lambda-calculus

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### Context
#### Type theory
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# The λ-calculus
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## Idea

The **lambda calculus** is:

* a simple [[programming language]];

* a *model of computation* (akin to [[Turing machine]]s and [[recursive function]]s), through which we can study the computability and complexity of [[functions]] and predicates; and

* an [[internal language]] for [[cartesian closed categories]] (for more on this see at _[[relation between type theory and category theory]]_).

It comes in both *typed* and *untyped* versions.

## Concepts

### Abstraction and application

The basic constructs of lambda calculus are *[[lambda abstractions]]* and *applications*.  If $t$ is an expression of some sort involving a free variable $x$ (possibly vacuously), then the lambda abstraction

$$ \lambda x. t $$

is intended to represent the function which takes one input and returns the result of substituting that input for $x$ in $t$.  Thus, for instance, $(\lambda x. (x+1))$ is the function which adds one to its argument.  Lambda expressions are often convenient, even outside lambda calculus proper, for referring to a function without giving it a name.

Application is how we "undo" abstraction, by applying a function to an argument.  The application of the function $f$ to the argument $t$ is generally denoted simply by $f t$.  Applications can be parenthesized, so for instance $f(t)$ and $(f)t$ and $(f t)$ all denote the same thing as $f t$.

Application is generally considered to associate to the left.  Thus $u v w$ denotes the application of $u$ to $v$, followed by application of the result (assumed to again be a function) to $w$.  This allows the representation of functions of multiple variables in terms of functions of one variable via "currying" (named for Haskell Curry, although it was invented by Moses Schönfinkel): after being applied to the first argument, we return a function which, applied to the next argument, returns a function which, when applied to the next argument, ... , returns a value.  For instance, the "addition" function of two variables can be denoted $(\lambda x. (\lambda y. x+y))$: when applied to an argument $x$, it returns a function which, when applied to an argument $y$, returns $x+y$.  This is so common that it is generally abbreviated $(\lambda x y. x+y)$.

### Evaluation and Reduction

*Evaluation* or *reduction* is the process of "computing" the "value" of a lambda term.  The most basic operation is called *[[beta reduction]]* and consists in taking a lambda abstraction at its word about what it is supposed to do when applied to an input.  For instance, the application $(\lambda x. x+1) 3$ reduces to $3+1$ (and thereby, presuming appropriate rules for $+$, to $4$).  In general, the beta reduction of a term $(\lambda x.t)(u)$ is defined as the [[substitution#avoiding variable capture|capture-avoiding substitution]] of $u$ for $x$ in $t$,

$$(\lambda x.t)(u) \to_\beta t[u/x].$$

Terms which can be connected by a [[zigzag]] of beta reductions (in either direction) are said to be *beta-equivalent*.

Another basic operation often assumed in the lambda calculus is *[[eta reduction]]/expansion*, which consists of identifying a function, $f$ with the lambda abstraction $(\lambda x. f x)$ which does nothing other than apply $f$ to its argument.  (It is called "reduction" or "expansion" depending on which "direction" it goes in, from $(\lambda x. f x)$ to $f$ or vice versa.)

A more basic operation than either of these, which is often not even mentioned at all, is *alpha equivalence*; this consists of the renaming of bound variables, e.g. $(\lambda x. f x) \to (\lambda y. f y)$.

More complicated systems that build on the lambda calculus, such as various [[type theories]], will often have other rules of evaluation as well.

In good situations, lambda calculus reduction is *[[confluent category|confluent]]* (the [[Church-Rosser theorem]]), so that every term has at most one *[[normal form]]*, and two terms are equivalent precisely when they have the same normal form.  A term is said to be _normalizing_ if its reduction is also terminating---more precisely, $t$ is said to be *weakly-normalizing* if there exists a finite sequence of reductions

$$ t \to t_1 \to t_2 \to \dots \to t_n \nrightarrow $$

terminating in a normal form $t_n$, and *strongly-normalizing* if _all_ sequences of reductions lead to a (by confluence, necessarily unique) normal form.  In general, there exist terms which do not normalize by any reduction sequence, although simply-typed lambda calculus and many other typed variants of lambda calculus are strongly-normalizing.

## Variants

### Pure lambda calculus

In the untyped (or "pure") lambda calculus, every term can be applied to any other term.  As an example of the sort of freedom this allows, we can form the term $u = \lambda x. x x$ which applies its argument to itself, and then define

$$ \omega = u(u) $$

as the self-application of $u$.  Performing beta-reduction on $\omega$ yields

$$ \omega \to_\beta x x[u/x] = u(u) = \omega \to_\beta \omega \to_\beta \dots$$

thus giving the classic example of a non-terminating program.

In pure untyped lambda calculus, we can define [[natural numbers]] using the [[Church numerals]]: the number $n$ is represented by the operation of $n$-fold iteration.  Thus for instance we have $2 = \lambda f. (\lambda x.f (f x))$, the function which takes a function $f$ as input and returns a function that applies $f$ twice.  Similarly $1 = \lambda f. (\lambda x.f x)$ is the identity on functions, while $0 = \lambda f. (\lambda x . x)$ takes any function $f$ to the identity function (the 0th iterate of $f$).  We can then construct (very inefficiently) all of arithmetic, and prove that the arithmetic functions expressible by lambda terms are exactly the same as those computable by Turing machines or (total) recursive functions.

The most natural sort of *model* of pure lambda calculus is a [[reflexive object]] in a cartesian closed category, that is, an object $U$ together with a pair of maps

$$
\array{U & \overset{a}{\underset{\ell}{\rightleftarrows}} & U^U}
$$

representing $a$pplication and $\ell$ambda, such that $a \circ \ell = 1$ (to validate the $\beta$ equation).  Such data thereby witnesses the [[exponential object]] $U^U$ as a [[retract]] of $U$.  Of course there are no nontrivial such models in sets, but they do exist in other categories, such as [[domain]]s. It is worth remarking that a necessary condition on such $U$ is that every term $f \colon U^U$ have a fixed-point; see [[fixed-point combinator]].

From the point of view of [[type theory]] and in particular the "types à la Church" perspective, such a reflexive object $U$ can also be seen as the intrinsic type of _every_ lambda term, and so the untyped lambda calculus is sometimes referred to (a bit cheekily) as really being "uni-typed".  On the other hand, from the "types à la Curry" perspective, it is also possible to begin with the pure lambda calculus as a foundation, and then try to ascribe individual terms more precise types.  For example, the normalizing terms of pure lambda calculus can be characterized precisely as those which are typable in an [[intersection type]] system.

### Simply typed lambda calculus

In *[[simple type theory|simply typed]]* lambda calculus, each variable and term has a [[type theory|type]], and we can only form the application $f t$ if $t$ is of some type $A$ while $f$ is of a [[function type]] $A \to B = B^A$ whose domain is $A$; the type of $f t$ is then $B$.  Similarly, if $x$ is a variable of type $A$ and $t$ is a term of type $B$ involving $x$, then $\lambda x. t$ has type $A\to B$.

Without some further type and term constructors, there is not much that can be done, but if we add a [[natural numbers object]] (that is, a type $N$ with constants $0$ of type $N$ and $s$ of type $N\to N$, along with a "definition-by-recursion" operator), then we can express many recursive functions.  (We cannot by this means express *all* computable functions, although we can go beyond [[primitive recursive function]]s; for instance we can define the [[Ackermann function]].  One way to increase the expressiveness to all [[partial recursive function]]s is to add a [[fixpoint]] [[combinator]], or an unbounded search operator).

Simply typed lambda calculus is the natural [[internal language]] of [[cartesian closed categories]].  This means that

* Every cartesian closed category gives rise to a simply typed lambda calculus whose basic types are its objects, and whose basic terms are its morphisms, while

* Every simply typed lambda calculus "generates" a cartesian closed category whose objects are its types and whose morphisms are its equivalence classes of terms.

These two operations are adjoint in an appropriate sense.

### Functional programming

Most [[functional programming|functional]] [[programming language|programming languages]], such as Lisp, ML, and Haskell, are at least loosely based on lambda calculus.

## Related entries

* [[combinator]], [[combinatory logic]]

  * [[fixed-point combinator]]

* [[type theory]]

* [[relation between type theory and category theory]]

* [[quantum lambda-calculus]]

## References

Textbook accounts:

* [[J. Roger Hindley]], [[Jonathan P. Seldin]], *Lambda-Calculus and Combinators -- An Introduction*, Cambridge University Press (2008) [[doi:10.1017/CBO9780511809835](https://doi.org/10.1017/CBO9780511809835)]

Introduction to [[simple type theory|simply typed]] $\lambda$-calculus:

* {#Girard89} [[Jean-Yves Girard]] (translated and with appendiced by [[Paul Taylor]] and [[Yves Lafont]]), *Proofs and Types*, Cambridge University Press (1989) [[ISBN:978-0-521-37181-0](), [webpage](http://www.paultaylor.eu/stable/Proofs+Types.html), [pdf](https://www.paultaylor.eu/stable/prot.pdf)]

The idea that untyped lambda calculus can be modeled internally to a cartesian closed category as a [[reflexive object]] (an object $U$ such that the exponential $U^U$ is a [[retract]] of $U$) was formulated explicitly by [[Dana Scott]] in

* Dana Scott. Relating theories of the $\lambda$-calculus. In _To H.B. Curry: Essays on Combinatory Logic, Lambda-Calculus and Formalism_ (eds. Hindley and Seldin), Academic Press, 403--450, 1980.

This followed his earlier work constructing an explicit such model in the category of domains:

* {#Scott70} [[Dana S. Scott]], *Outline of a mathematical theory of computation*, in: Proceedings of the *Fourth Annual Princeton Conference on Information Sciences and Systems* (1970) 169–176. [[pdf](https://ropas.snu.ac.kr/~kwang/520/readings/sco70.pdf), [[Scott-TheoryOfComputation.pdf:file]]]

* {#Scott76} [[Dana Scott]], *Data types as lattices*. SIAM Journal of Computing **5** 3 (1976) 522--587 [[doi:10.1137/0205037](https://doi.org/10.1137/0205037), [pdf](https://www.cs.ox.ac.uk/files/3287/PRG05.pdf)]

Another, more recent take on pure lambda calculus as a certain kind of [[algebraic theory]] (called a "[[lambda theory]]") can be found in

* {#Hyland13} [[Martin Hyland]]. _Classical lambda calculus in modern dress_ Mathematical Structures in Computer Science, 27(5) (2017) 762-781. doi:[10.1017/S0960129515000377](https://doi.org/10.1017/S0960129515000377). [arxiv:1211.5762](http://arxiv.org/abs/1211.5762)

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lambda-connection

## Idea

Deligne's $\lambda$-connection is a notion interpolating between [[flat connection]]s and Higgs fields (see [[Higgs bundle]]). Parameter $\lambda\in\mathbf{P}^1$.

## Literature

The notion has been intrduced in a series of letters of [[Pierre Deligne]] to [[Carlos Simpson]], realising the Higgs bundles as a degeneration of vector bundles with connections. Simpson further developed the notion and related moduli space of $\lambda$-connection which he called __Hodge moduli space__.

* C. T. Simpson, _The Hodge filtration on nonabelian cohomology_, in Algebraic geometry-Santa
Cruz 1995, 217--281
* Carlos T. Simpson, _A weight two phenomenon for the moduli space of rank one local systems on
open varieties_ [arXiv:0710.2800](https://arxiv.org/abs/0710.2800)
* C. T. Simpson, _Iterated destability modifications for vector bundles with connection, Contemp. Math. __522__ (2010) 183--206
* [[Carlos Simpson]], _On the notion of lambda-connection_, [slides](https://math.univ-cotedazur.fr/~carlos/slides/deligneconf/slideA.html)

* Zhi Hu, Pengfei Huang, _Simpson-Mochizuki correspondence for $\lambda$-flat bundles_, Journal de Mathématiques Pures et Appliquées 2022 [doi](https://doi.org/10.1016/j.matpur.2022.06.002) [arXiv:1905.10765](https://arxiv.org/abs/1905.10765)

> The notion of flat λ-connections as the interpolation of usual flat connections and Higgs fields was suggested by Deligne and further studied by Simpson. Mochizuki established the Kobayashi--Hitchin-type theorem for λ-flat bundles (λ≠0), which is called the Mochizuki correspondence. In this paper, on the one hand, we generalize Mochizuki's result to the case when the base being a compact balanced manifold, more precisely, we prove the existence of harmonic metrics on stable λ-flat bundles (λ≠0). On the other hand, we study two applications of the Simpson--Mochizuki correspondence to moduli spaces. More concretely, we show this correspondence provides a homeomorphism between the moduli space of (semi)stable λ-flat bundles over a complex projective manifold and the Dolbeault moduli space, and also provides dynamical systems with two parameters on the latter moduli space. We investigate such dynamical systems, in particular, we calculate the first variation, the fixed points and discuss the asymptotic behaviour.

