\'s paradox
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## Idea

**Russell's paradox** is a famous [[paradox]] of [[set theory]][^material] that was observed around 1902 by [[Ernst Zermelo]][^Schroeder] and, independently, by the logician [[Bertrand Russell]]. The paradox received instantly wide attention as it lead to a contradiction in Frege's monumental "[Foundations of Arithmetic](#Frege03)" (1893/1903) whose second volume was just about to go to print when Frege was informed about the inconsistency by Russell.

The paradox entangles a concept with its own extension in a **vicious circle**. The attempt to overcome this _circularity in set formation_ had a huge impact on subsequent forms of axiomatic set theory and in the aftermath mathematical logic became heavily focussed on consistency proofs for fully specified formal theories: the paradoxes triggered a shift in the foundation of mathematics away from the mathematics to the foundation itself.

>Doch zur Sache selbst! Herr Russell hat einen Widerspruch aufgefunden, der nun dargelegt werden mag.[^Frege]

[^material]: naive [[material set theory]] that is!

[^Schroeder]: Zermelo was apparently led to the paradox by considering a purported proof of Ernst Schröder in his _Algebra der Logik_ (1890) that Boole's concept of a universal class 1 was contradictory. Interestingly, the solution suggested by Schröder is reminiscent of [[type theory]] or even nested [[universe|universes]] where a universal class 1 recurs at each level.

[^Frege]: Frege ([1903](#Frege03), p.253).

## Statement

If one assumes a naive [[axiom]] of [[full comprehension]], one can form the [[set]]

$$ R = \{ x | x \notin x \}. $$

One then asks: is $R\in R$?  If so, then $R\notin R$ by definition, whereas if not, then $R\in R$ by definition.  Thus we have both $R\in R$ and $R\notin R$, a [[contradiction]].

## Related ideas

Russell's paradox is closely related to the classical [[liar paradox]] ("this sentence is false"), to Gödel's [[incompleteness theorem]], and to the [[halting problem]] --- all use a [[diagonalization argument]] to produce an object which talks about itself in a contradictory or close-to-contradictory way.

On the other hand, [[Cantor's paradox]] can be said to "[[beta-reduction|beta-reduce]]" to Russell's paradox when we apply [[Cantor's theorem]] to the supposed set of all sets. See [[Cantor's paradox]] for explanation.

Also related:

* [[Burali-Forti's paradox]]

* [[Curry's paradox]]

## Resolutions
 {#Resolutions}

There are a number of possible resolutions of Russell's paradox.

* Russell himself (1903,1908,1910) proposed the introduction of [[type theory]] as a solution e.g. in _[[Principia Mathematica]]_ (1910) an intricate system of [[ramified types]] tracks the variables of propositional functions in order to prevent circular propositions. This is inspired by Poincaré's ideas on [[predicative mathematics|impredicativity]] and can be viewed as a radical generalisation of Frege's ontological distinction between an argument as a satured object and a function or concept as an unsaturated object.

* The "[[classical mathematics|classical]]" solution, adopted in [[ZFC]] and thus by the mathematical "mainstream", is to restrict the axiom of [[comprehension]] so as to disallow the formation of the set $R$: one requires that the set being constructed be a subset of some already existing set.  The restricted axiom is usually given a different name such as the [[axiom of separation]].

* Another solution is to distinguish between sets and proper [[class|classes]] (= collections that are "too big" to be sets) as e.g. in [[NBG]] "set" theory.[^Cantor]  Here we may write down the definition of $R$, but from $R \notin R$ we may conclude $R \in R$ only if we already know that $R$ is a set; the $x$ in the definition must be a set.  So we have no contradiction, but only a proof that $R$ is a [[proper class]].

[^Cantor]: This solution proposed by J. von Neumann in the 1920s can be viewed as related to ideas of G. Cantor. The latter knew about similar phenomena concerning "the set of all sets" (in fact, Russell hit upon the paradox in a reflection on Cantor's proof of the inexistence of a largest cardinal number), and had already pointed out in 1885 in a review of Frege's "Grundlagen der Arithmetik" that not every concept has an extension. Therefore Cantor proposed in letters to Dedekind and Jourdain to differentiate between _'consistent multiplicities'_ ("compossible" multiplicities one could say) where things can coexist or compose to a consistent whole - 'ensemble' (fr.) which correspond to sets in the usual sense and, as completed collections, can in turn be elements in other sets, from _'inconsistent multiplicities'_ whose elements cannot consistently completed to a whole and cannot be member of other collections due to this lack of 'unity'.

* In the set theory called [[New Foundations]], the axiom of comprehension is restricted in a rather different way, by requiring the set-defining formula to be "stratifiable".  Since the formula $x\notin x$ is not stratifiable, the set $R$ cannot be formed.  This related to Russell's ideas on [[ramified types]].

* In most [[structural set theories]], the featurelessness of the elements of the structural sets secures the consistency of set formation. If sets cannot be elements of other sets, then the "definition" of $R$ is just a [[type]] error. The same is true in other structural [[foundations|foundational systems]] such as (modern, non-Russellian) [[type theory]].  However, Russell's paradox can be recreated in structural foundations with inconsistent [[universes]] by constructing [[pure sets]] within them.

* Alternatively, one can change the underlying logic.  Passing to [[constructive logic]] does not help: although there is a seeming appeal to [[excluded middle]] (either $R\in R$ or $R\notin R$), without using excluded middle we can obtain that $R$ is both not in $R$ and not not in $R$, which is also a contradiction.  However, passing to [[linear logic]] (or even [[affine logic]]) does help: there is an unavoidable use of [[contraction rule|contraction]] in the paradox.  There exist consistent [[linear set theory|linear set theories]] with the full comprehension axiom, in which $R\in R$ implies $R\notin R$ and vice versa, but we can never get both $R\in R$ and $R\notin R$ at the same time to produce a paradox. (See the Ph.D. thesis  Linear Set Theory of Masaru Shirahata, completed under Grigori Mints supervision, Stanford University, 1994.)

* Finally, and perhaps most radically, one can decide to *allow* contradictions, choosing to use a [[paraconsistent logic]].  There exist nontrivial paraconsistent set theories with full comprehension in which the set $R$ exists, being both a member of itself and not a member of itself.

## References

Zermelo's observation which is mentioned in a footnote of his 1908 paper on the [[well-ordering theorem]] is analyzed in

* B. Rang, W. Thomas, _Zermelo's Discovery of the "Russell Paradox"_ , Hist. Math. **9** no.1 (1981) pp.15-22. <https://doi.org/10.1016/0315-0860(81)90002-1>

together with a transcription of a note dated 16 April 1902 by Husserl describing Zermelo's proof.

Russell indicated the contradiction leading to the inconsistency of G. Frege's system of "Grundgesetze der Arithmetik" in a famous letter to the latter on June 16th 1902 which together with Frege's reply is reprinted pp.124-128 in

* J. van Heijenoort (ed.), _From Frege to Gödel - A Source Book in Mathematical Logic 1879-1931_ , Harvard UP 1967.

A re-typed version is available from the blog post [here](https://thehighergeometer.wordpress.com/2022/10/11/russells-paradox-the-original-letter-to-frege/).

For an account of Russell's encounter with the paradox:

* I. Grattan-Guinness, _How Russell Discovered his Paradox_ , Hist. Math. **5** no.2 (1978) pp.127-137.

The first published account is presumably in the appendix of

* {#Frege03}G. Frege, _Grundgesetze der Arithmetik II_ , Pohle Jena 1903.

Russell discusses the paradox extensively in chapter X of

* B. Russell, _The Principles of Mathematics_ , Cambridge UP 1903.

The influence of Poincaré's views shows in

* B. Russell, _Les paradoxes de la logique_ , Revue de métaphysique et de morale **14** no.5 (1906) pp.627-650. ([gallica](http://gallica.bnf.fr/ark:/12148/bpt6k11064n))

See also

* P. Várdy, _Some Remarks on the Relationship  between Russell's Vicious-Circle Principle and Russell's Paradox_ , Dialectica **33** (1979) pp.3-19.

Discussion of a paradox similar to Russell's in [[type theory]] with [[W-types]] is in

* [[Thierry Coquand]], _The paradox of trees in Type Theory_ (1991) ([web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.39.1953))

category: paradox

[[!redirects Russell paradox]]
[[!redirects Russell Paradox]]
[[!redirects Russell's paradox]]
[[!redirects Russell's Paradox]]
[[!redirects Russell's paradox]]
[[!redirects Russell's Paradox]]
[[!redirects Russell\'s paradox]]
[[!redirects Russell\'s Paradox]]


Russian constructivism

# Russian constructivism
* table of contents
{: toc}

## Summary

The Russian school of [[constructive mathematics]], associated principally with [[Andrey Markov Jr]], was (is?) a variety of constructive mathematics focussing on [[recursion theory]].

## Principles and results

*  Classically true principles (not always accepted by other constructivists):
   *  [[Markov's principle]] for [[natural numbers]]: if an [[infinite sequence]] of [[binary digits]] is not all $0$, then it has a least one $1$;
   *  [[dependent choice]] for natural numbers.
*  Classically false principles:
   *  every [[partial function]] from $\mathbb{N}$ to $\mathbb{N}$ is [[computable function|computable]];
   *  every set is a [[subquotient]] of $\mathbb{N}$.
*  Classically false results (false w.r.t. classical functions and sets):
   *  every [[total function]] from $\mathbb{R}$ (the [[real line]]) to $\mathbb{R}$ is pointwise [[continuous function|continuous]] ([[Ceitin's theorem]]);
   *  there exist continuous functions from $[0,1]$ (the [[unit interval]]) to $\mathbb{R}$ that are pointwise continuous but not [[uniformly continuous map|uniformly continuous]];
   *  there exist [[bounded subspace|bounded]] sets of real numbers with no [[supremum]] (given by [[Specker sequences]]).

[[!redirects Russian constructivism]]
[[!redirects Russian constructive mathematics]]
[[!redirects recursive constructivism]]
[[!redirects recursive constructive mathematics]]
[[!redirects constructive recursive mathematics]]
[[!redirects Russian recursive mathematics]]


Rust

Rust is an efficient programming language with compile time memory safety model based on the innovative notion of ownership and borrows, small runtime, modern concurency, zero cost abstractions, both object oriented and functional programming model. It's speed is about the speed of comparable C programs. Rust does not require garbage collector, hence it is useful for real time programming as garbage collectors periodically stall the system for order of 100 ms. It compiles both to native code (LLVM compilers are commonly used) and to [[WebAssembly]]. It is particularly useful for real time programming, systems programming, embedded systems, web programming (via WebAssembly), concurrent systems and safety critical applications including [[blockchain]] and [[smart contract]]s. It easily links to the languages in C/C++ (ABI) linking model.

* [rust-lang.org](https://www.rust-lang.org), the official rust [book](https://doc.rust-lang.org/book), [reference](https://doc.rust-lang.org/reference/index.html), [crates.io](https://crates.io)
* [what is rustdoc](https://doc.rust-lang.org/rustdoc/what-is-rustdoc.html#using-rustdoc-with-cargo)
* Jim Blandy, Jason Orendorff, _Programming Rust: fast, safe systems development_, O’Reilly Media 2017
* Steve Klabnik, Carol Nichols, _The rust programming language_, no starch press, xxvii+519 pp., 2018
* Rust crashcourse Rustlang (2hr) [yt](https://www.youtube.com/watch?v=zF34dRivLOw)
* yandex Rust course (in Russian, А. А. Кладов) [2019-spring](https://compscicenter.ru/courses/rustprogramming/2019-spring)
* double linked list in safe Rust [github](https://gist.github.com/pythonesque/5943252fb464b49123fa), std::collections::[LinkedList](https://doc.rust-lang.org/std/collections/struct.LinkedList.html), but use VectDeque and see Learn Rust With Entirely [Too Many Linked Lists](https://rust-unofficial.github.io/too-many-lists)
* _IOKH launches [[zoranskoda:Cardano]] Rust project_ [cardanorust.iohkdev.io](https://cardanorust.iohkdev.io), [news](https://www.the-blockchain.com/2018/10/02/iohk-launches-cardano-rust-project) Feb 2018
* Rust can also compile to [RISC-V](https://en.wikipedia.org/wiki/RISC-V), see github/[riscv-rust](https://github.com/riscv-rust)
* exonom[.com](https://exonum.com), [doc](https://exonum.com/doc/version/latest) -- extensible open-source framework for creating (permissioned) blockchain applications, smart contracts replaced by supposedly more flexible notion of [services](https://exonum.com/doc/version/latest/architecture/services) (see [comparision](https://exonum.com/doc/version/latest/architecture/services/#services-vs-smart-contracts)). Exonum services can be coded in Rust or Java.
* [rustsim](https://www.rustsim.org) numerical lib. and linear algebra for Rust
* list of crates on neural networks in Rust is [here](https://www.arewelearningyet.com/neural-networks)
* [[zoranskoda:Hyperledger]] Sabre is a transaction family/processor for Wasm on Hyperledger [Sawtooth](https://www.hyperledger.org/category/hyperledger-sawtooth) blockchain framework github/hyperledger/[sawtooth-sabre](https://github.com/hyperledger/sawtooth-sabre)

Linking to C/C++ is usually done via rust-bindgen

* github/[rust-lang/rust-bindgen](https://github.com/rust-lang/rust-bindgen); user guide [rust-lang.github.io/rust-bindgen](https://rust-lang.github.io/rust-bindgen)

Rust is commonly compiled either to the native code or optionally to [[WebAssembly]].