* Anoop Singh, _On the Moduli space of $\lambda$-connections_, (2020) [RG](https://www.researchgate.net/publication/339015385)
* C. C. Liu, S. Rayan, Y. Tanaka, _The Kapustin–Witten equations and nonabelian Hodge theory_ Eur. J. Math. __8__ (Suppl 1) 23--41 (2022) [doi](https://doi.org/10.1007/s40879-022-00538-4)

Simpson introduced shapes of a (smooth projective complex) curve $X$, and in particular Betti shape $X_B$, de Rham shape $X_{dR}$ and Dolbeaut shape $X_{Dol}$ of a scheme. Coherent sheaves on them correspond to local systems, flat vector bundles and Higgs sheaves. So called Deligne's shape $X_{Del}$ interpolates between de Rham and Dolbeaut shapes.

* [[Mauro Porta]], Francesco Sala, _Simpson’s shapes of schemes and stacks_, draft [pdf](https://www.salafrancesco.weebly.com/uploads/8/3/1/9/83195312/porta_sala_shapes.pdf)

This was used to obtain a categorified Hall algebra version of nonabelian Hodge correspondence in

* [[Mauro Porta]], Francesco Salla, _Two-dimensional categorified Hall algebras_, J. Eur. Math. Soc. __25__:3 (2023) 1113-1205 [doi](https://doi.org/10.4171/jems/1303)

category: algebraic geometry, analytic geometry

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Lambda-ring

> There is also a notion of [[special lambda-ring]]. But in most cases by ''$\lambda$-ring'' is meant ''special $\lambda$-ring''.

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# $\Lambda$-rings
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## Idea

A _$\lambda$-ring_ is a [[commutative ring]] which is in addition equipped with operations that behave as the operations of forming [[exterior powers]] (of [[vector spaces]]/[[representations]]) in a [[representation ring]]. The name derives from the common symbol $\Lambda^n$ for the $n$th [[exterior power]]. Hence $\lambda$-rings are one incarnation of the [[representation theory]] of the [[symmetric groups]].

Equivalently, it turns out ([Wilkerson 82](#Wilkerson82)) that a $\lambda$-ring is a commutative ring equipped with an [[endomorphism]] that lifts the [[Frobenius endomorphism]] after reduction mod $p$ at each [[prime number]] $p$. As such, $\lambda$-rings appear in [[Borger's absolute geometry]] and are related, in some way, to [[power operations]] (see there for more) in [[stable homotopy theory]].

### Motivation from representation theory

Typically one can form [[direct sums]] of [[representations]] of some [[algebra|algebraic]] [[structure]].  The [[decategorification]] to [[isomorphism classes]] of such [[representations]] then  inherits the structure of a [[commutative monoid]]. But nobody likes commutative monoids: we all have an urge to subtract.  So, we throw in formal negatives and get an [[abelian group]] --- the _[[Grothendieck group]]_.

In many situations, we can also take [[tensor product|tensor products]] of representations.  Then the Grothendieck group becomes something better than an abelian group.  It becomes a [[ring]]: the [[representation ring]].
Moreover, in many situations we can also take [[exterior power|exterior]] and [[symmetric power|symmetric]] powers of representations;  indeed, we can often apply any [[Young diagram]] to a representation and get a new representation. Then the [[representation ring]] becomes something better than a ring: it becomes a _$\lambda$-ring_.

More generally, the [[Grothendieck group]] of a [[monoidal category|monoidal]] [[abelian category]] is always a ring, called a [[Grothendieck ring]].   If we start with a [[braided monoidal category|braided monoidal]] abelian category, this ring is commutative.   But if we start with a [[symmetric monoidal category|symmetric monoidal]] abelian category, we get a $\lambda$-ring.

So, $\lambda$-rings are all about getting the most for your money when you [[decategorify|decategorify]] a [[symmetric monoidal category|symmetric monoidal]] [[abelian category]] --- for example the category of [[representations]] of a [[group]], or the category of [[vector bundle|vector bundles]] on a [[topological space]].

Unsurprisingly, the [[Grothendieck group]] of the free symmetric monoidal abelian category on one generator is the free $\lambda$-ring on one generator.  This category is very important in representation theory.  Objects in this category are called [[Schur functor|Schur functors]], because for obvious reasons they act as functors on _any_ symmetric monoidal abelian category.  The irreducible objects in this category are called 'Young diagrams'.  Elements of the free $\lambda$-ring on one generator are called [[symmetric function|symmetric functions]].

### In terms of universal algebra

A *$\lambda$-ring* $L$ is a [[P-ring]] presented by the [[polynomial ring]] $Symm=\mathbb{Z}[h_1,h_1\dots]$ in countably many indeterminates over the [[integer|integers]] or, equivalently, $Symm$ is the ring of [[symmetric function|symmetric functions]] in countably many [[variables]]. This means that (the underlying set valued functor of) $L$ is a [[copresheaf]] presented by $Symm$ such that

1. $L:\CRing \to CRing$ defines an endofunctor on the category of commutative rings.

2. $L$ gives rise to a [[comonad]] on $C Ring$.

A $\lambda$-ring is hence a commutative ring equipped with a [[co-action]] of this comonad. As always is the case with [[monad|monads]] and comonads this definition can be formulated in terms of an [[adjunction]].

## Definition

### The "orthodox" definition

+-- {: .num_defn}
###### Definition

A _$\lambda$-structure_ on a [[commutative ring|commutative unital ring]] $R$ is defined to be a sequence of maps $\lambda^n$ for $n\ge 0$ satisfying

1. $\lambda^0(r)=1$ for all $r\in R$

2. $\lambda^1=id$

3. $\lambda^n(1)=0$, for $n\gt 1$

4. $\lambda^n(r+s)=\sum_{k=0}^n \lambda^k(r)\lambda^{n-k}(s)$, for all $r,s\in R$

5. $\lambda^n(r s)=P_n(\lambda^1(r),\dots,\lambda^n(r),\lambda^1(s),\dots,\lambda^n(s))$ for all $r,s\in R$

6. $\lambda^m(\lambda^n(r)):=P_{m,n}(\lambda^1(r),\dots,\lambda^{m n}(r))$, for all $r\in R$

where $P_n$and $P_{m,n}$ are certain (see the reference for their calculation) [[universal polynomial|universal polynomials]] with integer coefficients. $R$ is in this case called a *$\lambda$-ring*. Note that the $\lambda^n$ are not required to be morphisms of rings.

A [[homomorphism]] of $\lambda$-structures is defined to be a [[homomorphism]] of [[rings]] commuting with all $\lambda^n$ maps.

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###### Example

There exists a $\lambda$-ring structure on the ring $1+ t R[ [t] ]$ of
[[power series]] with constant term $1$ where

a) addition on $1+t R[ [t] ]$ is defined to be multiplication of power series

b) multiplication is defined by

$$(1+\sum_{n=1}^\infty r_n t^n)(1+\sum_{n=1}^\infty s_n t^n):=1+\sum_{n=1}^\infty P_n(r_1,\dots,r_n,s_1,\dots,s_n)t^n$$

c) the $\lambda$-operations are defined by

$$\lambda^n (1+\sum_{m=1}^\infty r_m t^m)=1+\sum_{m=1}^\infty P_{m,n}(r_1,\dots,r_{mn})t^m$$

=--

([Hopkinson](#Hopkinson))

+-- {: .num_prop}
###### Proposition

Let $\Lambda$ denote the ring of [[symmetric function|symmetric functions]], let $R$ be a $\lambda$-ring.

Then for every $x\in R$ there is a unique [[homomorphism]] of $\lambda$-rings

$$\Phi_x:\Lambda\to R$$

sending $e_1\mapsto x$, $e_n\mapsto \lambda^n(x)$, $p_n\mapsto\psi^n(x)$ where $e_n$ denotes the $n$-th [[elementary symmetric function]] and $\psi^n$ denotes the $n$-th [[Adams operation]] (explained in the reference).

Equivalently this result asserts that $\Lambda$ is the [[free construction|free]] $\lambda$-ring in the single variable $e_1$.

=--

This is due to ([Hopkinson](#Hopkinson))

+-- {: .proof}
###### Proof

We define $\Phi_x(e_1)=x$, then the assumption on $\Phi_x$ to be a morphism of $\lambda$-rings yields $\Phi_x(e_n)=\Phi_x(\lambda^n(e_1))=\lambda^n(x)$.

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###### Theorem
([Hazewinkel](#Hazewinkel) 1.11, 16.1)

a) The [[endofunctor]] of the [[category]] of commutative rings
$$\Lambda:\begin{cases}C Ring \to C Ring\\A\mapsto 1 + A [ [t] ]\end{cases}$$
sending a commutative ring to the set of power series with constant term $1$ is representable by the polynomial ring $Symm \coloneqq \mathbb{Z}[h_1, h_2,\dots]$ in an infinity of indeterminates over the integers.

b) There is an [[adjunction]] $(forget\, \lambda\dashv \Lambda)$ where $forget\,\lambda: \lambda Ring\to CRing$ is the forgetful functor assigning to a $\lambda$-ring its underlying commutative ring.

The left inverse $\g_{S,A}$ of the natural isomorphism $q_{S,A}:hom(forget\,\lambda,A)\to hom(S,\Lambda(A))$ is given by the ghost component $s_1$.

(see also ([Borger 08, section 1.8](#Borger08)))

=--

An instructive introduction to the "orthodox"- and preparation for the "heterodox" view (described below) on $\lambda$-rings is Hazewinkel's survey article on [[Witt vectors]], ([Hazewinkel](#Hazewinkel)). There is also a [[Hazewinkel, Witt vectors|reading guide]] to that article.

### The "heterodox" definition
 {#HeterodoxDefinition}

There is a second, "heterodox" way to approach $\lambda$-rings with a strong connection to [[arithmetic]] discussed in detail in ([Borger 08, section 1](#Borger08)). A survey is in ([Borger 09](#Borger09)) where it says in the abstract:

> The theory of $\Lambda$-rings, in the sense of Grothendieck's Riemann--Roch theory, is an enrichment of the theory of commutative rings. In the same way, we can enrich usual [[algebraic geometry]] over the ring $\mathbf{Z}$ of integers to produce $\Lambda$-algebraic geometry. We show that $\Lambda$-algebraic geometry is in a precise sense an algebraic geometry over a deeper base than $\mathbf{Z}$ and that it has many properties predicted for algebraic geometry over the mythical [[field with one element]]. Moreover, it does this in a way that is both formally robust and closely related to active areas in arithmetic algebraic geometry.

First some standard notation:

+-- {: .num_defn #FrobeniusMorphism}
###### Definition

For $p$ a [[prime number]] write $\mathbb{F}_p$ for the [[finite field]] whose underlying [[abelian group]] is the [[cyclic group]] $\mathbb{Z}/p\mathbb{Z}$.

For $A$ an $\mathbb{F}_p$-[[associative algebra|algebra]], then the [[Frobenius endomorphism]]

$$
  F_p \colon A \longrightarrow A
$$

is that given by taking each element to its $p$th power

$$
  F_p \colon x \mapsto x^p
  \,.
$$

=--

+-- {: .num_defn #LambdaRingByFrobeniusLifts}
###### Definition

For $p$ a [[prime number]], then a _$p$-typical $\Lambda$-ring_ is

* a [[commutative ring]] $A$

* equipped with an [[endomorphism]] $F_A \colon A \to A$

such that under [[tensor product]] with $\mathbb{F}_p$ it becomes the [[Frobenius morphism]], def.\ref{FrobeniusMorphism}:

$$
  \mathbb{F}_p \otimes_{\mathbb{Z}} F_A
  =
  F_p
  \;
   \colon
  \;
  \mathbb{F}_p \otimes_{\mathbb{Z}} A
   \longrightarrow
  \mathbb{F}_p \otimes_{\mathbb{Z}} A
  \,.
$$

A _big $\Lambda$-ring_ is a commutative ring equipped with commuting endomorphisms, one for each prime number $p$, such that each of them makes the ring $p$-typical, respectively, as above.

=--

This is def. 1.7 in ([Borger 08](#Borger08)), formulated for the special case of example 1.15 there (which is stated in terms of Witt vectors) and translated to $\Lambda$-rings in view of prop. 1.10 c) (see [the adjunction](#AdjointTriple)) there.

> the following originates from revision 19 but needs attention

+-- {: .num_example}
###### Example
The $p$-th [[Adams operation]] $\psi_p$ is a Frobenius lift. Moreover given any two prime numbers then their [[Adams operations]] commute with each other.
=--

The following two theorems are crucial for the "heterodox" point of view. We will see later that in fact we do not need the torsion-freeness assumption.

+-- {: .num_theorem}
###### Theorem
([Wilkerson 82](#Wilkerson82))

Let $A$ be an additively torsion-free commutative ring. Let $\{\psi_p\}$ be a commuting family of Frobenius lifts.