* Rust & WebAssembly with Nick Fitzgerald [yt](https://www.youtube.com/watch?v=ZiiTRxWk8gA)

Parity supports [Kovan](https://kovan.etherscan.io) testnet for future candidate version of Ethereum which supports wasm (hence one can use Rust), see

* _Toward a brighter future for smart contracts_, at [troubles.md](http://troubles.md/posts/rust-smart-contracts)

Rust on Parity Substrate can be used for smart contracting via Ink eDSL package

* github/paritytech/[ink](https://github.com/paritytech/ink), [medium](https://medium.com/block-journal/introducing-substrate-smart-contracts-with-ink-d486289e2b59)
* Shawn Tabrizi, [substrate contracts workshop](https://shawntabrizi.com/substrate-contracts-workshop/#)

Some research papers

* Oxide: The Essence of Rust ([arXiv:1903.00982](https://arxiv.org/abs/1903.00982))
* _RustBelt: securing the foundations of the rust programming language_ [doi](https://doi.org/10.1145/3158154) (open access)
* B. Lamowski et al. _Sandcrust: automatic sandboxing of unsafe components in Rust_, [doi](https://doi.org/10.1145/3144555.3144562)
* Garming Sam, Nick Cameron, Alex Potanin, _Automated refactoring of rust programs_,  ACSW '17: Proceedings of the Australasian Computer Science Week Multiconference [doi](https://doi.org/10.1145/3014812.3014826)
* _POSTER: Rust SGX SDK: Towards Memory Safety in Intel SGX Enclave_ [doi](https://doi.org/10.1145/3133956.3138824)

category: computer science

[[!redirects rust]]


Rustam Sadykov

* [webpage](https://www.math.ksu.edu/~sadykov/)

## Selected writings

On [[surgery theory]]:

* [[Rustam Sadykov]], _Elements of Surgery Theory_ ([pdf](https://www.math.ksu.edu/~sadykov/Lecture%20Notes/Surgery%20Theory.pdf), [[SadykovSurgeryTheory.pdf:file]])

category: people


Ryan Grady

[Ryan Grady's Homepage](http://nd.edu/~rgrady)

category: people


Ryan Rohm

* [webpage](https://physics.unc.edu/people/rohm-ryan-m/)

* [Wikipedia entry](https://en.wikipedia.org/wiki/Ryan_Rohm)

## Selected writings

Introducing [[heterotic string theory]]:

* [[David Gross]], [[Jeffrey Harvey]], [[Emil Martinec]], [[Ryan Rohm]]:

  _Heterotic string theory (I). The free heterotic string_  Nucl. Phys. B 256 (1985), 253 (doi:10.1016/0550-3213(85)90394-3)

  _Heterotic string theory (II). The interacting heterotic string_ , Nucl. Phys. B 267 (1986), 75 (doi:10.1016/0550-3213(86)90146-X)

Introducing $H_3$-[[twisted de Rham cohomology]] (in discussion of the [[B-field]] in [[string theory]]):

* [[Ryan Rohm]], [[Edward Witten]], around (23) and appendix of: _The antisymmetric tensor field in superstring theory_, Annals of Physics Volume 170, Issue 2, September 1986, Pages 454-489 (doi:10.1016/0003-4916(86)90099-0)

category: reference


Ryan Thorngren

* [MO home page](http://mathoverflow.net/users/5312/ryan-thorngren)

category: people


Ryszard Engelking

Ryszard Engelking, Polish mathematician, born 1935.

* [Wikipedia entry](https://en.wikipedia.org/wiki/Ryszard_Engelking),

## Selected writings

On the notion of [[dimension]] for (in particular) [[separable metric spaces]]:

* {#Engelking78} [[Ryszard Engelking]], _Dimension Theory_, Mathematical Library **19**, North-Holland Publishing/Polish Scientific Publishers 1978 ([pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/engelking.pdf))

* {#Engelking95} [[Ryszard Engelking]], _Theory of Dimensions -- Finite and Infinite_, Sigma Series in Pure Mathematics **10**, Helderman 1995 ([pdf](http://www.gbv.de/dms/goettingen/187241074.pdf))

On ([[general topology|general]]) [[topology]]:

* _General Topology_, PWN, Warszawa 1977
* {#Engelking89} [[Ryszard Engelking]], *General Topology*, Revised edition, Sigma series in pure mathematics **6**, Heldermann 1989 ([ ISBN 388538-006-4](https://www.heldermann.de/SSPM/SSPM06/sspm06.htm))

## Related $n$Lab entries

* [[metrization theorem]]

* [[topology]]

category: people


Ryszard Nest

* [webpage](http://www.math.ku.dk/~rnest/)

## Related entries

* [[KK-theory]]

category: people


Ryszard R. Andruszkiewicz


Rzk

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[proof assistant]] implementing [[simplicial homotopy type theory]].

## Related entries

[[!include proof assistants and formalization projects -- list]]

## References

Named after [[Charles Rezk]]'s definition of [[complete Segal spaces]] (cf. *[[Rezk completion]]*).

Landing page:

* [github.com/rzk-lang](https://github.com/rzk-lang)

Formalization of the [[(infinity,1)-Yoneda lemma|$(\infty,1)$-Yoneda lemma]] via [[simplicial homotopy type theory]] (in [[Rzk]]):

* [[Nikolai Kudasov]], [[Emily Riehl]], [[Jonathan Weinberger]]. *Formalizing the $\infty$-categorical Yoneda lemma* (2023) [[arXiv:2309.08340](https://arxiv.org/abs/2309.08340)]

Overview talk:

* [[Nikolai Kudasov]]. *Rzk proof assistant and simplicial HoTT formalization*, Talk at [Homotopy Type Theory Electronic Seminar Talks (HoTTEST)](uwo.ca/math/faculty/kapulkin/seminars/hottest.html) (2023) [video](https://www.youtube.com/watch?v=IMxPu2agUNo)


Récoltes et semailles

**Récoltes et semailles** (Harvest and planting) is a set of  autobiographical reflections, written by [[Alexandre Grothendieck]].

* [[Alexandre Grothendieck]], _Récoltes et semailles: Réflexions et témoignages sur un passé de mathématicien_, [Grothendieck Circle page](https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/index.php), [pdf](https://agrothendieck.github.io/divers/ReS.pdf)
* Russian translation, _Урожаи и посевы_.  [url](http://www.mccme.ru/free-books/grothendieck/RS.html)
* English translation by Roy Lisker of the Preface, Introduction and first 18 sections. [url](http://matematicas.unex.es/~navarro/res/lisker1.pdf)
* Spanish translation by Juan A. Navarro González of the beginning. [url](http://matematicas.unex.es/~navarro/res/)

See also:

* [[Pierre Schapira]], *A truncated manuscript*, Inference-Review.com **7** 3 (Dec 2022) [[arXiv:2301.02898](https://arxiv.org/abs/2301.02898), [doi:10.37282/991819.22.61](https://doi.org/10.37282/991819.22.61)]

## Related entries

* [[The Rising Sea]]

* [[EGA]], [[FGA]], [[SGA]]

[[!redirects Recoltes et Semailles]]


Rémi Léandre

* [homepage](http://math.u-bourgogne.fr/IMB/leandre)

* _Path integrals in non-commutative geometry_, in Encyclopedia of Mathematical Physics (Elsevier, 2006)
* _Stochastic analysis without probability: study of some basic tools_, J. Pseudo-Differ. Oper. Appl. 1 (2010), no. 4, 389–400, preprint version [pdf](http://math.u-bourgogne.fr/IMB/prepub/prepub2010/leandre601.pdf)
* _Theory of distributions in the sense of Connes-Hida and Feynman path integral on a manifold_, [ps](http://www.iecn.u-nancy.fr/Preprint/publis/Textes/2002-44.ps)
* _Connes-Hida calclus in index theory_, [ps](http://www.iecn.u-nancy.fr/Preprint/publis/Textes/2003-30.ps)
* _A Lie algebroid on the Wiener space_, Adv. Math. Phys. 2010, Art. ID 146719, 17 pp. [MR2011j:58064](http://www.ams.org/mathscinet-getitem?mr=2594930)
* J D S Jones, Rémi Léandre, _$L_p$ Chen forms on loop spaces_,
In: Barlow M, Bingham N (eds.) Stochastic Analysis, 104–162, Cambridge Univ. Press 1991


Rémi Rhodes

* [institute page](http://www.i2m.univ-amu.fr/perso/remi.rhodes/english.html)

## Selected writings

On the rigorous construction of [[Liouville theory]] (from the [[Polyakov action]]):

* [[François David]], [[Antti Kupiainen]], [[Rémi Rhodes]], [[Vincent Vargas]], _Liouville Quantum Gravity on the Riemann sphere_, Communications in Mathematical Physics volume 342, pages869–907 (2016) ([arxiv:1410.7318](https://arxiv.org/abs/1410.7318))

* [[Colin Guillarmou]], [[Rémi Rhodes]], [[Vincent Vargas]], *Polyakov's formulation of 2d bosonic string theory*, Publ. Math. IHES **130** (2019) 111–185 [[arXiv:1607.08467](https://arxiv.org/abs/1607.08467), [doi:10.1007/s10240-019-00109-6](https://doi.org/10.1007/s10240-019-00109-6)]

* [[Antti Kupiainen]], [[Rémi Rhodes]], [[Vincent Vargas]], _Integrability of Liouville theory: proof of the DOZZ Formula_, Annals of Mathematics, Pages 81-166 from Volume 191 (2020), Issue 1, ([arxiv:1707.08785](https://arxiv.org/abs/1707.08785), [doi:10.4007/annals.2020.191.1.2](https://doi.org/10.4007/annals.2020.191.1.2))

* [[Antti Kupiainen]], [[Rémi Rhodes]], [[Vincent Vargas]], *The DOZZ formula from the path integral*, Journal of High Energy Physics volume 2018, Article number: 94 (2018) ([arXiv:1803.05418](https://arxiv.org/abs/1803.05418) doi:10.1007/JHEP05(2018)094)

and as a [[functorial field theory]] following [Segal 1988](conformal+field+theory#Segal88):

* [[Colin Guillarmou]], [[Antti Kupiainen]], [[Rémi Rhodes]], [[Vincent Vargas]], *Segal's axioms and bootstrap for Liouville Theory* [[arXiv:2112.14859](https://arxiv.org/abs/2112.14859)]

and via the [[conformal bootstrap]]:

* [[Colin Guillarmou]], [[Antti Kupiainen]], [[Rémi Rhodes]], [[Vincent Vargas]], _Conformal bootstrap in Liouville Theory_ ([arxiv:2005.11530](https://arxiv.org/abs/2005.11530))

category: people


Rényi entropy

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[information theory]], *Rényi entropy* refers to a class of measures,  of [[entropy]] that are essentially [[logarithms]] of diversity indices.

For special values of its parameter, the notion of Rényy entropy reproduces all of: [[Shannon entropy]], [[Hartley entropy]]/[[max-entropy]] and [[min-entropy]].

## Definition

Let $p$ be a [[probability distribution]] over $n \in \mathbb{N}$ elements, and let $\alpha$ be a non-[[negative number|negative]] [[real number]] not equal to 1:

$$
  \alpha
    \;\in\;
  \mathbb{R}_{\geq 0} \setminus \{1\}
  \,.
$$

The *Rényi entropy* of $p$ *at order $\alpha$* is:

$$
  H_\alpha(p)
    \;\coloneqq\;
  \frac{1}{1-\alpha}
  \log
  \left(
    \sum_{i=1}^n
    (p_i)^\alpha
  \right)
  \,.
$$

## Properties

### Relation to other notions of entropy
 {#RelationToOtherNotionsOfEntropy}

For various (limiting) values of $\alpha$ the Rényi entropy reduces to notions of entropy that are known by their own names:

* for $\alpha = 0$, Rényi entropy equals the *[[Hartley entropy]]*/*[[max-entropy]]*,

* in the [[limit of a sequence|limit]] $\alpha \to 1$, Rényi entropy equals the [[Shannon entropy]],

* for $\alpha = 2$, Rényi entropy equals the [[collision entropy]],

* in the [[limit of a sequence|limit]] $\alpha \to \infty$, Rényi entropy    equals the *[[min-entropy]]*.

[[!include Renyi entropies -- table ]]

### Monotonicity

The Rényi entropy is an anti-[[monotone function]] in the order-parameter $\alpha$:

$$
  \alpha_1 \;\leq\; \alpha_2
  \;\;\;\;\;\;\;
    \Rightarrow
  \;\;\;\;\;\;\;
  H_{\alpha_1}(p)
    \;\geq\;
  H_{\alpha_2}(p)
  \,.
$$

(e.g. [Ram & Sason 16, Fact 1](#RamSason16))

In particular, in terms of the [above](#RelationToOtherNotionsOfEntropy) special cases, this means that

$$
  HartlyEntropy
  \;\geq\;
  ShannonEntropy
  \;\geq\;
  CollisionEntropy
  \;\geq\;
  MinEntropy
  \,.
$$

## Related concepts

* [[Hartley entropy]], [[Shannon entropy]], [[collision entropy]], [[min-entropy]]

* [[holographic Renyi entropy]]

## References

Due to:

* [[Alfréd Rényi]], *On Measures of Entropy and Information*, Berkeley Symposium on Mathematical Statistics and Probability, 1961: 547-561 (1961) ([euclid](https://projecteuclid.org/ebooks/berkeley-symposium-on-mathematical-statistics-and-probability/Proceedings%20of%20the%20Fourth%20Berkeley%20Symposium%20on%20Mathematical%20Statistics%20and%20Probability,%20Volume%201:%20Contributions%20to%20the%20Theory%20of%20Statistics/chapter/On%20Measures%20of%20Entropy%20and%20Information/bsmsp/1200512181))

Textbook account:

* J. Aczél, Z. Daróczy, Chapter 5 of: *On Measures of Information and their Characterizations*, Mathematics in Science and Engineering **115**, Academic Press 1975 ([ISBN:978-0-12-043760-3](https://www.sciencedirect.com/bookseries/mathematics-in-science-and-engineering/vol/115/suppl/C))

See also

* Wikipedia, *[Rényi entropy](https://en.wikipedia.org/wiki/R%C3%A9nyi_entropy)*

* {#RamSason16} Eshed Ram, Igal Sason, *On Renyi Entropy Power Inequalities* ([arXiv:1601.06555](https://arxiv.org/abs/1601.06555))

On [[holographic Renyi entropy]] in relation to [[holographic entanglement entropy]] and [[quantum error correcting codes]]:

* [[Xi Dong]], *The Gravity Dual of Renyi Entropy*, Nature Communications 7, 12472 (2016) ([arXiv:1601.06788](https://arxiv.org/abs/1601.06788), [doi:10.1038/ncomms12472]( https://doi.org/10.1038/ncomms12472))

* [[Chris Akers]], [[Pratik Rath]], *Holographic Renyi entropy from quantum error correction*,  J. High Energ. Phys. 2019, 52 (2019) ([arXiv:1811.05171](https://arxiv.org/abs/1811.05171))

[[!redirects Rényi entropies]]

[[!redirects Renyi entropy]]
[[!redirects Renyi entropies]]


S-brane

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An _S-brane_ is a [[spacelike]] [[brane]], hence a brane [[instanton]].

For instance in [[11-dimensional supergravity]] S2-branes are [[membrane instantons]].

\begin{center}
\begin{imagefromfile}
  "file_name": "S-Branes.jpg",
  "width": 600
\end{imagefromfile}
\end{center}

> graphics grabbed from [Chen-Galtsov-Gutperle 02](#ChenGaltsovGutperle02)

## Properties

### Near horizon geometry

The [[near horizon geometry]] of an S$p$-brane involves [[hyperbolic manifolds]] ([Gutperle-Strominger 02 (5.20)](#GutperleStrominger02), [Chen-Galtsov-Gutperle 02, A.](#ChenGaltsovGutperle02), [Roy 02, p. 3-4](#Roy02)), the [[isometry group]] being $Iso(p+1) \times SO(d-p-2,1)$

## Related concepts

* [[M-theory on hyperbolic manifolds]]

[[!include table of branes]]

## References

The S-brane solutions in [[type II supergravity]] and [[11-dimensional supergravity]] were found in

* {#GutperleStrominger02} Michael Gutperle, [[Andrew Strominger]], _Spacelike Branes_, JHEP 0204:018, 2002  ([arXiv:hep-th/0202210](https://arxiv.org/abs/hep-th/0202210))

* {#ChenGaltsovGutperle02} Chiang-Mei Chen, Dmitri V. Gal'tsov, Michael Gutperle, _S-brane Solutions in Supergravity Theories_, Phys. Rev. D66 (2002) 024043 ([arXiv:hep-th/0204071](https://arxiv.org/abs/hep-th/0204071))

and by a different method in

* {#KruczenskiMyersPeet02} [[Martin Kruczenski]], [[Robert Myers]], [[Amanda Peet]], _Supergravity S-Branes_, JHEP 0205 (2002) 039 ([arXiv:hep-th/0204144](https://arxiv.org/abs/hep-th/0204144))

and the two formulations were shown to be equivalent in

* {#Roy02} Shibaji Roy, _On supergravity solutions of space-like Dp-branes_, JHEP 0208:025, 2002 ([arXiv:hep-th/0205198](https://arxiv.org/abs/hep-th/0205198))

[[!redirects S-branes]]

[[!redirects SM2-brane]]
[[!redirects SM2-branes]]


S-category

Depending what $S$ stands for various things can be called $S$-category. E.g. if $S$ is a category then we talk about categories over $S$, for them see [[overcategory]].

This entry is rather about another notion of $S$-category introduced in

* [[Tomasz Brzeziński]], _Notes on formal smoothness_, *in*: Modules and Comodules (series _Trends in Mathematics_). T Brzeziński, JL Gomez Pardo, I Shestakov, PF Smith (eds), Birkhäuser, Basel, 2008, pp. 113-124 ([doi](http://dx.doi.org/10.1007/978-3-7643-8742-6), [arXiv:0710.5527](http://arxiv.org/abs/0710.5527))

which is similar to the notion of $Q$-category of Rosenberg. It is called $S$-category as it is suitable context for a generalization of a [[separable functor]].

---
A little info is found in Toen: Homotopical and higher categorical structures in algebraic geometry. File Toen web unpubl hab.pdf. S-cats are closely related to Segal cats.

nLab page on [[nlab:S-category]]


S-duality

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+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Duality in string theory
+-- {: .hide}
[[!include duality in string theory -- contents]]
=--
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
#### Langlands correspondence
+--{: .hide}
[[!include Langlands correspondence -- contents
]]
=--
=--
=--

The term _S-duality_ can mean two different things:

* in [[mathematics]] it is short for _[[Spanier-Whitehead duality]]_ : [[dual object|monoidal duality]] in the [[stable homotopy category]];

* in [[physics]] it denotes a certain equivalence between [[quantum field theories]], this is what we discuss below

--

#Contents#
* tabe of contents
{:toc}

## Idea

In the original and restricted sense, _S-duality_ refers to the conjectured [[Montonen-Olive duality]] auto-[[equivalence]] of ([[super Yang-Mills theory|super]]) [[Yang-Mills theory]] in 4 dimensions under which the [[coupling constant]] is inverted, and more generally under which the combined [[coupling constant]] and [[theta angle]] tranform under an action of the [[modular group]]. At least for [[super Yang-Mills theory]] this conjecture can be argued for in detail.

There is also a [[duality in string theory]] called _S-duality_. Specifically in [[type IIB superstring theory]]/[[F-theory]] this is given by an action of the [[modular group]] on the [[axio-dilaton]], hence is, via the proportionality of the dilaton to the [[string coupling constant]], again a weak-strong coupling duality.

Indeed, at least for [[super Yang-Mills theory]] Montonen-Olive S-duality may be understood as a special case of the [[duality in string theory|string duality]] ([Witten 95a](#Witten95a), [Witten 95b](#Witten95b)): one may understand [[N=2 D=4 super Yang-Mills theory]] as the [[KK-compactification]] of the [[M5-brane]] [[6d (2,0)-superconformal QFT]] on the [[F-theory]] torus ([Johnson 97](#Johnson97)) to get the [[D3-brane]] worldvolume theory, and the remnant [[modular group]] action on the compactified torus is supposed to be the 4d Montonen-Olive S-duality ([Witten 07](#Witten07)).