Then there is a unique $\lambda$-ring structure on $A$ whose [[Adams operations]] are the given Frobenius lifts $\{\psi_p\}$.
=--

+-- {: .num_theorem}
###### Theorem
A ring morphism $f$ between two $\lambda$-rings is a morphism of $\lambda$-rings (i.e. commuting with the $\lambda$-operations) iff  $f$ commutes with the [[Adams operations]].
=--

+-- {: .num_cor}
###### Corollary
There is an equivalence between the category of torsion-free $\lambda$-rings and the category of torsion-free commutative rings equipped with commuting Frobenius lifts..
=--

Now we will argue that these statements hold for arbitrary commutative rings.

+-- {: .num_remark}
###### Remark
a)  The category $\lambda Ring$ of $\lambda$-rings is [[monadic]] and [[comonadic]] over the category of $C Ring$ of commutative rings.

b)  The category $\lambda Ring_{\neg tor}$ of $\lambda$-rings is [[monadic]] and [[comonadic]] over the category of $C Ring_{\neg tor}$ of commutative rings.
=--

+-- {: .num_prop}
###### Proposition

Let $\i:C Ring_{\neg tor}\hookrightarrow C Ring$ be the inclusion. Let $W^\prime$ denote this comonad on $C Ring_{\neg tor}$. Then

a) $W \coloneqq Lan_i i\circ W^\prime \colon C Ring\to CRing$ is a comonad.

b) The category of coalgebras of $W$ is equivalent to the category of $\lambda$-rings.

c) $W$ is the [[big-Witt-vectors functor]].
=--

+-- {: .num_remark}
###### Remark
The "heterodox" generalizes to arbitrary [[Dedekind domain|Dedekind domains]] with finite residue field.

 For instance over $F_p[x]$ (instead of $\mathbb{Z}$), we would look at families of $\psi$-operators indexed by the irreducible monic polynomials $f(x)$, and each $\psi_{f(x)}$ would have to be congruent to the $q$-th power map modulo $f(x)$, where $q$ is the size of $F_p[x]/(f(x))$.
=--

## Examples

+-- {: .num_example}
###### Example

The [[topological K-theory]] ring $K(X)$ of any [[topological space]] carries the structure of a $\lambda$-ring with operations induced from (skew-)symmetrized [[tensor products]] of [[vector bundles]].

=--

This is originally due to [[Alexander Grothendieck]]. See for instance [Wirthmuller 12, section 11](topological+K-theory#Wirthmuller12) and see at _[[Adams operations]]_.

Generalizing to [[equivariant K-theory]], the [[representation ring]] of a [[group]] inherits the structure of a Lambda-ring, see [there](representation+ring#LambdaRingStructure).

+-- {: .num_example}
###### Example

The [[equivariant elliptic cohomology]] at the [[Tate curve]]
$Ell_{Tate}(X//G)$ is a $\lambda$-ring (and even an "elliptic $\lambda$-ring").

=--

([Ganter 07](#Ganter07), [Ganter 13](#Ganter13))

## Properties

### Free and co-free $\Lambda$-rings -- Symmetric function and Witt vectors
 {#FreeAndCofreeLambdaRings}

+-- {: .num_prop #AdjointTriple}
###### Proposition

The [[forgetful functor]]
$U \;\colon\; \Lambda Ring \longrightarrow CRing$
from $\Lambda$-rings to [[commutative rings]] has

* a [[left adjoint]], given by forming the ring $Symm$ of [[symmetric functions]];

* a [[right adjoint]] given by forming the [[ring of Witt vectors]] $W$.

$$
  (Symm \dashv U \dashv W)
  \;\colon\;
  \Lambda Ring
   \stackrel{\overset{Symm}{\leftarrow}}{\stackrel{\overset{U}{\longrightarrow}}{\underset{W}{\leftarrow}}}
  CRing
  \,.
$$

Hence

* [[rings of Witt vectors]] are the _[[co-free functors|co-free]] Lambda-rings;

* rings of [[symmetric functions]] are the [[free construction|free]] Lambda-rings.

=--

This statement appears in ([Hazewinkel 08, p. 87, p. 97, 98](#Hazewinkel08)). The right adjoint in a more general context is in ([Borger 08, prop. 1.10 (c)](#Borger08)).

+-- {: .num_remark}
###### Remark

On the level of [[toposes]] ([[etale toposes]]) over these [[sites]] of rings, this statement reappears as an [[essential geometric morphism]]
from the [[etale topos]] of [[Spec(Z)]] to that over "[[F1]]"
in [[Borger's absolute geometry]] ([Borger 08](#Borger08), exposition in [Borger 09](#Borger09)).

=--

## Related concepts

* [[Grothendieck group]]

* [[p-divisible group|p-divisible groups]].

* [[field with one element]]

* [[blueprint|blueprints]], [[blueprint|blue scheme]]

* [[Witt vector]]

* [[Borger's absolute geometry]]

* [[Generalized Lambda-structure]]

## References

The $\lambda$-ring structure on [[topological K-theory]] goes back to [[Alexander Grothendieck]] in the 1960s.

The relation to lifts of [[Frobenius homomorphisms]] is due to

* {#Wilkerson82} Wilkerson 1982

Modern accounts include

* [[Michiel Hazewinkel]], _Formal groups and applications_

* {#Hazewinkel08} [[Michiel Hazewinkel]],  _[[Hazewinkel, Witt vectors|Witt vectors]]_, ([arXiv](http://arxiv.org/abs/0804.3888))

See also

* John Baez, [comment](http://golem.ph.utexas.edu/category/2007/12/this_weeks_finds_in_mathematic_19.html#c013821).

* John R. Hopkinson, _Universal polynomials in lambda-rings and the K-theory of the infinite loop space $tmf$_, thesis, [pdf](http://dspace.mit.edu/bitstream/handle/1721.1/34544/71011847.pdf?sequence=1)
  {#Hopkinson}

* Donald Knutson, $\lambda$-Rings and the Representation Theory of the Symmetric Group, Lecture Notes in Mathematics, Vol. 308, Springer, Berlin, 1973.

* [concretenonsense blog](http://concretenonsense.wordpress.com/2009/07/23/lambda-rings)
* Donald Yau, [LAMBDA-RINGS] (http://www.worldscibooks.com/mathematics/7664.html), World Scientific, 2010.

school/conference in Leiden: Frobenius lifts and lambda rings 5-10. October 2009 featuring

* [[Pierre Cartier]]: Lambda-rings and Witt vectors

* [[Lars Hesselholt]]: The de Rham-Witt complex

* Alexandru Buium: Arithmetic differential equations

* [[James Borger]]: Lambda-algebraic geometry

[conference site](http://www.lorentzcenter.nl/lc/web/2009/342/info.php3?wsid=342)

[participants](http://www.lorentzcenter.nl/lc/web/2009/342/participants.php3?wsid=342)

The $\lambda$-ring structure on [[equivariant elliptic cohomology]] is due to

* {#Ganter07} [[Nora Ganter]], _Stringy power operations in Tate K-theory_, Homology, Homotopy, Appl., 2013; arXiv:math/0701565

* {#Ganter13} [[Nora Ganter]], Power operations in orbifold Tate K-theory"; arXiv:1301.2754

Discussion in the context of [[Borger's absolute geometry]] is in

* {#Borger08} [[James Borger]], section 1 of _The basic geometry of Witt vectors, I: The affine case_ ([arXiv:0801.1691](http://arxiv.org/abs/0801.1691))

* {#Borger09} [[James Borger]], _Lambda-rings and the field with one element_ ([arXiv/0906.3146](http://arxiv.org/abs/0906.3146))

[[!redirects Lambda-ring]]
[[!redirects Lambda-rings]]
[[!redirects Lambda ring]]
[[!redirects Lambda rings]]
[[!redirects lambda-ring]]
[[!redirects lambda-rings]]
[[!redirects lambda ring]]
[[!redirects lambda rings]]
[[!redirects λ-ring]]
[[!redirects λ-rings]]
[[!redirects Λ-ring]]
[[!redirects Λ-rings]]


Lambert W-function

# The Lambert $W$ function
* table of contents
{: toc}

## Idea

In [[complex analysis]], the Lambert $W$-function is a [[multivalued function|multi-branched function]] [[inverse function|inverse]] to the function $f(z) = z e^z$. Among other things, it is important in the enumeration of tree-like structures, for instance [[tree|rooted trees]], and in the combinatorial study of the [[Lagrange inversion formula]].

We have $W(z) e^{W(z)} = z$. In a neighborhood of $z = 0$, we may develop $W(z)$ as a formal [[power series]], and there is a quite remarkable formula:

$$ W(z) = \sum_{n = 1}^{\infty} \frac{(-n)^{n-1} z^n}{n!} .$$

A proof of this formula is sketched below.

## Lambert $W$ and rooted trees

Let $R(X)$ be the [[species]] of rooted trees. Choosing a structure of rooted tree $T$ (on a finite set of vertices) amounts to

* Choosing one vertex to serve as the root of $T$;

* Choosing a structure of "rooted forest" (a disjoint collection of rooted trees) on the complement of the root.

The idea is that given a rooted forest, we get a rooted tree by adding in a new vertex (which serves as root) and drawing an edge from each root of the forest to the new vertex.

This observation leads to the structural isomorphism

$$ R(X) = X \otimes \exp(R(X))	$$

Let $R(x) = \sum_{n \geq 0} \frac{a_n x^n}{n!}$ be the formal power series corresponding to the generating function of $R(X)$, so that $a_n$ counts the number of rooted trees on $n$ vertices. We may call $R(x)$ the _cardinality_ of $R(X)$. One has the analogous decategorified formula

$$R(x) = x e^{R(x)}$$

or

$$R(x) e^{-R(x)} = x$$

which means, after a short series of algebraic manipulations, that $R(x)$ is related to the Lambert $W$ function as follows:

$$R(x) = - W(-x).$$

## Counting rooted trees

The number of possible rooted tree structures on $n$ labeled vertices is $n^{n-1}$. We sketch Joyal's beautiful proof of this fact; it follows that

$$\array{
W(x) & = & - R(-x) \\
 & = & - \sum_{n \geq 1} \frac{n^{n-1} (-x)^n}{n!} \\
 & = & \sum_{n \geq 1} \frac{(-n)^{n-1} x^n}{n!}
}$$

as claimed earlier.

In fact, Joyal counts the number of _bipointed_ trees (trees equipped with an ordered pair of vertices, possibly the same one), and shows this is the same as the number of endofunctions on an $n$ element set. Since this is $n^n$, and since the number of ordered pairs of vertices is $n^2$, it follows that the number of tree structures on an $n$-element set is $n^{n-2}$, a result referred to as **Cayley's theorem**. Therefore the number of _rooted_ trees is $n \cdot n^{n-2} = n^{n-1}$.

For a bipointed tree $T$, let the ordered pair of vertices be $(t, h)$ where $t$ is called a _tail_ and $h$ a _head_. The set of vertices on the unique path from $t$ to $h$ is endowed with a [[linear order]] (namely, the order in which they appear on the path, with $t$ first). We call this ordered set the _spine_ of the bipointed tree. At the same time, each vertex $v$ along the spine is the root of a rooted tree $T_v$: a full subgraph where a vertex $w$ of $T$ belongs to $T_v$ if the unique path from $w$ to $h$ first meets the spine at $v$.

A choice of bipointed tree structure on a finite set $S$ can thus be equivalently described as follows:

* Partition $S$ into one or more subsets (which are equivalence classes);

* Choose a rooted tree structure on each class (if the root is $v$, this tree structure will be $T_v$);

* Choose a way of linearly ordering the equivalence classes (which is really the same as linearly ordering the roots).

If $L$ is the species of linearly ordered sets, and $R$ is the species of rooted trees, this way of describing bipointed trees establishes an explicit bijection between $L \circ R$ and the species of bipointed trees.

For an endofunction $f \colon S \to S$, let $K$ be the intersection of images of iterates $f^{(n)}(S)$. It is immediate that the restriction $f\colon K \to K$ is a surjection and therefore a bijection (permutation). Picturing an endofunction as a graph (where an edge is drawn between $s$ and $f(s)$), each vertex $v \in K$ is the root of a rooted tree $T_v$: a full subgraph where a vertex $w$ belongs to $T_v$ if $f^{(n)}(w) = v$ for some $n$, but $f^{(k)}(w)$ does not belong to $K$ for $k \lt n$.

A choice of endofunction structure on $S$ can thus be equivalently described as follows:

* Partition $S$ into one or more subsets (equivalence classes);

* Choose a rooted tree structure on each class (if the root is $v$, this tree structure is $T_v$);

* Choose a permutation on the set of equivalence classes (tantamount to a permutation of the roots $v$).

If $P$ is the species of permutations, and $R$ the species of rooted trees, this establishes an explicit bijection between $P \circ R$ and the species of endofunctions.

Since the species $L$ of linear orderings and the species $P$ of permutations have the same cardinality, it follows that the species $L \circ R$ and $P \circ R$ also have the same cardinality, which is to say that the number of bipointed tree structures on $S$ equals the number of endofunctions, as was to be proved.