### In (super) Yang-Mills theory

#### General idea

In its original form, S-duality refers to **Montonen-Olive duality** , which is about the following phenomenon:

The [[Lagrangian]] of [[Yang-Mills theory]] has two summands,

$$
  S_{YM} : \nabla \mapsto \int_X \frac{1}{e^2} \langle F_\nabla \wedge \star F_\nabla\rangle
   + \int_{X} i \theta \langle F_\nabla \wedge F_\nabla \rangle
  \,,
$$

each pairing the [[curvature]] 2-form with itself in an [[invariant polynomial]], but the first involving the [[Hodge star operator]] dual, and the second not. One can combine the coefficients $\frac{1}{e^2}$ and $i \theta$ into a single _[[complex number|complex]] [[coupling constant]]_

$$
  \tau = \frac{\theta}{2 \pi} +  \frac{4 \pi i}{e^2}
  \,.
$$

Montonen-Olive duality asserts that the quantum field theories induced from one such parameter value and another one obtained from it by an [[action]] of [[SL(2,Z)]] on the [[upper half plane]] are equivalent.

This is actually not quite true for ordinary Yang-Mills theory, but seems to be true for [[N=2 D=4 super Yang-Mills theory]].

#### From compactification of the 6d (2,0)-SCFT and AGT correspondence
 {#ForSYMFromCompactification}

In ([Witten 95a](#Witten95a), [Witten 95b](#Witten95b), [Witten 07](#Witten07)) it was suggested that the above S-duality of [[N=2 D=4 super Yang-Mills theory]] may be understood geometrically by regarding the super Yang-Mills theory as the [[Kaluza-Klein compactification]] of the [[6d (2,0)-superconformal QFT]] -- that instead of a [[gauge field]] given by a [[principal bundle]] with [[connection on a bundle|connection]] involves a [[principal 2-bundle]] with [[connection on a 2-bundle|2-connection]] -- on a [[complex torus]]. The $SL(2,\mathbb{Z})$-invariance of the resulting 4-dimensional theory is then the [[modular group]] remnant of the conformal invariance of the 6-dimensional theory under conformal transformations of that torus.

Moreover, Witten has suggested that this S-duality secretly drives a host of other subtle phenomena, notably that the [[geometric Langlands duality]] (see there for more) is just an aspect of a special case of this.

The [[AGT correspondence]] refines this further and regards the [[6d (2,0)-superconformal QFT]] as something like a "[[2d SCFT]] with values in 4d super-Yang-Mills theories". This way the whole [[mapping class group]] of general 2d [[Riemann surfaces]] acts as a generalized S-duality on 4d super-Yang-Mills theory

### In string theory

In [[string theory]], S-duality is supposed to apply to whole string theories and make [[type II string theory]] be S-dual to itself and make  [[heterotic string theory]] be S-dual to [[type I string theory]].

#### Type IIB S-duality
 {#InTypeIIB}

##### General idea

[[Type IIB string theory]] is obtained by [[KK-compactification]] of [[M-theory]] on a [[torus]] bundle followed by [[T-duality|T-dualizing]] one of the torus cycles. This perspective -- referred to as [[F-theory]] -- exhibits the [[axio-dilaton]] of type IIB string theory as the fiber of an [[elliptic fibration]] (essentially the torus bundle that M-theory was compactified on ([Johnson 97](#Johnson97))).

The [[modular group]] acts on this [[elliptic fibration]], and this is S-duality for type IIB-strings. In particular the transformation $\tau \mapsto - \frac{1}{\tau}$ inverts the type II coupling constant.  See at _[[F-theory]]_ for more.

The type IIB [[F1-string]] and the [[D1-brane]] appear this way by [[double dimensional reduction]] from the [[M2-brane]] wrapping (either) one of the two cycles of the compactifying torus. S-duality mixes these strings by the evident [[modular group]] action on the $(p,q)\in \mathbb{Z}^2$ labels of the
[[(p,q)-strings]]. Here at least part of the S-duality action on $(p,q)$-strings may be seen as a system of autoequivalences of the [[super L-infinity algebras]] which defines the [[extended super spacetime]] constituted by the type II superstring ([Bandos 00](Bandos00), [FSS 13, section 4.3](#FSS13)).

Similarly the [[D5-brane]] and the [[NS5-brane]] are the [[double dimensional reduction]] of the [[M5-brane]] wrapping one of the two cycles of the compactifying torus, and hence the S-duality modular group also acts on $(p,q)$-5-branes, exchanging them.

Finally, the [[D3-brane]] is instead the [[double dimensional reduction]] of the [[M5-brane]], wrapping _both_ compactifying dimensions. Accordingly the worldvolume theory of the D3, which is [[super Yang-Mills theory]] in $d = 4$ has an S-self-duality. That is supposed to be the [[Montonen-Olive duality]] discussed above, which is thereby unified with type IIB S-duality.

##### Cohomological nature of type II fields under S-duality
 {#CohomologicalNatureOfTypeIIFieldsUnderSDuality}

While [[F-theory]] does capture much of this [[non-perturbative effect|non-perturbative]] S-duality, there currently remains a puzzle as to the correct [[differential cohomology]] nature of all the fields under S-duality: by the above S-duality mixes the [[Kalb-Ramond field]] $\hat B_{NS}$ with the degree-3 component $\hat B_{RR}$ of the [[RR-field]]. But the best available description of the fine-structure of these fields is (see also at _[[orientifold]]_) that $\hat B_{NS}$ is a [[cocycle]] in ([[twisted cohomology|twisted]]) [[ordinary differential cohomology]] while $\hat B_{RR}$ is (only) one component of a cocycle in ([[twisted cohomology|twisted]]) [[KU]] (or really: [[KR-theory]]).

This issue was first highlighted in ([DMW 00, section 11](#DMW00)). In ([DFM 03, section 9](supergravity+C-field#DFM)) it was observed that taking into account the [[cubical structure in M-theory]] on the [[higher dimensional Chern-Simons theory|11-dimensional Chern-Simons term]] of the [[supergravity C-field]] the conceptual mismatch is alleviated, but not quite resolved. See also ([BEJVS 05](#BEJVS05))

On the other hand, as discussed at _[[cubical structure in M-theory]]_, this structure plausibly relates to a [[generalized cohomology theory]] beyond [[ordinary cohomology]] and beyond [[K-theory]], namely to [[elliptic cohomology]]/[[tmf]]. Hints like this led in
([KrizSati 05](#KrizSati05)) to the [[conjecture]] that the right [[cohomology theory]] to capture the S-duality of [[type II superstring theory|type IIB]]/[[F-theory]] is [[modular equivariant elliptic cohomology]].

#### Heterotic/type I duality

> Something substantial should go here, for the moment the following is copied from a discussion forum comment by some Olof [here](http://physics.stackexchange.com/a/65546/5603):

For the Het/I relation, the first observation is that the massless spectra of the two models agree. Moreover, if we make the identification
$$
G^I_{\mu\nu} = e^{-\Phi_h} G^h_{\mu\nu} , \qquad
\Phi^I = - \Phi^h , \qquad
\tilde{F}^I_3 = \tilde{H}^h_3 , \qquad
A^I_1 = A^h_1
$$
the low energy effective supergravity actions of the two models match. Since the [[string coupling constants]] $g_s^I$ and $g_s^h$ are given as the expectation values of the exponentials of the dilatons $\exp(\Phi^I)$ and $\exp(\Phi^h)$, respectively, the above equations relates the type-I theory at strong coupling to the heterotic theory at weak coupling:
$$
g^I_s = \frac{1}{g^h_s} .
$$
From the relative scaling of the metric in (1) we also see that the string length in the two theories are related by
$$
l^I_s = l^h_s \sqrt{g^h_s}.
$$

As a non-perturbative check we can consider the tension of the type-I D1 brane. The brane is a BPS object, so for all values of the coupling $g_s^I$ the tension is given by the same formula
$$
T^I_{D1} = \frac{1}{g_s^I} \frac{1}{2\pi\left(l^I_s\right)^2} = \frac{g^h_s}{2\pi\left(l^h_s\sqrt{g^h_s}\right)^2} = \frac{1}{2\pi\left(l^h_s\right)^2}
$$
where I've used relations (2) and (3). But this is equal to the tension of the fundamental heterotic string
$$
T^h_{F1} = \frac{1}{2\pi\left(l^h_s\right)^2}.
$$
This indicates that it is sensible to identify the strong coupling limit of the type-I D1 brane with the heterotic string.

#### For type IIA

A priori [[type IIA superstring theory]] does not have S-duality, but by compactifying [[M-theory]] on a torus one can sort of read off what the non-perturbative additions to type IIA should be that make it have S-duality after all, see

* Gottfried Curio, Boris Kors, [[Dieter Lüst]], _Fluxes and Branes in Type II Vacua and M-theory Geometry with G(2) and Spin(7) Holonomy_, Nucl.Phys.B636:197-224,2002 ([arXiv:hep-th/0111165](http://arxiv.org/abs/hep-th/0111165))

## Overview

[[!include heterotic S-duality -- table]]

[[!include gauge theory from AdS-CFT -- table]]

## Related concepts

* [[Yang-Mills theory]]

* [[topologically twisted D=4 super Yang-Mills theory]]

[[duality in physics]], [[duality in string theory]]

* **S-duality**

  * [[electro-magnetic duality]]

    * [[Montonen-Olive duality]]

  * [[Seiberg duality]]

  * [[geometric Langlands correspondence]]

  * [[quantum geometric Langlands correspondence]]

* [[duality in string theory]]

  * [[T-duality]], [[U-duality]]

  * [[AdS/CFT correspondence]]

* [[type II string theory]], [[F-theory]]

  * [[(p,q)-string]]

  * [[modular equivariant elliptic cohomology]]

## References

* [[Mike Duff]], chapter 6 of _[[The World in Eleven Dimensions]]: Supgergravity, Supermembranes and M-theory_, IoP (1999)

### In (super-)Yang-Mills theory
 {#ReferencesInSYM}

It was originally noticed in

* {#GNO77} [[Peter Goddard]], [[Jean Nuyts]], [[David Olive]], *Gauge Theories And Magnetic Charge*, Nucl. Phys. B **125** (1977) 1-28 [doi:10.1016/0550-3213(77)90221-8, [spire:111435](https://inspirehep.net/literature/111435)]

that where [[electric charge]] in [[Yang-Mills theory]] takes values in the [[weight lattice]] of the [[gauge group]], then [[magnetic charge]] takes values in the lattice of what is now called the [[Langlands dual group]].

This led to the electric/magnetic duality conjecture formulation in

* [[Claus Montonen]], [[David Olive]], _Magnetic Monopoles As Gauge Particles?_ Phys. Lett. B72 (1977) 117-120.

According to ([Kapustin-Witten 06, pages 3-4](#KapustinWitten06)) the observaton that the Montonen-Olive dual charge group coincides with the [[Langlands dual group]] is due to

* {#Atiyah77} [[Michael Atiyah]], private communication to [[Edward Witten]], 1977

Discussion of the duality for abelian gauge theory ([[electromagnetism]]) is is

* [[Edward Witten]], _On S-duality in abelian gauge theory_ Selecta Mathematica, (2):383-410, 1995 ([arXiv:hep-th/9505186](http://arxiv.org/abs/hep-th/9505186))

* Jose Barbon, _Generalized abelian S-duality and coset constructions_, Nuclear Physics B, 452(1):313-330, 1995 ([arXiv:hep-th/9506137](http://arxiv.org/abs/hep-th/9506137))

* Gerald Kelnhofer, _Functional integration and gauge ambiguities in generalized abelian gauge theories_ J. Geom. Physics, 59:1017-1035, 200

See also the references at _[[electro-magnetic duality]]_.

The insight that the Montonen-Olive duality works more naturally in [[super Yang-Mills theory]] is due to

* [[David Olive]], [[Edward Witten]], _Supersymmetry Algebras That Include Topological Charges_, Phys. Lett. B78 (1978) 97-101.

and that it works particularly for [[N=4 D=4 super Yang-Mills theory]] is due to

* H. Osborn, _Topological Charges For $N = 4$ Supersymmetric Gauge Theories And Monopoles Of Spin 1_, Phys. Lett. B83 (1979) 321-326.

The observation that the $\mathbb{Z}_2$ electric/magnetic duality extends to an $SL(2,\mathbb{Z})$-action in this case is due to

* [[John Cardy]], E. Rabinovici, _Phase Structure Of Zp Models In The Presence Of A Theta Parameter_, Nucl. Phys. B205 (1982) 1-16;

* [[John Cardy]], _Duality And The Theta Parameter In Abelian Lattice Models_, Nucl. Phys. B205 (1982) 17-26.

* A. Shapere and [[Frank Wilczek]], _Selfdual Models With Theta Terms_, Nucl. Phys. B320 (1989) 669-695.