[[!redirects Lambert W function]]
[[!redirects Lambert W-function]]


Lance Dixon

* [webpage](http://www.slac.stanford.edu/~lance/)

## Selected writings

On [[scattering amplitudes]] in [[perturbative quantum gravity|perturbative]] [[D=4 N=8 supergravity]]:

* [[Zvi Bern]], [[John Joseph Carrasco]], [[Lance Dixon]], [[Henrik Johansson]], [[Radu Roiban]], _Amplitudes and Ultraviolet Behavior of $N=8$ Supergravity_, Fortschr. Phys. **59** (2011) 7-8 [[arXiv:1103.1848](http://arxiv.org/abs/1103.1848), [doi:10.1002/prop.201100037](https://doi.org/10.1002/prop.201100037)]

Summary of the history of on-shell methods for computing [[scattering amplitudes]]:

* Lance Dixon, _Calculuating Amplitudes_, December 2013 ([web](http://www.preposterousuniverse.com/blog/2013/10/03/guest-post-lance-dixon-on-calculating-amplitudes/))

category: people


Landau pole

## Idea

(...)

## Related concepts

* [[asymptotic freedom]]

## References

* Wikipedia, _[Landau pole](http://en.wikipedia.org/wiki/Landau_pole)_

* [[Paul Romatschke]], *Life at the Landau pole* [[arXiv:2212.03254](https://arxiv.org/abs/2212.03254)]

[[!redirects Landau poles]]


Landau theory of phase transitions

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#### Physics
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=--
=--
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#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concepts

* [[phase of matter]], [[phase transition]]

* outside of Landau theory: [[topological phase of matter]]

## References

Textbook accounts:

* [[John W. Negele]], [[Henri Orland]], Chapter 6 in: *Quantum Many-Particle Systems*, Westview Press (1988, 1998) [[doi:10.1201/9780429497926](https://doi.org/10.1201/9780429497926)]

Reviews:

* Alexander Umantsev, *Landau Theory of Phase Transitions*, in: *Field Theoretic Method in Phase Transformations*, Lecture Notes in Physics **840** (2012) 7–35 $[$[doi:10.1007/978-1-4614-1487-2_2](https://doi.org/10.1007/978-1-4614-1487-2_2)$]$

In contrast to [[topological phases of matter]]:

* [[Tudor D. Stanescu]], Section 4.2 of: *Introduction to Topological Quantum Matter & Quantum Computation*, CRC Press 2020 ([ISBN:9780367574116](https://www.routledge.com/Introduction-to-Topological-Quantum-Matter--Quantum-Computation/Stanescu/p/book/9780367574116))

See also:

* Wikipedia, *[Landau theory](https://en.wikipedia.org/wiki/Landau_theory)*

[[!redirects Landau theory]]


Landau's Fermi liquid theory

+-- {: .rightHandSide}
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### Context
#### Solid state physics
+-- {: .hide}
[[!include solid state physics -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[solid state physics]], *Landaus' Fermi liquid theory* is [[perturbation theory|perturbative]] [[theory (physics)|theory]] of [[metals]] regarded as seas of [[fermions]].

The model is highly successful, given its simplistic assumptions, for ordinary [[metals]], but fails for materials with [[non-perturbative effect|non-perturbative]] [[topological order]] thus known as *[[strange metals]]*. These are instead described by the [[AdS/CFT correspondence in condensed matter physics]].

## Related concepts

* [[fermion]]

* [[Pauli exclusion principle]], [[degeneracy pressure]]

* [[Pauli exclusion principle]]

* [[Landau theory of phase transitions]]

## References

Named after [[Lev Landau]] (following [[Enrico Fermi]]).

Introduction and survey:

* [[Andrea Amoretti]], Chapter 2 of: _Condensed Matter Applications of AdS/CFT: Focusing on strange metals_, 2017 ([spire:1610363](http://inspirehep.net/record/1610363), [[AmorettiStrangeMetals.pdf:file]])

  > (with an eye towards [[strange metals]] and [[AdS/CFT in condensed matter physics]])

* Erkki Thuneberg, *Introduction to Landau’s Fermi Liquid Theory* 2009, ([pdf](http://www.jiwaji.edu/pdf/ecourse/physics/The%20Fermi%20Liquid%20Theory.pdf))

See also

* Wikipedia, *[Fermi liquid theory](https://en.wikipedia.org/wiki/Fermi_liquid_theory)*

* Umang Mehta, *Postmodern Fermi Liquids* [[arXiv:2307.02536](https://arxiv.org/abs/2307.02536)]

[[!redirects Fermi liquid theory]]

[[!redirects fermion sea]]
[[!redirects fermion seas]]

[[!redirects Fermi sea]]
[[!redirects Fermi seas]]

[[!redirects Fermi liquid]]
[[!redirects Fermi liquids]]


Landau-Ginzburg model

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Quantum field theory
+-- {: .hide}
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=--
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

Originally, the _Ginzburg--Landau model_ is a mesoscopic [[model (physics)|model]] in [[solid state physics]] for [[superconductivity]].

Roughly this type of model has then been used as models for 2d [[quantum field theory]] in [[string theory]].
There, a _Landau--Ginzburg model_ (LG-model) is a 2-[[dimension|dimensional]] [[supersymmetry|supersymmetric]] [[sigma model]] [[QFT]] characterized by the fact that its [[Lagrangian]] contains a [[potential]] term: given a [[complex manifold|complex]] [[Riemannian manifold|Riemannian]] [[target space]] $(X,g)$, the [[action functional]] of the LG-model is schematically of the form

$$
  S_{LB} : (\phi : \Sigma \to X) \mapsto
  \int_\Sigma \left( \vert \nabla \Phi \vert^2 + \vert (\nabla W)(\phi) \vert^2
  + fermionic\;terms \right) d \mu
  \,,
$$

where $\Sigma$ is the 2-[[dimension]]al [[worldsheet]] and $W : X \to \mathbb{C}$ -- called the model's _superpotential_ -- is a [[holomorphic function]]. (Usually $X$ is actually taken to be a [[Cartesian space]] and all the nontrivial structure is in $W$.)

Landau--Ginzburg models have gained importance as constituting one type of QFTs that are related under [[homological mirror symmetry]]:

If the [[target space]] $X$ is a [[Fano variety]], the usual [[B-model]] does not quite exist on it, since the corresponding supersymmetric [[string]] [[sigma model]] is not conformally invariant as a quantum theory, and the axial $U(1)$ [[R-current]] used to define the B-twist is [[quantum anomaly|anomalous]]. Still, there exists an analogous [[derived category]] of B-branes. A Landau--Ginzburg model is something that provides the dual A-branes to this under [[homological mirror symmetry]]. Conversely, Landau--Ginzburg B-branes are homological mirror duals to the [[A-model]] on a Fano variety.
(...)

A mathematical definition of the category of B-branes in the LG model is given by [[Maxim Kontsevich]] (elaborated in  [Kapustin-Li, section 7](#KapustinLi)), the idea is that (at least in a certain class of cases, say affine varieties) the B-branes are not given by [[chain complexes]] of [[coherent sheaves]] as in the [[B-model]], but by _[[twisted complexes]]_ : for these the square of the [[differential]] is in general non-vanishing but rather a multiplication by _superpotential_ of the LG-model, which therefore supplies a way to twist the (derived) category of coherent sheaves. The $\mathbf{Z}$-grading breaks down to a $\mathbf{Z}_2$-grading. This category of D-branes of type B can be presented as the product of [[derived categories of singularities]] $\prod_\lambda D^b_{sg}(X_\lambda)$ (which is a finite product) where $X_\lambda$ is a fiber over $\lambda$ in $\mathbf{C}$, what can roughly be taken as a more general definition.

## Properties

### The $\infty$-categories of branes

A [[brane]] for a LG model is given by a [[matrix factorization]] of its superpotential.

(...) [[curved dg-algebra]]

(...) [CaldararuTu](#CaldararuTu)

## Related concepts

* [[2d TQFT]], [[TCFT]]

* [[holographic superconductor]]

## References

### In superconductivity

Discussion as a model for [[superconductors]]:

The original article:

* [[В. Л. Гинзбург]], [[Л. Д. Ландау]], _К теории сверхпроводимости_, ЖЭТФ 20 (1950), 1064.

English translation:

* [[Vitaly Ginzburg]], [[Lev Landau]], _On the Theory of Superconductivity_,  In: _On Superconductivity and Superfluidity_ Springer (2009) ([doi:1007/978-3-540-68008-6_4](https://doi.org/10.1007/978-3-540-68008-6_4))

Review:

* {#Chapman00} S. J. Chapman, Section 2 of: _A Hierarchy of Models for Type-II Superconductors_, IAM Review Vol. 42, No. 4 (2000), pp. 555-598 ([jstor:2653134](https://www.jstor.org/stable/2653134))

* {#Timm20} Carsten Timm, Section 6 of: _Theory of Superconductivity_, 2020 ([pdf](https://tu-dresden.de/mn/physik/itp/cmt/ressourcen/dateien/skripte/Skript_Supra.pdf?lang=en))

### In string theory

Discussion as a [[supersymmetry|supersymmetric]] [[sigma model]] in [[string theory]] [[KK-compactification|on]] [[Calabi-Yau manifolds]]:

* {#VafaWarner89} [[Cumrun Vafa]], [[Nicholas Warner]], _Catastrophes and the Classification of Conformal Theories_, Phys.Lett. B218 (1989) 51 ([doi:10.1016/0370-2693(89)90473-5](https://doi.org/10.1016/0370-2693%2889%2990473-5))

* [[Brian Greene]], [[Cumrun Vafa]], _Calabi--Yau Manifolds and Renormalization Group Flows_, Nucl.Phys. B324 (1989) 371 (doi:10.1016/0550-3213(89)90471-9)

* [[Paul Howe]], [[Peter West]], *Landau-Ginzburg Hamiltonians, string theory and critical phenomena*, in: *Proceedings of 25th International Conference on High Energy Physics*, Singapore (August 2-8, 1990) [[inspire:301119](https://inspirehep.net/literature/301119), [pdf](https://inspirehep.net/files/f9b5ac3ccd39ef8c30ca5c7b69b92fc8)]

* [[Edward Witten]], _Phases of $N=2$ Theories In Two Dimensions_, Nucl.Phys.B 403:159--222, 1993 ([arXiv:hep-th/9301042](https://arxiv.org/abs/hep-th/9301042))

Lecture notes:

* [[Edward Witten]], _Dynamical aspects of QFT_, Lecture 15: _The Landau--Ginzburg description of $N=2$ minimal models; Quantum cohomology and Kähler manifolds_,  in Part IV of _[[Quantum Fields and Strings]]_.

#### Branes

The [[branes]] of the LG-model are discussed for instance in

* [[Anton Kapustin]], Yi Li, _D-Branes in Landau--Ginzburg models and algebraic geometry_, J. High Energy Phys. 12 (2003), 005 [arXiv:hep-th/0210296](http://arxiv.org/abs/hep-th/0210296){#KapustinLi}

The [[derived category]] of [[D-brane]]s in type B LG-models is discussed in

* {#Orlov} [[Dmitri Orlov]], _Triangulated categories of singularities and D-branes in Landau--Ginzburg models_, Proc. Steklov Inst. Math. 2004, no. 3 (246), 227--248 ([arXiv:math/0302304](https://arxiv.org/abs/math/0302304))

* [[Dmitri Orlov]], _Derived categories of coherent sheaves and triangulated categories of singularities_, Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. II, 503--531, Progr. Math. __270,__ Birkhäuser Boston 2009 ([arXiv:math.ag/0503632](https://arxiv.org/abs/math/0503632))

* [[Dmitri Orlov|Dmitri O. Orlov]], _Triangulated categories of singularities and equivalences between Landau-Ginzburg models_, Sbornik: Mathematics __197__:12 (2006) 1827 [doi](https://doi.org/10.1070/SM2006v197n12ABEH003824)

* Andrei Caldararu, Junwu Tu, _Curved $A_\infty$-algebras and Landau--Ginzburg models_ ([pdf](http://www.math.wisc.edu/~andreic/publications/AinfinityLG.pdf))
 {#CaldararuTu}

#### Defects

General defects of B-twisted affine LG models were first discussed in

* [[Ilka Brunner]], [[Daniel Roggenkamp]], _B-type defects in Landau--Ginzburg models_, JHEP 0708 (2007) 093, ([arXiv:0707.0922](http://arxiv.org/abs/0707.0922))

The graded pivotal bicategory of B-twisted affine LG models is studied in detail in

* [[Nils Carqueville]], [[Daniel Murfet]], *Adjunctions and defects in Landau-Ginzburg models*, Advances in Mathematics **289** (2016) 480-566 [[doi:10.1016/j.aim.2015.03.033](https://doi.org/10.1016/j.aim.2015.03.033), [arXiv:1208.1481](https://arxiv.org/abs/1208.1481)]

and described in terms of [[linear logic]] and the [[geometry of interactions]]:

* [[Daniel Murfet]], _The cut operation on matrix factorisations_, Journal of Pure and Applied Algebra
**222** 7 (2018) 1911-1955 [[arXiv:1402.4541](https://arxiv.org/abs/1402.4541), [doi:10.1016/j.jpaa.2017.08.014](https://doi.org/10.1016/j.jpaa.2017.08.014)]

Orbifolds of defects are studied in

* [[Ilka Brunner]], [[Daniel Roggenkamp]], _Defects and Bulk Perturbations of Boundary Landau--Ginzburg Orbifolds_, JHEP 0804 (2008) 001, ([arXiv:0712.0188](http://arxiv.org/abs/0712.0188))

* [[Nils Carqueville]], Ingo Runkel, _Orbifold completion of defect bicategories_, ([arXiv:1210.6363](https://arxiv.org/abs/1210.6363))

* [[Ilka Brunner]], [[Nils Carqueville]], Daniel Plencner, _Orbifolds and topological defects_, Comm. Math. Phys. 332 (2014), 669--712, ([arXiv:1307.3141](https://arxiv.org/abs/1307.3141))

* [[Ilka Brunner]], [[Nils Carqueville]], Daniel Plencner, _Discrete torsion defects_, Comm. Math. Phys. 337 (2015), 429--453, ([arXiv:1404.7497](https://arxiv.org/abs/1404.7497))

A relation to [[linear logic]] and the [[geometry of interaction]] is in

* {#Murfet14} [[Daniel Murfet]], _Computing with cut systems_ ([arXiv:1402.4541](https://arxiv.org/abs/1402.4541))

#### TCFT formulation

Discussions of topological Landau--Ginzburg [[B-models]] explicitly as open [[TCFT]]s (aka open topological string theories) are in

* [[Nils Carqueville]], _Matrix factorisations and open topological string theory_, JHEP 07 (2009) 005, ([arXiv:0904.0862](http://arxiv.org/abs/0904.0862))

* [[Ed Segal]], _The closed state space of affine Landau--Ginzburg B-models_ ([arXiv:0904.1339](http://arxiv.org/abs/0904.1339))

* Nils Carqueville, Michael Kay, _Bulk deformations of open topological string theory_, Comm. Math. Phys. 315, Number 3 (2012), 739--769, ([arXiv:1104.5438](https://arxiv.org/abs/1104.5438))

[[!include elliptic genera as partition functions -- references]]

[[!redirects LG model]]
[[!redirects LG models]]
[[!redirects LG-model]]
[[!redirects LG-models]]

[[!redirects Landau Ginzburg model]]
[[!redirects Landau Ginzburg models]]
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[[!redirects Landau--Ginzburg models]]
[[!redirects Landau-Ginzburg theory]]


Landau-Lifschitz > history

see at _[[Course of Theoretical Physics]]_


landscape of string theory vacua

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+-- {: .hide}
[[!include string theory - contents]]
=--
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#Contents#
* table of contents
{:toc}

## Idea

### recalling the context ##

The undertaking called [[string theory]] started out as _perturbative string theory_ where the idea was to encode [[spacetime]] [[physics]] in [[perturbation theory]] by an [[S-matrix]] that is obtained by a sum of the integrals of the [[correlators]] of a fixed [[2d superconformal field theory]] over the [[moduli spaces]] of conformal structures on surfaces of all possible genera -- thought of as the [[second quantization]] of a [[string]] [[sigma-model]].

The [[S-matrix]] elements obtained this way from the [[string perturbation series]] could be seen to be approximated by an ordinary [[effective QFT]] (some flavor of [[supergravity]] coupled to [[gauge theory]] and [[fermion]]s) on [[target space]].

(The _first superstring revolution_ was given by the realization that this makes sense: the effective background theories obtained this way are indeed free of [[quantum anomaly|quantum anomalies]].)

Hence it is the choice of [[worldsheet]] [[2d SCFT]] which in [[perturbative string theory]] translates products of "field insertions" into [[scattering amplitudes]]. In [[perturbative AQFT]] it is the choice of [[vacuum state]] which does this, and therefore [[2d SCFTs]] are the [[perturbative string theory vacua]].

### narrowing in on the issue ##

The _second superstring revolution_ was given by the realization that all these background field theories seem to fit into one single bigger context that seems to exists independently of their perturbative definitions.

Aspects of this bigger non-perturbative context are known as [[M-theory]]. While one couldn't figure out what that actually is, the circumstantial evidence suggested that whatever it is, it has a low-energy limit where it also looks like an effective background field theory, this time [[11-dimensional supergravity]].

In a different but similar manner, other background field theories were found whose classical solutions are thought to encode "stable solutions" ("[[vacuum]] solutions") of whatever physical theory this non-perturbative definition of string theory is.

Here, when talking about a "stable solution" one thinks of solutions of these theories of [[gravity]] with plenty of extra fields that look like [[Minkowski space]] times something else, such that all these extra fields are constant in time (using the simple Minkowsi-space-times-internal-part-ansatz to say what "constant in time" means), hence sitting at the bottom of their corresponding effective potentials.

Solutions with this property, in particular for all the scalar fields that appear, are said to have _stabilized moduli_ : the scalar fields that encode various properties of the geometry of the solution are constant in time.

Since these geometric properties determine, in the fashion of [[Kaluza-Klein theory]], the effective physics in the remaining [[Minkowski space]] factor, it is these "moduli-stabilized" solutions that have a first chance of being candidate solutions of whatever that theory is we are talking about, which describe the real world.

### the landscape

At some point there had been the hope that only very few such solutions exist. When arguments were put forward that this is far from being true, the term **landscape** for the collection of all such solutions was invented.

So, to summarize in a few words, the landscape of string theory vacua is...

## Flux compactifications

One widely studied class of moduli-stabilized solutions to the string-theory background equations is that of **[[flux compactifications]]**.

These are classical solutions to the corresponding [[supergravity]] theory that are of the form $M^4 \times CY$ with $CY$ some [[Calabi-Yau manifold]] of six real dimensions such that the [[RR-field]] in the solution has nontrivial values on $CY$. Its components are called the _[[fluxes]]_ .

The presence of this [[RR-field]] in the solution induces an effective potential for the scalar moduli fields that parameterize the geometry of CY. Hence by choosing the [[RR-field]] suitably one can find classical solutions in which all these moduli have values that are constant in time.

A review of flux compactifications is for instance in ([Graña 05](#Grana05))

## Computer scans of Gepner model compactifications

Discussion of [[string phenomenology]] of [[intersecting D-brane models]] [[KK-compactification|KK-compactified]] with non-geometric [[fibers]] such that the would-be string [[sigma-models]] with these [[target spaces]] are in fact [[Gepner models]] (in the sense of _[Spectral Standard Model and String Compactifications](https://www.physicsforums.com/insights/spectral-standard-model-string-compactifications/)_) is in ([Dijkstra-Huiszoon-Schellekens 04a](#DijkstraHuiszoonSchellekens04a), [Dijkstra-Huiszoon-Schellekens 04b](#DijkstraHuiszoonSchellekens04b)):

>  A plot of [[standard model of particle physics|standard model]]-like [[coupling constants]] in a computer scan of [[Gepner model]]-[[KK-compactification]] of [[intersecting D-brane models]] according to [Dijkstra-Huiszoon-Schellekens 04b](#DijkstraHuiszoonSchellekens04b).

>  The blue dot indicates the couplings in $SU(5)$-[[GUT]] theory. The faint lines are NOT drawn by hand, but reflect increased density of Gepner models as seen by the computer scan.

## Related concepts

> at least one thing missing in the discussion here is the subtlety explained out by [[Jacques Distler]] in blog discussion [here](http://golem.ph.utexas.edu/category/2009/10/structural_foundations_of_quan.html#c028474)

* [[swampland]]

* [[cosmological constant]]

* [[axionic landscape]]

* [[string theory]]

  * [[heterotic string theory]]

    * [[Green-Schwarz mechanism]]

   * [[dual heterotic string theory]]

  * [[type II string theory]]

* **[[vacuum]]**

  * [[vacuum state]], [[Hadamard state]]

  * [[interacting vacuum]]

  * [[vacuum expectation value]], [[vacuum amplitude]], [[vacuum fluctuation]]

  * [[vacuum energy]]

  * [[vacuum diagram]]

  * [[vacuum stability]]

  * [[false vacuum]], [[tachyon]], [[Coleman-De Luccia instanton]]

  * [[theta vacuum]]

  * [[perturbative string theory vacuum]]

    * [[non-geometric string theory vacuum]]

    * [[landscape of string theory vacua]]

* [[moduli field]], [[multiverse]]

* [[model (in particle physics)]]

  * [[standard model of particle physics]]

  * [[Grand Unified Theory]], [[MSSM]]

* [[Brandenberger-Vafa mechanism]]

* [string theory FAQ - Does string theory make predictions?](string+theory+FAQ#HowDoesStringTheoryMakePredictions)

* [string theory FAQ -- What does it mean to say that string theory has a "landscape of solutions"?](http://ncatlab.org/nlab/show/string+theory+FAQ#WhatDoesItMeanToSayStringTheoryHasALandscapeOfSolutions)

## References

### F-theory flux compactification

Surveys of the general story of [[flux compactification]] in [[F-theory]] includes

* {#Denef08} [[Frederik Denef]], _Les Houches Lectures on Constructing String Vacua_, in _[[String theory and the real world]]_ ([arXiv:0803.1194](http://arxiv.org/abs/0803.1194))

### Landscape of Type II vacua

Scan of the moduli space of semi-realistic [[type II string theory|type IIB]] [[intersecting D-brane model]] [[KK-compactifications]] on [[orbifolds]] of [[Gepner models]] is in

* {#DijkstraHuiszoonSchellekens04a} T.P.T. Dijkstra, L. R. Huiszoon, [[Bert Schellekens]], _Chiral Supersymmetric Standard Model Spectra from Orientifolds of Gepner Models_, Phys.Lett. B609 (2005) 408-417 ([arXiv:hep-th/0403196](https://arxiv.org/abs/hep-th/0403196))

* {#DijkstraHuiszoonSchellekens04b} T.P.T. Dijkstra, L. R. Huiszoon, [[Bert Schellekens]], _Supersymmetric Standard Model Spectra from RCFT orientifolds_, Nucl.Phys.B710:3-57,2005 ([arXiv:hep-th/0411129](https://arxiv.org/abs/hep-th/0411129))

and scan [[type II string theory|type IIB]] [[intersecting D-brane model]] [[KK-compactifications]] on [[toroidal orbifolds]] is in

* [[Ralph Blumenhagen]], [[Florian Gmeiner]], [[Gabriele Honecker]], [[Dieter Lüst]], [[Timo Weigand]], _The Statistics of Supersymmetric D-brane Models_, Nucl.Phys.B713:83-135, 2005 ([arXiv:hep-th/0411173](https://arxiv.org/abs/hep-th/0411173))

* {#GBHLW05} [[Florian Gmeiner]], [[Ralph Blumenhagen]], [[Gabriele Honecker]], [[Dieter Lüst]], [[Timo Weigand]], _One in a Billion: MSSM-like D-Brane Statistics_, JHEP 0601:004, 2006 ([arXiv:hep-th/0510170](https://arxiv.org/abs/hep-th/0510170))

### Landscape of heterotic vacua

The origin of all [[string phenomenology]] is the [[top-down model building|top-down approach]] in the [[heterotic string]] due to ([Candelas-Horowitz-Strominger-Witten 85](heterotic+string+theory#CHSW85)).