The understanding of this $SL(2,\mathbb{Z})$-symmetry as a remnant conformal transformation on a 6-dimensional [[principal 2-bundle]]-theory -- the [[6d (2,0)-superconformal QFT]] -- compactified on a torus is described in

* {#Witten95a} [[Edward Witten]], pages 4-5 of _Some Comments On String Dynamics_, Proceedings of _String95_ ([arXiv:hepth/9507121](http://arxiv.org/abs/hepth/9507121))

* {#Witten95b} [[Edward Witten]], _On S-Duality in Abelian Gauge Theory_ ([arXiv:hep-th/9505186](http://arxiv.org/abs/hep-th/9505186))

* {#Witten07} [[Edward Witten]], _[[Conformal field theory in four and six dimensions]]_ ([arXiv:0712.0157](http://arxiv.org/abs/0712.0157))

On an [[algorithm]] for constructing [[S-duality|S-dual]] [[quiver gauge theories]]:

* Chiung Hwang, [[Sara Pasquetti]], Matteo Sacchi, *Rethinking mirror symmetry as a local duality on fields*, Phys. Rev. D **106** (2022) 105014 [[arXiv:2110.11362](https://arxiv.org/abs/2110.11362), [doi:10.1103/PhysRevD.106.105014](https://doi.org/10.1103/PhysRevD.106.105014)]

* Riccardo Comi, Chiung Hwang, Fabio Marino, [[Sara Pasquetti]], Matteo Sacchi, *The $SL(2, \mathbb{Z})$ dualization algorithm at work* [[arXiv:2212.10571](https://arxiv.org/abs/2212.10571)]

### In type II superstring theory

The suggestion of an $SL(2,\mathbb{Z})$-duality action in [[type II superstring theory]] goes back to

* [[John Schwarz]], [[Ashoke Sen]], _Duality Symmetries Of $4D$ Heterotic Strings_, Phys. Lett. 312B (1993) 105-114,

* [[Ashoke Sen]], _Dyon - Monopole Bound States, Self-Dual Harmonic Forms on the Multi-Monopole Moduli Space, and $SL(2,\mathbb{Z})$ Invariance in String Theory_ ([arXiv:hep-th/9402032](http://arxiv.org/abs/hep-th/9402032))

_Duality Symmetric Actions_, Nucl. Phys. B411 (1994) 35-63 ([arXiv:hep-th/9304154](http://arxiv.org/abs/hep-th/9304154))

* [[Chris Hull]], [[Paul Townsend]], _Unity of Superstring Dualities_, Nucl. Phys. B438:109-137,1995 ([arXiv:hep-th/9410167](http://arxiv.org/abs/hep-th/9410167))

* [[John Schwarz]], _An $SL(2,\mathbb{Z})$ Multiplet of Type IIB Superstrings_, Phys.Lett. B360 (1995) 13-18; Erratum-ibid. B364 (1995) 252 ([arXiv:hep-th/9508143](http://arxiv.org/abs/hep-th/9508143))

The geometric understanding of S-duality in [[type II superstring theory]] via [[M-theory]]/[[F-theory]] goes maybe back to

* {#Johnson97} [[Clifford Johnson]], _From M-theory to F-theory, with Branes_, Nucl.Phys. B507 (1997) 227-244 ([arXiv:hep-th/9706155](http://arxiv.org/abs/hep-th/9706155))

A textbook account is in

* {#West12} [[Peter West]], section 17.3 of _[[Introduction to Strings and Branes]]_, Cambridge University Press 2012

A [[2-loop]] test is in

* [[Eric D'Hoker]], Michael Gutperle, [[Duong Phong]], _Two-loop superstrings and S-duality_, Nucl.Phys. B 722 (2005) 81-118 ([arXiv:hep-th/0503180](http://arxiv.org/abs/hep-th/0503180))

S-duality acting on the [[worldsheet]] theory if [[(p,q)-strings]] is discussed for instance in

* {#Bandos00} [[Igor Bandos]], _Superembedding Approach and S-Duality. A unified description of superstring and super-D1-brane_, Nucl. Phys. B **599** 197-227 (2001) ([arXiv:hep-th/0008249](http://arxiv.org/abs/hep-th/0008249))

Closely related to this, S-duality in [[type II string theory]] as an operation on the [[extended super spacetime]] [[super L-infinity algebra]] is

* {#FSS13} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], section 4.3 of _[[schreiber:The brane bouquet|Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields]]_ ([arXiv:1308.5264](http://arxiv.org/abs/1308.5264))

The cohomological problem of the type II S-duality action on the 3-form flux was originally highlighted in

* {#DMW00} D. Diaconescu, [[Gregory Moore]], [[Edward Witten]], _$E_8$ Gauge Theory, and a Derivation of K-Theory from M-Theory_, Adv.Theor.Math.Phys.6:1031-1134 2003 ([arXiv:hep-th/0005090](http://arxiv.org/abs/hep-th/0005090))

The conjecture that with combined targetspace/worldsheet modular transformations the type IIB S-duality is reflected in [[modular equivariant elliptic cohomology]] is due to

* {#KrizSati05} [[Igor Kriz]], [[Hisham Sati]], _Type II string theory and modularity_, JHEP 0508 (2005) 038 ([arXiv:hep-th/0501060](http://arxiv.org/abs/hep-th/0501060))

See also

* {#BEJVS05} [[Peter Bouwknegt]], [[Jarah Evslin]], [[Branislav Jurco]], [[Mathai Varghese]], [[Hisham Sati]], _Flux Compactifications on Projective Spaces and The S-Duality Puzzle_, Adv.Theor.Math.Phys. 10 (2006) 345-394 ([arXiv:hep-th/0501110](http://arxiv.org/abs/hep-th/0501110))

Discussion of [[Montonen-Olive duality]] in [[D=4 super Yang-Mills theory]] via [[ABJM-model]] as [[D3-brane]] model after [[double dimensional reduction]] followed by [[T-duality]]:

* [[Koji Hashimoto]], Ta-Sheng Tai, [[Seiji Terashima]], _Toward a Proof of Montonen-Olive Duality via Multiple M2-branes_, JHEP 0904:025, 2009 ([arxiv:0809.2137](https://arxiv.org/abs/0809.2137))

See also:

* Surya Raghavendran, Philsang Yoo, _Twisted S-Duality_ ([arxiv:1910.13653](https://arxiv.org/abs/1910.13653))

### Relation to geometric Langlands duality

The relation of S-duality to [[geometric Langlands duality]] was understood in

* {#KapustinWitten06} [[Anton Kapustin]], [[Edward Witten]], _Electric-Magnetic Duality And The Geometric Langlands Program_ , Communications in number theory and physics, Volume 1, Number 1, 1–236 (2007) ([arXiv:hep-th/0604151](http://arxiv.org/abs/hep-th/0604151))

Exposition of this is in

* [[Edward Frenkel]],  _What Do Fermat's Last Theorem and Electro-magnetic Duality Have in Common?_ KITP talk 2011 ([web](http://online.kitp.ucsb.edu/online/bblunch/frenkel/))

[[!redirects S-dualities]]

[[!redirects Montonen-Olive duality]]

[[!redirects complexified coupling constant]]