A brief review of motivations for [[GUT]] models in [[heterotic string theory]] is in

* {#Witten02} [[Edward Witten]], _Quest For Unification_, Heinrich Hertz lecture at [SUSY 2002](http://www.desy.de/susy02/) at DESY, Hamburg ([arXiv:hep-ph/0207124](http://arxiv.org/abs/hep-ph/0207124))

The following articles establish the existences of exact realization of the [[gauge group]] and [[matter]]-content of the [[MSSM]] in [[heterotic string theory]] (not yet checking [[Yukawa couplings]]):

* [[Volker Braun]], [[Yang-Hui He]], [[Burt Ovrut]], [[Tony Pantev]], _A Heterotic Standard Model_, Phys. Lett. B618 : 252-258 2005 ([arXiv:hep-th/0501070](http://arxiv.org/abs/hep-th/0501070))

* {#BraunHeOvrutPantev06} [[Volker Braun]], [[Yang-Hui He]], [[Burt Ovrut]], [[Tony Pantev]], _The Exact MSSM Spectrum from String Theory_, JHEP 0605:043,2006 ([arXiv:hep-th/0512177](http://arxiv.org/abs/hep-th/0512177))

* [[Vincent Bouchard]], [[Ron Donagi]], _An SU(5) Heterotic Standard Model_, Phys. Lett. B633:783-791,2006 ([arXiv:hep-th/0512149](http://arxiv.org/abs/hep-th/0512149))

A computer search through the "[[landscape of string theory vacua|landscape]]" of [[Calabi-Yau varieties]] showed severeal hundreds more such exact heterotic standard models (about one billionth of all CYs searched, and most of them arising as $SU(5)$-[[GUTs]])

* [[Lara Anderson]], [[Yang-Hui He]], [[Andre Lukas]], _Heterotic Compactification, An Algorithmic Approach_, JHEP 0707:049, 2007 ([arXiv:hep-th/0702210](https://arxiv.org/abs/hep-th/0702210))

* {#AndersonGrayLukasPalti11} [[Lara Anderson]], James Gray, [[Andre Lukas]], [[Eran Palti]], _Two Hundred Heterotic Standard Models on Smooth Calabi-Yau Threefolds_ ([arXiv:1106.4804](http://arxiv.org/abs/1106.4804))

* [[Lara Anderson]], James Gray, [[Andre Lukas]], [[Eran Palti]], _Heterotic Line Bundle Standard Models_ JHEP06(2012)113 ([arXiv:1202.1757](https://arxiv.org/abs/1202.1757))

* [[Lara Anderson]], Andrei Constantin, James Gray, [[Andre Lukas]], [[Eran Palti]], _A Comprehensive Scan for Heterotic SU(5) GUT models_, JHEP01(2014)047 ([arXiv:1307.4787](https://arxiv.org/abs/1307.4787))

* [[Yang-Hui He]], Seung-Joo Lee, [[Andre Lukas]], Chuang Sun, _Heterotic Model Building: 16 Special Manifolds_ ([arXiv:1309.0223](http://arxiv.org/abs/1309.0223))

* {#CHE18} Andrei Constantin, [[Yang-Hui He]], [[Andre Lukas]], _Counting String Theory Standard Models_ ([arXiv:1810.00444](https://arxiv.org/abs/1810.00444))

The resulting database of compactifications is here:

* [[Lara Anderson]], James Gray, [[Andre Lukas]], [[Eran Palti]], _Heterotic standard model database_ ([web](http://www-thphys.physics.ox.ac.uk/projects/CalabiYau/linebundlemodels/index.html.))

Review includes

* [[Lara Anderson]], _New aspects of heterotic geometry and phenomenology_, talk at [Strings2012](http://wwwth.mpp.mpg.de/conf/strings2012/), Munich 2012 ([pdf](http://wwwth.mpp.mpg.de/members/strings/strings2012/strings_files/program/Talks/Wednesday/Anderson.pdf))

* {#He18a} [[Yang-Hui He]], _The Calabi-Yau Landscape: from Geometry, to Physics, to Machine-Learning_ ([arXiv:1812.02893](https://arxiv.org/abs/1812.02893))

* {#He18b} [[Yang-Hui He]], _Deep-learning the landscape_, talk at _[String and M-Theory: The new geometry of the 21st century](https://ims.nus.edu.sg/events/2018/wstring/wk.php)_ ([pdf slides](https://ims.nus.edu.sg/events/2018/wstring/files/yang.pdf), [video recording](https://www.youtube.com/watch?v=x3ThgBgkPlE))

Computation of [[metrics]] on these Calabi-Yau compactifications (eventually needed for computing their induced [[Yukawa couplings]]) is started in

* [[Volker Braun]], Tamaz Brelidze, [[Michael Douglas]], [[Burt Ovrut]], _Calabi-Yau Metrics for Quotients and Complete Intersections_, JHEP 0805:080, 2008 ([arXiv:0712.3563](https://arxiv.org/abs/0712.3563))

This "heterotic standard model" has a "hidden sector" copy of the actual standard model, more details of which are discussed here:

* [[Volker Braun]], [[Yang-Hui He]], [[Burt Ovrut]], _Supersymmetric Hidden Sectors for Heterotic Standard Models_ ([arXiv:1301.6767](http://arxiv.org/abs/1301.6767))

The issue of [[moduli stabilization]] in these kinds of models is discussed in

* Michele Cicoli, Senarath de Alwis, Alexander Westphal, _Heterotic Moduli Stabilization_ ([arXiv:1304.1809](http://arxiv.org/abs/1304.1809))

* [[Lara Anderson]], James Gray, [[Andre Lukas]], [[Burt Ovrut]], _Vacuum Varieties, Holomorphic Bundles and Complex Structure Stabilization in Heterotic Theories_ ([arXiv:1304.2704](http://arxiv.org/abs/1304.2704))

Principles singling out heterotic models with three generations of fundamental particles are discussed in:

* [[Philip Candelas]], [[Xenia de la Ossa]], [[Yang-Hui He]], Balazs Szendroi, _Triadophilia: A Special Corner in the Landscape_, Adv.Theor.Math.Phys.12:2,2008 ([arXiv:0706.3134](http://arxiv.org/abs/0706.3134))

See also

* Hajime Otsuka, _$SO(32)$ heterotic line bundle models_, ([arXiv:1801.03684](https://arxiv.org/abs/1801.03684))

### Moduli space of 2d SCFTs

Some general thoughts on what a [[moduli space]] of 2d CFTs should be are in

* [[Michael Douglas]], _Spaces of Quantum Field Theories_ ([arXiv:1005.2779](http://arxiv.org/abs/1005.2779))

The compactness results mentioned there are discussed in

* {#Soibelman11} [[Yan Soibelman]], _Collapsing CFTs, spaces with non-negative Ricci curvature and nc geometry_ in [[Hisham Sati]], [[Urs Schreiber]] (eds.) _[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]_ Proceedings of Symposia in Pure Mathematics, AMS (2011)

based on conjectures in

* [[Maxim Kontsevich]], [[Yan Soibelman]], _Homological Mirror Symmetry and torus fibrations_, ([math.SG/0011041](http://arxiv.org/abs/math/0011041))

### Phenomenological speculation
 {#ReferencesPhenomenologicalSpeculation}

Early and technical articles that amplified the existence of a finite but very large number of string theory compactifications are

* {#LercheLustSchellekens86} [[Wolfgang Lerche]], [[Dieter Lüst]], [[Bert Schellekens]], _Ten dimensional heterotic strings from Niemeier lattices_, Physics Letters B, Volume 181, Issues 1-2, 1986 ([pdf](http://cds.cern.ch/record/170910/files/198609340.pdf))

which says on p. 2

> Although the consistency requirements which string theories have to satisfy are quite restrictive, it has become clear that there are more solutions than one originally expected. [...] Although the possibility of making Lorentz rotations suggests a continuous infinity of new ten dimensional theories, there is actually only a discrete set of theories that makes physical sense, as we will explain below.

and

* {#LercheLustSchellekens87} [[Wolfgang Lerche]], [[Dieter Lüst]], [[Bert Schellekens]], _Chiral Four-dimensional Heterotic Strings from Self-dual Lattices_  Nucl. Phys. B 287, 477, 1987 ([pdf](http://lerche.web.cern.ch/lerche/papers/4dhetstrings.pdf))

which says in conclusion on page 45-46

> Although the number of chiral theories of this type is finite, our results suggest that there exist very many of them, so that a complete enumeration appears impossible.

A popular account of these observations was given in

* {#Schellekens98} [[Bert Schellekens]], _Naar een waardig slot_, inauguration speech ar University of Nijmegen, September 1998, ISBN 90-9012073-4 ([ubn:2066/18631](https://mobile.repository.ubn.ru.nl/handle/2066/18631))

a commented translation of which later appeared as

* {#Schellekens06} [[Bert Schellekens]], _The Landscape "avant la lettre"_ ([arXiv:physics/0604134](http://arxiv.org/abs/physics/0604134))

Similarly

* {#Schellekens08} [[Bert Schellekens]], _The Emperor's Last Clothes?_, Rept.Prog.Phys.71:072201,2008 ([arXiv:0807.3249](https://arxiv.org/abs/0807.3249))

* {#Schellekens16} [[Bert Schellekens]], _Big Numbers in String Theory_ ([arXiv:1601.02462](http://arxiv.org/abs/1601.02462))

The articles [Lerche-Lüst-Schellekens 86](#LercheLustSchellekens86), [Lerche-Lüst-Schellekens 87](#LercheLustSchellekens87), and the speech [Schellekens 98](#Schellekens98), did not cause much of excitement then. Also they did not discuss [[moduli stabilization]], which could still have been thought to reduce the number of vacua. Excitement was only later caused instead by more vague discussion of flux compactification vacua with moduli stabilization in type IIB string theory:

That there are $10^{hundreds}$ different flux compactifications was maybe first said explicitly in

* [[Raphael Bousso]], [[Joseph Polchinski]], _Quantization of Four-form Fluxes and Dynamical Neutralization of the Cosmological Constant_ ([arXiv:hep-th/0004134](http://arxiv.org/abs/hep-th/0004134))

The idea became popular in discussion of the [[cosmological constant]] with the alleged construction of a large set of metastable [[de Sitter spacetime]]-vacua in

* {#KKLT03} [[Shamit Kachru]], [[Renata Kallosh]], [[Andrei Linde]], [[Sandip  Trivedi]], _de Sitter Vacua in String Theory_, Phys. Rev. D68:046005, 2003 ([arXiv:hep-th/0301240](http://arxiv.org/abs/hep-th/0301240))

  ("KKLT", a good quick review is in [Danielsson-VanRiet 18 section 2.5.1](#DanielssonVanRiet18), also [Ibanez-Uranga 12, section 15.3.1](#IbanezUranga12))

and the amplification of the complication of the [KKLT 03](#KKLT03)-construction its alleged vastness in

* {#Susskind03} [[Leonard Susskind]], _The Anthropic Landscape of String Theory_, in B. Carr (ed.) _Universe or multiverse_, 247-266 ([arXiv:hep-th/0302219](http://arxiv.org/abs/hep-th/0302219))

  "The vacua in [KKLT 03](#KKLT03) are not at all simple.  They are jury-rigged, [Rube Goldberg contraptions](https://en.wikipedia.org/wiki/Rube_Goldberg_machine) that could hardly have fundamental significance." (p. 5)

* [[Joseph Polchinski]], _The Cosmological Constant and the String Landscape_ ([arXiv:hep-th/0603249](http://arxiv.org/abs/hep-th/0603249))

Review includes

* {#IbanezUranga12} [[Luis Ibáñez]], [[Angel Uranga]], section 15.3 of _[[String Theory and Particle Physics -- An Introduction to String Phenomenology]]_, Cambridge University Press 2012

(Beware that the approach of [KKLT 03](#KKLT03) is argued to be false in [DanielssonVanRiet 18](#Danielsson-VanRiet18) and is being abandoned in [Obied-Ooguri-Spodyneiko-Vafa 18](#ObiedOoguriSpodyneikoVafa18), [Danielsson et. al 18](#de+Sitter+spacetime#BDDGS18)).

The specific (but arbitrary) value  "$10^{500}$" for the typical number of flux compactification, which became iconic in public discussion of the issue, originates in

* {#Douglas04} [[Michael Douglas]], p. 4 of _Basic results in vacuum statistics_, Comptes Rendus Physique, vol. 5, pp. 965–977, 2004 ([arXiv:hep-th/0409207](http://arxiv.org/abs/hep-th/0409207))

* [[Ralph Blumenhagen]], [[Florian Gmeiner]], [[Gabriele Honecker]], [[Dieter Lüst]], [[Timo Weigand]], p.3 of _The Statistics of Supersymmetric D-brane Models_, Nucl.Phys.B713:83-135, 2005 ([arXiv:hep-th/0411173](http://arxiv.org/abs/hep-th/0411173))

* [[Michael Douglas]], [[Shamit Kachru]], p. 55 of _Flux Compactification_, Rev.Mod.Phys.79:733-796,2007 ([arXiv:hep-th/0610102](https://arxiv.org/abs/hep-th/0610102))

Previously

* [[Suyay Ashok]], [[Michael Douglas]], _Counting flux vacua_, JHEP, vol. 0401, p. 060, 2004

had considered $10^{120}$ and earlier [Lerche-Lüst-Schellekens 87](#LercheLustSchellekens87) had $10^{1500}$.

A review of the issue of flux compactifications is in

* {#Grana05} [[Mariana Graña]], _Flux compactifications in string theory: a comprehensive review_ ([arXiv:hep-th/0509003](http://arxiv.org/abs/hep-th/0509003))

General considerations on this state of affairs are in

* [[Frank Wilczek]], _Multiversality_ ([arXiv:1307.7376](http://arxiv.org/abs/1307.7376))

The fact that in principle all the parameters of the "landscape" of string theory vacua are dynamical (are moduli fields) and the idea that an [[eternal cosmic inflation]] might be something like an [[ergodic process]] in this landscape has led to ideas to connect this to [[phenomenology]] and the [[standard model of cosmology]]/[[standard model of particle physics]] by way of [[statistical mechanics]].

Summaries of this line of thinking include

* [[Raphael Bousso]], _The State of the Multiverse:
The String Landscape, the Cosmological
Constant, and the Arrow of Time_, 2011 ([pdf](http://www.ctc.cam.ac.uk/stephen70/talks/swh70_bousso.pdf))

For more on this see the references at _[[multiverse]]_ and _[[eternal inflation]]_.

### Landscape of de Sitter vacua (or not)
 {#ReferencesDeSitter}

On the other hand, discussion casting doubt on the existence of a large number of [[de Sitter spacetime]] [[perturbative string theory vacua]] includes the following:

* [[Tom Banks]], _Landskepticism: or Why Effective Potentials Don't Count String Models_ ([arXiv:hep-th/0412129](https://arxiv.org/abs/hep-th/0412129))

* [[Tom Banks]], _The Top $10^{500}$ Reasons Not to Believe in the Landscape_ ([arXiv:1208.5715](https://arxiv.org/abs/1208.5715))

* {#Banks19} [[Tom Banks]], _On the Limits of Effective Quantum Field Theory: Eternal Inflation, Landscapes, and Other Mythical Beasts_ ([arxiv:1910.12817](https://arxiv.org/abs/1910.12817))

  from pages 14-22:

  > these considerations lead to conclusions at odds with the seemingly similar arguments of $[$ the [[swampland conjectures]] $]$. $[\cdots]$  Perturbative moduli space completely distorts the true nature of the class of consistent models.

  > It’s important to realize that the entire procedure just outlined for finding (meta) stable AdS minima of a non-perturbative effective potential is purely hypothetical and has no basis in well founded string theory calculations. $[\cdots].$

  > The hypothesis of the String Landscape is entirely based on low energy effective field theory ideas about finding "vacua" by minimizing an effective potential.  Everything that’s been said above indicates that this idea has no validity in genuine models of quantum gravity. $[\cdots]$

  > The most serious issue,  in  my  opinion,  is  the  contention  that  one  can  make  the  AdS  radius  much  larger  thanthe size of the compact manifold.  All well established examples of large radius AdS/CFT havea compact manifold of dimension 2 or greater whose radius is comparable to that of the AdSspace.  In Appendix A we’ll present an argument based on the properties of AdS black holes,that this is in fact necessary.

  > The next step in the construction of "realistic" models involves "adding an anti-brane to break supersymmetry and make the c.c.  positive".  This is supposed to be a small modification of the model, calculable in low energy effective field theory, and that seems manifestly incorrect. $[\cdots]$.

  > even if one believes that the construction of meta-stable dS models is reliable, there is no clear argument about  what  the  proper  observables  of  the  model  are  nor  that different  dS  constructions are part of the same model.  Neither is there an interpretation of these correlators as transition amplitudes in a quantum mechanical model. $[\cdots]$

  > The conclusion  that  effective  field  theorists  should  draw  from  this is that unlike super-symmetric string models in flat or AdS space-time, many of which have at least perturbative definitions as mathematical models obeying the axioms of quantum mechanics, all literature on the String Landscape is speculation based on the unfounded notion that all string models with a given amount of SUSY are part of one single model and that it makes sense to define an effective action that encompasses all string models. Every single non-perturbative construction of string models contradicts this claim $[\cdots]$

* [Brennan-Carta-Vafa 17](#BrennanCartaVafa17)

* [[David Kutasov]], Travis Maxfield, Ilarion Melnikov, [[Savdeep Sethi]], _Constraining de Sitter Space in String Theory_, Phys. Rev. Lett. 115, 071305 (2015) ([arXiv:1504.00056](https://arxiv.org/abs/1504.00056))

* Jakob Moritz, Ander Retolaza, Alexander Westphal, _Towards de Sitter from 10D_, Phys. Rev. D 97, 046010 (2018) ([arXiv:1707.08678](https://arxiv.org/abs/1707.08678))

* {#Sethi17} [[Savdeep Sethi]], _Supersymmetry Breaking by Fluxes_ ([arXiv:1709.03554](https://arxiv.org/abs/1709.03554))

  re-amplified in:

  [[Savdeep Sethi]], _Hope or No Hope for a String Landscape?_, talk at [StringPheno2019](https://indico.cern.ch/event/782251/) ([pdf](https://indico.cern.ch/event/782251/contributions/3441886/attachments/1870138/3076971/3_Sethi.pdf), [[SethiLandscape2019.pdf:file]])

* {#DanielssonVanRiet18} [[Ulf Danielsson]], [[Thomas Van Riet]],  _What if string theory has no de Sitter vacua?_, International Journal of Modern Physics D, Vol. 27, No. 12, 1830007 (2018)  ([arXiv:1804.01120](https://arxiv.org/abs/1804.01120), [doi:10.1142/S0218271818300070](https://doi.org/10.1142/S0218271818300070))

* {#vanRiet18} [[Thomas Van Riet]], _Is dS space in the [[swampland|Swampland]]_, talk at [StringPheno18](http://sp18.fuw.edu.pl/) ([pdf slides](http://sp18.fuw.edu.pl/wp-content/uploads/participants-database/thomasvanriet.pdf))

* {#vanRiet18a} [[Thomas Van Riet]], _Status of KKLT_, talk at _[Simons summer workshop 2018](http://scgp.stonybrook.edu/archives/24870)_ ([recording](http://scgp.stonybrook.edu/video_portal/video.php?id=3730))

* Jakob Moritz, Ander Retolaza, Alexander Westphal, _On uplifts by warped anti-D3-branes_ ([arXiv:1809.06618](https://arxiv.org/abs/1809.06618))

### The Swampland
 {#ReferencesSwampland}

Discussion of aspects of [[effective field theories]] which might rule them out as having a [[UV-completion]] by a [[string theory vacuum]] (be in the "[[swampland]]") has been initiated in

* {#Vafa05} [[Cumrun Vafa]], _The String Landscape and the Swampland_ ([arXiv:hepth/0509212](http://arxiv.org/abs/hepth/0509212))

Comprehensive review is in:

* Eran Palti, _The Swampland: Introduction and Review_, lecture notes ([arXiv:1903.06239](https://arxiv.org/abs/1903.06239))

See also

* {#BrennanCartaVafa17} T. Daniel Brennan, Federico Carta, [[Cumrun Vafa]], _The String Landscape, the Swampland, and the Missing Corner_ ([arXiv:1711.00864](https://arxiv.org/abs/1711.00864))

* Ben Heidenreich, [[Matthew Reece]], [[Tom Rudelius]], _Emergence and the Swampland Conjectures_ ([arXiv:1802.08698](https://arxiv.org/abs/1802.08698))

Implications of the possible non-existence of de Sitter vacua in string theory are explored in

* {#ObiedOoguriSpodyneikoVafa18} Georges Obied, [[Hirosi Ooguri]], Lev Spodyneiko, [[Cumrun Vafa]], _De Sitter Space and the Swampland_ ([arXiv:1806.08362](https://arxiv.org/abs/1806.08362))

* {#AgrawalObiedSteinhardtVafa18} Prateek Agrawal, Georges Obied, Paul Steinhardt, [[Cumrun Vafa]], _On the Cosmological Implications of the String Swampland_ ([arXiv:1806.09718](https://arxiv.org/abs/1806.09718))

* {#Vafa18} [[Cumrun Vafa]], _Cosmology and the String Swampland_, talk at _[Strings 2018](https://indico.oist.jp/indico/event/5/)_ ([pdf slides](https://indico.oist.jp/indico/event/5/picture/96.pdf), [recording](https://www.youtube.com/watch?v=fU8sJRCRz24&t=1904s))

* [[Frederik Denef]], Arthur Hebecker, Timm Wrase, _The dS swampland conjecture and the Higgs potential_ ([arXiv:1807.06581](https://arxiv.org/abs/1807.06581))

[[!redirects landscape of perturbative string theory vacua]]


Landweber exact functor theorem

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###Context###
#### Algebraic topology
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[[!include algebraic topology - contents]]
=--
#### Cohomology
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[[!include cohomology - contents]]
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#### Higher algebra
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[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Every [[complex oriented cohomology theory]] induces a [[formal group law]] from its first [[Conner-Floyd Chern class]]. Moreover, [[Quillen's theorem on MU]] together with [[Lazard's theorem]] say that the [[cohomology ring]] $\pi_\bullet(M U)$ of [[complex cobordism cohomology]] [[MU]] is the classifying ring for formal group laws.

The _Landweber exact functor theorem_ ([Landweber 76](#Landweber76)) says that, conversely, forming the [[tensor product]] of [[complex cobordism cohomology theory]] ([[MU]]) with a [[Landweber exactness|Landweber exact]] [[ring]] via some [[formal group law]] yields a [[cohomology theory]].

By the [[Brown representability theorem]] this defines a [[spectrum]] and the spectra arising this way are called _Landweber exact spectra_. ([e.g. Lurie lect 17, p.2](#Lurie17)).

There is an analogue of the statement also for [[even cohomology theory|even]] 2-[[periodic cohomology theories]], with [[MU]] replaced by [[MP]] ([Lurie lecture 18, prop. 11](#LurieLect18)).

## Statement

By evaluation on $X = B U(1)$, the [[classifying space]] for [[complex line bundles]] (the [[circle 2-group]]), every [[multiplicative cohomology theory|multiplicative]] [[complex oriented cohomology theory]] $E$ gives rise to a [[formal group]] over the [[graded ring]] $\pi_\bullet(E)$ (see [there](complex+oriented+cohomology+theory#FormalGroupLaw)).

Conversely, given a formal group $F$ over a [[graded ring]] $R$, one can ask whether this arises from some $E$ in this way.

Now since the [[Lazard ring]] $L$ classifies formal groups, in that any formal group law $F$ over a [[ground ring]] $R$ is equivalently a [[ring]] [[homomorphism]] $L \longrightarrow R$, and since moreover [[Quillen's theorem on MU]] identifies $L$ with the [[cohomology ring]] of [[complex oriented cohomology theory]] $L \simeq MU_\bullet(\ast)$, one may consider the functor

$$
  X \mapsto E_\bullet(X) \coloneqq MU_\bullet(X) \underset{L}{\otimes} R
  \,.
$$

By construction this is such that $E_\bullet(\ast) \simeq R$. But this functor may fail to satisfy the axioms of a [[generalized homology theory]], hence may fail to be [[Brown representability theorem|Brown represented]] by an actual [[spectrum]] $E$.

Landweber exactness is a sufficient condition on $R$ for $E \coloneqq MU_\bullet(-)\otimes_L R$ to be indeed a [[generalized homology theory]]
(e. g. [Lurie, lect 15, theorem 9](#LurieLecture15)).

Notice that if it is, then the canonical map

$$
  MU_\bullet(X) \otimes_{MU_\bullet} E_\bullet \stackrel{}{\longrightarrow} E_\bullet(X)
$$

is an [[equivalence]]. (For $E = $ [[KU]] this was originally proven in [Conner-Floyd 66](#ConnerFloyd66).) In this form the statement has generalizations beyond [[complex oriented cohomology theory|complex orientation]]. See at _[[cobordism theory determining homology theory]]_.

## Properties of Landweber-exact spectra

### Phantom maps

Between Landweber exact spectra, every [[phantom map]] is already null-homotopic. (e.g. [Lurie lect 17, corollary 7](#Lurie)).

## Examples

* The special case of the Landweber exact functor theorem for [[complex topological K-theory]] [[KU]] is known as the _[[Conner-Floyd isomorphism]]_.

## Related theorems

* [[Lazard's theorem]]

* [[universal complex orientation on MU]]

* [[Quillen's theorem on MU]]

* [[Landweber-Novikov theorem]]

## References

For the special case of $E = $[[KU]] the statement ([[Conner-Floyd isomorphism]]) is originally due to:

* {#ConnerFloyd66} [[Pierre Conner]], [[Edwin Floyd]], Theorem 10.1 in: _[[The Relation of Cobordism to K-Theories]]_, Lecture Notes in Mathematics __28__ Springer 1966 ([doi:10.1007/BFb0071091](https://link.springer.com/book/10.1007/BFb0071091), [MR216511](http://www.ams.org/mathscinet-getitem?mr=216511))

The general result for [[complex oriented cohomology]] is due to:

* {#Landweber76} [[Peter Landweber]], _Homological properties of comodules over $MU^\ast(MU)$ and $BP^\ast(BP)$_, American Journal of Mathematics (1976): 591-610 ([jstor:2373808](https://www.jstor.org/stable/2373808))

Textbook accounts:

* {#TamakiKono06} [[Dai Tamaki]], [[Akira Kono]], Theorem 6.37 in: _Generalized Cohomology_, Translations of Mathematical Monographs, American Mathematical Society, 2006 ([ISBN: 978-0-8218-3514-2](https://bookstore.ams.org/mmono-230))

Lecture notes:

* {#LurieLecture15} [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_, Lecture series 2010, Lecture 15 _Flat modules over $\mathcal{M}_{FG}$_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture15.pdf))

* [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_, Lecture series 2010, Lecture 16 _The Landweber exact functor theorem_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture16.pdf))

* {#Lurie17} [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_, Lecture series 2010, Lecture 17 _Phantom maps_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture17.pdf))

See also

* Wikipedia, _[Landweber exact functor theorem](http://en.wikipedia.org/wiki/Landweber_exact_functor_theorem)_

For the even 2-periodic case:

* {#LurieLect18} [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_, Lecture series 2010 , Lecture 18 _Even periodic cohomology theories_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture18.pdf))

[[!redirects Landweber exact spectrum]]
[[!redirects Landweber exact spectra]]

[[!redirects Landweber-exact spectrum]]
[[!redirects Landweber-exact spectra]]


Landweber exactness

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#Contents#
* table of contents
{:toc}

## Idea

The _Landweber exactness criterion_ determines whether a given [[formal group law]] arises as the formal group law defined by a [[weakly periodic cohomology theory]].

Notice that since every [[formal group law]] over a [[ring]] $R$ is classified by a ring homomorphism $f : MP({\ast}) \to R$ where by [[Quillen's theorem on MU|Quillen's theorem]] $MP({\ast})$ is the [[Lazard ring]]. So for every formal group one obtains a contravariant [[functor]] on [[topological space]]s given by the [[extension of scalars]]-assignment

$$
  X \mapsto A_f^n(X) := MP^n(X) \otimes_{MP({*})} R
  \,,
$$

where $MP^\bullet$ denotes the [[complex cobordism cohomology theory]] and where the [[tensor product]] is taken using the $R$-[[module]] structure on $MP({*})$ induced by $f$.

The point of Landweber-exactness is that if $f$ is Landweber exact (i.e. if the corresponding [[formal group law]] is) then this construction defines a [[cohomology theory]] $A^\bullet(-)$.

## Definition

+-- {: .num_prop}
###### Proposition
**Landweber criterion**  Let $f(x,y)$ be a [[formal group law]] and $p$ a [[prime]],
$v_i$ the [[coefficient]] of $x^{p^i}$ in

$$
  [p]_f(x) \coloneqq \underset{p\,\text{summands}}{\underbrace{x+_f\cdots+_f x}}
  \,.
$$

If $v_0,\ldots,v_i$ form a [[regular sequence]] for all $p$ and $i$ then $f(x,y)$ is Landweber exact and hence gives a
[[cohomology theory]] via the the formula above.

=--

See at _[[Landweber exact functor theorem]]_

## Examples

+-- {: .num_example}
###### Example

Let $g_a(x,y)=x+y$ be the formal [[additive group]].
Then $[p]_a(x)= p x$ and so $v_0=p$, $v_i=0$ for all $i\ge1$.
The [[regular sequence|regularity]] condtions imply that
the [[zero map]] $R/(p)\to R/(p)$ must be [[injective]].  This implies that $R$ contains the [[rational numbers]] as a subring.

Note that the [[ordinary cohomology]] $HP^*(X,R)=\prod_k H^{n+2k}(X,R)$ is a [[cohomology theory]] over any [[ring]] $R$.

=--

+-- {: .num_example}
###### Example

$g_m(x,y)=xy$, $[p]_m(x)=(x+1)^p-1$, $v_0=p$, $v_1=1$, $v_i=0$ for all $i \gt 1$.
The regularity conditions are trivial.
Hence we know that $K^*(X)=MP^*(X)\otimes_{MP(\bullet)} \mathbb{Z}$ is a cohomology theory.

=--

## Related entries

* [[Landweber exact functor theorem]]

* [[A Survey of Elliptic Cohomology - formal groups and cohomology]]

## References

* [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_, Lecture series 2010,
Lecture 15 _Flat modules over $\mathcal{M}_{FG}$_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture15.pdf))
 {#LurieLecture15}

* [[Jacob Lurie]], _[[Chromatic Homotopy Theory]]_, Lecture series 2010, Lecture 16 _The Landweber exact functor theorem_ ([pdf](http://www.math.harvard.edu/~lurie/252xnotes/Lecture16.pdf))

[[!redirects Landweber exactness criterion]]
[[!redirects Landweber criterion]]


Landweber-Novikov theorem

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#### Stable Homotopy theory
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[[!include stable homotopy theory - contents]]
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#### Higher algebra
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[[!include higher algebra - contents]]
=--
=--
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#Contents#
* table of contents
{:toc}

## Idea

The _Landweber-Novikov theorem_ describes the structure of [[Hopf algebroid over a commutative base]] on the dual generalized [[Steenrod algebra]] of the [[complex cobordism cohomology theory]] [[spectrum]] [[MU]].

By [[Quillen's theorem on MU]] we have $\pi_\bullet(MU) \simeq L$, the [[Lazard ring]]. Moreover the dual [[Steenrod algebra]] is

$$
  MU_{\bullet}(MU) \simeq L B \coloneqq L \otimes \mathbb{Z}[b_1, b_2, \cdots]
$$

where on the right we have the [[polynomial ring]] on generators $b_i$ in degree $2i$.

Here $(L, L B)$ carries a natural structure of a [[Hopf algebroid over a commutative base]] (and apparently this example is what made [[Haynes Miller]] introduce Hopf algebroids in the first place). The Landweber-Novikov theorem asserts that
this is the [[Hopf algebroid]] structure on the dual [[MU]]-[[Steenrod algebra]]

$$
  (\pi_\bullet(MU), MU_\bullet(MU))
  \simeq
  (L, L B)
  \,.
$$

The analog for [[BP]] is the [[Adams-Quillen theorem]].

## Related theorems

* [[Lazard's theorem]]

* [[Quillen's theorem on MU]]

* [[universal complex orientation on MU]]

* [[Landweber exact functor theorem]]

## References

The original articles are

* [[Sergei Novikov]], _The metods of algebraic topology from the viewpoint of cobordism theories_, Izv. Akad. Nauk. SSSR. Ser. Mat. 31 (1967), 855–951 (Russian).

* [[Peter Landweber]], _Cobordism operations and Hopf algebras_, Trans. Amer. Math. Soc. 129 (1967), 94–110.

* [[Frank Adams]], Part II.11 of _[[Stable homotopy and generalised homology]]_, 1974

The Hopf algebroid structure on $(L, L B)$ is reviewed in

* [[Doug Ravenel]], appendix 2 of _[[Complex cobordism and stable homotopy groups of spheres]]_

The Landweber-Novikov theorem is reviewed in

* [[Doug Ravenel]], chapter 4, section 1 of _[[Complex cobordism and stable homotopy groups of spheres]]_


Lane P. Hughston

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

* [GoogleScholar page](https://scholar.google.com/citations?user=ASy_EF4AAAAJ&hl=en)

## Selected writings

On the [[symplectic formulation of quantum mechanics]]:

* {#Hughston96} [[Lane P. Hughston]], *Geometry of stochastic state vector reduction*, Proceedings of the Royal Society A **452** 1947 (1996) [[doi:10.1098/rspa.1996.0048](https://doi.org/10.1098/rspa.1996.0048), [jstor:52944](https://www.jstor.org/stable/52944)]

* {#BrodyHughston01} [[Dorje C. Brody]], [[Lane P. Hughston]], *Geometric Quantum Mechanics*, J. Geom. Phys. **38** (2001) 19-53 [[arXiv:quant-ph/9906086](https://arxiv.org/abs/quant-ph/9906086), doi:10.1016/S0393-0440(00)00052-8]

category: people

[[!redirects Lane Hughston]]


Langevin equation

__Langevin equation__ is the equation of motion which includes a contribution from a random force.

Related $n$Lab items include [[statistical physics]], [[stochastic process]], [[fluctuation-dissipation theorem]], [[anomalous diffusion]], [[dissipation]]

* [Langevin equation](https://en.wikipedia.org/wiki/Langevin_equation), Wikipedia

category: physics


Langlands analogies -- table

**[[analogies]] in the [[Langlands program]]:**

| [[arithmetic geometry|arithmetic]] [[Langlands correspondence]] | [[geometric Langlands correspondence]] |
|-----|------|
| [[ring of integers]] of [[global field]] | [[structure sheaf]] on [[complex curve]] $\Sigma$ |
| [[Galois group]] | [[fundamental group]] of $\Sigma$ |
| [[Galois representation]] | [[flat connection]]/[[local system]] on $\Sigma$ |
| [[idele class group]] mod [[integral adeles]] | [[moduli stack of line bundles]] on $\Sigma$ |
| nonabelian $\;$ "  | [[moduli stack of vector bundles]] on $\Sigma$ |
| [[automorphic representation]] | [[Hitchin connection]] [[D-module]] on bundle of [[conformal blocks]] over the moduli stack |


Langlands correspondence -- contents

**[[Langlands program]]**

* [[local Langlands conjectures]]

**[[geometric Langlands program]]**/**[[geometric Langlands correspondence]]**

* [[Hitchin fibration]]

* [[S-duality]]

  * [[electro-magnetic duality]]

    * [[Montonen-Olive duality]]

  * [[Seiberg duality]]

* [[quantum geometric Langlands correspondence]]

[Edit this sidebar](/nlab/edit/Langlands+correspondence+--+contents)


Langlands dual


Langlands dual group

(...)

See at _[[Langlands correspondence]]_.

(...)

Remark on terminology: [[Robert Langlands]] did not actually introduce or use the term "dual group". The original texts just speak of the "L-group" ([Gelbhart 84, page 29 (205)](#Gelbhart84)). Also, the role of ${}^L G$ in the number-theoretic [[Langlands program]] is not as symmetric as in the [[geometric Langlands program]], it essentially serves as an ingredient for the construction of [[automorphic L-functions]] from [[Galois representations]] in $G$. In the [comment to a message to Sarnak (2014)]() Langlands writes:

> This duality $[$ [[electric-magnetic duality]]/[[S-duality]] $]$ is quite different than the functoriality and reciprocity introduced in the arithmetic theory $[$ of the [[Langlands program]] $]$.

Further comments along these lines are in  ([Langlands 14, pages 6-7](#Langlands14)).

## References

* {#Gelbhart84} [[Stephen Gelbart]], _An elementary introduction to the Langlands program_,  Bull. Amer. Math. Soc. (N.S.) 10 (1984), no. 2, 177–219 ([web](http://www.ams.org/journals/bull/1984-10-02/S0273-0979-1984-15237-6/))

* {#Langlands14} [[Robert Langlands]], _[[Problems in the theory of automorphic forms -- 45 years later]]_, Oxford 2014

* Wikipedia, _[Langlands dual](http://en.wikipedia.org/wiki/Langlands_dual)_

[[!redirects Langlands dual groups]]


Langlands functoriality

#Contents#
* table of contents
{:toc}

## Idea

Langlands functoriality is a generalization of [[Langlands program|Langlands conjectures]] that essentially tells us that a $L$-homomorphism
$${}^L G\to {}^L H$$
between the [L-groups](http://en.wikipedia.org/wiki/Langlands_dual) of two [[reductive algebraic group|reductive groups]] over $\Q$ should induce a transfer map from automorphic representations for $G$ to automorphic representations of $H$.

## Known results

Langlands functoriality is now known in the case of the natural embedding $G\subset \GL_n$ of a classical ([[orthogonal group|orthogonal]], [[symplectic group|symplectic]] or [[unitary group|unitary]]) group in the [[general linear group]] through its canonical representation, following the work of [[James Arthur]] on the [[Arthur-Selberg trace formula]] and on endoscopy for classical groups.

It is also known (starting from Jacquet and Langlands' work for $\GL_2$) in the case of inner forms of a group: there is an equivalence between automorphic representations on the group and automorphic representations on its inner form. This was used by Harris and Taylor to prove instances of the local Langlands correspondence for $\GL_n$ by using unitary [[Shimura varieties]] associated to some twisted (unitary) forms of $\GL_n$.

James Newton and Jack Thorne proved [[symmetric power]] functoriality for holomorphic [[modular forms]] in [NewtonThorne19](#NewtonThorne19) and [NewtonThorne20](#NewtonThorne20).

##References

* [[Minhyong Kim]], _A superficial introduction to Langlands functoriality_, [slides](http://people.maths.ox.ac.uk/kimm/lectures/functoriality.pdf)

* Jae-Hyun Yang, _Langlands Functoriality Conjecture_, Kyungpook Math. J. vol. 49, no. 2 (June 2009), 355-387 ([arXiv:0808.0917](https://arxiv.org/abs/0808.0917))

* {#NewtonThorne19} James Newton and Jack Thorne, _Symmetric power functoriality for holomorphic modular forms_, [arxiv:1912.11261](https://arxiv.org/abs/1912.11261)

* {#NewtonThorne20} James Newton and Jack Thorne, _Symmetric power functoriality for holomorphic modular forms, II_, [arxiv:2009.07180](https://arxiv.org/abs/2009.07180)